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International Construction Specialty Conference of the Canadian Society for Civil Engineering (ICSC) (5th : 2015)

Evolutionary stable strategy for post-disaster insurance : a game theory approach Eid, Mohamed S.; El-adaway, Islam H.; Coatney, Kalyn T. 2015-06

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5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   EVOLUTIONARY STABLE STRATEGY FOR POST-DISASTER INSURANCE: A GAME THEORY APPROACH Mohamed S. Eid1, Islam H. El-adaway2, 4, and Kalyn T. Coatney3 1 Graduate PhD Student: Department of Civil and Environmental Engineering, University of Tennessee,-Knoxville, TN, United State. 2 Associate Professor of Civil Engineering and Construction Engineering and Management Program Coordinator, Department of Civil and Environmental Engineering, University of Tennessee - Knoxville, TN, United State. 3 Assistant Professor: Department of Agricultural Economics, Mississippi State University, Mississippi, United State. 4 Abstract: Natural disasters leave the impacted regions with financial burdens both on the individual and governmental levels. Thus, the goal of the associated stakeholders is to maximize the host communities’ welfare through minimizing their post-disaster financial burdens. Accordingly, this paper attempts to find a post-disaster insurance plans equilibrium so as to mitigate the financial impacts associated with the natural disasters. Utilizing an evolutionary game theory approach, the equilibrium is investigated between three different players including: resident families purchasing insurance plans; insurance companies offering different insurance plans; and the government agency that implements post disaster relief financial plans. The authors determined a set of decision actions as well as utility functions for the aforementioned stakeholders. Moreover, the authors created a hypothetical sample of 1,000 heterogeneous income level resident families, three insurance companies offering three unique and different insurance plans per company and two post disaster financial relief plans to be utilized by the government agency. The proposed model was implemented on NetBeans IDE 7.4 platform using JAVA programming language on the hypothetical case study simulating resident family evolutionary learning process in reaching an equilibrium. The results indicate that: (1) resident families tend to prefer insurance plans with the least premium value and coverage; (2) insurance plans with the most comprehensive coverage received the least demand; and (3) the evolutionary stable strategy path oscillates between chosen plans and insurers over time as a result of the stochastic and dynamics nature of the factors associated with disaster management. Currently, the authors are working to develop the model further to better account for simultaneous actions by all stakeholders (not only resident families), population growth and changes in financial and income standards. Ultimately, this evolutionary game theory model will be tested on real post natural disasters data representing physical damages in coastal Mississippi Counties post Katrina, so as to determine the significant increase in the host community welfare. 1 INTRODUCTION Mitigation of the financial impacts associated with natural catastrophes is becoming a focal issue at both the national and international levels as the rate and magnitude of natural disasters are increasing. According to Climatic Change Science Program (2008) and the National Association of Insurance 029-1 Commissioners (2005), recent examples in the United States include (1) Hurricane Andrew in 1992 which caused $20.9 billion in insured losses; (2) the Northridge earthquake in 1994 which resulted in insured losses of $15.9 billion; (3) the four Hurricanes Charlie, Ivan, Frances and Jeanne in 2004 which caused $21.9 billion in insured losses; (4) Hurricane Wilma and Rita in 2005 which resulted in insurance losses of $11.9 billion; (5) the devastating Hurricane Katrina in 2005 which caused economic losses approaching $125 billion; (6) Hurricane Ike in 2008 that resulted in $19.3 billion in property damages; and (7) Hurricane Sandy in 2012 which incurred damages more than $68 billion. Disaster insurance plays a significant role in the mitigation and preparedness phases of the emergency management. However, there are important gaps in the knowledge-base for emergency management and financial disaster mitigation. Past studies have primarily addressed the challenge of disaster risk management by segregation of the problem without concern of the integration of how these parts fit into a decision making tool that integrates the goals, objectives, perceptions, and beliefs of multiple agents in determining a set of social optimum strategies to mitigate the financial impact of future disaster damage. 2 GOALS AND OBJECTIVES  Using an evolutionary game theory approach, this paper aims to find an equilibrium profile of post-disaster insurance plans purchased by resident families, sold by insurance companies, and ex post disaster relief implemented by a government agency. This should identify the optimal balance between: (1) number of plans offered by insurance companies; (2) types of plans that should be selected by each type of resident family based on their income level; and (3) compensation ratio that the government will pay for each resident family to offset the post-disaster damages. This dynamic integrated assessment minimizes the total losses for the three aforementioned associated stakeholders, thus maximizing welfare within natural disaster host community systems. 