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Agent inactivity in the Evolutionary Minority Game Mitton, Michael D. 2001

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Agent inactivity in the Evolutionary Minority Game by Michael D . M i t t on B.Sc. (Physics) The University of British Columbia, 1996 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F Master of Science in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A Apri l 2001 © Michael D. Mitton, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada D E - 6 (2/88) Abstract In this work I have developed a model, based on the Evolutionary Minority Game (EMG), to examine the effect of agent inactivity within a group of agents competing for a limited resource, but governed by both supply-and-demand and a minority rule. The structure of the inactivity mechanism has been modeled after the strategy preference parameter, p, of the E M G . A parameter has also been introduced that models, in a very simple way, the effect of inflation on the system in order to motivate the agents to play the game. The behaviour of the model has been examined with the use of numeri-cal simulations over its entire phase space. The results focus on two aspects of the model: 1. how an agent's performance depends on his activity level; and 2. how the properties of the E M G are affected when agents are given the option of not playing the game. Results of these simulations demonstrate that an agent's performance is strongly dependent on his level of activity. Specifically, it has been shown that an agent's optimal level of activity is dependent on his strategy preference and, moreover, that this optimal activity level undergoes a first-order transition from a phase of optimal inactivity to a phase of optimal activity as the inflationary force is increased. Even though an agent's activity decision has been modeled as a process independent of his decision to "buy" i i or "sell" during a given round, results of simulations indicate that correlations between the two decisions have emerged from the collective dynamics of the group. A theory of the model has also been developed. The formulation of the theory is based on a mean-field approach in which the actions of a single agent are considered within a background field produced by the remainder of the group. It has been found that the results of the theory agree with those of the numerical simulations very well and appear to become exact in the thermodynamic limit. Furthermore, the discrepancy between theory and simulation at finite N indicate a possible breakdown of the non-local nature of the inter-agent interactions built-in to the model. i i i Contents Abstract ii List of Figures vii Acknowledgements x Chapter 1 Introduction 1 1.1 Financial asset prices under speculation 1 1.2 The physics of speculative asset pricing 4 1.2.1 Why should physicists study finance? 4 1.2.2 Complexity in finance 5 1.2.3 Physical models of asset price dynamics 9 1.3 The Minority Game 10 1.3.1 The model 10 1.3.2 The Evolutionary Minority Game 12 1.3.3 The M G as a realistic market model 13 1.4 Focus of thesis 19 1.4.1 Motivation for research 19 1.4.2 Present study 20 iv 1.5 Organization of thesis 21 Chapter 2 The Evolutionary Minority Game 22 2.1 The basic E M G model structure 22 2.2 Numerical results 25 2.2.1 Time evolution of the E M G 27 2.2.2 Equilibrium properties of L(p) 29 2.3 Discussion of results 34 Chapter 3 The Variable Activity Evolutionary Minority Game 37 3.1 Choice of activity structure 38 3.2 Model structure 39 3.3 Theory of the V A E M G 42 3.3.1 Derivation of an expression for the average lifetime dis-tribution, L(a,p) 42 3.3.2 Derivation of an expression for Pw(a,p) 44 3.3.3 Analytical forms for QN(Na) and FNa(n) 49 3.3.4 Solving for the analytical form of L(a, p) 50 Chapter 4 Results and Discussion 51 4.1 Setup of numerical routines 52 4.1.1 Experimental 52 4.1.2 Theoretical 53 4.2 Transient response of the V A E M G 54 4.3 Individual agent behaviour in the V A E M G 56 4.3.1 Introduction 56 v 4.3.2 Experimental results 57 4.3.3 Theoretical results 66 4.4 Global behaviour of the V A E M G 71 Chapter 5 Conclusions 77 5.1 Summary of results 77 5.2 Future directions 81 vi List of Figures 2.1 The 8 possible histories, , for M = 3 along with the 8 corre-sponding strategy elements, Sj, of a sample strategy, <S 24 2.2 The time evolution of \nd\ with pi = 0.25 Vi (—) and Pi ran-domly initialized (• • •) for N = 501, M = 3, Zc = -100, and R = l 27 2.3 The equilibrium average size difference between the two posi-tions for the E M G (o) and when the agents select their posi-tions randomly (•). Other parameters for the simulations are Zc = -100, M = 3, and R = 1 29 2.4 The distribution of average lifetimes, L(p), for N = 101, M — 3, T = 106 rounds, Zc = - 5 , and R = 2 30 2.5 L(p) for M = 3, T = 106, Zc = - 5 , and R = 2 with N=31 (—); JV=61 (—); and N=101 (—-) 31 2.6 L(p)/\ZC\ vs. p for Z c = -5 (—), -10 (—), -25 (—-), and -100 (• • •)• N = 101, M = 3, T = 106, and R = 2 32 2.7 A l l possible expected payoffs for agent 3 when N = 3 given that each agent can only access the states p =0, 1/2, and 1 33 vii 2.8 The mean payoff and corresponding variance for each of agent 3's gene values when N = 3 and each agent only has access to the states p =0, 1/2, and 1 33 4.1 Time series for \nd\ and |2N a - JV| for N = 301, M = 3, and I = —0.5 with both pi and a* evolutionary and randomly initialized. 