Forestry, Faculty of
DSpace
UBCV
Scorah, Hugh William
Doctor of Philosophy - PhD
University of British Columbia
Increases in wildfire suppression cost expenditures, an upward trend in burned area, increasing damages and evacuations, along with research pointing to a worsening trend in all of these areas as a result of climate change is motivating research into methods to limit the consequences of wildfire. My research is chiefly concerned with the use of economic tools to identify the best use of resources when evaluating fuel mitigation efforts both from a project perspective and at the landscape level. Using tools from Extreme Value Theory, I show that the distribution of wildfire costs is not likely to follow a normal distribution and instead is more likely drawn from fat-tailed distributions. This distribution implies that conventional tools of economic analysis, in particular, cost-benefit and cost-effectiveness analysis are not applicable and, if economic tools are to be applied, alternative approaches need to be developed. I present an alternative approach for analyzing individual projects, modeled after banking stress-tests, and apply it to four fuel-treatment case studies in the interior of British Columbia. This approach, called the Defensibility Heuristic, makes use of frontline experience (tacit knowledge), has few data and modeling requirements, and has the potential to quickly rank projects within a government budget and planning cycle that are most likely to protect the most important values. Following the development of this approach for picking fuel treatments in the short run, a tool for making decisions about longer-term forest management concerned with wildfire management is considered. This tool is community-centric as opposed to taking a provincial or federal perspective. Different approaches to ranking outcomes from decision theory were considered in the face of wildfire events that were drawn from a Power Law distribution. The existing decision rules were found wanting and I developed a rule called a Trauma Exposure. This decision rule was applied to the problem of setting annual harvest volumes to show that setting the harvest volume slightly below maximum sustained yield and organizing forest stands into a patchwork is likely to result in a more stable harvest volume over the long run.
ECONOMIC DECISION MAKING UNDER UNCERTAINTY FOR WILDFIRE FUELS MITIGATION by Hugh William Scorah B.A., The University of Victoria, 2008 M.A., The University of Victoria, 2010 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Forestry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) May 2022 © Hugh William Scorah, 2022 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Economic decision making under uncertainty for wildfire fuels mitigation submitted by Hugh Scorah in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Forestry Examining Committee: Harry Nelson, Associate Professor, Forest Resources Management, UBC Supervisor Lori Daniels, Professor, Forest and Conservation Sciences, UBC Supervisory Committee Member Mike Flannigan, Professor, Natural Resource Science, Thompson Rivers University Supervisory Committee Member James Vercammen, Professor, Sauder School of Business, Land and Food Systems, UBC University Examiner Mathieu Bourbonnais, Assistant Professor, EEGS, UBC Okanagan University Examiner iii Abstract Increases in wildfire suppression cost expenditures, an upward trend in burned area, increasing damages and evacuations, along with research pointing to a worsening trend in all of these areas as a result of climate change is motivating research into methods to limit the consequences of wildfire. My research is chiefly concerned with the use of economic tools to identify the best use of resources when evaluating fuel mitigation efforts both from a project perspective and at the landscape level. Using tools from Extreme Value Theory, I show that the distribution of wildfire costs is not likely to follow a normal distribution and instead is more likely drawn from fat-tailed distributions. This distribution implies that conventional tools of economic analysis, in particular, cost-benefit and cost-effectiveness analysis are not applicable and, if economic tools are to be applied, alternative approaches need to be developed. I present an alternative approach for analyzing individual projects, modeled after banking stress-tests, and apply it to four fuel-treatment case studies in the interior of British Columbia. This approach, called the Defensibility Heuristic, makes use of frontline experience (tacit knowledge), has few data and modeling requirements, and has the potential to quickly rank projects within a government budget and planning cycle that are most likely to protect the most important values. Following the development of this approach for picking fuel treatments in the short run, a tool for making decisions about longer-term forest management concerned with wildfire management is considered. This tool is community-centric as opposed to taking a provincial or federal perspective. Different approaches to ranking outcomes from decision theory were considered in the face of wildfire events that were drawn from a Power Law distribution. The existing decision rules were found wanting and I developed a rule called a Trauma Exposure. This decision rule was applied to the problem of setting annual harvest volumes to show that setting the harvest iv volume slightly below maximum sustained yield and organizing forest stands into a patchwork is likely to result in a more stable harvest volume over the long run. v Lay Summary This research advances economic analyses of wildfire by achieving three objectives, first, I described the state of knowledge about the costs of wildfire in Canada and the limits of economic analyses with respect to wildfire. This has important implications, as the economic tools we typically use to aid in decision-making are not applicable in the context of wildfire, requiring the development of new approaches. Second, I developed a novel tool for choosing between wildfire fuel treatment proposals with a finite budget when knowledge and resources are limited. Third , I present a new way of thinking about forest management and the setting of harvest levels and the patterns of those harvest levels when trying to limit wildfire events that can limit economic opportunity and damage the social fabric in rural forest-based communities. vi Preface I was responsible for the change in design that emphasized the use of Power Law distributions and the development of new decision tools to deal with the limitations that arise when conventional tools fail under Power Law conditions. Assistance on the literature review of cost-benefit analysis in the wildfire literature was provided by an undergraduate student Avery Maloney. I analyzed the data on suppression cost in Chapter 2 and developed the Defensibility Heuristic in Chapter 3 and did the work on the Case Studies in those chapters. In Chapter 4 I developed the idea of the Trauma Exposure and the computer simulation tools that were used to sketch out the implications of this idea for forest management as it relates to timber losses from wildfire. Work from Chapter 2 and Chapter 4 has been included in the published paper Nelson, Harry W; Scorah, Hugh William (2021). How Should We Sustain Future Forests Under Extreme Risk? Canadian Journal of Forest Research. I did the simulation work and wrote the model description. The remainder of the writing in this publication (not included in this dissertation) is the work of Dr. Harry Nelson but the ideas were developed in collaboration. vii Table of Contents Abstract..........................................................................................................................................iii Lay Summary .................................................................................................................................v Preface............................................................................................................................................vi Table of Contents .........................................................................................................................vii List of Tables .................................................................................................................................xi List of Figures..............................................................................................................................xiii List of Symbols .............................................................................................................................xv List of Abbreviations .................................................................................................................xvii Acknowledgements ...................................................................................................................xviii Dedication ....................................................................................................................................xix Chapter 1: Introduction ................................................................................................................1 1.1 Approach to Economics and Decision Making .............................................................. 3 1.2 Risk in Wildfire............................................................................................................... 5 1.3 The Costs of Wildfire in Canada................................................................................... 10 1.4 Prior Work on Economic Evaluation of Wildfire Fuel Treatments .............................. 11 Chapter 2: Fat Tails in Wildfire Cost Data ...............................................................................18 2.1 A Power Law model for British Columbia Suppression Cost Data.............................. 21 2.2 Implications of Fat-Tails for Modeling Risk and Cost-Benefit Analysis ..................... 28 2.2.1 Pareto-Distributed Damages and Cost-Benefit Analysis .......................................... 29 2.3 Marginal Thinking Under Fire ...................................................................................... 33 2.4 Implications of the Probability Model .......................................................................... 38 viii Chapter 3: A Stress Test Heuristic for Wildfire Fuel Treatment Decisions ..........................41 3.1 The Defensibility Heuristic ........................................................................................... 42 3.1.1 A Framework for Assessing Defensibility of a Fuel Treatment ............................... 43 3.1.2 Estimating Fuel Treatment Costs .............................................................................. 44 3.1.3 Probability of Fire Testing a Fuel Treatment ............................................................ 45 3.1.4 Stress Testing ............................................................................................................ 45 3.1.5 Calculating the Defensibility Threshold ................................................................... 47 3.1.6 A Simple Example Illustrating the Defensibility Heuristic ...................................... 50 3.1.7 Adjusting the Heuristic for a Regularly Maintained Fuel Treatment ....................... 50 3.2 A Description of the Potential Values at Risk in Fuel Treatment Decision Making .... 52 3.3 Case Studies Using the Defensibility Heuristic ............................................................ 61 3.3.1 A Description of the Fuel Treatment Data................................................................ 62 3.3.2 The Greata Ranch Case Study .................................................................................. 65 3.3.3 The Glenrosa Case Study .......................................................................................... 67 3.3.4 The 150 Mile House Case Study .............................................................................. 71 3.3.5 The South Lakeside – Williams Lake Case Study .................................................... 73 3.3.6 An Example with Perpetual Maintenance................................................................. 75 3.3.7 Changes in the Fire Weather Threshold.................................................................... 77 3.4 Limitations and Shortcomings of the Defensibility Heuristic ...................................... 77 3.4.1 Limited Value Inclusion............................................................................................ 78 3.4.1.1 Including Other Values ..................................................................................... 79 3.4.1.1.1 Greenhouse Gas Emissions ......................................................................... 79 3.4.1.1.2 Habitat Loss................................................................................................. 79 ix 3.4.2 Frontline Expertise .................................................................................................... 81 3.4.3 Implied Probabilities are not Intuitive. ..................................................................... 81 3.4.4 The Defensibility Heuristic is Not a General Theory ............................................... 82 3.5 On the ‘Business Case’ for Fuel Treatments ................................................................ 83 Chapter 4: The Long-Run Economics of Forest Management and Wildfire .........................86 4.1 A Stylized Model of Forest Growth.............................................................................. 87 4.2 Building an Exposure.................................................................................................... 91 4.3 Definition of an Exposure ............................................................................................. 97 4.3.1 Three Classes of Exposure ...................................................................................... 100 4.3.2 Formal Statement of a Trauma Exposure ............................................................... 102 4.3.3 Approaches to Decision Making and the Exposures .............................................. 103 4.3.3.1 Maximin .......................................................................................................... 105 4.3.3.2 Minimax Regret .............................................................................................. 106 4.3.3.3 Maximization .................................................................................................. 107 4.3.4 Things Fall Apart .................................................................................................... 110 4.3.4.1 Trauma Exposure as a Social Phenomenon .................................................... 110 4.3.5 The Trauma Exposure Decision Rule ..................................................................... 113 4.3.5.1 Estimating the Trauma Exposure .................................................................... 114 4.4 Sustainable Cut Revisited ........................................................................................... 116 4.4.1 Simulating Fires with Cellular Automata ............................................................... 118 4.4.2 Evidence for Increasing Friction............................................................................. 119 4.5 Finding Sustainable Yield with Asymmetric Shocks ................................................. 123 4.5.1 Combining the Worst-Case Burn Function and Harvest Dynamics ....................... 133 x 4.6 Hormesis and the Trauma Exposure ........................................................................... 140 Chapter 5: Conclusion ...............................................................................................................143 5.1 Concluding Thoughts on the Short-Run ..................................................................... 145 5.2 Concluding Thoughts on the Long-Run ..................................................................... 147 5.3 Final Thoughts ............................................................................................................ 150 Bibliography ...............................................................................................................................152 Appendices ..................................................................................................................................180 Appendix A The Bootstrapping Method for Estimating Power Law Parameter Estimates ... 180 Appendix B Variance of a Pareto Distribution ....................................................................... 182 Appendix C Finding the Pareto Index α ................................................................................. 185 xi List of Tables Table 1 Cost Categories and Damages ......................................................................................... 50 Table 2 Cost Categories, Data Availability and Difficulty of Estimation. Data Availability runs low (*), medium (**), high (***). Ease of estimation: difficult (*), moderate (**), easy (***). . 54 Table 3 Average mechanical fuel treatment costs in British Columbia. Data provided by the Forest Enhancement Society of British Columbia. ....................................................................... 63 Table 4 Average hand fuel treatment costs in British Columbia. Data provided by the Forest Enhancement Society of British Columbia. .................................................................................. 64 Table 5 Defensibility criteria for the Greata ranch area with three scenarios for potential loss of life from failed evacuation (10, 50, and 100 people). $2000/ha for treatment costs. ................... 67 Table 6 Defensibility criteria given a public expenditure perspective (property taxes only). Treatment costs averaging $2000/ha............................................................................................. 69 Table 7 Defensibility criteria with total value of homes lost. Treatment costs averaging $2000/ha........................................................................................................................................................ 69 Table 8 Defensibility criteria given a public expenditure perspective (property taxes only). Treatment costs averaging $4000/ha............................................................................................. 70 Table 9 Defensibility criteria with total value of homes lost. Treatment costs averaging $4000/ha........................................................................................................................................................ 71 Table 10 Defensibility criteria given a public expenditure perspective (property taxes only). Treatment costs averaging $2000/ha............................................................................................. 73 Table 11 Defensibility criteria with total value of homes lost. Treatment costs averaging $2000/ha........................................................................................................................................ 73 xii Table 12 Defensibility criteria given a public expenditure perspective (property taxes only). Treatment costs averaging $2000/ha............................................................................................. 74 Table 13 Defensibility criteria with total value of homes lost. Treatment costs averaging $2000/ha........................................................................................................................................ 75 Table 14 AAC decisions, stocking levels and wildfire risk responsiveness. .............................. 140 xiii List of Figures Figure 1 Sparhawk's Least Cost plus Damages model for pre-suppression expenditures. ........... 13 Figure 2 Individual fire suppression costs ordered from least costly to most costly.. .................. 21 Figure 3 Far from 'Normal', histogram of suppression costs in the Kamloops and Cariboo Fire Districts from 2003 – 2019.. ......................................................................................................... 22 Figure 4 A Zipf plot of BC fire suppression cost data. ................................................................. 24 Figure 5 Log-likelihood ratios of wildfire costs evaluated for a power law and log-normal distribution... ................................................................................................................................. 26 Figure 6 The ratio of the 1% largest cost events to the 99% bottom cost events over 1,000 simulations of 10,000 fire events. ................................................................................................. 32 Figure 7 Convergence of expectations under a Power Law driven process. ................................ 37 Figure 8 Individual mechanical fuel treatments by size and cost ($/ha).. ..................................... 65 Figure 9 Greata Ranch study region ............................................................................................. 66 Figure 10 Glenrosa study region. .................................................................................................. 68 Figure 11 150 Mile House study region ....................................................................................... 72 Figure 12 South Lakeside - Willams Lake study region............................................................... 74 Figure 13 The sigmoid growth (or logistic) curve. ....................................................................... 88 Figure 14 Deterministic model of growth and harvest. ................................................................ 94 Figure 15 Timber stock with Pareto distributed wildfires. ........................................................... 95 Figure 16 Timber stock with Pareto distributed wildfires with silviculture. ................................ 96 Figure 17 Stock vs Change in Stock. .......................................................................................... 117 Figure 18 Checkerboard model................................................................................................... 118 Figure 19 An example fire using a simplified Drossel and Schwabel model after 150 steps..... 126 xiv Figure 20 Examples of fire runs using the cellular automata model of Drosser and Schwabel at different densities.. ...................................................................................................................... 127 Figure 21 The relationship between fraction of stock burned, which can be multiplied by the stock, S, to get, H vs the density.. ............................................................................................... 128 Figure 22 Fraction of stock burned vs density under changes in x_0 and v ............................... 130 Figure 23 A run of the growth, fire and cut model with an H that is a function of the stock level..................................................................................................................................................... 134 Figure 24 Stock and cut when the worst fire happens every year. ............................................. 135 Figure 25 Annual burn and decline in stock at 96% of mean annual increment. ....................... 136 Figure 26 Annual burn at v= 25 and decline in stock at 92% of mean annual increment .......... 138 Figure 27 Aggressive harvesting at the beginning brings the worst case risk down to a level below the potential for a trauma exposure. ................................................................................. 139 xv List of Symbols 𝛼 – shape parameter of the Pareto distribution 𝐵𝑖𝑗 - benefits for the 𝑗𝑡ℎvalue at risk from the 𝑖 𝑡ℎ category of fire behaviour 𝑑 –the change in defensibility required to justify a fuel treatment 𝛿 – depreciation rate of the capital stock 𝜖 – random error in price estimation 𝐹𝑉 – Future Value 𝑔 – growth rate of the capital stock 𝛾 – linear factor that relates the maximum value of the truncated Pareto function to the forest stock 𝐻 – maximum value of the truncated Pareto function 𝜅𝑡 – harvest at time, t 𝐾𝑡 – capital stock at time, t 𝕀𝒜 – indicator function that takes a value of 1 if an observed value is in the set 𝒜 and zero otherwise. 𝐿 – minimum value of the truncated Pareto function 𝐿 𝑖𝑗 - losses for the 𝑗𝑡ℎvalue at risk from the 𝑖𝑡ℎ category of fire behaviour 𝑝 – future price 𝑝∗ - current efficient price 𝑝𝑖 – probability of event 𝑖 𝑝(𝐹𝑖) - the probability that the 𝑖𝑡ℎ fire behaviour encounters the 𝑗𝑡ℎ value Π – payoff xvi 𝑃𝑉 – Present Value 𝑟 – rate of growth 𝑆𝑡 – forest stock at time, t 𝑆̅ – maximum forest stock ∆𝑆 – change in forest stock 𝜏 – trauma threshold 𝑣 – measure of fuels fragmentation in model of the cellular automata fire outputs 𝜔𝑡 – random wildfire burn damage to forest stock at time t 𝜔(𝑥) – probability density that wildfire burn damage is drawn from 𝑥𝑖 – an exclusive member of the set of possible events in an expectation value 𝑥𝑚𝑖𝑛, 𝑥𝑚- smallest data point included for estimating the parameters of a pareto distribution 𝑥0 – centre of the fuels fragmentation model of the cellular automata fire outputs 𝜉 – strategy xvii List of Abbreviations AAC – Annual Allowable Cut CBA – Cost-Benefit Analysis MAI - Mean Annual Increment MSY – Maximum Sustained Yield xviii Acknowledgements I owe thanks to Dr. Harry Nelson for having patience through several changes of topic and allowing me to take this project in a very different direction and encouraging me to explore and develop new methods using unconventional approaches. I thank Dr. Lori Daniels and Dr. Mike Flannigan for having patience with my ignorance of wildfire, freely sharing their knowledge of the subject and asking questions through several iterations of these ideas that vastly improved the end product. My wife, Rebekah and son Max, put up with an absent minded and an occasionally less than genial husband and father through the process of writing. I will be forever grateful for their patience and the sacrifices they made to so that I could get through to the end. This research was made possible by funding from the Natural Resources Canada (NRCan) and the BC Ministry of Forests, Lands, Natural Resource Operations and Rural Development. xix Dedication To my wife, son and the baby on the way. I never would have finished without the motivation provided by the joy of this family. 1 Chapter 1: Introduction In the cultural history of fire, many traditions tell the story of humans stealing fire from the gods and putting it to work for human purposes. In these stories the theft of fire is coincident with or causative of the beginnings of technological civilization. Through fire we gain control over nature and mold it to our purposes (Pyne, 2012a). Despite our best efforts, after thousands of years there persists another side of fire: destructive, outside of human control, terrifying, awe inspiring. In the spirit of Prometheus, we still aspire to tame even the most wild and wrathful fires. This work attempts, perhaps in vain, to temper this hubris and point to the limitations of the resources available to us as well as the limits of what can be known and how limited resources and knowledge might be used in face of these fires. The original research proposal for this work was Promethean in spirit. It assumed we have well established tools for making decisions about the allocation of resources to wildfire fuel management projects and that it was a simple matter of tabulating costs and damages of wildfire in dollar terms. With this data I would construct a reference fire with costs and damages and a reference fuel treatment, and with that information be able to construct a cost-benefit ratio that would help inform budget allocation decisions. I ran into two, very fundamental, challenges with this approach. The first is epistemic. Given our observations about wildfire and wildfire costs what can be known, or what is generalizable? And more salient from the perspective of economics is what can be known at reasonable cost? In other words what information can be obtained where the benefits of data collection exceed the costs? This question is addressed in Chapter 2. 2 When the limits of what is possible, in practical terms, are identified it begs a further question: How do we make decisions with limited generalizable information? Or more specifically, how do we make decisions when appropriately calibrated models of statistical inference are too difficult or costly? It turns out that the traditional tools of Cost-Benefit and Cost-Effectiveness Analysis produce a mismatch for the underlying reality and have the potential to make poor use of public resources. Using wildfire fuel treatments as the starting point I explore two alternative approaches to thinking about decision making under these conditions when the basic assumptions of our mathematical models are violated and we can no longer make use of the tools that are conventionally applied by policy makers to understand reality. In Chapter 3 the short-run problem of choosing between already proposed fuel treatments with a finite budget is addressed1. A new method is proposed that limits the amount of statistical inference or other computational modelling required and makes use of the experience of frontline wildfire staff. Four case studies are evaluated using this approach. In Chapter 4, forest management is explored at the landscape scale over the long run under the same epistemic conditions identified in Chapter 2. The long run is defined as a time scale where capital investment decisions are made. A new definition of risk related to community well-being is introduced and the implications of this definition for decision making in forest management are sketched out. The argument draws on many fields, including mathematical decision theory, sociology, economics, and basic forestry. The problem is sufficiently complicated that I am only able to sketch the outlines of a new approach that places 1 There are four phases of emergency management: prevention and mitigation, preparedness, response and recovery(Tymstra et al., 2020). The applications of the ideas presented here are concerned only with prevention and mitigation. 3 the community scale at the centre of policy making. As a result, there are no case studies in this example and I only point to how practical decision-making tools might be developed. In the remainder of the introduction, I discuss the approach to economics that is taken in this work. In many ways it is out of fashion, a throwback to economics as it was done before the mathematical models from physics were introduced to the field and therefore needs some introduction. Following this I review ideas about risk in the wildfire literature, the costs of wildfire and the previous work that has been done in evaluating wildfire fuel treatments using more conventional methods. 1.1 Approach to Economics and Decision Making The history of economic thought prior to Adam Smith is scattered throughout works on moral philosophy. Economic activity is generally seen as means to developing a virtuous population and the productive activities of humans should be organized to that end. Early evidence of this is found in Xenophon’s Hiero (1925) a work from around 400 BC that discusses returns to scale and its uses to society, and Kautilya’s The Arthashastra(1992), which discusses around 300 BC the importance and administration of Autarky, or self-sufficiency for the virtue of the state. In the 13th century we see Aquinas discussing the dangers of usury and its consequences for the spiritual welfare of a people and he introduces the notion of a natural price, where prices realized through market power are unjust (Aquinas, 2018). Also, around this period the Nasirean ethics introduces the idea that there should be “the study of universal laws governing the public interest in so far as they are directed, through cooperation, toward the optimal perfection.”