3 BACKGROUND INFORMATION Natural hazards damages have reached a record level causing around 800,000 fatalities last decade as well as damages in the infrastructure of over a trillion dollars (Economics of Climate Adaptation Working Group 2009; Stern 2006).  Decision makers nationwide, in both public and private sectors, are concerned about the vulnerability of their economy to natural hazards. They face investment choices in a stochastic environment with overlapping risk factors. These risk factors consist of wind, flooding, fire, and earthquakes, as well as climate change and their effects on investments. Also, as population and economies continue to grow, the total value at risk from natural hazards will increase (Climatic Change Science Program 2008). Research has been conducted by governmental, private, non-profit and academic organizations and institutions to study, assess, and solve problems associated with disaster financial mitigation. Most of these valuable efforts generally fall into three main streams including loss estimation models, computational engineering approaches, and risk management using insurance. 3.1 Loss Estimation Models Since the 1980s, a number of major impact assessment models have been developed to support disaster preparedness and recovery efforts. For example, HAZUS-MH is a hazard prediction software program developed by the Federal Emergency Management Agency under a contract with the National Institute of Building Sciences to estimate potential losses from earthquakes, hurricane winds, and floods (HAZUS-MH 2007; Pradhan et al. 2007). Loss estimation models provide increasingly comprehensive estimates of regional risk, but offer little guidance about how to use that information to make damage mitigation resource allocation decisions (Dodo et al. 2005). These models can estimate losses in relation to structural damage, contents damage, time-based impacts, and only a small set of predefined mitigation alternatives can be considered (Grossi and Kunreuther 2006). However, they are not able to account for the stakeholders’ side including costs of the alternatives, the budget, and the specific objectives and priorities of each stakeholder (Federal Emergency Management Agency 2003; Dodo et al. 2005). 029-2 3.2 Computational Engineering Approaches  Computational engineering approaches have been used extensively for studying and mitigating the financial impacts from natural catastrophes including: • Deterministic Net Present Value (NPV): Altay et al. (2002); and Kuwata and Takada (2003) calculated the avoided loss as the difference between the losses estimated with and without implementation of the mitigation alternatives. • Stochastic NPV: Englehardt and Peng (1996) estimated probability distribution of the benefits associated with revising hurricane requirements and compared it with the cost of implementing the revision. Werner et al. (2002) compared various levels of proposed seismic design or upgrade on both means and standard deviations of losses.  • Multi-Attribute Utility Models: Nuti and Vanzi (1998) compared structural upgrading strategies based on various performance indices. Opricovic and Tzeng (2002) compared post-earthquake reconstruction plans using multi-criteria decision analysis approach.  • Optimization Models: Augusti et al. (1994) used dynamic programming to select structural mitigation alternatives. Researchers at the International Institute for Applied Systems Analysis (IIASA) developed a spatial-dynamic stochastic optimization model to select the insurance policy design that maximizes profits and minimizes risk of insolvency for insurance companies (Ermoliev et al. 2000; Ermolieva et al. 2001; Brouwers et al. 2001). Dodo et al. (2005) developed linear program for resource allocation in earthquakes that incorporates spatial correlation among a set of mitigation alternatives, associated probabilities, and decision timing.  3.3 Risk Management Using Insurance Insurance is utilized to spread the financial risk of loss resulting from low frequency-high consequence disastrous events (Kunreuther and Michel-Kerjan 2007). Insurance companies have made significant changes in their approaches to provide coverage for natural hazards (Muller 2008; Mills 2007). Capital market participants developed catastrophe bonds, which are a type of securities that can be purchased by institutional investors to cover certain insurer risks (Cardenas 2006). Proposals have been made to Congress and regulatory agencies to take additional steps in changing the U.S. tax laws and accounting standards to allow insurers to set aside funds on a tax deductible basis and establish reserves for hazards (Smetters and Torregosa 2008; Cardenas 2006). However, these reserves lower federal tax receipts and do not necessarily bring about a meaningful increase in the capacity of the insurance industry.  