55 4.2 Average lifetime distributions for I — -0.05 (—), -0.10 (— -), -0.25 (—-), and -0.75 (• • •) with N = 101, Zc = -100, and Ra = l 58 4.3 Normalized average lifetime distributions, L(a)/LQ(a), for T = 2 x 107,1 x 108 and 1 x 109 when the penalty, 7 = 0. Other parameters are N — 101, Zc = —100, and Ra = 1 59 4.4 Average lifetime distributions for N=31 (---), 101 (—), 301 (— - ) , and 501 (• • •) when the penalty, I is -0.1. Other parameters are Zc = -100 and Ra = 1 60 4.5 Time evolution of Na for N = 31,51, and 101 for / = -0 .1 . The time series have been averaged over a window of 10000 rounds. 61 4.6 Average lifetime distributions for Zc=-10 (—), -25 (---), -100 (---), and -200 (• • •) with I = -0.1 and N = 101 62 4.7 Dependence of the optimal activity level, a*, on the inflation rate, |7| for N = 101 63 4.8 Average lifetime distributions, L(a), near the active/inactive phase transition, for N = 101 64 4.9 Active/inactive phase transitions for N — 31(—), 101(—), 301(-•—), and 1001(- • •) 66 viii 4.10 (a) Average lifetime distribution, L(a), for 7. = -0.05 (o), -0.1 (A) , -0.25 (•), and -0.75 (o) with N = 101. The solid lines are the respective theoretical distributions, (b) The relative error, 5(a) = (Lsimuiation(oi) - Ltheory(a))/Lsimulau0n{a), of the theoretical distributions in (a) 68 4.11 Comparison of "experimental" phase transition plots (—) (from figure 4.9) with theory (—) for N = 31, 101, 301, and 1001. . 69 4.12 QN(Na) distributions for N = 31 (a) and N = 101 (b) for / = -0.01(o), -0.02 (A) , -0.05 (•), -0.1 (o), and -0.2 ( y ) 70 4.13 (a) The distribution of average lifetimes, L(a,p), for N = 101, J = -0.05, M = 3, Zc = -100, and Ra = Rp = 2. (b) Relative error of the corresponding theoretical distribution 72 4.14 Measure of the p-symmetry in L(a,p) for the distribution shown in figure 4.13 (a) 73 4.15 The distribution of average lifetimes, L(a,p), for / = 0.0(a) and I = -0.013(b). Other parameters were N — 101, Zc = -100, M = 3, and Ra = Rp = 2 74 4.16 Optimal performance, L(l,p*), of an purely active agent vs. penalty, / . Other parameters are N — 101, Zc — —100, M = 3, and Ra = Rp = 1/2 76 ix Acknowledgements Firstly, I would like to thank Dr. Birger Bergersen for taking me on as his stu-dent, providing me with access to the new and exciting field of "econophysics", and allowing me the freedom to choose and develop this thesis topic. I would also like to thank Dr. Alan Kraus for providing me with helpful suggestions and keeping my feet on the ground with regards to the financial validity of the Minority Game model. Finally, I would like to thank Dr. Hendrik Blok for his constant input throughout the duration of this work and his patience with the many, semi-prepared, late-night "reference" visits that kept him from a warm dinner, ps - a post-doc does hold weight M I C H A E L D . M I T T O N The University of British Columbia April 2001 Chapter 1 Introduction 1.1 Financial asset prices under speculation A financial market is a structured organization designed to centralize, and therefore facilitate the trade of similar types of financial assets, such as stocks, bonds, commodities, and derivative securities. The specific mechanism by which a trade occurs varies from market to market, but in general there are traders (which I shall term agents) who submit bids to buy an asset at a certain price and therefore create demand for that asset. Opposing this flow of demand is the supply created by agents who submit offers to sell the asset. In very general terms, a trade is then executed in one of two ways [3]: in a centralized market, a central third-party, or market maker, weighs the total supply and demand and sets a price to clear the market; in a decentralized market, individual agents simply meet face-to-face and agree on a price at which to exchange the asset. ' From the above description it might seem that the process of asset price 1 formation is relatively simple: agents simply determine the value of the asset by the value of the underlying (ie. the value of one non-dividend-paying stock of a company would simply be the value of the company divided by the number of stocks outstanding) and buy (sell) when its price is lower (higher) than this value. In this scenario, only the change of exogenous factors, such as interest rates, the Gross Domestic Product, and foreign exchange rates, or the release of news concerning the underlying should cause a change in the asset's price (although it should be noted that even a change in the value of the underlying will not necessarily lead to trading of the asset). When none of these factors are present each agent should hold the same opinion for the value of the asset and no trading should occur. In reality, we know that trades of most assets occur at a high frequency not only during the release of news, but also in between these events. This observation is indicative of a very complicated mechanism of asset price for-mation and is mainly a result of two factors. First, not all agents have the same opinion about the "true" value of an asset and about how it will change in the future. These individual beliefs can result from an agent's risk prefer-ence, social conscience, or simple irrationalities; since they are not necessarily tied to anything tangible, these speculative effects are typically very hard to