(Samuels et al., 2008, pp.39). Adam Smith continued this tradition with his Theory of Moral Sentiments 4 (Smith, 1759) preceding the Wealth of Nations(Smith, 2000) by 17 years wherein the Wealth of Nations describes the means by which government lays the material foundation for the moral society. Up to this point in the history of economic thought, the study of economics falls squarely in the domain of moral reasoning. Productive work is a means to a moral or virtuous end. This orientation towards the subject of economics mostly disappears with work of Alfred Marshall (1920) and Stanley Jevons (1905). These works attempted to put economics on a scientific foundation. Rather than assuming a moral grounding for economic activity these authors sought to describe human behaviour in the markets using mathematical models. These models made use of the notions of equilibrium and stationary points from late 19th century physics. These models are still dominant today. Morality was stripped from the subject. Data, of course, has been marshalled to support the mathematical theory but this approach was not subject to real testing until the latter half of the 20th century. Few of these models stand up to scrutiny and the basic assumptions about human rationality and the behaviour of the firm are violated in practice(Eichner, 1983). Besides the repeated failure of economic theory to withstand empirical investigation we find morality creeping back into the subject. When politicians, bureaucrats and business owners make decisions about the allocation of resources in society they make some people better off and other worse off. They organize and direct human efforts to ends that suit their purposes. These are inherently questions of morality. This work is best read as an attempt to revive the spirit of economics as a form of moral reasoning. It is not to be confused with science, whether you take a process view of science, wherein a hypothesis is proposed, and tested against observed data or a view that science must 5 past epistemic tests of explaining all or as many of the observations as possible while being parsimonious and being difficult to falsify(Deutsch, 2011). The goal is not to make falsifiable statements and test them but to provide a tool for reasoning about the allocation of resources for the well-being of members of a society. In the case of the short-run problem of allocating resources to already proposed fuel-treatments (Chapter 3) we are constrained by existing centralized institutions and the moral calculus is reduced to the question of “How should the money be spent so as to minimize waste on projects that are unlikely to provide benefit?”. An admittedly low bar in regards to the need to make any moral judgments. In Chapter 4 the degrees of freedom are opened up so we can explore ideas about risk management at the community scale and the question becomes “How might we chart a course of action in the face of wildfire when the survival of a community is at stake?”. These are normative, not positive questions and should be treated as such. Their utility cannot be determined with a hypothesis test but depend on the values of the individual reader and the discussions between them. In the tradition of moral reasoning the arguments are evaluated on whether or not the conclusions follow from a reasonable chain of logic starting with a set of assumptions or axioms and whether or not the decision mechanism results in positive improvements in people’s lives. The latter can only really be evaluated with the implementation of the ideas in the real world which has not been done in this case. 1.2 Risk in Wildfire The formal study of risk has its beginning in mathematical study of games of chance(Peters, 2011). In these games all of the possible outcomes can be enumerated and whether flips of coins, drawing balls from urns or rolling dice a frequency of each outcome can be observed and 6 calculated. The calculated frequency is the probability. In these models there are two important assumptions: (1) The enumerated outcomes are mutually exclusive (2) The probabilities of each outcome sum to one This approach of enumerating the possibilities and assigning probabilities to the possibilities based on frequency was adopted in the insurance industry which was the first practical application of risk theory in the real world(Dionne, 2013). A key assumption in the application of these tools is that the probabilities of events stay constant over the duration that the risk is being insured (stationary). These three assumptions, mutually exclusive possibilities, that probabilities sum to one, and probabilities are stationary over time are important to remember as more complicated problems are addressed with the tools of risk management. The challenge of using these probability models becomes apparent as difficulties arise in enumerating events and calculating the probability of the event based on observed frequencies. The late frequentist interpretation of a probability is the frequency of a mutually exclusive event in the limit as the number of observations gets very large(von Mises, 1941). In other words, we converge on the true probability as the number of observations goes to infinity. In applications where it is not possible to collect a large number of observations the frequentist approach loses its applicability in applied settings. This is true for wildfire fuel mitigation where the process that generates fire events differs substantially depending on local climate, plant species and available firefighting resources. Solutions to this problem have included subjective probabilities or thinking about a probability as a degree of belief (Ramsey, 2016) or an information theoretic approach where a probability is related to the notion of entropy, where the probability measures 7 the amount of information a new piece of information is expected to carry(Cover & Thomas, 2006). Regardless of the approach, for practical applications, we need mutually exclusive events, probabilities of the set of events that sum to one and stationarity. If these assumptions do not hold true it is not reasonable to use mathematical expectations as an input to a decision tool. In the wildfire risk literature this basic model is used as a building block to assess wildfire risks. As an example of this, it is worth quoting the abstract of a paper by Joe H. Scott called An Analytical Framework for Quantifying Fire Risk and Fuel Treatment Benefit at length. This abstract sums up the basic approach that is used with minor adjustments in large parts of the wildfire risk management, cost-benefit and cost-effectiveness literature. Wildland fire risk is quantified as expected annual loss ($ yr-1or $ yr-1ac-1). Fire risk at a point on the landscape is a function of the probability of burning at that point, the relative frequency of fire behaviors expected if the point does burn, and the response of various resources to those expected fire behaviors (net value change). The probability of fire burning at any point on the landscape is a function of the spatial arrangement of fuel, weather, topography and ignition locations surrounding the point of interest, but not characteristic of the point itself. Relative frequency of fire behavior is a function of the local fire environment and the likelihood of burning at various portions of an assumed elliptical fire. Fire loss is assumed to be a function of fire behavior characteristics. Fire behavior can be measured by the Fire Intensity Index (FII), the common logarithm of fireline intensity. A risk reduction treatment is 8 an investment of capital today for a benefit to be reaped in the future. The benefit of a risk reduction treatment is the present value of the difference in risk with and without treatment. Cost is the present value of current year and future treatment expenditures(J. H. Scott, 2006). Right from the beginning, risk is defined as an expectation. That is the sum of a mutually exclusive set of events (𝑥𝑖) multiplied by their respective probabilities(𝑝𝑖), ∑ 𝑥𝑖𝑝𝑖𝑖 . In this case the events are losses measured in dollar terms. To find a value for this measure of risk we need to identify all of the relevant things that could incur losses and put a dollar value on those losses. In wildfire besides trees and built infrastructure, which have a market value, we may wildlife, aesthetic, public health values etc… that are more d ifficult to price. Economists have methods for doing this but before considering probabilities we already have an expensive problem in terms of deciding which losses to be included and finding the data and appropriate methodology to price them. To assign a probability to each of the loss events Scott outlines a method that involves compounding probabilities. To calculate this probability, we need to calculate a probability of a fire burning around the point of the loss event, the probability of a dangerous fire event occurring at that point, which itself is a function of environmental variables and then the probability that given a fire of sufficiently high Fire Intensity Index damage will be incurred to the value at risk (this itself is a matter of chance). All of this is data and computationally intensive. Real world applications of Scott-type analytical framework requires reducing the details of the model to only their most salient features. For instance, modelling each tree or piece of fuel and the spread of fire between them is too resource intensive so the fire behaviour is modeled at high resolution (e.g. behaviour is aggregated over cells 10x10m, 30x30m, 100x100m, etc). It is also necessary to choose only the highest value assets at risk to limit the cost of 9 modelling. Whether or not this model is still a reasonable way of defining risk and an effective decision-making tool depends on how much of the underlying reality is lost in this aggregation process. An examination of this model lossiness and its implications for decision making is the subject of Chapter 2. The consensus in the literature so far is that the Scott-style approaches, using expectations, is reasonable and it has been implemented in the United States (Fairbrother & Turnley, 2005; Scott et al., 2013), Australia (Tolhurst et al., 2008) and Spain (Alcasena et al., 2017). These expectation models use simulation models to calculate probabilities largely because of the spatial aspect of the problem. Finney (2005) points out that these probabilities are dependent on nearby ignitions, fuels, topography, and weather. This makes frequency statistics not useful for this kind of problem. A Canadian review (Johnston et al., 2020) of the subject of wildfire risk presents a standardized measure of risk that draws on the papers cited above. This again, takes the form of an expectation: 𝑅𝑖𝑠𝑘 = ∑ ∑ 𝑝(𝐹𝑖)[𝐵𝑖𝑗 − 𝐿 𝑖𝑗]𝑛𝑗𝑁𝑖 (1) Here, B and L are benefits and losses for the 𝑗𝑡ℎvalue at risk from the 𝑖𝑡ℎ category of fire behaviour. 𝑝(𝐹𝑖) is the probability that the 𝑖𝑡ℎ fire behaviour encounters the 𝑗𝑡ℎ value. This captures all of the characteristics of Scott’s verbal description. The Canadian review identifies a further modelling challenge in the need to assess the vulnerability of an impact, (𝐵𝑖𝑗 − 𝐿 𝑖𝑗) to 10 fire behaviour. For instance, changing the roof of a house from cedar shakes to metal roofing will reduce the vulnerability of the house to losses from fire behaviour. This necessitates a transfer function from the fire to the asset at risk, and another complication is now added. This requires parameterizing the transfer function and its interactions with the other transfer functions. This means collecting more data. This is costly in terms of time and resources and , in the case of wildfire, depending on the value at risk the parameters do not generalize to other regions. For example, if parameters are estimated for a transfer function that relates fire behaviour in Florida to the probability of home loss, it is not appropriate to use these parameter estimates in Montana. The review identifies research that has been done on how to calculate these probabilities using simulation methods and impact transfer functions for a wide variety of different values that include property, ecological, business and cultural values. This thesis takes a different approach to this attempt at completeness in modelling. One way of understanding the work in Chapter 3 is as an answer to the questions: “Are there alternative ways of thinking about uncertainty besides the gambling model? What is the minimum amount of information necessary to make a resource allocation decision on wildfire fuel treatments? Do we need to calculate risk at all, as defined by an expectation of loss or net value change?” 1.3 The Costs of Wildfire in Canada Interest in fire mitigation efforts is, at least partially, motivated by a desire to limit increases in wildfire management expenditures. Stocks and Martell (2016) collected data and found that Canadian national expenditures (in 2013 Canadian dollars) have grown from $290 million in 1970 to $900 million in 2013 and the average over the decade from 2006-2015 exceeded $800 11 million. Expenditures on recent memorable fires have also made the headlines. During the 2017 fire season in British Columbia the estimated wildfire fighting cost alone was $568 million (Abbott & Chapman, 2018) and insurance claims from the Horse River fire in northern Alberta that burned through neighborhoods in Ft McMurray and a vital oil producing region amounted to $3.7 billion (Insurance Bureau of Canada, 2017). Looking to the future, fire management expenditures in Canada are expected to continue to increase as a consequence of climate change. In one analysis making use of climate change projections and estimated changes in the Climate Moisture Index calibrated to the expenditure data from the Stocks & Martell paper the authors estimated that annual average expenditures will increase to nearly $1 billion under a low-emissions scenario and $1.4 billion under a high-emissions scenario by the end of the decade (Hope et al., 2016). 1.4 Prior Work on Economic Evaluation of Wildfire Fuel Treatments One general description of wildfire fuel treatments includes reduction of surface fuels, increasing the height to live crown, decreasing crown density, and retaining large trees of fire-resistant species (Agee & Skinner, 2005). The economic evaluation of these treatments involves finding whether or not the benefits of these activities outweighs the costs and ranking different treatment options where benefits do exceed costs. Before focusing exclusively on wildfire fuel treatments, it is worth understanding how wildfire in general has been treated in the economics literature as it helps to illuminate how the literature came to evaluate wildfire fuel treatments as it currently does. Consistent with textbook economic theory, the wildfire economics literature is generally concerned with finding optimal expenditures under equilibrium conditions of suppression and 12 pre-suppression efforts (Milne et al., 2014). This approach to analyzing wildfire expenditures begins with the work of Sparhawk (1925) who introduced the idea of Least Cost Plus Damage. The idea was to model a production function that takes pre-suppression expenditures as an input and the output is the decrease in fire damage and suppression costs. Pre-suppression in this case can refer to staffing fire watchtowers, purchasing water bombers or hiring seasonal staff prior to the fire season. An analyst using this model should set the marginal pre-suppression expenditure increase equal to the marginal decrease in damages to get the optimal expenditure. Any increase in pre-suppression expenditures beyond this point provides less benefit than the costs. This model is represented graphically in Figure 1. 13 Figure 1 Sparhawk's Least Cost Plus Damages model for pre-suppression expenditures. Total Costs to society are minimized when the marginal pre-suppression cost is equal to the marginal damage and suppression costs. Least Cost plus Net Value Change was an addition to this model that recognized that fire can have positive effects for wildlife and ecosystems generally. Donovan and Rideout (2003) updated this model to account for the fact that suppression is not a direct output of pre-suppression expenditures (it is an input itself that is limited, but not determined by pre-suppression expenditures) and these expenditures are not, in fact, negatively correlated. In a busy fire year an increase in pre-suppression expenditures could make even higher variable suppression expenditures possible because more resources are available to be used. In practice, these models are impractical because we never have enough information to know where marginal benefits are equal to marginal costs so we cannot find the optimal allocation of 14 pre-suppression expenditures. Despite repeated efforts to improve this model Least Cost plus Damage approach has found little utility in the field. Martell (2001) noted that it is likely impossible to parameterize a production function that relates expenditures on fire towers, aircraft, firefighters etc. to fire damages. He also notes that the model is deterministic and does not account for the stochastic nature of fire behaviour and damages. Despite this, much of economics literature on fuel treatments works with models in the spirit of Least Cost plus Damages and tries to find the optimal expenditures where expected marginal benefit is equal to expected marginal cost. Most of these studies focus on a single benefit like reduced fire intensity or increase in deer harvest (Loomis et al., 2003; Thompson et al.,2015,2017; Gannon et al., 2019). Hesseln (2000) completed a review of the literature on the economic analysis of prescribed burning from the 80s and 90s in which she describes four requirements of a framework for use in fuel management: 1. A description of the relationship between fuel management strategies and the change in fire behaviour. 2. A description of the relationship between observed fire behaviour and economic variables over the longer-run. These could include suppression costs, timber and property values, tourism and recreation as well as non-market values. 3. A consideration of the risk management aspect of fuel management decisions. 4. Collecting and storing data in a way that is conducive to analyses of the economic efficiency of fuel management strategies. None of the studies Hesseln reviewed met these criteria in full. While there has been some improvement since the year 2000, the studies that have been published since Hesseln’s review 15 continue to have shortcomings with respect to these criteria. It is worth noting that the first three of these criteria correspond to the expectation-value definition of risk presented by Scott (J. H. Scott, 2006) and in the Canadian wildfire risk management review (Johnston et al., 2020). In particular, the first criterion relates to how probabilities of an area burning with a given severity change with a change in fuel structure and the second criterion related to notion of a values susceptibility or vulnerability in the Johnston et al. paper. In my review of recent literature, there were twenty-three (23) studies examining fuel treatments or fuel management that included a cost-effectiveness or cost-benefit analysis of fuel treatments since Hesseln’s review. Since 2000, considerable effort has been made to improve the understanding of how fuel mitigation changes fire behavior (Barnett et al., 2016; Fernandes, 2015; Kalies & Yocom Kent, 2016). In some cases, these improvements have been integrated into later economic analyses in the literature. Eleven of the studies made use of a fire behavior model to describe the relationship between fuel management strategies and change in fire behavior. These included studies in Australia (Bradstock et al., 2012; Florec et al., 2019; Penman et al., 2014), China (Liu et al., 2010), Italy (Elia et al., 2016), and the United States (Jones et al., 2017; Mercer et al., 2007a; Prestemon et al., 2012; Stockmann et al., 2010). The remaining studies (Deloitte Access Economics, 2014; Heines et al., 2018; Keegan et al., 2004; Loomis et al., 2003; Mason et al., 2006; Mercer et al., 2007b; Snider et al., 2006) did not make use of simulation modelling and instead made use of case studies and linear regression estimates to model change in fire behaviour as a function of fuel mitigation efforts to evaluate economic tradeoffs. With the exception of case study approaches both simulation and regression approaches make use of the expectation model in costing risk. 16 In general, there have not been improvements in understanding the relationship between wildfire behavior and economic variables over the longer run. I did not find any papers that addressed economic returns to forest activity beyond initial losses, changes in the capital structure of the industry or the labor market equilibrium. The studies almost exclusively consider short-run costs and use observed case-study values or simple correlations to describe these relationships. There is a good reason for this. The second-order effects (e.g., changes in long-run employment, differences in tree growth rates) and non-linear changes in values-at-risk (e.g., regime changes in ecosystems, or the long-term shutdown of a mill) under some wildfire behaviour scenarios make it very difficult or impossible to parameterize a functional relationship between fire behaviour metrics and measurable changes in the values-at-risk. It is interesting to note that there appears to be a trade-off in this literature between more realistic efforts at simulating wildfire behaviour and addressing multiple values-at-risk. Simulation intensive approaches tended to only look at a single variable like timber damages (Liu et al., 2010), property (Bradstock et al., 2012; Elia et al., 2016; Penman et al., 2014; Stockmann et al., 2010), suppression costs (Prestemon et al., 2007, 2012) and erosion (Gannon et al., 2019; Jones et al., 2017). The case study approaches (Deloitte Access Economics, 2014; Mason et al., 2006) are able to include greater numbers of values-at-risk and obtain a bigger picture of the potential magnitude of benefits from fuel treatments but they are limited in their generalizability as the sample size is only a single treatment and single fire. An important take away from this trade-off observed in the literature is that modelling is expensive. With a finite budget it is necessary to choose between capturing the impacts of fire on a wide variety of values-at-risk or more accurately simulating fire’s impacts on a single value of interest. 17 An additional weakness of these studies is the implicit assumption in most models that once a treatment has been completed it has an infinite life, it maintains effectiveness for all times and that a fuel treatment has a 100% likelihood of being tested. Two studies made use of Bayesian Network analysis to model the interaction between uncertain ignitions and fire spread and damage to property (Penman et al., 2014, 2020). These models captured the likelihood that a fuel-treatment would be tested. Perhaps the most complete implementation of the Scott approach to estimating wildfire risk is to be found in the Thompson et al. (2017) model. Three measures of effectiveness of fuel mitigation were evaluated in the simulation model: avoided area burned; avoided suppression cost; and avoided damages. This model required pre-existing model inputs that included maps of fuel treatment constraints, biophysically driven fuel treatment prescriptions and spatial risk assessment models as well as a pre-existing model of suppression costs and fuel hazards. The spatial suppression cost model (Hand et al., 2016) that was used as input to the Thompson et al. model made use of a least squares regression model under normality assumptions. This approach has been used elsewhere in forecasting suppression costs (Fitch et al., 2018) and points to one of the chief problems with using expectations as measures of risk. Many variables in the wildfire literature, including burned area and individual fire suppression costs exhibit extreme behaviour in a probabilistic sense. This limits the applicability of the usual probability models used in decision theory. This is a significant shortcoming of these approaches and what is meant by extreme behaviour and how its consequences for applications in decision theory is explored in the next chapter. 18 Chapter 2: Fat Tails in Wildfire Cost Data When studying risk, the most common assumption is that events and their probability can be modelled using a normal or Gaussian distribution. While the Gaussian distribution does appear to model some natural phenomena nearly exactly (e.g. a microscopic particle moving in still air (Brezezniak & Zastawniak, 1999), it finds greater use because of its mathematical convenience. Statistics calculated from a data generating process that is Gaussian have the property of converging quickly and the Central Limit Theorem allows that if ANY set of random variables that are identically and independently distributed with a mean zero and finite and stable variance, it will converge to a Gaussian distribution (Greene, 2008 pp.1050-1056). Strict Gaussian behaviour or even the assumptions of the Central Limit Theorem are rarely observed outside of pure physics; however, there are also many cases in nature where the generating process of events does appear to be approximately normal and careful use of statistics as it is taught in undergraduate programs is appropriate (Lyon, 2014). There are many phenomena both in the natural world and social science where the Gaussian assumption is unreasonable and the assumptions of the Central Limit Theory do not hold, even approximately. The study of wildfire and its impacts on social phenomena represent a case where the assumptions do not hold. We have to turn to tools of probability that deal with the non-normal. In the study of probability, a fat-tail is a characteristic of a probability distribution that is skewed (non-symmetrical) and/or leptokurtic (more probability mass/density in the tails and less near the centre of the distribution) relative to a normal or Gaussian distribution. Distribution functions that are fat-tailed generate extreme values that would be unexpected if the distribution function being used to model the random variable is Gaussian. In the financial news, you may read or hear that a pricing event was “10 sigma!”. This means that the modelers generally use a 19 normal or log-normal distribution of changes in price and that the change in price was 10 standard deviations from the mean. In frequentist terms, this is an event that would be expected 1 in 5.249e+20 years. Vanishingly unlikely if you believe the price movements follow a normal distribution. Yet large sigma events, market crashes, occur every few years. This suggests that we have the wrong model for price movements (Mandelbrot & Hudson, 2006). If a series of events occur that are extremely unlikely under the Gaussian model it is reasonable to explore the idea that maybe a different distribution function needs to be selected to model the data generating process. The study of these different distribution functions and their applications is called Extreme Value Theory (Embrechts et al., 1997). If a fat-tailed distribution function captures the extreme nature of a dataset it creates problems for the use of standard models of statistical inference such as least squares and the expected value models in risk modeling. These challenges include non-convergent estimates of distribution parameters or probabilities (Taleb, 2020). In the case of the Bayesian approach used in two of the papers described in the review (Penman et al., 2014, 2020), the priors can be shown to be not observable if the data generating process is fat-tailed and the necessary conditions for a convergent Bayesian calculation are violated (Diaconis & Freedman, 1986). Unfortunately for making decisions about wildfire fuel mitigation, there is evidence that many datasets related to wildfire events follow fat-tailed distributions. Specifically, there is evidence that a class of power law distributions, some of the most extreme distributions, are a better fit to many wildfire data series than a normal or log-normal distribution. Several papers have already mentioned or explored this topic. The Wildland Fire Leadership Council observed that 94% of suppression costs were caused by 1.4% of fires on US Forest Service lands between 1980 and 2002 (Strategic Issues Panel on Fire Suppression, 2004). In Canada 4% of fires exceed 200 ha in 20 size but they account for 99% of the area burned (Tymstra et al., 2019). Small numbers of events accounting for very large amount of cumulative outcomes is characteristic of fat-tailed distributions (Lorenz, 1905). Distributions in the power law family have been used to analyze droughts and fire size extremes in Ontario Boreal Shield (Beverly & Martell, 2005), wildfire sizes in Portugal (Bermudez et al., 2009) and Los Angeles County over 50 years (Schoenberg et al., 2003) , and area burnt throughout the United States since 1825 (Ramesh, 2005). With the exception of work by Wildland Fire Leadership Council these results address area burned and not costs. In the next subsection I describe the results for data from British Columbia that support the idea that wildfires here follow a similar power law distribution. I then subsequently model that distribution and discuss some of the implications of fat-tailed cost data in wildfire modelling. 21 2.1 A Power Law model for British Columbia Suppression Cost Data Figure 2 Individual fire suppression costs ordered from least costly to most costly. The vast majority of fires result are less than $500,000 but there are very few at the end that are orders of magnitude more expensive. Here 78 of the 3167 fires in the dataset, or 1.8% of observations result in 60% of the Suppression Costs. This is indicative of fat-tailed distributions. In the suppression costs data (2017-2019) provided by the BC Wildfire Service, 1.8% of the fires accounted for 60% of costs (Figure 2). This is not as extreme as the data from California but is still indicative of a less extreme parameterization of a power law distribution. In Figure 3, we can see the histogram of suppression costs for the period 2003-2019 in the Cariboo and Kamloops fire district. In this data 0.8% (99 of 12,080) of fires resulted in 60% of suppression costs. In absolute terms, that is $364 million over 17 years for just 99 fires. If we take the average $- $5,000,000 $10,000,000 $15,000,000 $20,000,000 $25,000,000 $30,000,0000 500 1000 1500 2000 2500 3000 3500Cumulative ObservationsSuppression Costs - BC (2017-2019)22 suppression cost for a fire in this data set, it is about $60,000. Clearly this average is not informative if we are trying to make decisions that will reduce costs from fires that are not well-controlled using existing suppression methods. Figure 3 Far from 'Normal', this histogram shows a very long tail for suppression costs in the Kamloops and Cariboo Fire Districts from 2003 – 2019. We can safely reject the null hypothesis that a normal distribution is a suitable fit for modeling this data. These figures are indicative of non-normality and fat-tailedness being reasonable models for this data but more can be done to assess the appropriate probability model for this data. I was provided with a limited data set from BC Wildfire Service that has a collection of cost data from wildfires from the period 2000-2019. This data is only for the fire-prone Cariboo and Kamloops fire centres in British Columbia. Of the 12,080 fires included in this data set , 11,145 fires were 0100200300400500600700$500,000 $1,500,000 $2,500,000 $3,500,000 $4,500,000 $5,500,000 $6,500,000 $7,500,000 $8,500,000 $9,500,000 $10,500,000 $11,500,000 $12,500,000 $13,500,000 $14,500,000 $15,500,000 $16,500,000 $17,500,000 $18,500,000 $19,500,000 $20,500,000 $21,500,000 $22,500,000 $23,500,000 $24,500,000 $25,500,000 More # FIRESBINHistogram of Fires with Greater than $60,000 in Suppression Costs 2003-2019Kamloops and Cariboo Fire Districts23 included in the analysis. Many fires were reported as having costs of $1 or $0. In private communications with members of the BC Wildfire Service they suggested that these were fires either with no data or where the team leads had not recorded costs above the baseline expenditure. After discussions with the BC Wildfire Service we decided to set the minimum bound at $100 (Personal Communication, Research and Innovation Officer, BC Wildfire Service, October 13, 2020). For legal reasons some of the most expensive fires in British Columbia history (2017, 2018) were removed as well. This does not change the general result of this analysis. To evaluate potential probability models of this suppression cost data, I made use of the poweRlaw R package (Gillespie 2014) . This package builds on the work of Clauset and Shalizi in using bootstrap methods to estimate fat-tailed distribution parameters (Clauset et al., 2009). The Pareto distribution and, more generally, power law distributions are known to be ‘scale-free’. One of the consequences of this is that we can look at different sub-sets of the data and see a similar pattern in the data. The defining characteristic of a power law is that the probability of observing a given quantity varies inversely as a power of the observed value (Newman, 2005). Even with this limited data set we should still be able to see if this data roughly fits a power law. The Pareto distribution is normally used as a model for data in the tail of a distribution. poweRlaw provides a maximum-likelihood estimator for estimating where the tail begins for the ‘tail’ of the data. For the suppression cost data for the Cariboo and Kamloops fire centres the estimated 𝑥𝑚𝑖𝑛 value is $60,000. 