This is because insurers may substitute their reserves for other types of capacity (Shear 2005).  More analytically, Picard (2008) investigated the equity-efficiency trade-off faced by policy makers under imperfect information about individual prevention costs. His research highlighted the complementary relationship between individual incentives tax cuts and collective incentives grants to the local jurisdictions where natural hazard insurance plans are enforced. Chen et al. (2008) studied the determinants for the short-run position resulting from ex-ante insufficient premium and the long-run position resulting from ex-post insurance supply reductions. Greiving et al. (2006) studied the spatial limitations of the Natural Hazard Index for Mega Cities as well as the Total Place Vulnerability Index, and developed an integrated hazards map that combines regional hazards and vulnerability. Also, research revealed that while catastrophe insurance is more price elastic than non-catastrophe insurance in cities like New York, responsiveness to price is inelastic in the coastal areas because price increases only with mandatory purchase by mortgage borrowers (Grace et al. 2004; Kriesel and Landry 2004). The aforementioned research illustrate several models that asses disaster damages and how to financially mitigate the impacts of disasters on the existing environment and host community. However, few to none discuss the social and individual decision process for selecting an insurance company given their preference for different disaster insurance plans. To this end, this research aims to utilize Evolutionary game theory to simulate residences’ post disaster learning to better guide the disaster 029-3 insurance plans selection. This research will also guide insurers on how to determine an optimal array of plan premiums and coverages that will eventually be accepted by the community over time. 4 METHODOLOGY In order to attain the aforementioned goal and objectives, the authors utilized the following three-step research methodology: (1) determined the set of possible actions and utility functions that govern the strategy profiles of the associated stakeholders including resident families, insurance companies, and government; (2) utilized the evolutionary stable strategy profile amongst the aforementioned players using game theory; and (3) applied the proposed model to a hypothetical dataset as a proof of concept. 5 ASSOCIATED STAKEHOLDERS: ACTIONS AND UTILITY FUNCTIONS As previously mentioned, the stakeholders are resident families, insurance companies, and the government. The resident families and insurance companies will be represented by a population of players while the government is represented as a single player. Thus, it is worth noting that selecting a specific insurance plan will affect the resident family player through determining the amount of money spent on the premiums and the compensation obtained from insures in case of an occurrence of a natural disaster. This will also affect the insurer in term of earned revenue (i.e. the amount of premiums collected) and the amount of compensation paid out in case of a natural disaster event occurrence. Moreover, after calculating the post disaster damages for the residential sector, and taking into account the compensation by the insurer, the government compensation can be likewise calculated.  5.1 Resident Families Each property owner player has a set of actions to choose from, A = {a(n,i)}, where A is the set of possible actions, and a is the chosen insurance coverage plan n offered by the associated company i. In selecting a plan at each iterative step t, each resident considers their current wealth, the indemnity received from the insurance company if a natural hazard causes damages to the residence building, the amount of tax paid and the compensation paid by the government post disaster as shown in Eq. (1). [1] Wp(t) = Wp(t+1) – P(n,i) – T – D(t) + C(n,i) + G           Where Wp(t+1) is the amount of wealth of a property owner p at time t +1. P(n,i) is the insurance premium paid by a property owner to insurance company i using plan n, T is the taxes paid to the government, D is the damage cost by natural disaster at time t, C(n,i) is the compensation paid by the insurance company i if the property owner is using plan n, and G is the compensation paid by the government. Also, it is worth noting that utility does not govern actions. Generally speaking, players maximize their utility by choosing their optimal actions subject to their beliefs of the actions taken by their rivals. In evolutionary games, the players observe the payoffs of others and mimic those with superior outcomes. 