quantify. Second, the vast communication structure connecting agents within and across markets further complicates the dynamical behaviour of speculative asset pricing. With the progress of information technology in recent years, the physical barriers within financial markets are crumbling and the network of interactions between agents is increasing rapidly resulting in trading decisions 2 that are now rarely made independent of other agents' opinions and decisions. Empirical properties of speculative asset prices The above mechanisms produce very interesting properties observed in asset price time series. Let Xt be the asset's price at time t and rtAt = (Xt ~ Xt-At)/Xt-At be its rate of return at time t and over the time interval At. It has been well documented [4-6,9,14,15,17,18] that the probability density function, fTt At(x) = Prob[rtAt = x], is distinctly leptokurtic for time-scales, At < 1 month; as At —> oo, this leptokurtic form converges to a Gaussian. It has also been shown that interesting correlations exist between suc-cessive asset price movements [4-6,11-14,16]. Let E[] be the expectation operator and px = E[xt] be the mean of some time series xt. Specifically, the autocorrelation function of returns /, ^ ^ _ E[{rtlAt - (ir)(rt2At - Mr)] n n Pr{ti,t2) = , , [i-i) yjE[(rtlAt-^y]^E[(rHM-fir)2] for an asset within a well-established and highly liquid market decays rapidly (specifically, with half-life r w 4 min) to the level of system noise within ap-proximately 30 minutes. This result is somewhat expected since it reflects the time for new information to be reflected in the asset price. More interestingly, though, is the very slow decay of the autocorrelation function of volatility (where the volatility, or standard deviation, vst{t), is defined as the R M S re-turn over a time, 8t » At) Pv{h,t2) = , , = (1.2) y/EftvsM - pv)y E[(vst(t2) - pvf] 3 Even for t 2 — > several years, p„(<i, £2) is still significantly above the level of noise. This effect, known as volatility clustering, simply means that an asset price movement tends to be followed by one of a similar size; it has received much attention since it shows that there are long-range correlations within |uat(£)| and it might therefore be possible to predict the asset price's future movements. 1.2 The physics of speculative asset pricing 1.2.1 W h y should physicists study finance? Although numerous variations have been developed, to date, no model has been put forward that is able to accurately generate all of the known em-pirical characteristics of speculative asset price time series. Historically, the majority of this effort has come from the financial economics community; sev-eral popular examples of these models are Mandlebrot's Stable Paretian Hy-pothesis [1], Clark's Subordinated Stochastic Process (SSP) model [27], and the (Generalized) Autoregressive Conditional Heteroscedasticity ((G)ARCH) class of models (see, for example [28-30]). While it is a problem that none of these econometric models are able to fully capture all of the stylized facts of the empirical data, much progress has been made since their introduction and it is feasible that this goal will be met in the future. Of a much greater concern is the fact that econometric models typically give very little cause-and-effect information about how the underlying market mechanisms generate the empirically observed stylized facts. Without knowl-4 edge of how the model produces the desired asset price dynamics our use of it would be "blind" and it would not be known whether its assumptions were valid from one time to the next. As physicists, it is our natural inclination to seek out the fundamental cause to a problem, which, in this case, should also provide us with both practical and academic benefits. Over the past few years, physicists have shown a lot of interest in spec-ulative asset pricing and financial systems in general. The idea behind the physics of finance, or econophysics, is that financial markets act in a very simi-lar way to certain statistical physical systems and therefore should be amenable to the tools developed to treat them. There is no question that researchers in the fields of finance and economics have been studying these problems for some time and have had a great deal of success. The goal of the econophysics field is not to replace this work, but rather to provide an approach to complement it. 1.2.2 Complexity in finance Along with the goal of producing a model that gives insight into the cause of an asset price's behaviour, the motivation to study finance has, in large part, come from the realization that financial systems (ie. markets) are in many ways similar to the general class of "complex" physical systems such as frustrated magnetic systems, neurological networks, and surface growth. 