24 Usually, the first step in exploring fat-tailed data is generating a Zipf plot of the data above the x_min threshold. This is a plot of the logarithm of the frequency of binned data versus the logarithm of the sorted x values themselves. The data should roughly follow a straight-line over most of the observations for a Pareto distribution to be considered. It is common for the less frequent data to not sit exactly on the line as there are so few observations at the far end of the tail. As can be seen in Figure 4, we do see a roughly straight line and it is worthwhile evaluating parameters and using some of the limited tests that are available. Figure 4 A Zipf plot of the fire suppression data. This plot shows the logarithm of the sorted suppression cost data versus the logarithm of the frequency of the events. Data that follow a roughly linear pattern suggest that a power law or pareto distribution might be a reasonable model for the data. 25 Using a bootstrap method (this method is described in Appendix A), we can estimate the shape, or 𝛼 parameter of the distribution. The 𝛼 gives the slope of a fitted line in log-log space like the one depicted in Figure 3. The bootstrapping procedure converges on a value of 𝛼 = 1.764 after 1000 iterations2. Using this same bootstrapping procedure, we can also generate a p-value for the following test: 𝐻𝑜 : 𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟 𝑙𝑎𝑤 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑛𝑜𝑡 𝑏𝑒 𝑟𝑢𝑙𝑒𝑑 𝑜𝑢𝑡; 𝐻1: 𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟 𝑙𝑎𝑤 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑢𝑙𝑒𝑑 𝑜𝑢𝑡. The p-value given through this procedure is 0.431 which suggests we cannot rule out a power law or pareto model for the data generator. We can never ‘accept’ the hypothesis that the true data generator is a power law but we can test it against another model with a fat tail, with the most common being the log-normal (Clauset et al., 2009). The test is a ratio of the log-likelihoods of the two distribution models developed by Vuong (1989). First an 𝑥min is set for both models and we use the model parameters set by the bootstrap model for the power law and re-evaluate the log-normal parameters given the 𝑥minthat has been set. The test only works when the 𝑥min are equal. For each value in the data set greater than the 𝑥min the log-likelihood is calculated for each distribution and the ratio of the two numbers taken. In this case, the power law is the numerator and the log-normal value the denominator. If the log-likelihood ratio is positive, the power law is the better fit and if it is negative the log-normal is 2 Using a maximum likelihood estimator also gives 𝛼 = 1.76. I use the bootstrapping procedure here because it offers the possibility of doing the hypothesis test at the same time. 26 the better model. These ratios can be graphed over the range of values from the 𝑥min threshold to the maximum value in the dataset. Figure 5 shows these results. Figure 5 Log-likelihood ratios of wildfire costs evaluated for a power law and log-normal distribution. A positive result favors the power law and a negative result favours the log -normal distribution. A convincing case for either model would require that the log-likelihoods be positive or negative over the full range of fire cost values. Instead, we see that power law performs better over many of the values just above the x_min=60,000, the log-normal does better over the mid-range, and then for extreme values the power-law is the better model. A strong case cannot be made for either model. A convincing case for either model would require that the log-likelihoods be positive or negative over the full range of fire cost values. Instead, we see that power law performs better over many of the values just above the 𝑥𝑚𝑖𝑛 = 60,000, the log-normal does better over the mid-range, and then for extreme values the power-law is the better model. A strong case cannot be made for either model over the full range of data. We do know, however, that from the perspective of improving the safety of our values at risk that we are interested in the most extreme fire events, 27 the ones not handled well with the current suppression efforts. For these fires, the ones most likely to result in evacuations, large timber losses and lost structures the power law is convincingly the better model for the data. We also know that there are some very large fires, with very extreme suppression costs even farther out in the tail that have been excluded from the dataset for legal reasons. This data would likely push things even farther in favour of the power law as events above $25 million have extremely low likelihood of occurring under a log-normal assumption. Suppression costs in British Columbia appear to be reasonably modeled by an extreme, fat-tailed distribution that looks nothing like the normal or Gaussian distribution that is usually used when conducting cost benefit analyses. When one adds in evacuation costs, timber losses and public health impacts from smoke on the largest fires, one can see that the largest, most damaging fires are many times more damaging than the most frequent fires and that it is meaningless to talk about averages. In the following sections I assume that non-suppression costs and damages follow a similar power-law distribution. This is a strong assumption based on a handful of observations. This assumption does not receive the rigor it deserves because of the lack of data availability. I make this assumption because we know that between 1980 and 2019 each year saw somewhere between 4000 and 12000 fires in Canada (Canadian Wildland Fire Information System, 2019) . One of these fires, the 2016 Fort McMurray wildfire, was the most expensive natural disaster in Canadian history in terms of insurance claims (Mamuji & Rozdilsky. 2019). Most of these fires are extinguished with minimal damages to other values, even timber values. In between we have the fires with high suppression costs that also result in large damages. In 2017, just two of the fires in BC resulted in $127 million in insurance claims (CatIQ, 2020). In 28 addition to these observations, we know that suppression efforts are prioritized based on proximity to critical infrastructure, property etc. (BC Wildfire Service, 2010). Given these observations it would be surprising if other wildfire costs could not be reasonably modeled with the same distributions as suppression costs. 2.2 Implications of Fat-Tails for Modeling Risk and Cost-Benefit Analysis In the Organization for Economic Cooperation and Development’s guide to Cost-Benefit Analysis (CBA) for environmental problems (Pearce et al., 2006) the authors describe two general approaches to dealing with uncertainty in a CBA. The first category of uncertainty was what they described as ‘risk’. Risk, in their terminology, is uncertainty sufficiently well-behaved that each state of the world can be assigned a probability and the estimates of the probabilities do not move around very much as new data comes in. In other words, uncertainty can be described as risk when the Central Limit Theorem applies and probability estimates converge with each new observation (Feller, 1968). In the context of ‘risk’ Pearce et al. recommend using mean values or expectations of the observed costs and benefits used in the CBA. For fat-tailed distributions, the risk assumption is not appropriate. Depending on what the true, underlying data-generating process is, estimates of mean and variance fail to converge, so further analysis is not possible. Expectations are undefined and cannot be used. In general, for data that can be described with fat-tailed distributions, new observations occasionally change the parameters used to best fit the distribution. In other words, probability estimates don’t necessarily converge and as a result, statistical estimators of variance and mean also do not converge (Taleb, 2020). Another way of describing this is that the average benefits and costs are 29 not a meaningful measure of the possible outcomes. Using the average to summarize data that is drawn from a fat-tailed distribution ignores the extreme possibilities that can be several orders of magnitude larger than the average as we saw in the BC wildfire suppression cost data above. The other problem with this is that taking the expectation or average discounts catastrophic or ruinous events assumes that we can take every future path and choose the best one. This is not how it works in reality; we often only get one chance at getting things right in life. In the most extreme instance, if I make a poor decision that leads to my death I don’t get to go back in time and make the decision again. If death is one of the mutually exclusive outcomes in the probability model it gets the same weight in the calculation of an expectation even though it prevents all future decisions from being taken. Likewise in a wildfire context, if we calculate event frequencies to estimate average statistics when there are path dependencies, we ignore the fact when a community is burned over or the loss of timber supply shuts down a local mill there is no going back in time. If you die, or go bankrupt, there are no second chances. 2.2.1 Pareto-Distributed Damages and Cost-Benefit Analysis If the underlying distribution of random variables is not normal and instead follows an asymmetric distribution with fat tails, decision tools can break down. Here I illustrate the nature of these difficulties with non-convergent parameter estimates under Pareto distributed costs. In cost-benefit analysis the standard distributional assumption is Gaussian or Normal. Under this assumption, estimators of expectation and variance are well-behaved (finite) and sample estimates converge on the ‘true’ value quickly. This convergence is a consequence of the Central Limit Theorem (Feller, 1968). The most common alternatives are the log-normal, exponential and Pareto models, with the Pareto being the most extreme model, in the sense that the extreme 30 high-magnitude, low probability events in the far tails are nigh impossible under a Gaussian assumption. If the data is Pareto-distributed, it has implications for decision making using averages, as is normally the case in cost-benefit analysis. The limited available data on suppression costs point to the Pareto/Power Law model being appropriate, but even if it is not, other fat-tailed models lead to the same conclusions about decision making. To see the effects of these tail events on cost-benefit-type calculations, I modelled 10,000 fire events 1,000 times using the Pareto distribution with α=1.005. For each of these runs, I split the events into the 99% of events with the lowest costs and 1% of events with the highest costs and take the ratio of the 1% draws to the 99% draws. In Figure 6, I have plotted the value of this ratio for each of the 1,000 simulations of 10,000 fire events. There are two things to note: first the sum of the top 1% is always greater than the sum of the 99% events and second, it never seems to settle down. This second point is a consequence of the fact that for a Pareto distribution with α<2, the variance is infinite. Government budgets must be set at some amount less than infinity and available resources are eventually exhausted so this raises a question as to the validity of the model, but we have yet to reach a point where a government refuses to spend another dollar on firefighting, so this seems like a reasonable assumption for argument’s sake. One implication of an underlying process that follows a Pareto distribution is that the next observation may be orders of magnitude higher than the previous sample mean and greater than any previous observation with a much higher probability than is possible under a normal distribution. How can we use a financial decision-making model that depends on the mass of observation being around some measure of central tendency (usually the average) when most of the pain occurs with only 31 1% of the events? The central tendency is not relevant to making decisions in this kind of environment. Fundamentally, in this model, if we do not get a handle on the top 1% of the fire events, we have not made good use of the taxpayer dollar. Put another way, if the fuel management efforts are not limiting damages in the most extreme fire weather then the damages will overwhelm any of the benefits that have accrued in the bottom 99% of the distribution. I could choose a less extreme Pareto distribution or model events with a log-normal distribution and the result could be less extreme, but we would still find the same thing. A small percentage of the events will completely dominate and, if they are not managed, efforts are being misdirected. 32 Figure 6 The ratio of the 1% largest cost events to the 99% bottom cost events over 1,200 simulations of 10,000 fire events. This is a simple probabilistic model of fire damages, but it makes an important point. Very few fires cause the vast majority of damages. Examples of extreme fire behaviour show us that these rare fires can be quite insensitive to one-time fuel reduction (Graham et al., 2009; Jain et al., 2012) and we should focus on the 1% that make an impact to the worst-case fires, if that is possible. Most of the policies and fuel treatments being suggested in the news and the cost-effectiveness literature are not a good use of resources in many ecosystems under most distributions of fire weather when applied on their own. Docking the tail of the distribution in most fire-prone ecosystems requires on-going management (Hoffman et al., 2018; Jain et al., 2012; Kalies & Yocom Kent, 2016). This has implications for the use of our traditional decision-33 making tools in economics. This simple model implies that the use of marginal economic decision making, using cost-effectiveness or cost-benefit analysis as an example doesn’t make sense with fat-tailed risks. No incremental increase in the removal of wood fibre from the forest is going to eliminate these catastrophic events. There is a threshold that has to be managed to. Any effort below that threshold counts for little. In economic terms, the elasticity of expected damages to treatment effort is going to be zero until the threshold is reached and then extraordinarily high. Under these circumstances, we need to think about opportunity costs differently. 2.3 Marginal Thinking Under Fire If an economist is faced with an externality that is best solved by government intervention that economist is next going to ask "What is the most efficient use of tax payer dollars for managing the externality?" The answer to this question is always, "The best use of government dollars is found when the marginal benefits of an intervention equal the marginal costs." This is at the heart of the justification for the use of cost-benefit and cost-effectiveness analyses. Here the 'margin' is the increment of doing or not doing a project. If the cost-benefit ratio is positive, then the marginal benefits must exceed the marginal costs and the project should go ahead. In 1925, Sparhawk had already developed such a theoretical model to find the point at which expend itures allocated to the pre-suppression budget produced at least as much benefit as the avoided costs (Figure 1). In this model pre-suppression costs included equipment, planning and pre-season staffing requirements. It is assumed that within season suppression costs are not a control variable and that damages plus suppression costs are best modeled as a monotonically decreasing 34 function with decreasing marginal returns to pre-suppression expenditures. The optimal expenditure on pre-suppression activities that minimizes total costs is, of course, when the marginal cost of more pre-suppression activities is equal to the marginal damage plus suppression cost. This minimizes total expenditures on wildfire. One stand-out simplification in this model is that it assumes we know the damage and suppression costs as a function of pre-suppression costs with certainty in advance of the fire season. This is clearly not the case. In order to account for the uncertainty in these models, a rational expectations type argument is sometimes made (Martell, 2001). The rational expectations model makes the argument that the market has a price, 𝑝∗ , for something, say an interest rate, that incorporates all the current information about that thing. The actual realized price at some time in the future is the current price plus some random error, 𝑝 = 𝑝∗ + 𝜖. If we assume that the error component has mean zero, 𝐸(𝜖) = 0 then our current price is our best guess of the future price, 𝐸(𝑝) = 𝑝∗. If you accept a weak form of the efficient markets hypothesis (Fama, 1970), you could take this argument and apply it to doing cost-benefit analyses for government projects and assume that the price of inputs or outputs for the project should be today's prices because they are the best guess we have and we think the mean error is zero. This logic is implicitly or explicitly extended to wildfire costs when a cost-benefit analysis is performed. Even if you do accept efficient markets it really is not an acceptable assumption because there is no market for suppression, pre-suppression or damages from wildfire. There are no market participants collecting information 35 and trading in a market attempting to profit off of a counter-party's ignorance of a jurisdiction's fire suppression capabilities. This sounds silly and it is. The alternative justification for using mathematical expectations is making the less restrictive assumption that our models for forecasting these costs in all the cost categories are so good that 𝐸(𝜖) = 0 and we are just as likely to be wrong to the upside as to the downside in forecasting these costs. I am going to return now to the Pareto model to show why that is very unlikely to be the case for wildfire cost modelling. The expected value of a random variable with a Pareto distribution is given 𝐸(𝑋) =(𝛼𝑥𝑚)𝛼 − 1, 𝛼 > 1 (2) To estimate the expected value one usually finds the sample mean 𝑋 =1𝑁∑ 𝑥𝑖𝑁𝑖=1 . If the mean and variance of the random variables from which the 𝑥𝑖 are drawn are finite, the distribution of 𝑋 will converge to a Gaussian distribution as 𝑁→ ∞ by the Central Limit Theorem (Feller, 1968). This is standard, introductory probability. Unfortunately, for our purposes, the 𝑥𝑖 are drawn from a Pareto random variable with 𝛼 ≤ 2. The necessary conditions of the Central Limit Theorem do not hold because we know that the variance is defined as being infinite when 𝛼 ≤ 2 (See Appendix B for why this is). Fortunately, we can still work with this as Gnedenko and Kolmogorov (1954) have shown that there is a stable limit for random variables that are a function of random variables that are distributed with random variance and have a shape parameter 1 < 𝛼 < 2, but this distribution is not Gaussian. Instead, the distribution of the sample 36 mean statistics will converge to another fat-tailed distribution with infinite variance (Mandelbrot (1960) has a nice discussion of this in the context of income distributions). All of this to say, that if the Pareto distribution with shape parameter 1 < 𝛼 < 2 is a reasonable model of wildfire cost and damage events, the expected value will not provide a lot of useful information for decision making. To visualize this, I simulated 10,000 fire cost events. For every incremental event, I calculated the sample mean and plotted it against the expected value of the underlying distribution, 𝐸(𝑋) =(𝛼𝑥𝑚)𝛼 −1. The results of this simulation are in Figure 73. From the figure, we see that the sample mean both underestimates E(X) and converges very slowly. After 10,000 events the sample mean still has a long way to go. The defender of cost-benefit-style analyses would say that the finite sample mean seems to consistently underestimate the expected value and that, as a result, any future benefits would actually be underestimated. Unfortunately, this is also true of costs. Ultimately, in a cost-benefit analysis, we are asking whether there is a meaningful difference in two statistics, the expected benefits and expected costs. Further, we are comparing a ratio of these statistics to other ratios of statistics to find the preferred options when choosing between multiple projects. And even more, there are many values that we care about when it comes to wildfire risks (property and infrastructure damage, smoke exposure, ecosystem damage, timber values, just to name a few). Many of these are likely to follow some fat-tailed distribution because of the nature of fire and the way people are distributed on the landscape. Because of the bias in the estimators, we can expect errors to compound, not cancel. 3 The parameter selection for this model is described in Appendix C, these parameters are re -used for modelling in the discussion of long-run decision making in the second half of the thesis. 37 Figure 7 The expectation value (top blue line) and the slow convergence of ten sample means as 𝑵 → 𝟏𝟎, 𝟎𝟎𝟎 with the events being drawn from the Pareto distribution 𝜶 = 𝟏. 𝟎𝟎𝟓 and 𝒙𝒎 = 𝟏. There is very slow, uneven convergence to the expectation value. What makes these estimations even more suspect is that in applied settings the sample sizes are going to be quite small because these analyses have to be done on a community by community basis and there are few or none of the parameter estimates by which we can generalize our models. Different ecosystems are going to exhibit different fire behaviour under similar fire weather conditions. Even in the same ecosystem with similar fire weather, the human settlement patterns will be different with different values at risk. We are unlikely to have a data set with hundreds of fully costed fires, never mind tens of thousands. Using established methods, we simply have no reasonable means of finding the best use of taxpayer dollars. It is impossible to 38 say what the marginal value of any given project is at any given time with any confidence. It is necessary to consider an alternative way of thinking about the problem. 2.4 Implications of the Probability Model Given the impact of fat-tailed risks on using expectations can we make use of the second piece of the Pearce’s CBA risk framework discussed above? When we cannot assign stationary probabilities to events, Pearce et al. (2006) describe this as uncertainty and argue that a scenario planning approach should be taken. They offer that many practitioners build a matrix of likely scenarios, estimate the costs and benefits for those states of the world and assign them weights to come up with a weighted average of these states to estimate the cost-benefit ratio for the project or policy. There are two challenges with taking this approach for the problem of wildfire fuel treatments. First, we have to choose the scenarios and second we have to translate the things we can model (fire behaviour) into measurable changes into the things we care about (values-at-risk). In terms of choosing the scenarios, we start by noting that most of the fires in our dataset have suppression costs less than $500,000. These fires are extinguished before they escape suppression efforts and incremental management effort in the form of fuels mitigation is unlikely to offer significant benefit in limiting suppression costs or other damages from wildfire. A consequence of this is that we should focus on the larger fires that are not well-handled by suppression efforts. Because these large and difficult to control fires are rare and surprising, we 39 cannot get convergent estimates of their likelihood. In this case, as I argue below, it makes sense to take a ‘stress-test’ approach. This means choosing the most damaging fires as our scenarios that need to be protected against with fuels mitigation treatments. If a fuel treatment is justified under those conditions, then it is justified under all of the other potentially less-damaging outcomes as well. Finally, we come to the problem of choosing scenarios in which we can minimize the risk of wildfire or at least order them in terms of effectiveness. In order to do so, we would need to be able to translate fire behaviours that are induced by fuel treatment efforts into differences in outcomes of the values-at-risk that we choose to measure. What does success look like for a fuel treatment in relation to protecting infrastructure, homes, evacuation routes or environmental values or any other number of potential values-at-risk? In discussions with members of the BC Wildfire Service it came up repeatedly that there is no ‘sure thing’. No fuel treatment is going to 100% protect an asset under all conditions. What we get from a fuel treatment is greater ‘defensibility’. In poking at the notion of defensibility we did not settle on a quantitative metric that could be used as proxy for defensibility. The outcomes we really care about, the change in defensibility after a fuel treatment cannot be measured. This makes it difficult to create scenarios in the sense that Pearce et al. (2006) suggest. To get around this problem we recommend inverting the usual Cost-Benefit problem and rather than estimating probabilities we do not and cannot know or guessing at changes in defensibility that cannot be measured it is better to make use of the implicit knowledge of the wildfire fighters and estimating defensibility threshold. That is, given the cost of a treatment and the value of the 40 assets that are protected, we can estimate the likelihood that the treatment will succeed in making the assets more defensible in order to be worthwhile. This approach is described in the next Chapter. 41 Chapter 3: A Stress Test Heuristic for Wildfire Fuel Treatment Decisions When a data generating process is best modelled with a power law or other fat-tailed distribution versus normality, the theory tells us that assumptions about convergence of parameter estimates no longer applies. In probabilistic simulations, when we know what the underlying parameters are, hundreds of thousands of samples can be taken and we still do not get convergence to the true mean. In the real world, if we are not aware that we are dealing with power laws, it can be tempting to believe that we just need more or better data to improve understanding of the problem. This will never come in a power-law world. Wildfire is one of these worlds. This means we must accept that we are making decisions where statistical estimations of expectations as in the Scott (2006) model never converge. When we use simulation models to estimate these parameters and we do get convergence, this is likely to be a side-effect of underlying model assumptions that force convergence and do not reflect reality. This is a special kind of uncertainty that is not well addressed by the risk models described in Johnston (2020). Instead, we have to think about, how should one make decisions when expectations cannot be taken or the estimated parameters cannot be trusted? To answer this question a decision-making heuristic is developed that draws on the stress-test literature in banking. I first present the heuristic, then discuss some of the values that could be included in the model and the relative difficulty of collecting data and modelling those values and conclude with four case studies and discuss some of the shortcomings of the model. 42 3.1 The Defensibility Heuristic In trying to make decisions with uncertainty that cannot be parameterized or are difficult to parameterize in the context of wildfire mitigation strategies, I rely on a notion of ‘defensibility’. In discussions with firefighters, this idea of improving or creating defensible positions against a fire came up repeatedly. There is no guaranteed way of stopping a wildfire but it is possible to create defensible positions that improve the chances of success. In a modeling exercise, the defensibility idea gets at the change in probability of a fire damaging an asset , conditional on a fuel treatment being done. Unfortunately, we do not have a reliable means of estimating this change in probability for most values. The most reliable guide is the experience of the firefighters themselves. This is implicit knowledge, an idea introduced by Friedrich Hayek in his essay The Use of Knowledge in Society (Hayek, 1945). Implicit knowledge is knowledge that is particular to an individual’s experience in a specific context that either cannot be parameterized at all or cannot be parameterized at a sufficiently low cost that benefits of collecting the data outweigh the costs. The literature evaluating fuel treatments discussed in the introduction frames the question of fuel treatments as if we are getting a return on investment. This is an unfortunate analogy. When we evaluate a new highway or dam using cost-benefit analysis, the underlying assumption is that these investments will raise the total factor productivity of society (we will be able to generate more economic output per unit of inputs) and the cost-benefit analysis is a way of comparing results between different public investment options. This is not what we achieve with fuel treatments. There is no return on the investment. A well-designed fuel treatment is going to 43 reduce the risk of damage to productive assets. This is more like insurance. Expenditures on a fuel-treatment can be thought of as a buffer against the risks posed by catastrophic wildfire. With this in mind, we propose that fuel treatments should be assessed on whether or not they provide an adequate ‘buffer’ in the event of a wildfire. If a wildfire represents a risk to the community that we can put a price on, does the reduction in risk from a fuel treatment justify its price? We call this buffer, defensibility. Since security of a value can never be guaranteed by fuel treatments, we are concerned with how much more defensible the value has been made by the fuel treatment. 3.1.1 A Framework for Assessing Defensibility of a Fuel Treatment The goal here is to assess whether or not a fuel treatment increases the defensibility and reduces risks sufficiently to justify its price. Ideally, the process should be repeatable at low cost to evaluate hundreds or thousands of potential fuel treatments in a reasonable amount of time at reasonable cost. This is a simple, four-step process: 1. Identify the boundaries and methods used for the fuel treatment and their estimated costs. 2. Estimate the life of the fuel treatment and the likelihood that it will be challenged over its useful life or the costs of maintaining the treatment in perpetuity. 