5.2 Insurance Companies A successful strategy for post disaster damage mitigation should decide on the type of coverage provided by the insurer and the premium structure (Jaffee et al. 2008). However, there are two main concepts that may negatively affect the optimum strategy profile. First, adverse selection as the pool will contain mostly high risk resident families and so the insurance company will keep the premium at a fair rate (Janssen and Karamychev 2005). It is noted though that insurers can change rates to overcome the problem of adverse selection. Second, moral hazard as losses will always be not in the favor of the insured pool and thus the insurance will not change the situation or mitigate the damage for the insured party (Lee and Ligon 2001; Breuer 2005; Doherty and Smetters 2005). This emphasizes the need of an optimum post-disaster insurance plan strategy profile where a selective value of premiums and coverage values should be determined as well. These issues can be handled by allowing the insurer to be myopic in their product offerings and learn from their rivals given the distribution of population types per contract. A decision for 029-4 each company is to determine the distribution and pricing of plans to offer the population of resident families. Accordingly, the insurer utility function is shown in Eq. (2).   [2] Wi(t+1)= Wi(t)+ ∑ R            Where, Wi(t+1) is the insurance company i wealth at time t + 1. R is the aggregate monetary utility gained by an insurance company by calculating the difference between the sum of the premiums paid by the residents and the sum of the indemnities paid to the residents when a natural hazard event occurs. Thus, R is equal to the P(n,i)– C(n,i) for every resident that purchased the insurance plan n from company i.  5.3 Government State protectionism is essential for a post natural disaster relief. This can be achieved by subsidizing the insurance costs on families or financially aiding families in reconstructing their damaged homes and reconstructing the state damaged infrastructure during the disaster event. The government action will determine the financial compensation for damaged houses post a natural disaster event. The government utility function is simplified to the difference between its current wealth, obtained through tax payments, and compensation paid to the families as showed in Eq. (3).  [3] WG(t+1)= WGt+ ∑(Tp - Gp )              6 EVOLUTIONARY GAME THEORY  Game theory is the study of the ways in which strategic interactions among economic agents produce outcomes with respect to the preferences (or utilities) of those agents, where the outcomes in question might have been intended by none of the agents (Samuelson 1997). Since Von-Neumann and Morgenstern (1944), game theory has been used in many different research areas (i.e. economics, biology, engineering, political science, computer science, and philosophy) because of its advantage of a natural and plausible representation of strategic interaction between individuals, organizations, and countries (Son and Rojas 2011). In evolutionary games, a large population of individuals, each having their own actions and strategies, meet in an environment to determine their optimum strategy profiles depending on their payoffs (Samuelson 1997). In other words, evolutionary game theory allows imperfect players to learn from observations. The dynamics are based on the assumption that each strategy is played by a certain fraction of individuals at each moment of the game (Turocy et al. 2001). Inspired by Darwin theory of survival of the fittest, stakeholders having better than average payoff will be more successful and more likely to survive in the next round. Those players who chose strategies that result in less than average payoffs update their strategic choices to mimic (replicate) those making above average payoffs (Samuelson 1997). The replicator dynamics govern the law of motion for the game, and are unique to each stakeholder group. Thus, one would not expect a resident final payoff to equal that of the insurance company or government. A basic requirement in evolutionary games is that a set of strategies is evolutionary stable if for any mutant strategy (perturbation) in the game; the non-mutates must result in a higher payoff than the mutant strategy (Weibull 1995; Smith and Price 1973). Evolutionary game theory has been applied in Economics (Cressman 1995, Friedman 1998), and explored by mathematicians (Hofbauer and Sigmund 2003).  However, as far as the authors’ knowledge, there is no implementation for the Evolutionary game theory in the construction management research.  7 PROPOSED MODEL: SOLUTION PROCESSES The solution processes for the proposed model consider: (1) the post-disaster insurance plans selected by each resident family from the different insurance companies, (2) the premium value charged as per the distribution of contract types offered by each insurer for each plan, and finally (3) the government post-disaster damage compensation ratio. At this stage, as an initial step, the data associated with the three main stakeholders is entered into the model. For resident families, the model requires the population size, the ratio between different income families (i.e. poor, medium, and high), income and current wealth, and a random initial set of selected plans and insurance companies. For the insurance companies data about 029-5 the number of associated insurers, different plans offered, premiums, compensation ratios, and wealth of the insurance companies including relationship with reinsurance companies. For the government, the tax rate should also be set as well as an initial percentage of the collected tax amount to be dedicated to post natural disaster mitigation plan. Last, the nature, as a Pseudo player, should also be set in this step including determination of type of hazard accompanied with its characteristic parameter such as severity, frequency, and return period. This information will help the model to create an initial random population of players that have their own actions, and measuring their utility function after the disastrous event, and choose the fittest parents of the population for future evolution.  