5 Structural similarities The most obvious similarity between a financial market and one of these phys-ical systems is in their underlying structures. Complex systems always possess a large number of interacting subunits. While the form of this subunit is hu-man in the case of a financial system (as opposed to an atom or molecule typical of physical systems), markets are very large structures with an even larger communication network connecting one another making the form of the macroscopic systems very similar. There are, however, two significant structural differences between the two classes of systems. First, in a financial market each subunit is a living organism and therefore a complex system in its own right. While the internal structure can be very complicated for some physical systems (ie. the individual phospholipids of a biological membrane), it pales in comparison to that of a human being. Second, in a physical system, the laws governing the interaction and evolution of the microscopic components are well-defined fundamental forces; in a financial system, however, many of these forces are the result of poorly understood cognitive factors. It is interesting to wonder how well techniques designed for systems with particle-like subunits and which evolve under physical laws will adapt to social systems governed by less well-defined forces. Empirical similarities As the number of subunits within a macroscopic system is increased, it is intuitive to reason that the complexity of its behaviour will increase propor-6 tionately. It is a well documented fact that this is not the case. As the system's size increases, regular and distinctive macroscopic properties actually emerge from the complex microscopic dynamics. To illustrate these properties, consider for example the magnetization, M, of a macroscopic magnetic material [20] at a temperature, T immersed in an external magnetic field, H. If H = 0 and the temperature is increased, it is found that at some critical temperature, Tc, the system enters a critical state in which: 1. correlations between individual "spins" of the magnet become very long-range; 2. power law behaviour (indicating scale-invariance) is observed; and 3. the magnetization undergoes a transition and falls sharply to zero. Rescaling the temperature so that r = (T — Tc)/Tc and then measuring M vs. r for several values of H yields a very interesting result. It is found that, if M is rescaled by some power of H, and r by some different power of H, plots of the rescaled M versus the rescaled r all collapse onto a single curve. This scaling law (the specific values of the scaling exponents for a given system) and data collapse for this magnetic system are traits common to complex physical systems. Another interesting characteristic of such systems is termed universal-ity. To demonstrate this effect, imagine the above M = M(H,T) measure-ments are repeated on several different types of magnetic materials. From the measurements it is found that, ignoring the necessary material constants, the same scaling law holds for each material; it is said that all of these systems belong to the same universality class which means that, as the critical state is approached, their macroscopic properties behave in a similar manner. 7 Since financial markets posses a similar structure to traditional complex systems it is interesting to wonder whether they also share the same empirical characteristics such as scaling behaviour and universality. Many studies have been performed along these lines [1,2,7-9,16-24] and two key results have been found. First, the cumulative distribution of returns, Fn At(x) = Prob[rtAt > x] is found to asymptotically decay as a power law with exponent, a ~ 3. When the rate of return is scaled with the appropriate exponent, FrtAt(x) is also found [1,2,4,5,7-9,16-19] to be stable for time scales A t < 20 days (for individual stocks). Second, it is also now well known that the asymptotic power law decay exponent, a « 3 and the appropriate scaling exponent hold for individual stocks, whole-market indices, and even across various equity markets suggesting universality of the price formation process in these markets. Finally, the fact that long-range autocorrelations exist in an asset price's volatility time series and its cumulative distribution of returns exhibits power law behaviour suggest that a financial market exists normally near its critical state. This behaviour differs from that observed in the magnetic example above in that the magnet will only reach a critical state if its parameters, r and H, are tuned to their critical values. One possible explanation for this phenomenon is the Self-Organized Criticality hypothesis proposed by Bak et al. [50] in which a state of dynamical equilibrium is reached once the system has evolved to the point where it can no longer propagate a perturbation over the correlation length of the system. 8 1.2.3 Physical models of asset price dynamics The evidence is quite strong that the mechanism underlying the process of speculative asset price formation is very similar to that of various "complex" physical systems. The question is now How do we use the standard tools of statistical mechanics to develop a useful model of speculative asset prices? The number of different physical models of speculative asset price dy-namics is very large (see for example [32-49]) and to give a description of a "general" method would do none of them any justice. Instead I shall briefly describe the structure (and its motivation) of one of these models which I believe to be a good representation of the efforts of this approach. The model I shall discuss is the Cont-Bouchaud (CB) model [31]. In the C B model, correlations are introduced by incorporating a communication structure between the N individual agents who are situated at the vertices of an iV-dimensional random graph. Each agent is allowed to randomly es-tablish binary links with other agents and, through