44 3. Choose ‘stress-test’ damages for the included cost categories.4 4. Calculate a defensibility threshold given the damages and cost of treatment. The defensibility threshold identifies the likelihood of success that a fuel treatment needs to have in order to be worthwhile. If the result of the calculation is 10%, the project evaluator needs to believe that the fuel treatment is going to protect the identified values-at-risk at least 1-in-10 times that the treatment is tested under extreme fire weather conditions. The BC Wildfire Service Fuel Management Prescription document prescribes that “prescribed fuel reduction targets should be sufficient to be effective to meet treatment objectives of reduced fire behaviour under 90th Percentile Fire Weather Index Conditions.” (BCWS Fuel Management Prescription Guidance 2019, n.d.) . It makes sense to be judging defensibility according to this standard because it will be the standard faced by practitioners, but the stress test can re-evaluate other values easily. We do this in one of the case studies to illustrate how other fire weather conditions can be evaluated. 3.1.2 Estimating Fuel Treatment Costs This is generally going to be straight-forward in the evaluation of most proposals. In practice, a project proponent submits their estimated costs along with a project proposal. For the purposes of our test cases we will be using historical fuel treatment cost data from the Forest Enhancement 4 The included cost categories will be the ones that stand out as the most important for the area of interest. 45 Society of BC (FESBC) and completed or proposed and accepted fuel treatment boundaries in the case study areas5. 3.1.3 Probability of Fire Testing a Fuel Treatment The probability of a fuel treatment being tested by a wildfire depends on two things: how long the fuel treatment is expected to do its job; and the likelihood of a fuel treatment getting tested by fire in any given year. Plants keep growing every year and so fuel treatments need regular maintenance in order to maintain their usefulness. Different ecosystems have different rates at which biomass returns to dangerous levels (Boer et al., 2009; Lydersen et al., 2017; Prichard et al., 2010; Prichard & Kennedy, 2014; van Mantgem et al., 2016). One study from the Black Hills of South Dakota in ponderosa pine that is similar to the low-elevation forests of the Okanagan study area found that it was necessary to re-burn treatment areas at an interval shorter than 15 years to maintain efficacy (Battaglia et al., 2008). In the case studies below, we use a sensitivity analysis to test different probabilities of a wildfire testing a fuel treatment over its lifespan. The values used in the sensitivity analysis are 0.05, 0.10 and 0.15. In modelling a fuel treatment that is maintained in perpetuity the probability that fire tests the treatment is assumed to be 1. 3.1.4 Stress Testing After the mortgage backed security crisis, as part of the Dodd-Frank Act (PwC Financial Services Regulatory Practice, 2014), banks were required to survive stress tests to show that they 5 The Forest Enhancement Society of BC is organization that distributes government funding to forestry projects that are designed to reduce risks from Climate Change and wildfire. FESBC funded the vast majority of the wildfire fuel treatment projects over the period of time that this study examines. 46 had sufficient assets on hand to survive a drop in the markets or a credit market freeze. This idea of a stress test is useful for fuel treatments. Does a cost-category or value-at-risk do better under the fuel treatment or not? In the case of a bank stress test, we know that if there are sufficient liquid assets on hand, a bank will survive a stress with certainty. In the case of wildfire, we do not have that certainty, we only have changes in defensibility. Adequacy of a wildfire fuel treatment in the event of an actual fire will depend on the prevailing fire weather and availability of firefighting resources. We address this in the final step. The costs that make sense to include in the stress test should be the ones that are appropriate to the area of interest. In the 150 Mile House example, there is a natural gas compressor station with a minimum $125 million value (its replacement cost). This critical infrastructure value dwarfs all other potential values that could be included in the analysis, so we include that as the primary focus of the stress test. As has been emphasized throughout this document, it is more productive to get the first order costs and benefits modeled. The net benefit of including every value we can enumerate is usually minimal for the purpose of deciding which fuel treatments to fund.6 There are critical assets that are essential for keeping the power grid online and food and essential services online that still need protecting. When this is true, the use of a heuristic is more sensible than using sophisticated computer modelling for decision making. We will run out of budget before we need to begin sophisticated modelling to choose between marginal projects. 6 And as I argue at the end of the chapter, this is likely true for making the case for funding these programs. At least during the early stage of deciding what to fund. 47 The point of stress testing is to choose numbers that reflect costs arising from a catastrophic fire in the tail of the cost distribution. From that starting point, costs can be scaled to the limit (potential damages are not infinite) to see what happens under the worst possible case. For each category, it is necessary to estimate the costs with and without the fuel treatment. 3.1.5 Calculating the Defensibility Threshold The final step is calculating the Defensibility Threshold. Again, the defensibility threshold identifies the likelihood of success that a fuel treatment needs to have in order to be worthwhile. If the result of the calculation is 10%, the project evaluator needs to believe that the fuel treatment is going to protect the identified values-at-risk at least 1-in-10 times that the treatment is tested under extreme fire weather conditions. To find this number we first calculate the difference in estimated costs, with and without treatment. We will call this Sum of Avoided Costs. It is possible that the fuel treatment does not get tested by fire over its useful life. So, the Sum of Avoided Costs needs to be discounted by the probability of getting tested by fire. Multiply the Sum of Avoided Costs by this probability. 𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐴𝑣𝑜𝑖𝑑𝑒𝑑 𝐶𝑜𝑠𝑡𝑠 = 𝑝𝑟.𝑏𝑢𝑟𝑛 ∗ 𝑆𝑢𝑚 𝐴𝑣𝑜𝑖𝑑𝑒𝑑 𝐶𝑜𝑠𝑡𝑠 (3) As discussed above there is limited research on what this probability should be so in the case studies below, we use a sensitivity analysis to test different probabilities of a wildfire testing a 48 fuel treatment over its lifespan. The values used in the sensitivity analysis are 0.05, 0.10 and 0.15. If we divide the estimated cost of the fuel treatment by the Adjusted Avoided Costs we have an implied probability that we denote d for defensibility. 𝑑 =(𝐹𝑢𝑒𝑙 𝑇𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝐶𝑜𝑠𝑡)(𝐴𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝐴𝑣𝑜𝑖𝑑𝑒𝑑 𝐶𝑜𝑠𝑡𝑠) (4) This implied probability is interpreted as the incremental improvement in defensibility that would justify the calculated avoided costs. The actual improvement in defensibility is estimated by the project evaluator because we do not have a cost-effective means of estimating changes in defensibility using computer simulation or any other tools. If the project evaluator believes that the improvement in defensibility is at least as good as d in percentage terms then the project should go ahead. If d is greater than 100% the project automatically fails the stress test and should not go ahead. A value for d greater than 100% suggests that even a guaranteed defensibility of the values-at-risk is insufficient to justify the expenditures on the fuel treatments. This will occur when the cost of the fuel treatments exceeds the Adjusted Avoided Costs. Different projects within the same study area can be ranked using the method. A lower d is better than a higher one. 49 It is intuitive that we want the adjusted avoided costs to be larger than the fuel treatment costs. So, all else being equal a smaller d is better than a larger d. We can get a smaller ratio, either by increasing the amount of avoided costs through the increase of the avoided costs or the decrease of fuel treatment costs. Suppose we have a 1 ha treatment that costs $10,000. This treatment is intended to protect $1,000,000 in assets. This gives a p=0.01. One would think that this is rather good! The project evaluator has to believe that the fuel treatment will improve the defensibility of the assets by at least 1%. In other words, if there were to be 100 extreme weather fire events that test the fuel treatments, the fuel treatment would have to protect against at least one more of them relative to business as usual. If the evaluator realizes that this 1 ha treatment is useless for improving defensibility of the assets included in the evaluation, then the project is rejected. Now, if you have two projects, one with a treatment cost of $200,000 protecting $1,000,000 in assets and another that costs $300,000, also protecting $1,000,000 in assets then their respective ds are going to be 0.2 and 0.3. If the evaluator agrees that both projects will make the values-at-risk defensible (relative to the base case) at least 2/10 and 3/10 more of the time, then the lower cost project is the one to go with. It provides the same insurance at lower cost. You may protest that the $300,000 project is provided protection at least 1 out of 10 times more than the $200,000 project but this ignores the fact that by choosing the $200,000 project we have $100,000 more to work with and we could, presumably, protect assets of an even greater value. The value at risk per dollar spent is reduced more with the $200,000 project. 50 3.1.6 A Simple Example Illustrating the Defensibility Heuristic To get a sense of how the cost adjustments and discounting of future value might work, imagine we had a large landscape level fuel treatment that was going to cost $1,000,000. The estimated avoided damages for this project are as follows: Table 1 Cost Categories and Damages The life of the project is 15 years, and we believe that the likelihood of the fuel treatment being tested during that time with a high severity fire is 30%. The Adjusted Avoided Costs are then $19,000,000*0.3=$5,700,000. Then we can calculate d. 𝑑 =$1,000,000$5,700,000= 0.18 (5) We can read this as saying that the fuel treatment is required to provide the conditions for keeping the stress test wildfire at bay at least 1/5 more times than the status quo without the fuel treatment in order to justify the expenditures. 3.1.7 Adjusting the Heuristic for a Regularly Maintained Fuel Treatment What if we wanted to account for the fact that the fuel treatment will be managed continuously with regular prescribed burning? Treat the costs of maintenance as a Perpetual Bond and discount the costs to the present at the discount rate r. Extending the example above, if we have 51 to pay $50,000/year in perpetuity and the discount rate is 5%. The new fuel treatment cost would be 𝑓𝑢𝑒𝑙 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡 = $1,000,000 +$50,0000.05= $2,000,000 (6) We also no longer have to adjust avoided costs. If a fuel treatment is being maintained in perpetuity, it will be tested at some point in its life. Our new d for the perpetually maintained fuel treatment is 𝑑 =$2,000,000$19,000,000= 0.11 (7) If the fuel treatment maintenance is only done every five years, we can adapt the Land Expectation Formula (Zhang & Pearse, 2011) to calculate the present value of the costs 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝑀𝑎𝑖𝑛𝑡𝑒𝑛𝑎𝑛𝑐𝑒 𝐶𝑜𝑠𝑡(1 + 𝑟)𝑡 − 1 (8) Where t is the maintenance interval and, in this case, takes the value 𝑡 = 5. We can recalculate the fuel treatment cost: 𝑓𝑢𝑒𝑙 𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 𝑐𝑜𝑠𝑡 = $1,000,000 +$50,000(1.05)5 − 1= $1,180,974 (9) And the defensibility threshold: 𝑑 =$1,180,974$19,000,000= 0.06 (10) 52 If the fuel treatment only needs to be maintained every 5 years, the required improvement in defensibility drops to only 6% in this example (Eq. 5). Without maintenance of the fuel treatment, the required improvement in defensibility was 18% (Eq. 10). This is justified by the fact that with the passage of time extreme wildfire will eventually test the fuel treatment. It is worth spending money on maintaining fuel treatments. 3.2 A Description of the Potential Values at Risk in Fuel Treatment Decision Making Before turning to the case studies implementing the Defensibility Heuristic it is worth discussing some of the values that could evaluated using this method and the difficulty or ease with which one would expect to be able to include them in the analysis. Historically, cost-benefit analyses of fuel treatments have primarily focused on suppression costs as the primary cost avoided in these exercises (Thompson et al., 2015). This is likely due to the fact that suppression costs are some of the more immediate costs realized during these crises and these costs are usually easier to document. However, in measuring the effectiveness of fuel treatments solely against avoided suppression costs, one excludes many of the other costs, and this potentially understates the effectiveness of any interventions evaluated in the cost-benefit framework. It is also possible for given areas of interest that suppression costs are not the most important risk. Obstacles to accurate cost accounting of wildfires arise when costs are incurred over a long period of time (e.g. employment insurance), or there are no market prices. As will be shown in the case studies, for the methodology described below, we decided to include only the values that are most likely to result in the greatest damages for the area of interest. It is the facts on the ground that are driving what is included, not a pre-determined set of values. This is justified because we 53 ultimately are not trying to find the optimal allocation (something we cannot find under the dynamic uncertainty we are faced with) but trying to rank projects based on how cost-effectively the fuel treatments improve defensibility of critical assets. Nonetheless, in the review below I look at different cost categories and how easy or difficult it is to collect data and model or estimate damages as a reference for future modelers. These categories came out of discussions with staff at the BC Ministry of Forests, Lands, Natural Resource Operations and Rural Development. I initially chose to filter cost categories on the basis of two criteria, availability of data, and ease of estimation. This is summarized in Table 2. The justification for the categorization is as follows: 54 Table 2 Cost Categories, Data Availability and Difficulty of Estimation. Data availability is low (*), medium (**), high (***). Ease of estimation: difficult (*), moderate (**), easy (***). Property, Infrastructure and Structural Damages and Loss Property valuations data can be obtained from the Catastrophe Indices and Quantification (CatIQ, 2020) database and the BC Assessment Authority (BC Assessment Authority, n.d.). Cost Category Data Availability Difficulty of Estimation Property and Structural Damages and Value Loss *** ** Suppression *** ** Destruction of Habitat ** *** Loss of Life * *** Clean up, Forest Restoration, Rehabilitation *** *** Flooding and Changes in Hydrology * * Fire Evacuation ** ** Smoke Costs * * Lost Recreation Value *** ** Productivity/Business Income Losses * * Timber and Forest Product Loss *** ** 55 This data is good down to the resolution of a forward sorting area in a postal code. Estimating the total potential loss is straight-forward with this data. To estimate a number for total property and infrastructure value-at-risk we will make use of the BC Assessment Authority data. Each jurisdiction in the BC Assessment data has value estimates for a variety of property classes that include Residential, Utilities, Major and Light industry, Commercial and Agricultural. Every property class has an estimate of the total bare land and land improvement value for each jurisdiction as well as the number of properties in that property class. We can calculate the average value for each property subclass for each jurisdiction in the study area. With these averages we will have to count individual structures in the study area and multiply the number by the average values to give the total value-at-risk. For the purposes of sensitivity analysis, we can adjust the number of structures that are damaged or destroyed. Suppression Costs Historical suppression cost data has been provided by the BC Wildfire Service. Regression approaches to estimating suppression costs have been used in the wildfire literature (Butry et al., 2001; Canton-Thompson et al., 2006; Fitch et al., 2018; Gebert et al., 2007; Hand et al., 2016). As discussed in the literature review, these methods have ignored the fat-tailed nature of the data which violates some of the fundamental assumptions in statistical analysis (e.g. The Central Limit Theorem). For the purposes of the methodology below the most straight-forward way of including suppression costs would be by using comparable fires in the region as case studies. In the Okanagan for instance, it might make sense for using the 2003 Okanagan Mountain Fire as 56 an analogue for suppression efforts under extreme fire weather conditions. Suppression costs ultimately were not included in the case studies below because it was determined with BC Wildfire Service staff that the specific fuel treatments being evaluated in the case studies are not likely to significantly reduce the costs of suppression for a fire that starts out of town and burns its way in. They are, however, likely to improve the defensibility of the key assets that are considered. Clean-Up, Forest Restoration, Rehabilitation Data was provided by the Forest Carbon Initiative7 on reforestation. In the Summer of 2020, the all-in costs of site preparation and planting were estimated to be $1.10/seedling. For the purposes of the Forest Carbon Initiative they were planting at 2000 stems/ha, giving a cost per hectare of $2200. Fire Evacuation Evacuation is difficult to model because the consequence of failing to evacuate may be death. Creating scenarios around failed or aborted evacuation is challenging because we have few examples. For the purposes of the methodology described below we use the Value of Statistical Life (VSL). This is described in the Canadian Regulatory Cost-Benefit Guide put out by the Canadian Treasury Board (Jenkins et al., 2007) In discussion with Ministry staff we have settled on a VSL value of $8 million dollars. The second challenge with this is deciding how many 7 The Forest Carbon Initiative is a joint effort between the Canadian federal government and the provincial government of British Columbia to fund projects that manage carbon through reforestation, fertilization, fibre utilization, road rehabilitation and improvement of tree genetics. 57 deaths to include under a failed evacuation scenario. This is treated, simply, with a sensitivity analysis. Examples where deaths did occur can be read about in descriptions of the 2018 Camp Fire in California (CalFire, 2018) and the 2009 Victoria Bushfires in Australia (Victorian Bushfires Royal Commission, 2009). Productivity/Business Income Losses After the 2017 fire season in British Columbia the Thompson-Nicola Regional District (TNRD) and the Kootenay Rockies Tourism Association (KRTA) estimated business losses from the wildfires (O’Leary and Associates, 2018; Peak Solutions Consulting, 2018). Both groups used survey methods of business owners to estimate losses. In the TNRD the estimated loss was $31 million and the KRTA estimated the losses to the tourism industry were $38 million. Economists generally do not consider these to be sound measures of economic losses as they miss several important things about the economy. Generally, after a disruption there is a bump in the economy as society recovers, there is also tends to be relative changes in prices both during and after the event and there are frequently substitutions to other parts of the economy. For example, instead of a trip to the Kootenays, tourists decided to take a drive up the Dunster Highway. The region is affected but the entire economy is not, the trip is still taken and jobs are created, they just get moved elsewhere in the economy. To capture these effects a computable general equilibrium model is generally used (Kajitani & Tatano, 2017). Building these models is a data intensive and specialized endeavor that is not practical for the purpose of economic evaluation of fuels mitigations and is especially difficult to do at so fine a scale because of data limitations. 58 Timber Loss It is possible to use the latest available provincial inventory data and log pricing reports to generate a reasonable estimate of timber losses. Using geophysical modelling of wildfires, we believe it is possible to use model outputs such as hectares burned, and fire intensity, for given wildfire management regimes, that will in turn allow us to produce accurate estimates of timber lost in a given area (likely in timber volume). The amount of assessed value lost depends on whether or not a tree is exposed to fire and what the intensity of the fire is when it is burned over. Work has been done in the boreal forest (Heĺy et al., 2003) and in California (Regelbrugge & Conard, 1993) to estimate regression equations that relate fire intensity/char height to a probability of mortality for a given size class. There is also empirical work on the probability of tree survival after fuel treatments (Outcalt & Wade, 2004; Ritchie et al., 2007). These values can be included in the economic analysis tool, FAN$IER, of the provincial stand dynamics model TIPSY (Financial Analysis of Silviculture Investment & Economic Returns (FAN$IER) - Province of British Columbia, n.d.). These values have not been modeled in the case studies below because the proposed fuel treatments provide limited or no improvement of the defensibility of timber values. Damage and Destruction of Habitat Some attempts have been made to estimate loss of habitat by area for particular species like the spotted owl (Ager et al., 2007) but there has been less attention paid to putting a dollar value on these estimates. For non-market values, the US Forest Service has employed a methodology they call implied minimum value or IMV(Calkin et al., 2008). The example given in their explanatory 59 paper is one of protecting the Bitterroot flower. If a treatment reduces the probability of extirpation of the flower from 50% to 20% and costs $5000 then they calculate a probability adjusted cost is $50000.5−0.2=$50000.3= $16,667. This is the implied minimum value. Policy makers must then make the decision whether the reduction in risk to the survival of the Bitterroot Flower is worth the $16,667. Alternatives like contingent valuation have received thorough criticism elsewhere as being unreliable and logically inconsistent (Hausman, 2012). In general, valuation of habitat, species and ecosystems is expensive, difficult and prone to error. Venn & Calkin (2011) provide a review of the use of Non-Market Valuation in the study of wildfire economics. These values have not been included in the case studies below because there are high value assets at risk that take priority over these values given budget constraints. Loss of Life Death from wildfire in Canada has been relatively rare in Canada in the past century. While there is data collected by StatsCan on deaths in structural fires there is no similar data on wildfire losses. It is possible to use the Value of a Statistical Life used in Cost-Benefit Analyses for the Canadian federal government (Jenkins et al., 2007) to assess a value but it is not possible to estimate probability of loss for a given fire because there have been very few wildfire caused deaths in Canada in the past century. This has been tackled in the context of evacuation above with sensitivity analyses. 60 Flooding and Changes in Hydrology It is well established that wildfire can have an impact on hydrology and increase the risk of flooding after major wildfire (Agee, 1993; Meixner & Wohlgemuth, 2004; Neary et al., 2003). It is not clear that we have the data to model this. I know of no database in British Columbia that would allow us to estimate this in an expedient fashion. Estimating the effects of fire on hydrology is difficult. Changes in surface soils depends on the intensity of the fire and the probability of damaging floods will depend on the weather the following season. Modelling this would require sophisticated probabilistic models. An example of an application of this modelling can be found in Gannon et al. (2019) . Lost Recreation Value It is relatively straight-forward to obtain a first pass estimate of losses from campsite and hotel revenues for a given fire scenario. Other declines in tourism revenue could be estimated using StatsCan data for a given fire size. For changes in other recreational values, such as mountain biking, snowmobiling and hiking it would be possible to make use of travel cost methodologies (Hesseln et al., 2003). Travel cost estimations are significantly more time consuming and data hungry. We opted not to include this in the analysis because it is not high on the list of priorities when it comes to fighting wildfire and these values were not relevant for the case studies that were selected. Smoke Costs There is a substantial literature on the health effects and costs of illnesses due to wildfire smoke (Kochi et al., 2010). Costs tend to be measured in terms of sick days, medication, lost 61 productivity or units similar to Quality Adjusted Life Years (Torrance & Feeny, 1989). The values that have been arrived at are from historical fires. The challenge here is estimating the difference in smoke outcomes between no treatment and different treatment scenarios. If there are prescribed fires, how significant is the reduction in lost productivity, sick days, or medication? If it reduces the probability of future mega-fires, how far in the future does that occur? How should one discount these two streams of costs and benefits? Doing this properly would be its own project. There also business implications where wildfire smoke may impact recreation or tourism opportunities, an important contributor to many local economies in BC. 3.3 Case Studies Using the Defensibility Heuristic Together with the BC Wildfire Service, we selected four case studies in two regions of the province for this study. In the central Cariboo we selected an area around 150 Mile House and the South Lakeside neighborhood of Williams Lake. In the Okanagan, we selected the Glenrosa neighborhood in West Kelowna and the Greata Ranch area just south of Peachland. In all cases we are working with data for proposed or completed fuel treatments. The different treatments areas were selected to highlight different scenarios and different types of values at risk. The South Lakeside – Williams Lake scenario was selected to look at treatments in the wildland-urban interface (WUI) that primarily protected residential homes. The 150 Mile House scenario includes residential homes and a new natural gas compressor station that is being constructed on an Enbridge gas pipeline. The Greata Ranch scenario is intended to capture the costs of not being able evacuate and the Glenrosa scenario contains a combination of residential values, a middle school and a water treatment plant. The Glenrosa scenario is also useful for 62 looking at how different real-estate valuations impact decision making, relative to the South Lakeside example where land improvements are valued by the market at about half the amount of Glenrosa. For each of the scenarios, we consider sensitivity analyses on the probability that wildfire under extreme fire weather conditions, will test the fuel treatment as well as number of homes burned, or the number of people trapped when evacuation routes are not safe. In addition to this, for the scenarios that included residential houses, two different perspectives are considered. The first is a purely public expenditure perspective which only accounts for foregone property taxes; and the second, a more holistic perspective that accounts for the entire replacement value of the home based on its assessed value taken from the BC Assessments Roll Totals (BC Assessment Authority, 2020). 3.3.1 A Description of the Fuel Treatment Data In the case studies described below, a standard cost for mechanical fuel treatments of $2000/ha was used. This number was settled on after examining data provided by the FESBC and discussing the trends in fuel treatments with one of the staff members from FESBC who is responsible for evaluating proposed fuel treatment projects. The data provided by FESBC contained information on 83 different fuel treatments in seven regions of the province. A summary of the average cost for mechanical treatments is found in Table 3. 63 Region Avg. Cost / ha Cariboo $ 1,784.03 Kootenay $ 3,125.38 Kootenay-Boundary $ 2,480.31 Okanagan $ 1,975.40 Omineca $ 2,087.18 South Coast $ 8,558.76 West Coast Region $ 14,033.64 All Regions $ 2,348.92 Table 3 Average mechanical fuel treatment costs in British Columbia. Data provided by the Forest Enhancement Society of British Columbia. The way in which the specifics of the fuel treatments were recorded meant that individual cost components (e.g. type of thinning, machinery used, etc.) were not consistently reported. Because of this the data could only be separated into two groups, mechanical and hand treatments for the different regions in the data set. Data for hand-treatments are provided in Table 4. Region Avg. Cost / ha Cariboo $ 5,105.18 Kootenay-Boundary $ 3,313.82 Okanagan $ 6,394.29 South Coast $ 15,200.79 64 All Regions $ 6,302.83 Table 4 Average hand fuel treatment costs in British Columbia. Data provided by the Forest Enhancement Society of British Columbia. The individual fuel treatment costs are plotted in Figure 8. At this time, the operators have not found an efficient operating frontier. We would expect there to be clear returns to scale in the fuel treatments and we could draw a line that shows that there has been some drop in costs as a function of treatment size but there is a great deal of noise in the dataset. In the bottom left of the figure we can observe many treatments of small size with similar costs to the largest treatments in the dataset. In discussions with staff at FESBC the project evaluators were confident that operators have learned a lot about fuel treatments and project proponents are starting to settle on an appropriate mix of equipment and it would be reasonable to see mechanical fuel treatment costs average $2000/ha and on large projects the interviewee expected to see $1200/ha in the near future. 65 Figure 8 Individual mechanical fuel treatments by size and cost ($/ha). Region indicated by colour and a triangle indicates that there was overstory removed which is correlated with removal of merchantable timber. Projects that removed overstory are shown as a triangle in Figure 8. This is correlated with removing more merchantable timber. Examination of the figure does suggest that sites where overstory removal is permitted lowers the cost per hectare as we would expect. 