7.1 Updating Utility Functions for Associated Players This step depends firstly on the occurrence of disaster or not, and the damage rate for each family residence. Determining that, the model can estimate the loss and calculate the associated compensation ratio by the insurers and the government. The total change of any player’s utility will be equal to the difference of the utility function prior and after the disaster occurrence as shown in Eq. (4-6): [4] For resident families:    ∆Wp= Wp(t+1)- Wp(t)         [5] For the insurance companies:  ∆Wi= Wi(t+1)- Wi(t)                [6] For the local government:   ∆WG= WG(t+1)- WG(t)                Through Eq. (4-6), the relative fitness of each player for every stakeholder can be calculated as shown in Eq. (7) & (8): This is carried out by dividing the player’s (resident family or insurer) change in the utility function’s value by the total change in the utility values for all the players of the same type. Thus, the players with higher positive changes in the utility values will have higher relative fitness values, so as other (lower relative fitness value) players will chose to mimic them – via replicator dynamics – by duplicating their actions and decisions in the next time steps. It is worth noting that there is no relative fitness for the local government as there is only one government player in the game. [7] For resident families:                 Relative Fitnessp= (∆Wp)/∑ (∆Wp)        [8] For the insurance companies:  Relative Fitnessi= (∆Wi)/∑ (∆Wi)        8 MODEL IMPLEMENTATION The authors created a hypothetical sample including: (1) 1,000 resident families taking into account the different level of income; (2) three insurance companies with three different insurance plans available per company, each with different premium percentage to the family house value and different compensation ratio; and (3) two different type of government compensation plans for post disaster damage mitigation. Realizing how complex the evolutionary game theory model between the three associated stakeholders can be, and in order to focus more on the foundational and fundamental steps associated with the model development, the authors treated the resident families as the principal controller of the game’s environment and that insurance companies and the government are supportive players for analysis. The model was implemented on NetBeans IDE 7.4 platform using JAVA programming language. To this effect, the resident family population is randomly created with 20% under poverty level, 60% of average income and 20% of high income level. The initial insurance plan and insurance company are created randomly for each family, where the three generated insurance companies’ premiums percentages as well as the coverage compensation ratios are given in Table 1. In addition, the government offers plan (A) of compensation percentages of 10%, 15% and 20% or plan (B) of a compensation percentage 15%, 20% and 25% of the damages to the property for the high, medium and poor level income families, respectively. Also, for the sake of simplicity for this model, the probability of windstorm occurrence per time period in the implementation process is set to 95% with damages 029-6 each resident family to offset the post-disaster damages. To this end, the authors developed a computer model on NetBeans IDE 7.4 platform using JAVA programming language, and applied it to a hypothetical case study that involves resident families, insurance companies, and the government. Realizing how complex the evolutionary game between the three associated stakeholders can be, and in order to focus more on the foundational and fundamental steps associated with the model development, the authors treated the resident families as the principal controller of the game’s environment and that insurance companies and the government are supportive players for analysis. This proof-of-concept analysis revealed that: (1) resident families tend to prefer insurance plans with the least premium value and coverage; (2) insurance plans with the most comprehensive coverage received the least demand; and (3) the evolutionary stable strategy path oscillates between chosen plans and insurers over time as a result of the stochastic and dynamics nature of the factors associated with disaster management. Based on the results of the hypothetical case study, the authors will develop the model further to take into account simultaneous actions by all stakeholders as well as changes in the social parameters. Furthermore, more investigation will be carried out on the stakeholders’ preferences and how they approach risk. Also, an effort will be directed towards integrating input from existing natural hazard prediction software systems (for example, HAZUS-MH) with the new evolutionary game theory model for a more precise simulation of the hazard characteristics. 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