successive links between existing clusters (two or more agents linked together), large clusters can form; in this way the communication structure is analogous to bond percolation in an infinite-dimensional space. Within a single cluster it is assumed that all agents hold the same position towards the asset (ie. buy, sell, or hold) and so the price change (which is assumed to be proportional to the the excess demand) is simply proportional to the cluster weighted total demand. A key aspect of the model is an agent's ability to form coalitions; in the C B model this is controlled by a constant, user-set parameter. For a range of values of this parameter, the 9 model generates frtAt (x) in a manner that is very similar to what is observed in real markets. Specifically, it predicts both exponentially truncated tails (with a > 2) and a — 1.5 in the region of small-to-moderate \r&t\ which is very close to what is observed empirically. Finally, it is found that, as the clustering parameter approaches some critical value, very large coalitions form and a market crash occurs. As well as generating quantitatively accurate data, the C B model is useful for showing how a specific mechanism (in this case the inter-agent com-munication structure) can result in the leptokurtic form of frt&t(x)-1.3 The Minority Game 1.3.1 The model The Minority Game (MG) was developed by Challet and Zhang [55] as a simplification of Arthur's El-Farol Bar-attendance problem [54]. The model is designed to examine the behaviour of a group of individuals who all compete for a limited resource and are governed by the rule of supply-and-demand. The M G consists of a non-spatially oriented, odd number of agents who repeatedly attempt to out-guess one another and earn a reward. On each round of the game, the agents take one of two possible positions; with foresight in hand I will label these buy and sell. After the agents have made their choices, the size of each position is tallied; those agents who chose the position taken by the minority of agents are rewarded with +1 point, while those who took the opposite (majority) position are penalized —1 point. 10 To make his decision, an agent has a set of strategies. Each strategy predicts a specific position for the agent to take given each unique history of the game. The history consists of a certain number of the most recent winning positions of the game and is common to all agents within the group. Finally, in order to incorporate adaptability into the game, each agent rates the performance of his strategy (ies) and makes his strategy selection based on this performance. Many details were omitted from the above description of the M G since there are several formulations of the model which are all slightly different. In the original M G (which I will refer to as the Standard Minority Game), each agent possesses a finite and (possibly) unique set of strategies randomly drawn from some large simplex space. A virtual score is kept for each strategy and a virtual point is awarded on any round in which the strategy would have predicted the minority position. In the Standard M G agents use the simple strategy selection rule of choosing their strategy on each round with the highest virtual point score. The Standard M G is useful for examining global properties (such as system efficiency) of the group of agents within a M G framework. In a slightly generalized version of the Standard M G , agents have the ability to select any of their strategies (not just the one with the highest virtual score) according to the Boltzmann weight of their virtual scores. Because of the ever-changing dynamic of the game and since the agents' strategy sets can overlap, playing what is thought to be a "sub-optimal" strategy may in fact be beneficial if enough of the other agents play their similar "optimal" strategies. In this way, 11 this Thermal Minority Game is useful for examining whether-or-not using a sub-optimal strategy is actually irrational in this complex environment. 1.3.2 The Evolutionary Minor i ty Game While the agents within the Standard and Thermal M G models are adaptive (the agents can select from a variety of strategies depending on the current conditions), they are not strictly evolutionary since both the population of agents and each agent's strategy set is fixed throughout the duration of the game. In reality, we would expect that an underperforming agent or strategy would be replaced with one that appears to be better equipped to compete within the group dynamics. Another popular version of the Standard M G , designed to include evo-lutionary effects, is the Evolutionary Minority Game [70]. In this model there are only two strategies available to the agents; one which contains the minor-ity positions the last time each of the possible histories was encountered and the other with the opposite positions (ie. the majority positions); in this way the two strategies are dynamic and change constantly throughout the game. Each agent then has a unique parameter (called his gene value) that predicts which of the two strategies he will play. Evolution is introduced by removing from the population agents whose performance falls below a certain level and replacing them with agents having different gene values. 