3.3.2 The Greata Ranch Case Study Greata Ranch is a large winery-vineyard south of Peachland along Highway 97 in the Okanagan (Figure 9). There are few residences here but this is a critical evacuation route for the region. I have included the replacement value of the vineyard, but the focus is on the potential lives lost when evacuation routes cannot be kept open. I had tried to avoid putting a price on life, but it 66 was unavoidable for the purposes of attempting to value the potential loss of life from not being able to keep an evacuation route open. Life is priced using the Value of a Statistical Life (VSL) method recommended by the Canadian Federal Government’s Treasury Board of Canada Secretariat’s Canadian Cost-Benefit Analysis Guide (Jenkins et al., 2007). The specific value of $8 million used for a VSL was determined in discussion with economists at the Ministry. As loss of life from unsuccessful evacuation, even in extreme wildfire conditions, is rare we do not have a good sense of what the correct number of lives is to use in the model and I have used a sensitivity analysis with 10, 50 and 100 lives lost. Figure 9 Treatment area for the Greata Ranch area just south of Peachland, British Columbia The replacement stock for the vineyard is valued at $75,861 / ha (BC Ministry of Agriculture, 2014) totaling roughly $3.5 million for the entire vineyard and is fixed for all scenarios. This value is dwarfed by the total VSLs for the lives at risk. The area of the fuel treatment is 22.4 ha for a cost of $44,800 at $2,000/ha. The defensibility values are given in Table 5 below. 67 Probability of Treatment Being Tested 0.05 0.1 0.2 Loss of Life 10 1.07% 0.54% 0.27% 50 0.22% 0.11% 0.06% 100 0.11% 0.06% 0.03% Table 5 Defensibility criteria (%) for the Greata ranch area with three scenarios for potential loss of life from failed evacuation (10, 50, and 100 people). $2000/ha for treatment costs. Given these estimates, even with the most conservative assumptions the fuel treatment is justified if it succeeds in improving defensibility of the highway only 1% of the time. Given the small size of the fuel treatment (22 hectares) it is a legitimate question whether or not this fuel treatment would succeed in keeping the highway open. If we flipped this problem around and asked, how big of a treatment could be justified if the treatment is only tested by extreme fire once in twenty years and there are only 10 lives at risk? The answer is about 2200 hectares. This area is found by solving for the treatment size that would give a required defensibility improvement of 100%. In this case we have estimated adjusted total costs of $8.35 million with a 5% probability of being tested by extreme fire. At $2000/ha this is 2200 hectares, which is a large treatment by the standards of what has been funded so far. 3.3.3 The Glenrosa Case Study The Glenrosa neighbourhood (Figure 10), part of West Kelowna is the second set of treatments to be considered. The Values at Risk here include a middle school, a water treatment plant and residential homes. 68 Figure 10 The neighbourhood of Glenrosa is surrounded by the fuel treatments marked in yellow. The total area treated in this case study is 405 ha. I have included two fuel treatment cost scenarios for this case study. One at the $2000/ha that has been used for the other case studies (Tables 6 & 7) and another scenario using $4000/ha (Tables 8 & 9). The $4000/ha scenario has been included because some of the planning that has been done in the area described some of these treatments being done by hand (Forsite Consultants Ltd., 2020). In the dataset provided to me on fuel treatment costs, hand treatments averaged $6300 / ha. I have assumed a roughly half - half balance of hand treatments to mechanical treatments for the second treatment scenario. The sensitivity analysis for these scenarios are included in Tables 8 & 9. Total treatment costs for the $2000/ha scenarios are about $800,000 and for the $4000/ha scenario they are $1.6 million. Two different viewpoints for valuing lost home values are considered. An exclusively public expenditure perspective which only makes use of the foregone property taxes (0.6% of property 69 value) and whole home replacement cost perspective. The average land improvement assessment for this area is $391,000, about twice that of the Williams Lake area. Probability of Treatment Being Tested 0.05 0.1 0.2 Homes at Risk 50 50.6% 25.3% 12.6% 100 50.4% 25.2% 12.6% 200 50.0% 25.0% 12.5% Table 6 Defensibility criteria (%) given a public expenditure perspective (property taxes only). Treatment costs averaging $2000/ha. Probability of Treatment Being Tested Homes at Risk 0.05 0.1 0.2 50 31.5% 15.7% 7.9% 100 22.8% 11.4% 5.7% 200 14.7% 7.4% 3.7% Table 7 Defensibility criteria (%) with total value of homes lost. Treatment costs averaging $2000/ha. The Helen Gorman Middle School was costed at replacement value of $9.9 million based on square footage and a Canadian public sector construction costing index for schools (bty ltd., 2020). The Powers Creek Water Treatment plant replacement value was taken from costing data published by a local newspaper at the time of construction(“Powers Creek Plant,” 2007). It was valued at $18,000,000 at the time of construction in 2007 and inflated to 2020 dollars which is roughly $22,000,000. Table 6 contains results for a public expenditure perspective with residential homes valued with lost property tax revenue only and an assumption that treatments are done using mechanical means only. Given the most conservative set of assumptions where there are only 50 homes at 70 risk and a1 in 20 year chance that the fuel treatment will be tested under extreme fire weather conditions the treatments have to make the assets defensible 50% of the time to make it worthwhile. As the number of homes at risk is increased, we don’t see much of a change in the defensibility criteria because it is valued only as the lost property taxes. In Table 7 the entire value of the homes are included and this improves the desirability of the fuel treatment. As a result of including the full value of the home when the number of homes at risk is increased to 200 the fuel treatment only needs to work about 1 in 8 times to be worthwhile. When the average cost of the fuel treatments is doubled to account for potential hand treatments the treatments appear less economic. From the property tax only perspective, the treatments are not worth doing at all if they are only being tested once in twenty years (Table 8). Probability of Treatment Being Tested 0.05 0.1 0.2 Homes at Risk 50 101.1% 50.6% 25.3% 100 100.8% 50.4% 25.2% 200 100.0% 50.0% 25.0% Table 8 Defensibility criteria (%) given a public expenditure perspective (property taxes only). Treatment costs averaging $4000/ha. If the full cost of the home is included, and 50 homes are at risk, the fuel treatment needs to be effective 2/3rds of the time (Table 9). Unsurprisingly, the more area that can be treated mechanically the smaller the chance of success it needs to have to be worthwhile. Probability of Treatment Being Tested Homes at Risk 0.05 0.1 0.2 50 63.0% 31.5% 15.7% 100 45.6% 22.8% 11.4% 71 200 29.4% 14.7% 7.4% Table 9 Defensibility criteria (%) with total value of homes lost. Treatment costs averaging $4000/ha. 3.3.4 The 150 Mile House Case Study The 150 Mile House region (Figure 11) was chosen largely because it is home to an Enbridge gas compressor station, critical infrastructure for the gas supply of the southern half of the province. When a pipeline exploded in 2018 on the same gas transport system, it resulted in curtailed gas use for Canadian customers with the Fortis gas system running at 40% capacity (Brend, 2020), American refineries and power plants had to shut down, 750,000 residential power prices increased more than two orders of magnitude (“BC Natural Gas Pipeline Explosion Affects US Energy Markets,” 2018) and, in personal conversations with energy traders, two utilities nearly lost their entire gas systems from loss of pressure and large parts of the electrical grid in the pacific northwest were close to extended blackout. Total additional costs from the loss of that transport line relative to business as usual likely ran in excess of a billion dollars. Providing an accurate valuation of this outage would be as large a project as this entire thesis so I have limited the value-at-risk to the replacement value of the 2020 Enbridge compressor station upgrades at $125 million (Lamb-Yorski, 2020). I do this because the investment cost of the plant is the minimum bound for the social value of the plant. If Enbridge was not going to see present value returns at least as much as $125 million it would not be making the investment. 72 Figure 11 150 Mile House study region This is the largest treatment area in the 4 case studies. The fuel treatments shown in Figure 11 total 1,577.5 hectares. At $2000/ha this totals over $3 million. In addition to the compressor station there are homes at risk. Once again two scenarios are included with Table 10 including only foregone property taxes and Table 11 including the full value of the homes. Property taxes are calculated at 1.14% of assessed value and the mean land improvement is $178,218 for the region. Probability of Treatment Being Tested 0.05 0.1 0.2 Homes at Risk 50 50.4% 25.2% 12.6% 100 50.4% 25.2% 12.6% 200 50.3% 25.2% 12.6% 73 Table 10 Defensibility criteria (%) given a public expenditure perspective (property taxes only). Treatment costs averaging $2000/ha. Probability of Treatment Being Tested Homes at Risk 0.05 0.1 0.2 50 47.1% 23.6% 11.8% 100 44.2% 22.1% 11.0% 200 39.3% 19.6% 9.8% Table 11 Defensibility criteria (%) with total value of homes lost. Treatment costs averaging $2000/ha If the fuel treatments are to be tested under extreme fire weather conditions once in twenty years then the fuel treatment needs to make the assets more defensible 50% of the time. If the true cost of a gas line outage is in fact closer to $500 million this would drop to 12.5% and at a billion dollars in costs the fuel treatment only needs to make the asset more defensible 1 in 20 times. Consistent with the Wildfire Service placing a high priority on critical infrastructure, the economics of this also place a high priority on the protection of these assets. 3.3.5 The South Lakeside – Williams Lake Case Study The South Lakeside case study (Figure 7) is the most straight-forward in its valuation. Only residential home values have been included. Both the foregone property tax and full-assessed values are included. 74 Figure 12 South Lakeside - Willams Lake case study area Mean land improvements are assessed at $173,671 dollars and property taxes assessed at 1.14% of assessed value. The treated area is 315ha at $2000/ha for a total cost of $630,000. In Table 12, we can see that if only the foregone property taxes are included no combination of the assumptions comes anywhere close to justifying the expenditure on the fuel treatments. From the public expenditure perspective this absolutely should not be done. Probability of Treatment Being Tested 0.05 0.1 0.2 Homes at Risk 50 12728% 6364% 3182% 100 6364% 3182% 1591% 200 3182% 1591% 796% Table 12 Defensibility criteria (%) given a public expenditure perspective (property taxes only). Treatment costs averaging $2000/ha. 75 In Table 13, if we account for the full value of the homes the fuel treatments still cannot be justified if under conservative assumptions of 50 homes being at risk of being burnt down and the fuel treatments being challenged 1 in 20 years. If the fuel treatments were expected to be tested 1 in 5 years and the number of homes at risk is 200 then the fuel treatments only need to work 1/10 times to be worthwhile. Probability of Treatment Being Tested 0.05 0.1 0.2 Homes at Risk 50 145% 73% 36% 100 73% 36% 18% 200 36% 18% 9% Table 13 Defensibility criteria with total value of homes lost. Treatment costs averaging $2000/ha. This case study shows that the economics suggest prioritizing expenditures on critical infrastructure and other public infrastructure, like schools and water treatment plants before residential homes. This is consistent with existing risk priorities. 3.3.6 An Example with Perpetual Maintenance In each of the case studies above we include a probability of treatment being tested because fuel treatments have a finite life if they are not maintained. As fuel treatments are currently funded on a project-by-project basis this approach makes sense but what if funding was available for maintaining fuel treatments? To treat this case, I will return to the 150 Mile House example with the compressor station and assume that the fuel treatments are being maintained at two-year intervals (likely an aggressive assumption) in perpetuity. I have only one example of a pure prescribed burn in my fuel 76 treatment dataset and over 182 ha the treatment costs came to $74/ha. To provide some padding I will round up to $100 /ha. These costs will be incurred every two years forever. We can modify the Land Expectation Value formula to account for these costs: 𝑃𝑟𝑒𝑠𝑒𝑛𝑡 𝑉𝑎𝑙𝑢𝑒 = 𝑅𝑥 𝐵𝑢𝑟𝑛 𝐶𝑜𝑠𝑡(1 + 𝑟)𝑡 − 1 (11) If we use a social discount rate of 3% as recommended in the Canadian Cost-Benefit Analysis Guide (Jenkins et al., 2007) with a prescribed burn repeated every 2 years at a cost of $100/ha we get: 𝑃𝑉 = $100(1.03)2 − 1= $1,642 (12) If we add this present value of the perpetual prescribed burns to the initial mechanical treatment costs at year 0, we can recalculate the defensibility threshold. If I take the most conservative example for 150 Mile House with the $125 million value for the compressor station and 50 homes at risk, only counting foregone property taxes the present value of prescribed burning all 1,577 hectares is $2,590,311 and a probability of 1 that the treatment will get tested over its lifetime8 we get d = 4.6%. We interpret this to mean that the fuel treatment only needs to work 4.6% of the time to be justified. Under the one-time treatment model with a 1 in 20-year likelihood of the treatment being tested over its life the same treatment 8 If the treatment is maintained in perpetuity, it will eventually meet with a wildfire under extreme fire weather conditions. 77 resulted in a defensibility threshold of 50%. The prescribed burning cost at two-year intervals would have to be $2300/ha in order to make the defensibility threshold that high. The message from theses analyses is: It really makes sense to maintain these fuel treatments! 3.3.7 Changes in the Fire Weather Threshold All of these studies were assumed that the treatment should withstand a fire under 90 th percentile fire weather conditions. What changes when that is changed to the 85th percentile or the 99th percentile? From the perspective of the person working with the spreadsheet there is no change. If the defensibility heuristic gives a needed change in defensibility of 30% to justify expenditures, it now needs to be justified under 99th percentile fire weather instead of 90th percentile and it is up to the experienced firefighting professional to determine if it still works-not the modeler. 3.4 Limitations and Shortcomings of the Defensibility Heuristic Below I discuss four shortcomings of this approach from the perspective of the economist as a modeler: 1. Limited number of values are included. 2. Requires expertise of frontline wildfire operators. 3. Defensibility as an implied probability may not be intuitive. 4. The Defensibility Heuristic is not a general theory of fuel treatments. 78 3.4.1 Limited Value Inclusion Limiting the number of values included is not an inherent failure of the Defensibility Heuristic nor is it unique to this approach. To a greater or lesser extent, all of the other approaches have also suffered from this shortcoming. The number one challenge with modeling this problem is that the interactions between variables are complicated and highly uncertain. The costs of modelling additional variables within acceptable error bounds is very expensive. It is sufficiently expensive that it interferes with the advancement of the fuel treatment projects themselves in a timely manner. The greatest let-down in this limitation is that completing a project evaluation in a cost-effective and timely manner does not allow for the evaluation of the long-term effects of catastrophic wildfire. There are important and difficult questions to be answered around the long-term viability of some of the forest-based communities in British Columbia in the face of catastrophic wildfire. Can landscape-level fuel treatments protect against the loss of their economic lifeblood? This modelling approach does not have anything to say about that. It focuses only on the near-term effects. New theory is presented in the next chapter that attempts to address these questions. One upside of this approach is that it quickly identifies the assets with the greatest value at risk for easier prioritization. We also find that the economic prioritization aligns with the priorities of the BC Wildfire Service. 79 3.4.1.1 Including Other Values The case studies only deal with a handful of values that are relevant to the areas of interest and the priorities of the BC Wildfire Service. In discussions with BC Ministry of Forests staff two values came up repeatedly as being of interest for inclusion in an analysis. These are greenhouse gas emissions and habitat loss. A process for estimating these values is sketched out in the following subsections. 3.4.1.1.1 Greenhouse Gas Emissions Continuing to work from the perspective of a stress test, estimating initial emissions from an extreme fire weather scenario without a fuel treatment should not be too difficult. If we make use of the timber value loss model to build an estimate of above ground carbon losses and then add on an estimate of surface fuel combustion the initial emissions could be estimated for both with a treatment and without treatment scenario. To fully account for the differences in emissions though would require modelling stand dynamics with and without the treatment. We would have to assume that the burned area is being replanted immediately and calculate the differences in stored carbon over the lifetime of the stand discounted to the present. Are there differences in growth between burned and unburned? Changes in the costs of silviculture between the two scenarios? Finally, the modeler would also have to account for lower stand density in the stands that have been treated to reduced risk. 3.4.1.1.2 Habitat Loss 80 There are two challenges with modelling the avoided damages to habitat loss. The first is the complicated interactions between ecosystem components and fire and the second is the valuation of ecosystem values. Many species in British Columbia ecosystems are adapted to low-to-moderate severity fire, but high-severity fire often does not provide the same opportunities for species to thrive. Modelling the with treatment and without treatment effects of fire would require modeling the distribution of fire severity in the geography of interest. There may be trade-offs between organisms within the study-area boundary. Some species are going to do better after fire than others. How do we choose which ones should be valued? To address the explosion of possibilities here a focused model would consider species-at-risk alone. How is their habitat affected given changes in the severity of fire? The valuation of these factors is another challenge. In general, we want to keep people away from species-at-risk, even as tourists. This makes the use of hedonic pricing methods like travel costing unusable. This leaves us with contingent valuation. To my mind, the willingness-to-pay, willingness-to-accept literature has been thoroughly put to rest as unreliable and inconsistent (Hausman, 2012). Unfortunately, this is the only way forward for species and ecosystem values that have no instrumental value to society in this kind of analysis which demands everything be evaluated in terms of dollar values. For ecosystem services related to hydrology there are more established methods. For instance, researchers have developed models for the economic estimation of post-wildfire flood effects (Neary et al., 2003). These are involved models however, that require at least as much effort as estimating the impacts of fire to the carbon cycle. 81 3.4.2 Frontline Expertise The Defensibility Heuristic requires decision-making by experienced frontline staff, unlike traditional cost-benefit analysis. Most economists do not have the experience to judge whether or not a fuel treatment is sufficient to protect assets for some fraction of the exposures to extreme fire. One way of looking at this is that the Defensibility Heuristic is not intended to be computationally model driven approach. It is a systematic way for experienced frontline professionals to think their way through the economic problem of evaluating potential fuel treatments, hopefully with the help of economists to do some of the leg work on the valuations. Compared to other approaches the expertise demands are in-fact lower. The more advanced approaches used in Scott et al. (2013) and Thompson et al. (2017) are very time consuming and expensive relative to this approach. In speaking with wildfire fighters over the past few years, the experienced ones can describe how changes in forest structure and fuels density improves their ability to defend a position. They are also able to rank fuel treatment options in terms of defensibility. More work would have to be done to validate the idea of fire-fighters being able to translate their intuitions into a metric9 as is theoretically required by this method. 3.4.3 Implied Probabilities are not Intuitive. Trying to understand the ratio of costs as an implied probability is not terribly intuitive for most people. I think this can be overcome by simply asking the question, “How many t imes in 10 9 Metric is used in the mathematical sense here. A nonnegative function, that satisfies the triangle inequality, 𝑓(𝑥, 𝑦) + 𝑓(𝑦, 𝑧) ≥ 𝑓(𝑥, 𝑧) , and that gives a distance between two points. In this case, the distance is a measure of how much more defensible a forest area is relative to the status quo. 82 would this fuel treatment have to protect these values in the event of a fire in the 95th percentile of fire weather?” If that number exceeds the implied probability, then it is worth doing the project. 3.4.4 The Defensibility Heuristic is Not a General Theory Even when CBA is not appropriate, one of the common justifications for doing CBA anyway is that a cost-benefit ratio is, in theory, universally comparable. That is, we can take a cost-benefit ratio from public health expenditures and compare it to expenditures on transportation infrastructure or any other public expenditure and make constrained budget decisions. The defensibility heuristic is only suited to making decisions between fuel treatments under similar conditions. It does not allow for comparing expenditures on a fuel treatment and a renovation of the local radiology department at the hospital. The severity of this shortcoming depends on your philosophical outlook on government and governance. If the technocratic dream of aggregating preferences and allocating resources to individual members of society without the messy process of politics is appealing this is a substantial shortcoming. If one believes that we have a process for electing officials to make difficult decisions about the trade-offs in public expenditure, this is less of a shortcoming. The political process can allocate funds to general categories and heuristics like the one described here are potentially of use. 83 3.5 On the ‘Business Case’ for Fuel Treatments This effort has focused on evaluating and ranking individual WUI level fuel-treatments, not comparing them to FireSmart efforts or other government expenditures. It does not allow us to compare a dollar spent on a new hospital or school versus a dollar spent on a fuel treatment or any other attempt to reduce risks from wildfire. The question of whether or not it would be possible to do this has come up repeatedly in this research. If we were to tackle those questions, how should they be approached? The fact that the evaluation of individual landscape fuel treatments does not make sense in a cost-benefit framework has already been identified. This is not an investment that provides future returns to society. It is an expenditure that manages the risk of future losses. No one makes the argument that national defense expenditures or even wildfire service expenditures should exceed some hurdle return rate. Policy makers recognize that these are necessary expenditures to protect the future viability of the economy. When comparing landscape-level fuel treatments to FireSmart, it should be made on this basis. Does the landscape level fuel treatment lower the cost of passing a stress test relative to the alternatives? In terms of regional economic viability what is the stress test? Certainly, it is not finding the lowest cost means of lowering the average fire damage. Instead, we should be thinking about protecting the viability of forest-based communities. Following the 2017 wildfires in the Cariboo, the Quesnel, Williams Lake, and 100 Mile House Timber Supply Areas saw 84 losses of 22.5%, 17.7% and 12% of their Timber Harvesting Landbase, respectively (Forest Analysis and Inventory Branch, 2018). How many forest industry communities can come out alright from losses to the resource base like that? Where is the threshold beyond which economic recovery becomes difficult or impossible? This should define the stress test. Do landscape-level fuel treatments make it less costly to survive this tail event relative to FireSmart or equivalent expenditures on increased suppression? This is a distinct way of thinking about this problem that differs from cost-benefit analysis. Rather than trying to improve the efficiency of expenditures by measuring the average benefit/avoided damages per dollar spent we are trying to find the most effective way of protecting against the worst-case events. One way of doing this is checking project adequacy using a defensibility threshold as described in the sections above. In practice, there is no need to do this for all values-at-risk at once. In justifying expenditures start at the top of the Resource Sharing Wildfire Allocation Protocol (RSWAP)(BC Wildfire Service, 2010). The BC Wildfire Service has already identified and mapped evacuation choke points where lives are at higher risk of being lost. Fuel treatments that increase the defensibility of these chokepoints can evaluated as a package for a funding request. Moving down the RSWAP priorities next is critical infrastructure. There are several pieces of infrastructure in the gas distribution network in British Columbia that are critical to the functioning of the entire Western North American power grid. These could be evaluated as a package and so on. As we move down the RSWAP priority list the economic justification for the work is likely to be weaker, the work will have to be more careful and require greater use of complicated modelling. Given the extent of the fuel treatments required, this may never happen but there is utility in 85 using these economic analyses in the context of the RSWAP to prioritize which ones should take place first. From a theoretical perspective, we do not have effective tools for answering some of the questions asked above: “Where is the threshold beyond which economic recovery becomes difficult or impossible? “ What does a stress-test look like in this context? How do we decide what actions to take? These are not questions for the short-term. An outline of how to address these questions is the subject of the next chapter. 86 Chapter 4: The Long-Run Economics of Forest Management and Wildfire In the previous chapters I have dealt with uncertainty that arises out of the underlying data generating process itself and the lack of convergence in parameter estimates when trying to make decisions using traditional measure of risk. I showed how this limits the traditional tools we would apply to evaluate fuel treatments, developed an alternative heuristic but also showed its limits in regard to scope and time. In this chapter I turn to examining the implications of non-Gaussian uncertainty at the landscape level over a longer time horizon. Over the long run it is even more difficult to estimate convergent probabilities as there are interactions that are hard to identify and model. For instance we might need to ask, what is the relationship between an increase in post-traumatic stress disorder among the work age population and the long-term financial viability of local mills? In this chapter we focus on the scenario where a catastrophic fire event undermines the ability of a local community or region to sustain itself in the way that is had been accustomed to. I define a catastrophic event as one that undermines steady employment, community participation and healthy relations within and between households. I elaborate on this in the following sections. Is it possible to understand what happens to the long-run economic prospects of a community after a significant proportion of the working forest is subjected to high severity fire as was the case with Williams Lake in 2017? Can this be done in a tidy way? Can we or should we change forest management over the long-run (moving beyond a project-by-project approach to fuel treatments) to lower the probability of catastrophic fire events? I explore these questions by first building a simple model of forest growth that incorporates losses to the forest stock that are Pareto distributed, then argue for a conceptualization of risk that is better suited for describing the 87 consequences of these losses than is currently available. Following this I describe the consequences of this conceptualization of risk for forest management. 4.1 A Stylized Model of Forest Growth It is easy to get caught up in the details when attempting to answer these questions, but I will start by offering a very simple model of forest growth, harvest, and capital accumulation. With this model we can start to introduce disturbances to the model that may affect the ability of industry to conduct their business. This simple model provides insights that could direct future research on the economics of wildfire and forest management. The first step on our way of making simplifying assumptions is adopting a stylized model of forest growth. At risk of offending some, I am going to begin at the simple end of the modeling spectrum and start with the logistic or sigmoid growth curve. One of the earliest uses of this model in studying growth dynamics of plants was studies by Ludwig von Bertalanffy (Pienaar & Turnbull, 1973). The sigmoid growth curve is a simple mathematical model of growing living things, individuals and organized collections of individuals and adoption of technology among other phenomena that feature in complex adaptive systems with replication or reproduction (Figure 13). 88 Figure 13 The sigmoid growth (or logistic) curve 𝒇(𝒕) =𝟏𝟏+𝒆−𝒙=𝒆𝒙𝒆𝒙+𝟏 provides a reasonable deterministic model growth in living things for our purposes. The S-curve makes intuitive sense as a model of growth. Living things need energy and nutrients to survive and grow. In the beginning, life has minimal apparatus to obtain these necessities, it puts a lot of the nutrients and energy into creating more of the structures necessary to collect more of the needed inputs. At a certain point, there is a snowball effect where the return on investment results in rapid, exponential growth. This continues until the limits of a growth factor have been reached. A nutrient in the soil has been exhausted or the available sunlight is being utilized. The curve flattens out, growth slows and eventually stops, perhaps reverses before death. In an isolated chamber with no competition, a plant will put on biomass in a pattern close to an S-curve. 89 Living things do not exist in isolation. There is competition for resources and often uses other living things as feedstock. In some ways, fire behaves like a living thing (Pyne, 2012). It requires oxygen, fuel and heat to exist and grow. It starts and grows on a roughly exponential growth curve until it is limited by some factor, often fuel of sufficiently low moisture content. Of course, a fire is not building metabolic machinery to exploit available nutrients and energy. It has an analogous process of (1) preheating and distillation, (2) distillation and combustion of volatile fractions and (3) burning of residual charcoal (Agee, 1993; Byram, 1959). Fire size can also be roughly modelled using a logistic growth curve. At the overlap of the stage (1) and (2) there is little growth in the fire. Fire growth can be a slow process at the beginning until the exothermic reaction is sufficient to heat nearby fuels, causing further distillation and combustion until there is a chain reaction that eventually consumes the available dry plant matter. Humans do not have much control over heat and oxygen so we look to managing fuels to limit the growth of fires. As discussed in earlier chapters, the approach to managing wildfire fuels in British Columbia has focused on single projects and the literature on the economic evaluation of fuel treatments has focused on single projects (Hesseln, 2000; Huang et al., 2013; Mason et al., 2006; Snider et al., 2006). This amount to shifting the 𝑓(𝑡) in Figure 13 downwards. The effect on fuel availability is of limited duration. Growth in forest biomass will continue on its merry way. The benefits from one-time treatments are short lived. These fuel treatments (with the exception of prescribed burns) focus largely on the removal of large woody biomass. We know that brush and grass under extreme fire weather conditions can contribute to damaging and lethal fires (CalFire, 2018; Cruz et al., 2012; Minnich et al., 1983) These fuels need to be managed in 90 on-going manner to meaningfully reduce the risks of catastrophic fire through time. This is a never-ending process. I showed in the earlier chapters that the tools we have for evaluating policy decisions, like cost-benefit and cost-effectiveness analysis, are static. These tools are used to evaluate a single project (represented as an incremental or marginal change from the status quo), under a specific model of a risk (I use risk very loosely here to capture all the uncertain changes in costs, interest rates, fuels, biomass growth and fire weather changes that a modeler needs to consider when undertaking a cost-benefit analysis) at a single point in time. These tools use discounting and mathematical expectations to reduce an entire set of possible outcomes to a single number. This approach of cost-benefit analysis works under a narrow set of conditions but is not appropriate to evaluate on-going changes in the way that we manage forests. If cost-benefit analysis and the language of investment are not appropriate for choosing between different fuel management processes, what should we use to evaluate different approaches? How do we choose between potential fuel growth curves that determine what fuels are available to fires? How should we think about opportunity costs over the long run when prioritizing fuel management? What is the appropriate way of modeling uncertainty in wildfire management questions? To answer these questions, I have tried to represent growth and uncertainty in the simplest way possible while still telling a story that has meaningful things to say about these questions. The 91 goal is illustrating a change in the way we think about the decision process, not in the minutiae of executing it. 4.2 Building an Exposure The S-curve is a deterministic model of forest growth but in this model, I am interested in what happens to growth when it is met with fire events that reduce the quantity of available timber for harvest. This requires that we introduce uncertainty. In what follows, we will add to the deterministic model of forest growth, a random fire process that reduces cut according to a Pareto distribution and cut level determined by available capital and the forest yield. We start with a stock of timber in cubic meters 𝑆. This stock of timber grows according to a logistic function (S-curve). ∆𝑆 = 𝑆𝑡+1 − 𝑆𝑡 = 𝑟𝑆𝑡 (1 −𝑆𝑡𝑆̅) (13) The change in stock (the term Mean Annual Increment (MAI) will be used interchangeably with the change in stock in the latter parts of this document), ∆𝑆 = 𝑆_(𝑡 + 1) − 𝑆_𝑡 depends on the growth rate 𝑟, the current stock and the maximum achievable stock 𝑆̅. When humans enter the picture they cut a certain volume of the forest every period 0 ≤ 𝜅𝑡 ≤ 𝑆𝑡. Now the change in stock is ∆𝑆 = 𝑟𝑆𝑡 (1 −𝑆𝑡𝑆̅)− 𝜅𝑡 (14) 92 Humans harvest trees to do useful things with them and they use tools to do it. We will call this collection of tools the capital stock, 𝐾𝑡. The amount that can be cut in any period is a function of the capital stock and the MAI. 𝜅 = 𝑓(𝐾𝑡 , ∆𝑆) (15) For now, we assume that one unit of capital allows us to cut one unit of forest stock and the most we will cut in any given year is the change in stock 𝜅 = 𝑀𝐴𝑋(𝐾𝑡 ,Δ𝑆). Our forest is publicly owned, and the government has decided that industrial development will be encouraged so that the harvest rate equals the change in stock or MAI. To achieve this, the capital stock grows at a rate 𝑔, less depreciation, 𝛿, if the cut is less than the MAI, otherwise no investment occurs, and capital depreciates at the rate 𝛿, 𝑔 > 𝛿. Depreciation is included because it allows capital to decline when there is a forest disturbance and the Annual Allowable Cut (AAC) drops10. Δ𝐾 = (𝑔 − 𝛿)𝐾𝑡 𝑖𝑓 𝜅𝑡 < Δ𝑆 (16) Δ𝐾 = (−𝛿)𝐾𝑡 𝑖𝑓 𝜅𝑡 ≥ Δ𝑆 (17) 10 The Annual Allowable Cut is the amount of timber that can be harvested annually over a 5 -year planning horizon in a specified area as determined by the Chief Foresters Office in British Columbia. 93 Now we have all the pieces of a simple, deterministic forestry model. When we step through this model using a computer simulation, we approach a steady state amount of capital equal to the MAI. We can see this in Figure 14. This all works as we would expect but living in a deterministic universe is not interesting. What happens to this model if we introduce damages from wildfire that follow a Pareto distribution? Wildfire damages are given by 𝜔𝑡 , which is a random event drawn from a bounded Pareto distribution so that the highest possible damage is equal to some fraction of the total forest stock. The change in stock is now: ∆𝑆 = 𝑟′𝑆𝑡 (1 −𝑆𝑡𝑆̅)− 𝜅𝑡 − 𝜔𝑡 (18) And the probability density function that describes the distribution of risks from fire is given by: 𝜔(𝑥) =𝛼𝐿𝛼𝑥−𝛼−11 − (𝐿\𝐻)𝛼, 𝐿 ≤ 𝑥 ≤ 𝐻 (19) Where 𝐿 ≥ 0 is the lower bound of the distribution and 𝐻 = 𝛾𝑆𝑡 with 0 ≤ 𝛾 ≤ 1. 94 Figure 14 The deterministic model of growth and harvest. The blue line is natural re-growth and the redline is with silvicultural treatments. Equilibrium is reached when the annual cut equals the mean annual increment. As expected, in this model without disturbance, silviculture increases the equilibrium cut level. The parameter values used in the model: 𝑲𝟎 = 𝟏, 𝒓 = 𝟎. 𝟎𝟏, 𝜻 = 𝟎. 𝟎𝟎𝟓, ?̅? = 𝟏𝟎, 𝟎𝟎𝟎, 𝑺𝒔𝒊𝒍𝒗̅̅ ̅̅ ̅̅ = 𝟏𝟐, 𝟎𝟎𝟎,𝒈 =𝟎. 𝟏, 𝑺𝟎 = 𝟓, 𝟎𝟎𝟎. In Figure 15 we see a single, example run of the model with fire disturbances. No matter how many times this model is run, the forest stock always heads down to the minimum value set in the model where a forest industry is no longer viable. This always happens because once the capital stock is built up the cut is equal to the MAI. When the MAI drops because the stock has dropped, the capital drops with depreciation to accommodate the reduced MAI but the cut is still equal to the new growth each year. Without a refractory period where cutting is less than growth 95 it is impossible for the system to recover to previous harvest levels. The system inevitably trends towards the minimum values. Figure 15 In this model, we have the same starting parameters except the stock is exposed to fires each year. The fire events are drawn from a Pareto distribution with 𝜶 = 𝟏. 𝟎𝟎𝟓, 𝑳 = 𝟏. 𝟎 and 𝑯 equal to a fraction of 𝑺𝒕 equal to 0.25. The stock inevitably travels to the minimum as we continue to cut all of the growth in the forest while facing repeated random negative shocks in total volume. This is common sense. As long as there are downward shocks to the cut level without balancing shocks upwards or a capital re-balancing there will be continuous crisis in BC forests. The capital re-balancing could happen in a steady, negotiated manner that could reflect depreciation or all at once. In 2019-2020 we are observing rapid, painful change as forestry companies try to rebalance their asset portfolios in the face of abrupt changes to the cut level that are a 96 consequence of biotic and abiotic disturbances (Kurjata, 2019). The current pain is a direct consequence of trying to maximize the value of stands damaged by disturbances without giving much consideration to the consequences for the dynamics of the capital stock, future cut and the impacts this would have on communities. If we introduce silviculture again in this Pareto random world, we get some recovery but continue to trend down to the minimum bound (See Figure 16). Figure 16 Using the same growth model as Fig. 15 this introduces shocks drawn from the same distribution as in Figure 12. The redline includes silviculture and blue line is without. No matter what is done the stock will always fall to the minimum. Even with silviculture there is no avoiding a refractory period. In the real world, this is what we are likely to see. Not because government policy demands a refractory period, but because the 97 mills are unable to operate without access to fibre11. We are most likely to see this when there are discontinuous changes in operating conditions. In this model the only operating condition that matters is Δ𝑆. Small Δ𝑆 are easy to adjust to on the margin. Staffing can be changed, and product mixes can be adjusted over time. It is the large, discontinuous changes that really disrupt the industry and forest-based communities. When high volatility events, like catastrophic wildfire, occur a very large gap can suddenly open up between marginal benefits and marginal costs. Investors who had calculated a positive return on investment given current harvest levels find their discounted cash flow models to be a poor fit for realities of large loss events. In other words, we are long ways from equilibrium analysis being applicable (Farmer & Geanakoplos, 2009). This brings us to the idea of a Trauma Exposure as an alternative decision tool. 4.3 Definition of an Exposure In the formal, mathematical modeling of risks there are often two categories of decision making that depend on how uncertainty is modeled. The first category can be described by analytically tractable (i.e. linear) models of economic activity or public investment where prices or events (or any changes in variables) are distributed normally and continuously. In this regular world, decisions are made on the margin and, in finance, opportunity cost is modeled as a trade-off between risk and return (Dixit & Pindyck, 1994; Markowitz, 1952). It is even common to use 11 In Québec these risks are being accounted for in the establishment of their AAC for two management units in the Nord-du-Québec region where fire risk is high. They approach the wildfire risk by running the usual deterministic growth and yield models and then they use expert opinion to estimate the necessary reduction in AAC(Personal communication with a modeler in the Bureau du Forestier en Chef in Quebec’s Ministère des Forêts, de la Faune et des Parcs). It is interesting that this is consistent with the general approach taken to risk in the Defensibility Heuristic. 98 this kind of thinking in the evaluation of new pharmaceuticals (Hauber et al., 2013) and crop technologies (Andow & Hilbeck, 2004). Every time a cost-benefit or cost-effectiveness analysis is done and expectations are taken, there is an implicit handling of the uncertainty in terms of normality (in the probability distribution sense) and making decisions on the margin. These approaches are consistent with the mathematical description of risk used in the Canadian Wildfire risk review(Johnston et al., 2020). The other description of risk that we have explored in Chapter 2 in the section on Pareto distributed risks on cost-benefit analyses and the use of averages or expectations on decision making. When fat-tailed distributions are the appropriate model we observe low-probability, high magnitude events distant from the calculated mean. If the socio-biological system gives rise to a distribution of events with these low-probability, high-magnitude events it is not appropriate to equate marginal benefits and costs. Instead we commonly use the precautionary principle. A strong form of the precautionary principle argues that when there is an activity or policy that has the potential for catastrophic, irreversible consequences there is no amount of benefit from the activity or policy that will make up for this and the risk should be avoided if possible ( Taleb et al., 2014). A key feature of problems where the Precautionary Principle is the appropriate framework is the risk of ruin: an event that a person, company, community ecosystem or society cannot get back up from again. We call these ruin events by a variety of names: death, bankruptcy, extinction, passing over the event horizon. In the study of random walks, these states are called absorbing barriers (Kac, 1945). In the language of stochastic processes this is a state from which the 99 probability of exit is almost surely zero. Often, the probability of hitting an absorbing barrier is small, but when you are participating in an activity where there is an absorbing barrier, the decision process is simple: avoid participating if at all possible. If the probability of ruin is small and positive, given enough time you will eventually find the ruin state, with probability one. Decisions that involve wildfire and other natural disasters often do not fit either of these approaches to thinking about risk. These events are unavoidable (the probability of an event occurring is always greater than zero and we play repeatedly whether we want to or not) but they also tend not to be entirely ruinous. For the communities, companies and ecosystems there is still the possibility of recovering and carrying on in most cases. As a result, the Precautionary Principle and the tools used for analyzing it are not entirely appropriate. And, as we have seen above, the standard tools from analyzing risk-return trade-offs are also not appropriate. Big fire events happen rarely, have an enormous impact, can permanently or semi-permanently impair the healthy functioning of the system and tend to result in large discontinuous changes in system variables (e.g. employment, harvest levels, mental health, animal and plant life). Thinking on the margin is not helpful in this world because sudden changes in conditions unexpectedly open up wide gaps between marginal benefit and marginal cost. In short, these are out of equilibrium conditions. I believe that these wildfire risks represent a third category of risk that needs a modified approach to thinking about decision making when we face this class of risks. This third category of risk exposure occurs when, (a) we are faced with low-probability, high-magnitude events, that (b) unfold over a short period of time and (c) are not absorbing but do result in a long-term loss of function to the community. That is, the community makes it to the next round but are unable 100 to offer the same possibilities for human well-being they did before the event. This loss of function can be measured in terms of economic activity, social participation (e.g. volunteerism and participation in community events), public health measures, or trust in institutions to name just a few. In the process of introducing this class of risk and the decision tools needed to think about it I will review the history of decision theory and define the idea of an exposure. 4.3.1 Three Classes of Exposure The two classical types of exposure are Gaussian and Ruinous (these correspond to Taleb’s Mediocristan and Extremistan (Taleb, 2007). With a Gaussian exposure, random changes to measured variables are balanced around the mean. Over time, once the trend is subtracted, you expect that random increases cancel out random decreases. There is no chance of ruin or absorbing barriers. In a Ruin exposure, the changes are systemic, usually fat-tailed and there is a state with non-zero probability that the agent, or system of interest hits an absorbing barrier. I am proposing that there is a third class of exposure that is described by large, discontinuous changes in state variables but there is no ruin, no absorbing barrier. There is, however, a permanent decrease in functional capacity. I call this a trauma exposure. I introduce this by analogy to physical tissue damage. The key feature of a trauma exposure is a discontinuous shock in the environment causing a significant reduction in functional capacity that is not reversible (or at least not over the period of time that is of concern to the modeler). Sufficiently long exposure to freezing temperatures results in frost bite. There is a point beyond which tissues cannot repair themselves. A frostbitten finger or toe is not going to end a life, but it may affect the functioning of that limb for the 101 reminder of the person’s life. Elsewhere this has been described as fragility (Borio et al., 2014; Panteli et al., 2017; Taleb & Douady, 2013). A roof collapsing under snow is an event where a threshold is crossed and the damage is irreparable. Thresholds are nice to model because it is simple when working with probabilities. It is often reasonable to model these decision problems with thresholds in terms of survival probabilities (Embrechts et al., 1997). Once we run into the absorbing barrier the game is over. There may be consequences from wildfire (like measures of ecosystem health) that can be described in terms of fragility, but I think there is another approach that better captures the exposures that characterize wildfire risks. There is another kind of physical exposure where there is a rapid change, when an external variable changes and results in ongoing pain and loss of function: whiplash. There is no physical discomfort when a person is sitting in a motionless vehicle, nor when the vehicle is moving at 120km/h. In fact, they feel nearly the same. There is a problem, however, when there is an abrupt deceleration from 120km/h to 0km/h. I think this is a reasonable metaphor for what happens to a resource economy hit by catastrophic wildfires (or policy changes). It is not immediately apparent how much pain is coming, but over time we find that businesses shut down, economic opportunity disappears, and social outcomes worsen (Carroll, 2018; Cordial et al., 2012; Kirschner, 2010; Perdue & Pavela, 2012; Shandro et al., 2011; Venkataramani et al., 2020) . This is not unlike the onset of soreness, lack of mobility, enduring pain and the potential for unrelenting migraines with severe whiplash. 102 4.3.2 Formal Statement of a Trauma Exposure To better define the statement of exposures generally and the meaning of a trauma exposure I define some relations to distinguish between different kinds of exposures. Suppose we have a payoff Π that depends on a system state, 𝑆𝑡 which can be influenced by a strategy 𝜉 and random shocks 𝜔: Π(𝑆(𝜉, 𝜔, 𝑡)) Π is increasing in 𝑆, 𝜕Π𝜕𝑆> 0 for 0 < 𝑆 < 𝑆.̅ For a Gaussian exposure we have that 𝜔~𝒩(0, 𝜎) and there is no state that causes the payoff to go to zero. The strategy can change the state space and the distribution of risks. A fragile discontinuity exists when Π = 0 𝑖𝑓 𝑆 = 𝑠 ∗ where 𝑠∗ is an absorbing barrier that there is some non-zero probability of reaching. This is ruin. In the event of this kind of exposure we want to minimize the exposure to ruin and we do this by following the Precautionary Principle. More interesting for the case of wildfire, we consider a whiplash-like trauma exposure. A trauma exposure occurs when the rate of change in the input variable exceeds a threshold ∆𝑆𝑡> 𝜏 which changes the achievable payoff so that it is permanently reduced, or the payoff function responds differently to the system state than it did before. In our notation, there is a new ceiling to the payoff Π̅ or there is a new payoff function Π′ such that 𝜕Π′𝜕𝑆<𝜕Π𝜕𝑆. Below, I show an example using the growth function in the growth and harvest model. From a decision-making perspective we want to avoid ∆𝑆𝑡> 𝜏. There are two ways of doing this. We can try and change 𝜏 by getting stronger or more robust. This would be a bit like 103 strengthening our neck in preparation for a car accident. Or the alternative is trying to change the factors that increase the likelihood of exceeding the threshold. In the wildfire case, we have already identified fuels as being the target. Wildfire exposure is clearly not Gaussian, which precludes the use of cost-benefit analysis, risk-return analyses and other expected benefit tools. Instead, we can choose to model the decision problem using the Precautionary Principle or by thinking about the new trauma exposure. In all cases, we want to choose the best option by changing the strategy, 𝜉, in a way that changes the risks, 𝜔, that determines the exposure to the payoff. The history of how we think about 𝜉 is discussed in the following section to inform decision making around the trauma exposure. 4.3.3 Approaches to Decision Making and the Exposures The theories of decision making proposed in the 20th century were an odd combination of practical philosophy, subjective perceptions of risk combined with a sometimes forbidding technical apparatus. I review the most popular approaches for making the “right” decision given uncertainty from the past century to justify my position that the appropriate metric for making decisions about wildfire management and other similar classes of problems is not maximizing expected net benefit but minimizing trauma exposure. In formal theories of decision making under uncertainty we start with two steps: first, identify the possible consequences of an action; and second, assign probabilities to these consequences (Arrow, 1951). There is an ongoing controversy over how these probabilities should be 104 interpreted that goes back to the beginnings of formal decision theory12. Different strategies result in different sets of consequences with each consequence having its own probability. The set of consequence-probability pairs can be described with a probability measure13. Choosing the right measure of consequence is a difficult problem. In engineering the problem will be defined by the physical properties of the system in question. If we want to minimize the costs of a bridge that meets the structural requirements to keep people safe the shocks and their distributions are more readily measured. In human matters, the description of consequences is fraught with slippery ideas. Most humans do not assign any inherent value to logs. We care about logs because they can be transformed into lumber and subsequently homes. And, generally, we do not care much about the home itself, we care about the joy that the relationships build in the home bring us14. These good feelings are given the name utility by moral philosophers (Bentham 1748-1832, 1843; Mill, 2014) and all but the most stubborn economists. Economists try not measure consequences in terms of net-income or collections of things or measurable physical properties but in terms of the utility we get from those things. I am in the stubborn category, as while it is obviously true that it is the feelings that drive our behaviour not the 12 In general, the theorists fall into two camps. Those who believe that probabilities can only be interpreted as relative frequencies (von Mises, 1941) and those who interpret probabilities as degrees of belief about the world (Keynes, 1921). Modern information theoretic version, a probability says something about the amount of new information that the arrival of an event carries (Cover & Thomas, 2006). The practitioner carries on giving little attention to what probabilities mean. The techniques from all camps of the debate are used in practice when a given technique has the greatest functional utility. 13 This is a bit hand-wavy. More formally, we can define a sample space, Ω, representing all of the mutually exclusive consequences. A set of zero or more of these consequences is called 𝔉 and there is a function 𝒫 that assigns events in 𝔉 a probability. This triple (Ω, 𝔉,𝒫) defines a probability space. 𝒫 is a probability mass function when the outcomes are discrete and a probability density function when the outcomes are continuous. Each change in strategy offer a different probability space. The rigorous foundations for this modeling approach were developed by Kolmogorov (Kolmogorov, 1950) and an accessible treatment is given by Feller (Feller, 1968). 14 Or the capacity of the home to signal our social status to the people around us. 105 objects themselves, trying to observe these feelings, measure them and model them using mathematics results in all kinds of problems (Boettke, 2012; Robinson, 1981; Simon, 1969). As a result, I have avoided utility in what follows and for the purposes of developing the models I have used measures of physical consequences, like annual cut and tonnes of greenhouse gas emissions. I leave it to the reader to imagine what kinds of feelings these physical goods furnish in the mind of the end user. An exercise in modelling empathy. If we have our sets of consequences and their probabilities, we can start go about making decisions by ranking different distributions of outcomes according to our preferred criteria. Different decision theorists have different criteria for ranking outcomes depending on the type of problem they face. I will describe three that were popular in the twentieth century and explain why I have decided to go with minimizing trauma exposure for the problem of wildfire. These three are Maximization of Expected Value, Minimization of Maximum Regret (MINIMAX regret) and Maximization of the Minimum Value (Maximin). I will address them in reverse order. 4.3.3.1 Maximin Maximin was introduced by Abraham Wald (Wald, 1945) This decision ranking approach chooses the strategy with the set of outcomes that has the least bad possible outcome in the set. That is the Maximin selects the strategy that minimizes the maximum loss for all the strategies15. A nice property of this rule is that you only need a list of the consequences for each strategy, and 15 The defensibility heuristic could be thought of as following the spirit of a Maximin strategy. 106 it is not necessary to assign probabilities to the outcomes. For this reason, it has been recognized as being useful for making decisions under extreme uncertainty about the appropriate probability distribution model This has proven useful in some engineering applications where failure is not an option or is highly undesirable or expensive. In these applications the use of Maximin has developed into a sub-field of operations research called Robust Optimization (Ben-Tal & Nemirovski, 2002). Outside of engineering, Leonard Savage (1951) and Kenneth Arrow (1951) have criticized Wald’s Maximin for being logically and morally indefensible in social decision making. Maximin is fundamentally pessimistic about the world. Wald described choice problems as a player facing off against nature where nature was out to get player 1 and the job of the player was to avoid the worst outcomes nature could throw at them. Under certain conditions (discussed more below) Maximin leaves a lot on the table. That is, there are strategies where we are ‘objectively’ better off with a different approach to ranking consequences. 4.3.3.2 Minimax Regret Savage (1951) proposed an alternative ranking approach that he called minimizing maximum regret. In Savage’s formulation players at the game of life should not try and manage downside risks all the time but accept that they are going to lose sometimes and instead focus on avoiding missed opportunities. This notion incorporates the idea of opportunity cost in a way that Maximin does not. Minimizing maximum regret asks, “What do we have to give up to achieve the Maximin optimum?” Calculating the regret functions can get complicated when there are many possible consequences, but the core argument is intuitive. If regret is defined in terms of expected opportunity cost we then try to minimize the opportunity cost of any decision. This works regardless of whether not you have a well-defined probability assignment to consequences 107 but if you do have a well-defined probability measure this approach is equivalent to maximization of expected net benefit. Savage convinced most social decision theorists and Maximin sees little play when downside is well understood or there is no risk of ruin. As long as the potential downside consequences of a strategy are limited, Maximin looks bad as a decision tool. We walk away from the exciting parts of life because of our fears. In a safe world, with no lasting negative consequences you are going to do a lot better in life taking the risks and minimizing opportunity cost by using the Minimax Regret rule. 4.3.3.3 Maximization If you are convinced that Maximin is inappropriate, and you think it is possible to estimate stationary probabilities16 for all consequences, you may as well just Maximize Expected Net Benefits. This is what most of decision theory has focused on for the last half of the 20th century. Maximization is the most straight-forward (least computationally intensive) approach to ranking outcomes but it requires a lot of assumptions to hold. Rankings of consequences are generated by taking the probability weighted average or expected value, 𝐸(𝑋) = ∑ 𝑝𝑖𝑥𝑖𝑛𝑖=1 . The strategy with the largest expected value is the one that should be selected. Cost-Benefit uses this same ranking approach but instead of individuals it assumes that policy makers can aggregate and calculate the distribution of net-benefits for society as a whole. This is a big ask. 16 There are two possibilities for dealing with non-stationary probabilities. If there is a predictable trend in the non-stationary data, this trend can be subtracted from the data to get a stationary series and the trend in changes in probabilities can sometimes be added back into the values as a growth rate in the benefits and cost series. The other option is to assume that over the timeframe of the analysis the probabilities are sufficiently stationary to be able to use expectations as if they were stationary. If neither of these are possible Maximization is likely not a good candidate for the problem that is trying to be solved. 108 Kenneth Arrow states that “No matter how complicated a game of chance is we can represent it by a single probability distribution of final outcomes.” (1951) Cost-Benefit Analysis is structured to model exactly this type of game and it does this by compressing consequences through time in a single probability distribution in two steps. First, we take the future stream of net-benefits and discount them to the present at specific interest rate17. This step compresses benefit flows in time into a single number. Second, the present value of the benefit flow is assigned a probability. The final step is taking the expected value of all of the present values. This provides a single number that summarizes all future possibilities for a given decision and can be ranked relative to the other outcomes from other projects or strategies. There are problems with this approach when one of the possible states along these time paths takes a value of 0. These states are called absorbing barriers in probability theory (Kac, 1945). Absorbing barriers are states you can’t get out of: bankruptcy, death, black holes. Things that really should be avoided at all costs when sober. In the expected value, these states are assigned a value of zero and assigned a weight 𝑝𝑖 so that 𝑝𝑖 ∙ 0 = 0 and depending on where 𝑝 is in [0,1] this state will not have much of an effect on the expected value. Using the maximization of expected value can generate questionable rankings when there are absorbing barriers. If I am faced with a decision between two strategies and the first strategy has a higher expected value than the second, 𝐸(𝑆1) > 𝐸(𝑆2), but in strategy one I am at significant risk of dying and in the second strategy I am not at risk of dying then we would expect most people to contradict the 17 The formula for present value is 𝑃𝑉 = ∑𝐹 𝑉𝑡 1(1+𝑟)𝑡𝑇𝑡 =1 109 maximization prediction. In these instances when there is an absorbing barrier the Maximin approach to ranking strategies corresponds to most people’s decision making. Another way of thinking about the relative appropriateness of Maximization vs. Maximin is in terms of the cost of experimentation. Suppose I am seeking information about the world that I suspect will make me better off. I am trying to decide where or not it will be worth the expense of running several different experiments to get more information. If the downside seems small (i.e. I spend a small of money or the risks to society are low) for all of the options, then it makes sense to rank all of the experiments according to the expected value measure. If, on the other hand, I am faced with the possibility that one of the outcomes is catastrophic (perhaps it prevents me from running any experiment ever again or my capacity for future experimentation is severely limited) then it no longer makes sense to rank the experiment by expected value. We should turn to Maximin. This seems to create a clear separation between the decision problems that are properly addressed with Maximin and those that are best address with Maximization or Minimizing Maximum Regret. Absorbing barriers, like bankruptcy and death, are clear and hard stops on future possibilities and they should be treated differently than other possible states of the world. Absorbing barriers deserve a different weight in your decision making than their probabilities alone would suggest. For these ruin exposures we choose Maximin and for Gaussian exposure we choose maximization as our decision rule. What rule should be used for a Trauma exposure? 110 4.3.4 Things Fall Apart In addition to absorbing barriers there is another kind of possibility that needs to be considered when deciding which ranking tool is the best. Up to now, we have only dealt with the value of different states (or sets of states unfolding through time) of the world. How we transition between states is not relevant to the discussion. This is a mistake in situations when the rate of change matters as much or more than the change itself. Returning to the whiplash example, when we abruptly decelerate in a vehicle from 90km/h to 0km/h, if death is avoided, there is a high risk of permanent tissue damage that will limit quality of life permanently. Decelerating over a long period of time causes no discomfort at all. We are sensitive to the rate of change. I think this is also true for economic activity and social relations. A rapid decline in economy activity (say over a few days or weeks from a mill or mine closure) can cause long lasting damage to individual mental health and economic prospects as well as damaging relations between business partners and members of communities that will not be observed if the same decline were observed over a period of a year or a decade. 4.3.4.1 Trauma Exposure as a Social Phenomenon Does rapid decline in economic activity limit future capacity for economic output, well-being and social cohesion in measurable ways? Yes, but not always. There seem to be three major factors that contribute to decline in community capacity for economic sustainability after a negative shock to the economy. The first is labor mobility of the youngest, healthiest and most-skilled members of society (Alston, 2004; 111 Simmons & Ramiro, 1972; Stalker & Phyne, 2014). The second factor is the decline in well-being of those left behind. Crime increases (de Blasio et al., 2016; Krüger, 2011), mental health declines (Kirschner, 2010; Paul & Moser, 2009), all-cause mortality goes up and life-expectancy goes down (Beckett, 1988), in many cases driven by deaths related to substance abuse (Perdue & Pavela, 2012; Shandro et al., 2011; Venkataramani et al., 2020), and domestic violence increases (Pattavina et al., 2015). Those left behind are clearly much less well-off and this manifests itself socially in less volunteering and community activities (Stalker & Phyne, 2014) . Which leads us to the third factor, people lose trust in the institutions that organize their lives when these downturns occur making governance more difficult (Sani & Magistro, 2016; Stevenson & Wolfers, 2011; Tonkiss, 2009) In British Columbia these factors are exacerbated by the fact that since the industry restructuring driven by the mountain pine beetle harvests, firms have attempted to lower costs by hiring contractors from out of town and successfully lobbying to have their municipal taxation lowered (Martin, 2014). As a result, the labour market is even more fragile, and the communities have had their capacity for responding reduced through lower incomes. Given these factors, skilled-labor mobility, declines in health in the remaining population and loss of trust in institutions, we can see how a rapid decline in economic activity could reduce economic capacity over the long run. A catastrophic wildfire could cripple a town indefinitely. Does this always happen to a community in the face of economic shocks? No. Jane Jacobs describes the rapid turn-around of the Los Angeles economy after the second world war. Los Angeles was a critical region for producing munitions during the war. Overnight the factories were shut down with the signing of the armistice with Japan. Within months the city had started 112 to reorganize itself for the production of domestic consumer goods and was registering positive economic growth in short-order (Jacobs, 1970). Why did Los Angeles see better outcomes then rural communities? The young people stuck around to build businesses, there was strong trust in the institutions and business networks in the city and subsequently there was no massive increase in addiction deaths or physical capacity of older workers. These social effects of rapid economic decline in rural resource-based communities are useful to think about because they allow us to select the right decision-making tool. In the case of wildfire, we can capture the complexity of changes in income, wealth, mental health, life expectancy, trust, and institutional resilience in a single measure because we have found a choke point in the system: discontinuous shocks to fibre flow. This is the ‘lifeblood’ of these communities. In a tourist-based town like Tofino, it could be tourism dollars so this is a more general idea but will focus on the loss of economic viability of a community due to wildfire. In communities with diverse economies, like Greater Los Angeles, there is not going to be a choke point like this, and the trauma exposure will not apply. The exposure in economically diverse communities does not fall into any of the three exposure categories. The risk is not Gaussian, but an extreme fire event is unlikely to result in ruin or trauma as it is def ined here. Only ruin and trauma exposure models are appropriate when there is a fragile local economy threatened by wildfire. If there is an absorbing barrier then now we need to apply the Maximin rule and the Precautionary Principle. What about the trauma exposure? 113 4.3.5 The Trauma Exposure Decision Rule With catastrophic wildfire threatening a community we have a situation where there is no absorbing barrier, but we do have the potential for reduced functioning in the local economy following a rapid decline in the cut or severe infrastructure damage (say to a mill). The appropriate decision tool in this scenario is not maximizing or applying the Maximin rule but to choose a level but instead choosing the strategy, 𝜉, that either avoids the trauma if that is an option or changes the distribution of outcomes in a way that lowers or eliminates the probability of a trauma exposure. As we develop the model further I continue to use annual cut as a proxy for economic activity. A trauma exposure occurs when the change in cut changes at an unacceptable rate. An unacceptable rate would be one where the mill managers cannot adjust mill assets in a way that avoids wide-spread lay-offs and reduced investment in the community. This rate is defined as 𝛿. So that we want to choose a forest management strategy, that minimizes the probability of exceeding this critical rate: 𝑚𝑖𝑛𝜉 𝑝(Δ𝑆Δ𝑡> 𝛿) (19) This is an optimization problem, but we are not minimizing or maximizing a level but the probability of observing a negative rate of change beyond a certain threshold. In the case of mill towns, we would ask, what can the local economy take given extant finished product prices? 114 4.3.5.1 Estimating the Trauma Exposure Before continuing it should be noted that this is purely theory and, for the same reason we have trouble getting a convergent expectation value for the purposes of maximizing the level of expectations we will have trouble minimizing the probability of change above a threshold. The modelling tools don’t exist and the data follows fat-tailed distributions so estimations of expected values and variances are not convergent. Following the logic of this theory, however, allows us to ask interesting questions about how we should arrive at a ‘sustainable yield’ for forest harvest and so we will continue down this path. Continuing on with the stylized model of forest growth, we begin by describing an estimation procedure for 𝑝 (Δ𝑆Δ𝑡> 𝛿). Steps in the Analysis of a Trauma Exposure 1. Identify the class of exposure, make sure it is a Trauma Exposure (i.e. the rate of change matters and there is no ruin state) 2. Identify a growth model for the forest1 and the cut level that results 3. Describe the pay-off for a given growth path to the local economy (Π) 4. Choose a model for the disturbances (𝜔) 5. Identify the 𝛿 that defines an exposure 𝑝(|Δ𝑆Δ𝑡| > 𝛿) and describe the change in Π under whiplash conditions 6. Identify strategies 𝜉 that change, either change the level of 𝛿 or minimize 𝑝(|Δ𝑆Δ𝑡| > 𝛿) 7. 𝑚𝑖𝑛𝜉 𝑝 (Δ𝑆Δ𝑡> 𝛿) 115 Remember that the change in stock Δ𝑆𝑡 is a function of the current stock and the random disturbance 𝜔𝑡 and the cut level (which in equilibrium is equal to the mean annual increment). Δ𝑆𝑡 = 𝑟𝑆𝑡 (1 −𝑆𝑡𝑆̅) − 𝜅𝑡 − 𝜔𝑡 (20) There is an interaction between the growth in the forest stock, the growth in capital and the random shocks. As a result, estimating 𝑝 (Δ𝑆Δ𝑡> 𝛿) is not a simple matter of looking up the probability of events that exceed 𝛿 given the parameters that describe the disturbance 𝜔. To get an estimate of 𝑝 (Δ𝑆Δ𝑡> 𝛿) we would have to run many Monte Carlo simulations until we begin to see convergence. Depending on how extreme the distribution of 𝜔 is this could be hundreds of thousands or millions. This is even with our extremely simple model of stand and capital dynamics. The most straight-forward way of doing this would be to run simulations that draw the different growth paths of 𝑆𝑡 and their impact on capital. For each growth path if there is an event where the change in stock is greater than the threshold there is an indicator function which takes the value one. To find the probability of exceeding this threshold we sum the values of the indicator function and divide it by the total number of runs. Formally, there is a set 𝒜 that includes all paths where 𝑝 (Δ𝑆Δ𝑡> 𝛿). And there is an indicator function that takes the value 1 if a path is in 𝒜 and 0 otherwise. 𝕀𝒜 ≔ { 1 𝑖𝑓 𝑥 ∈ 𝒜0 𝑖𝑓 𝑥 ∉ 𝒜 (21) 116 The probability 𝑝 (Δ𝑆Δ𝑡> 𝛿) is given by the sum of the 𝕀𝒜divided by the number of growth paths, N, that are simulated. 𝑝 (Δ𝑆Δ𝑡> 𝛿) = p(𝕀𝒜 = 1) =∑ 𝕀𝒜𝑁 𝑎𝑠 𝑁 → ∞ (22) The strategy that results in the smallest 𝑝 (Δ𝑆Δ𝑡> 𝛿) is the best bet for avoiding economic trauma in the wildfire case. We would no longer be using expected value or a level of output as our decision variable but instead this probability of change beyond a threshold. Unfortunately, this strategy falls to the same difficulties of convergence that expected value does. 𝑁 must be very large, even for exceedingly simple models, for there to be a convergence in 𝑝 (Δ𝑆Δ𝑡> 𝛿) when the uncertainty is not normally distributed. Is there a useful way to model this in the wildfire context? The description so far treats stock as a uniform aggregate, 𝑆𝑡, at time t. In practice the stock of trees has a non-uniform distribution on the landscape. By modeling this non-uniform distribution in a simple fashion using cellular automata we can learn some interesting things about the conditions that can give rise to different types of exposures and how we might change the stock level and its distribution on the landscape to reduce wildfire caused trauma exposures and limit the opportunity cost of doing so. 4.4 Sustainable Cut Revisited The current means of calculated mean annual increment does not work in the face of asymmetric shocks. If we return to the original forest stock dynamic ∆𝑆 = 𝑟𝑆𝑡 (1 −𝑆𝑡𝑆 ̅) and plot change in 117 stock as a function of stock level we can see that without uncertainty, we would want to choose the stock level that maximizes annual growth and harvest continuously at that level. In Figure 17 this is at 5000. Figure 17 Stock vs Change in Stock. For a maximum stock of 10,000 and a growth rate of 2% /year. We have seen that if we introduce the asymmetric risk of wildfire and we maintain harvest at the mean annual increment we eventually drive the harvest to zero. In a world where large scale catastrophic wildfire and pest epidemics are possible this traditional approach to resource management cannot be correct. We need a new sustainable yield rule that accounts for arrival of large-scale forest losses and the potential mitigation efforts that could reduce the size of these losses. We need to take a short detour through cellular automata to motivate what follows. 118 4.4.1 Simulating Fires with Cellular Automata Figure 18 Imagine the forest on a checkerboard space. Each cell may contain a tree or an empty cell. If a fire ignites in the centre cell it can move to any other cell surrounding it that contains a tree. Image from Flavio Sousa (Sousa, 2015) Imagine a grid like the one shown in Figure 18 as a set of potential growing sites for trees extending over the horizon. If every site has a tree on it the stand is maximally dense. Under dry conditions, fire can travel to every tree on the grid. As we start to remove trees from this grid it slows fire because it has to travel around the empty cell. If trees were removed in a cell on this grid the fire would only burn in a cluster. If a certain percentage of trees, p, were removed at random from this grid then it would reduce the potential size of the fire clusters by at least p times the number of cells in the grid. Below a critical level, 𝑝𝑐 18, the trees are separated into many clusters and the fire can no longer travel through the forest (Barabasi, 2016) . This is a simple model but it has been proposed repeatedly as a model for wildfire (Bak et al., 1990; Drossel & Schwabl, 1992; Grassberger, 1993). As p changes, the distribution of fire outcomes 18 In the simple cellular model in Figure 18, 𝑝𝑐 = 0.593. 119 changes. For a given p, if I run thousands of random ignitions on the grid I will get a distribution of fire sizes that is approximately Pareto distributed (Malamud et al., 1998). Analysis of real-world fires suggests that percolation models may be appropriate as well (Caldarelli et al., 2001). In the real world, the passage of fire from one cell to another is not going occur with a 0 or 1 probability. Fuel moisture, the amount of fuel burning in the cell next door and the wind can all contribute to the likelihood of a percolation from one cell on the grid to the next. In a contiguous forest of a pine species with heavy surface fuels developing into ladders reaching into the canopy we can imagine that an ignition in one square has relatively little ‘friction’ when it travels into the adjacent square. That is, it will percolate quickly through the network. With no breaks and no fire suppression efforts we can imagine that the upper end of the burn area distribution (we use the notation 𝐻 in the Pareto model) is going to be all of the cells in the grid. The maximum fire area is the whole area under consideration. By increasing the friction between cells we reduce both the maximum burn area, 𝐻, and the shape of the distribution of outcomes. The tail becomes less fat and there is a possibility that the probability of a trauma exposure can be reduced this way. 4.4.2 Evidence for Increasing Friction This abstraction of fire spreading as a cellular automata is a fair distance from reality and I should support the frictional assumptions I indulge in above with a review of some of the work that has been done on real-world changes in forest fuel structure and fire behaviour. Studies using fire behaviour simulation tools have been the most skeptical of fuel thinning and prescribed burning. These are models where a virtual landscape is populated with fuels. Wind 120 and temperature is simulated. The modeler starts a fire and sees how it burns according to a fire behaviour rule-set or set of equations. One study finds that fuel-treatment effort explains little of the area burned by moderate-to-high severity fire (Cary et al., 2017) or fire extent or intensity in a simulation of prescribed burning strategies in Tasmania (Furlaud et al., 2018). Another simulation study across several US states finds mixed results. Some simulated areas showed a decrease in burn severity and burn area and others saw an increase (Cochrane et al., 2012). The authors note that forest practices around the treatment matter for the type of outcome observed but that the simulation model does not have high enough resolution to capture these differences. These results are not always consistent with other types of evidence from the field and there is, at least one, simulation paper that finds that fuel treatments do have an effect on burn severity (Krofcheck et al., 2018). It is possible that the models are not able to capture the nuances of what is happening on the ground. The observations that are most consistent with the simulation results are reports from wildfire fighting teams. A report on the Cascade Mountain Complex in Idaho (R. T. Graham et al., 2009) found that prescribed burning from 1996 had no impact on fire spread in the 2007 fire. Another report from the Fourmile Canyon fire in Colorado (R. Graham et al., 2012) found that the fuel treatments completed in the prior seven years had not been maintained and had no discernible effect on fire behaviour. The authors implied that this lack of effectiveness was likely more a failure of on-going management than a shortcoming of the fuel treatments themselves. A study of fuels fragmentation in the Boreal forest of Saskatchewan showed that forested areas with higher fuel fragmentation had fires with smaller burned area (Lehsten et al., 2016). Similar 121 results have been found for the high-elevation sub-alpine forests in the Sierra-Nevada (Collins et al., 2009). Another study in the Sierra-Nevada has shown that fire that burns over an area burned in the prior 15 years tends to burn at lower severity. If the fire interval is longer, burn severity tends to be in the same class as the previous fire (Harris & Taylor, 2017). In Grand Canyon National Park the evidence shows a trend toward lower severity fire when burned for the second time with a wide distribution of outcomes that seems to be dependent on dominant tree species/ecosystem(Hoff et al., 2014). A study in the Gila National Forest using satellite data finds that low-severity fire tends to follow low-severity fire and high-severity fire tends to follow high-severity fire. The authors speculate that the high-severity followed by high-severity phenomenon may be recent and caused by changes in climate (Holden et al., 2010). The team of Park et al. studied the effect on severity and burn area of prior burns in the Frank Church wilderness in Idaho and the Gila Wilderness in New Mexico and found that prior fire limited fire spread for 6-18 years and decrease burn severity for up to 22 years (Parks et al., 2016). A further study in other wilderness areas in the US by the same team found similar results with a longer lasting down-regulating effect in wetter ecosystems (Parks et al., 2016). Although there are differences between ecosystems, a global review found that prior fire will lower fire spread and severity for a time (Prichard et al., 2017). This evidence is consistent with the theory being proposed here. A single fuel-management study from the Black Hills of South Dakota in Ponderosa Pine forest (similar to those found in the Okanagan Valley, British Columbia) found that prescribed burning worked to limit understory fuels for 10-20 years and recommended that burning on an interval shorter than 15 years was likely necessary to maintain the understory in a lower risk condition (Battaglia et al., 2008). 122 Some of the more convincing evidence for fuel treatments comes from studies of fuel treatments that have been studied after being tested by fire and evaluated for severity (generally measured by tree mortality as a proportion of the stand). A study of the 2006 Tripod Complex in Washington State found that 57% of the stand survived in areas that were thinning and burned vs 19% in stands with just thinning and even worse outcomes for no treatment (Prichard et al., 2010). Further study of the fires in the complex revealed that more recent prescribed burns were the most effective in limiting stand mortality (Prichard & Kennedy, 2014). A study of the 2013 California Rim Fire (Lydersen et al., 2017) found similar results and found that stands that received both a thinning treatment and prescribed fire performed better than either treatment on their own in terms of the proportion that suffered from high severity fire. One longitudinal study out of Southwest Australia looking at 52 years of fire in Eucalypt forests found that prescribed burning is effective but loses efficacy, on average, 6 years after treatment (Boer et al., 2009). More substantially, a review of 56 studies of fuel treatments being test by wildfire and measured in terms of a decline in tree mortality and/or crown scorch found that a mechanical treatment following by prescribed burning is the most effective approach with less benefit from either approach on their own (Kalies & Yocom Kent, 2016). Guides to fuel treatments have adopted the view that mechanical fuel treatment should be followed by prescribed fire and treatments need to be maintained for effectiveness (Hoffman et al., 2018; Jain et al., 2012). This evidence supports the idea that we can reduce losses to harvestable timber, the ‘lifeblood’ we try to protect in the exposure models, with mechanical fuel treatments and the regular use of prescribed burning. This is consistent with the abstraction of increasing friction between cells. 123 4.5 Finding Sustainable Yield with Asymmetric Shocks Again, the basic premise here is that if we insist on harvesting at maximum MAI (the peak of the curve in Figure 17) without a refractory period in the face of asymmetric shocks we eventually get no harvest at all. Instead, we need to choose a harvest level below max MAI that has a buffer built in. How do we go about figuring out what this yield should be? If we are in a low friction scenario and the upper limit on the distribution, 𝐻, and the entire area under consideration19 we can examine the curve in Figure 17 and note that there is no safe harvest level over the long run. Given enough time the entire board will burn over and the forest dependent community will suffer whiplash (in fact, the community would suffer whiplash at much lower levels of damage).20 What happens if we could apply mechanical fuel treatments and prescribed fire to increase friction to fire spread and lower 𝐻 while sacrificing the total potential for harvestable biomass. In our model this means sacrificing some of the stock, 𝑆𝑡 , for a decrease in 𝐻 (and likely a change in the shape of the distribution so that there is less mass in the tail of the distribution). The economic question now becomes, what is the trade-off between changing the forest structure, lowering 𝑆𝑡, and decreasing the level of the worst-case scenario, 𝐻, that sustains the greatest harvest. 19 In this simplified model we assume that a burned square reduces harvest to 0. We could make assumptions about differential burn severity but that is not salient to the point being made here. 20 In practice, fires that do not have full coverage would fragment the forest, increase friction and lower 𝐻 but the potential for economic damage is substantial. 124 This implies finding the most efficient means of introducing friction in fire spread to make the best use of lowering 𝑆𝑡 and changing the distribution of the forest areas on the landscape. This is a kind of fire-landscape selection problem that could correspond to the ideas of Leverkus’ fire absorbent landscapes (Leverkus et al., 2017) or the patchwork landscape idea out of the Pacific Northwest Research Station (Hessburg et al., 2005). In this model of forest growth, harvest, and fire-loss, when we choose a growth curve, we are also choosing a pattern of the forest on the landscape and a set of management activities that create and maintain that pattern on the landscape. The arrangement of tree stands/fuels determines the maximum loss, H, as well as the shape of the distribution of losses, and the available regular harvest or sustained yield. In the model described above the potential loss of timber units has an upper bound, symbolized as H. All forests are of finite size, so the worst-case scenario is that 𝐻 = 𝑆𝑡 at all times. In practice, we don’t see fires that destroy the entire timber harvesting land -base which implies that, generally, 𝐻 < 𝑆𝑡. We also suppose that modifications to the fuel structure can influence the worst-case scenario fire so that H is a function of the stock, 𝐻(𝑆𝑡). We can then ask, which stocking level (and in practice, arrangement of fuel on the landscape) should we choose to limit the possibility of a trauma exposure, or avoid it altogether, and what does this stocking level mean for the forest yield over time? To answer these questions we need to, (a) find the decline in 𝑆𝑡, that will result in trauma exposure, and (b) determine the relationship between 𝐻 and 𝑆𝑡. For the first question, the answer would depend on the number of mills in town, the kind of fibre they require to operate and what the minimum profitable throughput of the mill is. For the purposes of the simple model here, we will assume that a decline in Mean Annual Increment of 125 25% results in the shut-down of the local forest products industry and a trauma exposure for the community. To give shape to 𝐻(𝑆𝑡) we will return to the simple grid models of fire spread making use of the Drossel and Schwabel model (1992). In this model we have a set of cells arranged in a grid like a chessboard. The cells are either empty or occupied by a tree. The ratio of cells occupied by trees to the total number of cells is called the density. In the first step of the model a tree is set alight by a lightning strike. Any cells in the Moore neighborhood 21 (Moore Neighborhood -- from Wolfram MathWorld, n.d.) of the tree that was set alight that also contain a tree will start to burn. We do this again until all of the trees contiguous with the first one have burned. For an example model run see Figure 19. 21 Choose any cell, all the cells that surround that cell, even on the diagonal, are in the Moore neighborhood of that cell. 126 Figure 19 An example fire using a simplified Drossel and Schwabel model after 150 steps. Red is fire, green is unburned trees. If we run this model many times at different densities, we can map out approximations of the maximum fire size for a given density. Using this simple model, we will take this curve as defining our relationship between maximum fire size, 𝐻, and 𝑆𝑡 . Figure 20 shows examples of the fire runs at different densities. 127 Figure 20 Examples of fire runs using the cellular automata model of Drosser and Schwabel at different densities. The growth in burned area in non-linear as the tree-occupied cells become more densely packed. For the curve of maximum burn sizes see Figure 21. This curve is constructed by plotting the maximum burn sizes after 10,000 fires at each density and then fitting a logistic curve to the results. This model gives another S-curve! The equation for this logistic curve is given by 𝑓(𝑥) =11+𝑒−𝑣 (𝑥−𝑥0) with parameter values 𝑣 = 43 and 𝑥0 = 0.5525 and 𝑥 takes the values of the densities. In Figure 21 we see that the curve has a very aggressive slope. This model is clearly an extreme simplification or abstraction. Once a fire has started nearby trees ignite with probability 1. There is no accounting for spotting, terrain, windspeed or direction, or any other fire weather variable. This is a model where the ‘friction’ only arises because of changes in the 128 forest density. Further on we can use changes in 𝑣 and 𝑥0 to model friction from changes in fuel structure in the model and see its effects. Figure 21 The relationship between fraction of stock burned, which can be multiplied by the stock, S, to get, H vs the density. The density multiplied by the total carrying capacity, ?̅?, gives us the stock. Using this curve we get our functional relationship H(S). For this model in a 100x100 grid we can fit a logistic curve and get the equation 𝒇(𝒙) =𝟏𝟏+𝒆(−𝒗(𝒙−𝒙𝟎)) with parameter values 𝒗 = 𝟒𝟑 and 𝒙𝟎 =0.5525 and 𝒙 takes the values of the densities. With a 100x100 grid we have a maximum potential stock of 10,000 units, similar to the difference equation model we used previously. We can use this to relate the stock to H using the curve that we have fitted by multiplying the density by the maximum potential stock to get 𝑆𝑡 and the maximum burned area fraction by 𝑆𝑡 to get 𝐻, the upper truncation of our burn area distribution. With the logistic curve, 𝑓(𝑥) =11+𝑒−43 (𝑥−0.5525) the lower the stock, the lower the 𝐻 but it does not happen in a linear fashion. The potential for burned area changes very quickly between a density of 0.4 and 0.6 in this model. The slope is steep. 129 We could adjust this model by changing the parameters 𝑣 and 𝑥0. Figure 22(a) shows an example where 𝑣 is adjusted to change the slope to make the decrease in density result in a less aggressive decline in maximum burned area and 22(b) shows an example where the centre of the curve is shifted along the density axis. These variables would roughly correlate to how worst -case burn area responds to fuel treatments, or fuel fragmentation in the case of 𝑣 and how quickly we reach a critical fuel density with the shift in 𝑥0. 𝑣 relates to the probability that a treed cell catches fire when a neighbor cell is on fire. In the real world, 𝑣 could be a function of the fraction of deciduous trees, the growing habits of the conifers (do they self-prune or do the lower 130 branches brush the forest floor like black spruce) and other characteristics of the ecosystem and 𝑥0 would really be affected by the probability of spotting. Figure 22 (a) shows an example where v is adjusted to change the slope to make the decrease in density result in a less aggressive decline in maximum burned area and (b) shows an example where the centre of the curve is shifted along the density axis. These variables would roughly correlate to how worst-case burn area responds to fuel treatments, or fuel fragmentation in the case of v and how quickly we reach a critical fuel density with the shift in x_0. v relates to the probability that a treed cell catches fire when a neighbor cell is on fire. 131 This abstraction leaves out a lot of detail to model a simple relation between forest density and the distribution of burned area. Before continuing it is worth recapitulating what has been done: (1) To represent the geographic distribution of the stock, 𝑆𝑡, I have used a cellular automata model with trees distributed on a grid. (2) Fire spreads on the grid with probability one when there are cells in the Moore Neighborhood that also have trees. (3) The size of the fire is determined by the density of the trees on the grid. Where density is defined by the number of cells with trees divided by the total number of cells in the grid. (4) By running thousands of simulations at each density and measuring the total burned area for each simulation we can estimate the maximum fire size at each density. (5) This maximum fire size corresponds to the upper bound of the fire distribution, in the aggregated model before the cellular automata was introduced. This allows us to fit a function that describes the relationship between the maximum fire size and the stock. Where the stock is the density multiplied by the number of cells in the model. When we examine this function that relates the maximum fire size (H) to the stock (S) we see two interesting things. There is a critical stocking density (𝑥0) above which the maximum fire size quickly approaches the entire stock and there is a parameter 𝑣 that changes the slope of the function. These parameters are of interest because we can map them on to real world features of the forest system that helps to identify trade-offs that should be analyzed in more detail. We can think of the critical stocking density (𝑥0) as an abstraction of the real-world arrangement of fuels 132 that is very amenable to fire spread. In some ecosystems, for instance where we grow spruce for commercial purposes or we are looking to maximize carbon sequestration the density of the forest fuels also leads to a propensity for rapid fire spread under extreme fire weather conditions. The v, or slope parameter corresponds to how fire spread is affected by changes in the density of fuels. If arrangement of fuel is changed from a high density, yield maximizing plantation to a patchwork or fire absorbent landscape, v, determines how quickly fire spread is reduced. In the cellular automata model this is very straightforward because a tree burns with probability one when a neighbor tree is on fire. There is no differentiation between species, dry matter on the forest floor or ladder fuels. We can use our imagination though for the following sections and imagine that we can also affect v by planting different species, removing ladder fuels, etc.… One of the shortcomings of this model is that as 𝑣 is adjusted and slope becomes less steep on the part of the curve below 𝑥0the worst-case burn area as a function of density actually increases (Fig 133 22b). These worst burn scenarios are still below the trauma-exposure threshold and do not affect the general results of the model but they are not realistic. 4.5.1 Combining the Worst-Case Burn Function and Harvest Dynamics We can combine this model of the relationship between maximum burn size and stocking density with the growth model described previously to explore what happens to harvest and capital stock as stock densities are changed. The growth model was described by the difference equation: ∆𝑆 = 𝑟𝑆𝑡(1 −𝑆𝑡𝑆 ̅) with 𝑟 = 0.02 and 𝑆̅ = 10,000 (which corresponds to the number of cells in the cellular automata model). The maximum sustained yield in this model is found at 5000 units. This corresponds to a Mean Annual Increment of 50 units. A 25% loss (the trauma exposure) would correspond to a decline in MAI of 37.5 which corresponds to a stock level of 2500 units. The maximum sustained yield of 5000 is close to the centre of the maximum burn model where the mid-point of the curve is found at a density of 0.5525 or 5525 units. It turns out that this matters for avoiding a trauma exposure. Consider what happens with the original fitted curve from the worst-case burn model from Figure 21 and using it to set the upper bound for burn damages in the sustained yield model. An example of the output from this model is given in Figure 23. Notice that the disturbances get smaller over time. As the stock level declines so do the worst-case fires. The worst-case fire damages are self-limiting. Still because the harvest continues at the mean annual increment there 134 is no room to recover from the annual fires. The fires become limited in size though so no trauma exposure occurs and it takes a very long time for the stock to decline. We can see that, with the parameters used in this model (𝑣 = 43 and 𝑥0 = 0.5525) a trauma exposure will never occur and the fires are self-limiting. Figure 23 A run of the growth, fire and cut model with an H that is a function of the stock level, where 𝒗 =𝟒𝟑 and 𝒙𝟎 = 𝟎. 𝟓𝟓𝟐𝟓. Notice that the disturbances get smaller over time. As the stock level declines so do the worst-case fires. The worst-case fire damages are self-limiting. Still because the harvest continues at the mean annual increment there is no room to recover from the annual fires. The fires become limited in size though so no trauma exposure occurs and it takes a very long time for the stock to decline. This is even true if the worst-case fire happens every year (Figure 24). 135 Figure 24 Stock (S_t) and Cut (k_t) when the worst fire happens every year, k=43, x0=0.5525. Stock will decline indefinitely but it happens at a very slow rate. We can adjust for this annual burn and stop the decline by slightly reducing cut to 96% of the mean annual increment. This allows for an equilibrium to arise between what burns every year and stock left behind that leads to more growth. For the parameters selected here it settles down around a stock of 4000 for a cut of 96% of the mean annual increment. An example run can be seen in Figure 25. 44.045.046.047.048.049.050.035003700390041004300450047004900110 19 28 37 46 55 64 73 82 91100109118127136145154163172181190199Cut Level Stock LevelTime Period (t)Stock Level and Cut When the Worst Case Fire Happens Every Year S_t k_t136 Figure 25 In this figure we adjust for the annual burn and stop the decline in stock by slightly reducing cut to 96% of the mean annual increment. This allows for an equilibrium to arise between what burns every year and stock left behind that leads to more growth. For the parameters selected here i t settles down around a stock of 4000 for a cut of 96% of the mean annual increment. So far, the fire is self-regulating in this model, and does not threaten a trauma exposure and we can stop the monotonic decline in MAI by cutting slightly below the annual growth. Is this true for all possible parameters 𝑣, 𝑥0 in the worst-case burn model? The answer is no and it gives some insight into prioritizing fire hazards and designing fire-absorbent or patchwork landscapes that might avoid trauma exposures. 137 We can think about this in terms of what happens when the worst-case burn is more or less responsive (the slope of the curve) and where the stock is relative to the mid -point of the worst-case burn curve as a function of density. First consider what happens when the slope variable, 𝑣, changes so that worst-case burn is less sensitive to changes in the density and stocking level (Figure 22b). Together with tree spacing (lower density) changes in the slope parameter introduce the idea of friction discussed above into the model. In figure 26 in both (a) and (b), the slope parameter v has been set equal to 25. This results in worst-case burns that are less sensitive to declines in stock. At the outset, the worst-case burn is less than the level of a trauma-exposure so aggressive fuels mitigation is unnecessary. In (a) we continue to harvest at the MAI and the decline in cut is more aggressive than in Figure 23. In (b) the cut is reduced to 92% of the MAI and a rough equilibrium is met. Harvests can continue indefinitely at the 4600 stock level. Again fire takes care of itself and is self-limiting without threatening the viability of the forest-based community in this model. The outcome is different when the stocking level is above the mid-point of the worst-case burn function. Suppose we go back to the sensitive slope with 𝑣 = 43 but reduce the midpoint density to 𝑥0 = 0.45. This means that above a stocking level of 4500 the worst-case fire quickly becomes large and dangerous and trauma exposure becomes a possibility. If our starting point is still 𝑆𝑡 = 5000, at this level the worst-case fire is at 4478. Potentially wiping out about 90% of the cut. If the distribution of fire outcomes really does follow a Pareto distribution this is an unlikely outcome but it can still be avoided in this model by aggressively cutting down below 4500. In Figure 27 the cut is aggressive at the beginning, above the MAI, until the stock level is 138 driven below the critical density, after which point the AAC is set to 0.94% of the MAI to find an equilibrium where the cut can truly be contained. Figure 26 In both (a) and (b), the slope parameter 𝒗 has been set equal to 25. This results in worst-case burns that are less sensitive to declines in stock. At the outset, the worst-case burn is less than the level of a trauma-exposure so aggressive fuels mitigation is unnecessary. In (a) we continue to harvest at the MAI and the (a) (b) 139 decline in cut is more aggressive when than in Figure 23. In (b) the cut is reduced to 92% of the MAI and a rough equilibrium is met. Harvests can continue indefinitely at the 4600 stock level. Figure 27 𝒗 = 𝟒𝟑, 𝒙𝟎 = 𝟎. 𝟒𝟓. Aggressive harvesting at the beginning brings the worst case risk down to a level below the potential for a trauma exposure. Finally, there is a third outcome, where the fuels are unresponsive to fuels mitigation, the stocking level is well above the critical point in the worst-case burn curve and getting the stocking level below the mid-point means that the annual growth does not result in a viable industry. The logical route here is to liquidate the timber asset and compensate residents to move and support them in re-training wherever they land. This can be summarized in the two by two matrix in Table 14. 140 Worst-Case Responsive to Treatment Worst-Case Unresponsive to Treatment Stocking Level Above Critical Cut to below critical stocking level. Reduce AAC sufficiently to stop slow degradation of resource from fire. Liquidate Timber-Asset, Compensate Residents and Re-Train Stocking Level Below Critical Fire takes care of itself. Slightly reduce AAC to account for small fire loss every year. Fire still takes care of itself. Reduction in AAC needs to be more aggressive to account for larger annual fires. Table 14 AAC decisions, stocking levels and wildfire risk responsiveness. In summary if natural fires will fragment fuels and not result in a trauma exposure. Let it go. Reduce the Annual Allowable Cut a bit so that the subsequent fires don’t continuously drive down the stock and mean annual increment. If the worst-case fire will cause a trauma exposure, fragment fuels mechanically until the worst-case fire will no longer result in trauma-exposure and then management can be treated as above. If the worst-case fire size can be limited, annual allowable cut can be reduced by an amount equal to these limited worst-cases and an equilibrium can be achieved that yields more volume over the long-run. 4.6 Hormesis and the Trauma Exposure Hormesis is a theory from toxicology that describes non-linear dose response to toxins in living organisms. At low-doses the organisms respond to environmental toxins with an increase in 141 growth or other measure of vitality and at high-doses the toxin is harmful, limiting, or lethal to the organism. Examples of Hormesis have appeared in diverse fields such as neuroscience, immunology, wound healing, aging and plant biology (Calabrese, 2014; Parsons, 2001). It also appears to be a way of summarizing the results of these thought experiments on fire, fuel fragmentation and industrial activity. There is a biphasic dose response to fire in this toy model of a socio-ecological system. When the forest density is below a critical threshold (in the hormetic zone), the fire serves as a self-regulating mechanism that improves the health of the system and allows for a perpetual or ‘sustainable’ harvest. Above the critical zone the fires become so damaging that it can be debilitating over the long-run to the social organizations that depend on extracting timber from the forest and are expected to contribute to forest management (the trauma exposure). Of course, one interesting aspect of the hormesis in this system that does not seem to appear in the toxicology literature is that if the fires in the hormetic zone are suppressed it ultimately leads to fires that are trauma exposure inducing. The hormesis analogy begs some interesting questions: 1. Can we measure these effects in the real world? 2. Is it possible to identify where the thresholds are? 3. Can a forest structure in the hormetic zone sustain a viable forest products industry? 4. For forest ecosystems that are outside of hormetic zone and cannot sustain industry within it, what is the appropriate rate of liquidation or forest activity? 142 Answering these questions would require adding substantial amounts of detail to the models. This chapter is just an exploration of a thought experiment instantiated in code that highlights some interesting questions and possibilities for forest management and industrial activity. It could of course be extended to carbon analysis. If a hormetic theory is reasonable, one could imagine that forcing the forest structure out of the hormetic zone with the intention of accelerating carbon uptake or increasing total carbon stock could have counterproductive results. There is possibly a trade-off between stability in carbon stocks at lower densities and attempting to maximize carbon sequestration. In general, I think it is useful activity to run these algorithmic experiments to explore how economic activity responds inside of the natural world’s feedback loops. Malthus(London, 1998) and Ehrlich(Ehrlich & Holdren, 1971) dealt in absolute limits but it is perhaps more interesting to understand how industrial activity interacts with feedback loops in the environment and what is possible for managing and maintaining ecosystem structure to sustain human and other life. The upper-bound on the risky event distributions, H, and its limiting consequences to human activity are likely better indicators of what is possible than drawing lines in the sand as has been done in the past. 143 Chapter 5: Conclusion And new philosophy calls all in doubt, The element of fire is quite put out, The sun is lost, and th’earth, and no man’s wit Can well direct him where to look for it. And freely men confess that this world’s spent, When in the planets and the firmament They seek so many new; they see that this Is crumbled out again to his atomies. ‘Tis all in pieces, all coherence gone … (An Anatomy of the World by John Donne | Poetry Foundation, n.d.) In the quoted section of John Donne’s poem, An Anatomy of the World, the disorientation caused by the Copernican revolution from a geocentric to a heliocentric cosmology was being referenced but it is true of any model where the old assumptions no longer hold that we get a sense that all is in pieces and the coherence is gone. This is the sense I have had throughout the research process in studying economics and decision making in the wildfire literature. While researchers and analysts persist in their use of expected value models for modeling risk and decision making, research reviews give indications of things being in pieces without a unifying framework that makes sense. The frameworks used in Australia, Europe and the United States do 144 not make sense to me when wildfire data violates fundamental assumptions that are necessary conditions for those frameworks to be useful. We default to dollar values or rank-orders of values that already in enshrined in law without questioning whether the moral reasoning is adequate (this is also the approach taken in Chapter 3). One piece of reasoning that has stood the test of time in the study of economics is that to move forward and make decisions in the face of uncertainty, even in matters of social and environmental decision making, it is necessary to focus on the margin (Uzawa, 2005). As described in Chapter 2, dealing with fat-tailed data generating processes means we cannot think traditionally in terms of margins, but it is still helpful to ask the questions: What values face the biggest risk? What values are immediately affected by fire? On the margin, exploring the consequences of wildfire on carbon emissions, for instance, for fuels mitigation treatments is not meaningful when there are more immediate impacts to health and safety that already justify action. The Defensibility Heuristic was developed to address these short-run issues. Chapter 3 showed how the tool could be applied using real-world data and examples from proposed treatments and values-at-risk in BC. It is harder to identify what the margin might be from the perspective of longer-term planning at the provincial or national scales but at the community scale survival or avoiding trauma to the social fabric can become an issue when the impacts to the timber harvesting land base become large enough. An approach to modelling decision making over the long run was presented in Chapter 4. This involved both reconceptualizing risk from this perspective and the appropriate decision-making metric by which to evaluate the risk, including the implications that follow form this approach. 145 5.1 Concluding Thoughts on the Short-Run In Chapter 3, a new approach to evaluating risk management projects is presented in the form of a Defensibility Heuristic. This approach is intended to overcome three limitations of the existing approaches. The first limitation of the existing approaches to economic modelling in wildfire economics do not take the irreducible uncertainty of fat-tailed distributions seriously and reduce the problem by modelling with traditional tools that require an assumption normality. This effectively throws out information carried by the data to simplify the problem. The second problem is that of describing the relationship between wildfire behaviour modelling outputs and potential damages to values-at-risk. The defensibility threshold bypasses both problems by relying on the implicit knowledge of people who have worked in the field with wildfire. Finally, there are modelling efforts that have attempted to get around the above through very thorough simulation efforts and translating expert feedback into spatial response functions (Scott et al., 2013; Thompson et al., 2017). I have argued that from the perspective of scaling real-world fuel treatment efforts, the cost in time and money of these models is too high. There needs to be a simpler approach that offers a quick filter for projects. The core of the idea is captured in a metaphor from optimization. In the mathematics of optimization one way of categorizing algorithms that find minima and maxima is those that guarantee an optimal solution and those that search for the optimum in incremental steps. We can describe the entire world of study with equations that we know have a geometry that guarantees an optimal solution. We might lose important details of the world in this description, but we are also guaranteed a unique answer. Often when we take this approach in a complex environment, 146 we find that we have made assumptions that make the model unactionable or when action is taken it is disastrous (Scott, 1998). The temptation is to invest more resources in developing the model to make it better rather than acting. The other approach is to look around in the immediate environment and to use a heuristic to find the path that is likely to take us in the direction we want to go and take the first step. After we take a few steps, we evaluate whether we have moved closer to our goal. If not, we change direction and take another path, if we have moved closer to the goal then we keep moving. In a dynamic and complex operating environment this leads to a tighter feedback loop with faster learning and better use of resources. The goal of the defensibility threshold approach is to provide a decision tool in this spirit. Practically speaking this means using the BC Wildfire Service Resource Sharing Wildfire Allocation Protocol (RSWAP) as a tool for identifying priorities for protection of values-at-risk. This has already been done as part of the mapping effort in the Provincial Strategic Threat Analysis (PTSA). We start here with an economic analysis. Canada-wide work by Johnston and Flannigan (2018) identified 32.3 million hectares of wildland-urban interface (WUI) at risk, 10.5 million hectares of industrial WUI and 109.8 million hectares of infrastructure WUI. From an economic perspective, there are parts of the infrastructure WUI that represent critical risks for the entire provincial economy because of the role they play in energy production and distribution. Starting with the improvement of the defensibility of these critical assets there is great opportunity for learning and study of these systems that cannot be had through building 147 elaborate simulation and optimization models. These are low-hanging fruit. Simple analyses, like the ones included in these case studies, can show whether these projects are worth undertaking. Finally, the simple discounting exercise above, shows that maintenance of fuel treatments is more cost-effective than funding individual projects and leaving them without maintenance funding. These projects have a finite life if they are not kept up and there is no guarantee that they will get to do their job over the lifespan of the fuel treatment. In using the models of tail-risks and stress testing, I offer the Defensibility Heuristic for the short-run budget-allocation decisions in Chapter 3 that is applicable to a broad set of challenges beyond wildfire. Any situations that involve fat-tailed risks in a complex, difficult-to-model environment could benefit from this approach – focus on prioritizing and taking action – as there are likely to be assets and values that are obviously important under extreme risk and are, potentially, going to be ignored by the conventional tools. An important side-effect of this is that we act quickly and learn in ways that we can’t from computer simulations. 5.2 Concluding Thoughts on the Long-Run Economic outcomes that may be fine at a national or provincial/state level can be traumatic or even terminal at the regional or community level. At the national or provincial/state level there may not be an absorbing barrier and so the use of expected value tools in decision making may 148 seem appropriate. As James C. Scott (1998) has convincingly argued in Seeing Like A State, at the regional and community level this can be catastrophic. In Chapter 4, I offered a new way of categorizing risks that provides a new window into decision making under uncertainty with the new risk category of a trauma exposure. Scale is important here. What is traumatic regionally or at the community scale may be of limited relevance to the national decision maker. This is part of the cruel indifference of modern policy tools. If we are genuinely striving to be more inclusive of values in policy making a tool like trauma exposure could be used to try and capture the effects of national or provincial policy decisions. The series of abstractions used to explore these ideas in Chapter 4 cannot be directly applied to real-world problems but they do point toward some general principles: 1. Consider scale in evaluating risks; 2. Pay attention to the ‘shape’ of the risk at each scale; and 3. The biophysics determine what can and cannot be done. In the toy cellular automata model of the forest, depending on forest density and how fire spreads, the risks may take care of themselves with minor adjustments to harvest levels relative to the MAI or they may require aggressive action to get to a stable equilibrium or no action can reduce the risk of trauma exposure or even a terminal event. In some ecosystems I believe that some attention should be paid to this latter possibility. Aggressive fuels management may be able to bring the risks down but the action may also make industry non-viable. If this is the case, it requires a quite different kind of action on the part of government, money is better spent on 149 liquidating the asset and effectively funding a slow-moving evacuation and re-training of residents to avoid the worst social outcomes when the inevitable occurs. Future research should focus on developing tools that identify the type of exposure at relevant scales. An important part of this is including the incentives and behaviour of industry within the biophysical dynamic. This does not have to be complicated. Once capital is deployed it wants to be used until it is no longer profitable to do so. In some systems there is a kind of natural risk dampening that occurs. In the simple cellular automata model, fire fragments the fuels and lowers the size and damage of the worst-case fire, capital can be deployed to management harvest levels and manage the landscape to reach a dynamic equilibrium. When this is not achievable on its own and larger adjustments are required through aggressive harvesting (i.e. when there is a risk that fire can cause sufficient damage to induce a trauma-exposure) research should focus on characterizing the response of the tail risks to fuels reductions. The economics of this the trade-offs still need to be described. This largely determines whether or not the risks fall into the manageable or inevitable destruction categories. If the worst-case risks cannot be reduced at a reasonable cost then we may be into inevitable destruction territory and our society may have to face the reality that we simply cannot operate some of these communities at the industrial scale they have been. In other words, these forests cannot sustain the necessary harvests over the long-run because of the nature of the worst case fires. 150 I have suggested that hormesis, a biphasic dose response observed in toxicology, is a reasonable analogy for the outcomes in this model of fire, harvest, and industrial development. The first step in future research would be validating this analogy and, if this analogy is reasonable, beginning research on effective means of identifying which phase different forest structures are in under expected extreme fire weather conditions. Beyond this, when there is the possibility for action to bring fire risks below critical – trauma exposure – levels at reasonable cost we have to describe and prioritize the available methodologies. This may involve steering fires (Donovan & Brown, 2005), changes in silviculture with more deciduous trees (Fechner & Barrows, 1976) and uneven-age stand management (Hyytiäinen & Haight, 2010), changes in land-use with more grazing (Davies et al., 2015; Varela et al., 2018) and patches of grassland and less contiguous forest. Each of these land-use patterns will come with a different risk profile and set of economic outcomes that require different sets of skills besides managing for maximum timber yields. 5.3 Final Thoughts It is my hope that this work motivates the reader to think critically about the use of mathematical tools used in policy making. There is a need for more epistemic humility and time and resources need to be invested in developing tools for making decisions when we admit we don’t know or can’t know important features of the problem at reasonable cost and it is more cost effective to rely on the intuitions of those with hands on experience. 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Estimating the parameters that define a Pareto distribution 𝑥𝑚𝑖𝑛, 𝛼 using a bootstrapping procedure can be done with a Maximum Likelihood Estimator (MLE) and the Kolmogorov-Smirnoff statistic. The MLE estimator for 𝛼 is given: 𝛼 = 1 + 𝑛 [∑ 𝑙𝑛𝑥𝑖𝑥min𝑛𝑖=1]−1(23) Proof is given in (Muniruzzaman, 1957) . This estimator requires that 𝑥𝑚𝑖𝑛 be assumed. This is frequently done by inspection of the Zipf plot to see where it settles in to a straight-line. This is considered to be unreliable and the alternative approach of using a Kolmogorov-Smirnoff statistic (Massey, 1951) to make decide on the appropriate 𝑥𝑚𝑖𝑛 has been proposed (Clauset et al., 2009). The Kolmogorov-Smirnoff statistic is given as a distance: 𝐷 = max𝑥≥𝑥𝑚𝑖𝑛|𝑆(𝑥) − 𝑃(𝑥)| (24) Where 𝑆(𝑥) gives the CDF at the observed value and the 𝑃(𝑥) gives the CDF at the modelled value. The best estimate of 𝑥𝑚𝑖𝑛 is the one that minimizes the distance between the observed and modelled values. We can search the data for the best 𝑥𝑚𝑖𝑛 by sampling with replacement from the observed cost data and checking each possible 𝑥𝑚𝑖𝑛 over a range. In the case of the wildfire data the values $100-100,000 were selected. To test the Power Law hypothesis, 181 𝐻𝑜 : 𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟 𝑙𝑎𝑤 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑛𝑜𝑡 𝑏𝑒 𝑟𝑢𝑙𝑒𝑑 𝑜𝑢𝑡; 𝐻1: 𝑡ℎ𝑒 𝑝𝑜𝑤𝑒𝑟 𝑙𝑎𝑤 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑢𝑙𝑒𝑑 𝑜𝑢𝑡. Use the following algorithm: 1: Calculate point estimates for 𝑥𝑚𝑖𝑛 and 𝛼 2: Calculate the Kolmogorov-Smirnoff statistic, 𝐾𝑆𝑑, for the original data set 3: Set 𝑛1 equal to the number of values below 𝑥𝑚𝑖𝑛 4: Set 𝑛2 = 𝑛 − 𝑛1 and P = 0 5: for i in 1:B: 6: Simulate 𝑛1 values from a uniform distribution: 𝑈(1, 𝑥𝑚𝑖𝑛) and 𝑛2 values from a power law distribution (with parameter 𝛼). 7: Calculate the associated Kolmogorov-Smirnov statistic, 𝐾𝑆𝑠𝑖𝑚 8: If 𝐾𝑆𝑑 > 𝐾𝑆𝑠𝑖𝑚, then 𝑃 = 𝑃 + 1 9: end for 10: P=P/B P giving the p-value with the usual interpretation. 182 Appendix B Variance of a Pareto Distribution The tools for developing this are found in (Feller, 1968). For a maximum value 𝑥𝑚 and a shape parameter 𝛼 the probability (Lorenz, 1905) density function of a pareto distribution is given 𝑓(𝑥) =𝛼𝑥𝑚𝛼𝑥 𝛼+1, 𝑥 ≥ 𝑥𝑚 (25) The integral of this density should sum to one for shape parameters greater than 0. ∫ 𝑓(𝑥)𝑑𝑥 =∞𝑥=𝑥𝑚 𝛼𝑥𝑚𝛼 ∫ 𝑥−(𝛼+1)∞𝑥=𝑥𝑚𝑑𝑥 (26) Taking the limit on the right-hand side we do in fact get one. This will be useful later. Take the 𝑘𝑡ℎ raw moment as given by 𝐸[𝑋𝑘] = ∫ 𝑥 𝑘𝛼𝑥𝑚𝛼𝑥 𝛼+1𝑑𝑥∞𝑥=𝑥𝑚 (27) 𝐸[𝑋𝑘] = 𝛼𝑥𝑚𝛼 ∫1𝑥 𝛼+1−𝑘𝑑𝑥∞𝑥=𝑥𝑚 (28) 𝐸[𝑋𝑘] =𝛼𝑥𝑚𝛼(𝛼 − 𝑘)𝑥𝑚𝛼−𝑘∫(𝛼 − 𝑘)𝑥𝑚𝛼−𝑘𝑥 𝛼+1−𝑘𝑑𝑥∞𝑥=𝑥𝑚 (29) 183 The final integral here is the same as 𝑃𝑎𝑟𝑒𝑡𝑜(𝛼 − 𝑘, 𝑥𝑚), so it equals 1 and we get that the 𝑘𝑡ℎ raw moment is 𝐸[𝑋𝑘] =𝛼𝑥𝑚𝛼(𝛼 − 𝑘)𝑥𝑚𝛼−𝑘 (30) The first raw moment or mathematical expectation is 𝐸[𝑋𝑘] =𝛼𝑥𝑚(𝛼 − 1) (31) And the second is 𝐸[𝑋2] =𝛼𝑥𝑚2𝛼 − 2(32) The first moment is defined for 𝛼 > 1 and the second moment for 𝛼 > 2 only. The variance is defined 𝑉𝑎𝑟[𝑋] = 𝐸[𝑋2] − 𝐸[𝑋]2 (33) 𝑉𝑎𝑟[𝑋] =𝛼𝑥𝑚2𝛼 − 2−𝛼2𝑥𝑚2(𝛼 − 1)2 (34) 𝑉𝑎𝑟[𝑋] = 𝛼𝑥𝑚2(𝛼 − 1)2(𝛼 − 2) (35) 184 This is clearly not defined for 𝛼 = 1,2 and if we go back to the derivation of the 𝑘𝑡ℎ raw moment we see that the integrals are divergent for 𝛼 ∈ (1,2). By convention the variance is defined as taking the value of ∞ over this interval. 185 Appendix C Finding the Pareto Index 𝜶 We start with the observation that in one US Forest Service study 1.4% of the fires resulted in 94% of the suppression costs (Strategic Issues Panel on Fire Suppression, 2004) One way of generalizing this observation is using the Pareto Law (Hardy, 2010). The shape parameter for the Pareto distribution can be backed out using the Lorenz Curve (Gastwirth, 1972; Lorenz, 1905). The Lorenz curve was developed to measure the difference between a society with perfect equality and the realized distribution of wealth or income. For our purposes, we are interested in the distance between a world where all wildfires cause equal damages and a realized distribution where 1.4% of the fires resulted in 94% of the suppression costs. The Lorenz Curve, in terms of a cumulative distribution function (CDF), 𝐹(𝑥) and its inverse 𝑥(𝐹) is given by: 𝐿(𝐹) =∫ 𝑥(𝐹 ′)𝑑𝐹 ′𝐹0∫ 𝑥(𝐹 ′)𝑑𝐹 ′10 (36) If we think that a good model for the underlying fire costs data might be a Pareto distribution that we assume an inverse CDF for the Pareto distribution of: 𝑥(𝐹) =𝜃(1 − 𝐹)(1−1𝛼)(36) Where 𝜃 is the minimum bound of the distribution and α is the shape parameter of the Pareto distribution. This gives the Lorenz curve: 𝐿(𝐹) = 1 − (1 − 𝐹)1−1𝛼 (37) 186 Given the assumption of the Pareto model we can calculate α\alphaα when 94% of the wildfire suppression costs are caused by 1.4% of the fires, or in terms of the Lorenz curve, when 𝐿(0.94) = 0.014 (38) 0.014 = 1 − (1 − 0.94)1−1𝛼 (39) Solving for 𝛼 gives 𝛼 ≈ 1.005. And a Pareto density function 𝑓(𝑥; 𝛼, 𝜃) =𝛼𝜃𝛼𝑥 𝛼+1, 𝑥 ≥ 𝜃 (40) 𝑓(𝑥; 1.005,1) =1.005𝑥 2.005, 𝑥 ≥ 1. (41)
Thesis/Dissertation
eng
Forestry
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Nelson, Harry W., 1961-
Economic decision making under uncertainty for wildfire fuels mitigation
Text