12 1.3.3 The M G as a realistic market model As should be clear from the above description of the M G , the structure of the game is very simplistic. One might ask (justifiably) why the M G should be included in the discussion of speculative asset price models. To be perfectly honest, none of the versions of the M G presented thus far are remotely close to being accurate representations of a real financial market. Some problems with using the M G as a realistic model are: • Each agent is constrained to be active on every round of the game. • The size of the group is fixed. • There is no external information present in the model. • The binary payoff function (+1 for taking the minority position and —1 for taking the majority position)is overly simplistic. While these details definitely restrict the applicability of the Minority Game to financial systems, they are simply structural components of the model and can be removed and replaced (fairly easily) by more realistic constructions. For instance, in a real market the total sales and purchases of an asset must equal; this fact implies that the total wealth is conserved within the group trading in the asset (neglecting commissions, spreads, e tc . ) . The binary payoff function of the M G is inaccurate since it does not preserve the system's wealth. Given there are n m ,„ agents in the minority group and nmaj majority agents, then one simple method of incorporating this conservation law while maintaining the model's basic minority structure would be to give a reward of +1 point to 13 each of the minority agents and a penalty of —nmin/nmaj points to each of the majority agents. Of a far greater concern are the components of the M G that are funda-mentally incompatible with a financial market; these include: • In the M G there is no motivating factor for the agents to play the game. Even though speculation is a crucial component of a real market, for the agents to trade they must at least believe that the asset has some intrinsic value. In the M G this does not exist; the agents simply play the game because we force them to. • The M G has no market making mechanism; agents simply "buy" and "sell" the asset. It is assumed that there is an unlimited quantity of the asset and all of those sold are then bought and the quantity of those to be bought are present in the market to begin with. • The M G neglects the effect of an agent's actions on his long-term per-formance. If there are more buyers than sellers during a round then we should expect the price to rise. While selling the asset during this round will earn an agent a good short-term profit, that agent will then own less of the asset when the price rises which is not beneficial in the long run. The M G essentially models how agents compete under supply-and-demand pressures, but neglects any effects due to the arrival of external information (which result in the long-term movements of the asset price). With its construction, the M G was not intended to act as a model of a financial market; it was simply meant to model how a group of heterogeneous 14 individuals behave when competing for a limited resource and governed by a minority rule. For instance, in its basic form the M G acts as a fairly realistic model of: 1. commuters competing for the less crowded of two possible routes; or 2. foraging animals searching for a limited food source. Because of the reasons listed above the M G is too simplistic to be considered as a realistic market model. However, some researchers [56,57,62,66,69] have discussed the M G in terms of a market model. Neglecting the above problems, the M G does have features that make it attractive for this use. Minority rule It was stated above that the movements of an asset's price usually follow the actions of the majority. While this effect is central in determining the long-term performance of an agent, under certain circumstances the minority rule effect is present and significant. Imagine the situation of a group of fund managers competing for profits through ownership of two stocks, x\ and x2 • Assume it is a commonly held belief that both stocks will increase in value, but that xi will increase at a slightly higher rate than x2- If we further assume that : 1. there are equal amounts available of both X\ and x 2 ; 2. each manger can only invest in one of X\ or x2; 3. the total quantity of each asset is shared equally among those who buy it: and 4. all resources not used to purchase X\ or x2 are held in the form of some more secure, but lower return asset such as a government bond, then it might seem logical for each manger to ignore X2 and buy as much of X\ as possible. But if this happens then none of the 15 managers will do very well since they will have to share the profits amongst a very large group. If the commonly held belief holds true and x2 earns slightly less per share, then a small (minority) group of the managers will outperform those in the majority group if they buy x2 since they will be able to buy more and earn a greater profit. Inductive reasoning Traditional financial economics has always assumed the use of a deductive agent and the Efficient Market Hypothesis (EMH); given any economic situation, an agent makes a logical decision based upon a perfectly defined environmental state in which all information about an asset is reflected in its current price. For instance, within an equity market, such a perfectly rational agent would weigh all information (such as the stock's price history and the "true" value of the underlying company) and trade according to the relationship between the current price and this information. In real financial systems these conditions are often very complex and sometimes ill-defined; assuming that an agent can decipher all of the informa-tion perfectly is not reasonable. It is known [54], however, that humans are very good at pattern matching and therefore reasonable to expect that, under complex circumstances, we will use inductive reasoning and base our decisions on our past experience. As opposed to standard models of speculative asset pricing, the dynam-ics of the Minority Game are formed through the use of inductive reasoning. On any round, an agent selects a position based on the current history of the 16 game and his personal experience with that history. Using this experience, the agent can then adapt (in a Lamarkian way in the Standard and Thermal MG) or evolve (in the genetic sense of the E M G ) in order to be more competitive within the ever-changing dynamic of the game. The Minority Game approach I believe the Minority Game is a convenient starting point to develop a market model which is capable of generating empirically accurate results. Although many details of the structure of a financial market are not present in the model, it does provide an attractive model (with its use of inductive reasoning) of how a group of agents compete for a limited resource when governed by a minority rule with dynamics resembling the process of price formation in many situations. The idea central to the M G approach for market modeling is to first understand the dynamics underlying the price formation process and then to build a more realistic model upon this base. There are two steps behind this approach. 1. Characterize the basic M G model. 2. Using this model as a base, successively remove its simplifying assump-tions and gradually build-up the more realistic model. Actually, the simplicity of the M G permits this approach; using such a simple model provides a tractable base that is relatively easy to treat analytically. While this approach is time-consuming, it has several benefits. First, 17 beginning with a simple base and then building the model up component-by-component should allow us to determine which components (market mecha-nisms) are necessary for the model to produce results in agreement with those observed empirically. Second, it is also hoped that this type of approach will allow us to understand the specific effect (s) of each component and provide us with a cause-and-effect map between these microscopic components and the macroscopic dynamics of the asset price. Finally, from the vast number of current speculative asset pricing models it is clear that the structure of these components can be modeled in a wide variety of ways (for example the payoff function can be modeled as a simple step function as in the basic M G , or it could be modeled (slightly) more realistically as a fixed level of resource split evenly between each member of the minority group); with this bottom-up ap-proach we should be able to gain insight into how the specific structure of each component affects the asset price dynamics. Progress to date Since its introduction, much work has been done on the Minority Game (see, for example [55-72]). This has resulted in both a thorough understanding of the dynamical behaviour of the models and in accurate analytical theories of both the Standard M G [61] and the E M G [71]. Many extensions of the models have also been studied. First, a more realistic payoff function (similar to the one suggested in 1.3.3) that conserves the wealth of the system has been studied for the Standard M G [66]. It has been found that the properties of the model are largely independent of the 18 specific structure of the payoff function. In addition, the effect of various strategy distributions on both the Standard [63] and Evolutionary [72] M G models has been investigated as well as the size and structure of the agent's strategy space [67,68]. Along with many other studies investigating single extensions of the basic M G model, there have also already been two attempts at large-scale extensions of the model with the goal of creating price dynamics with empiri-cally accurate features. Removing only 3 or 4 of the simplifying assumptions of the basic M G , these papers [57,62] have already produced results that are in good qualitative agreement with both the empirical distribution of returns, frt,At(x)> a n d the autocorrelation of returns, pr(^i;^2)-1.4 Focus of thesis 1.4.1 Motivat ion for research Apart from a brief treatment in [62], the effect of agent inactivity on the M i -nority Game model has largely been ignored. The lack of attention to this phenomenon is puzzling; from both day-to-day experience and the results of academic research, inactivity within a group of competing individuals seems to be an important component of the price formation mechanism. In their pa-per [31], Cont and Bouchaud state that allowing the agents to sit out at various times is crucial to obtaining the leptokurtic / r j A t (x) and volatility clustering characteristic of empirical data. From real-world experience it should be obvi-ous that: 1. the number of people trading in any particular asset is constantly 19 changing; and 2. investors are not forced to trade continuously and so their trading frequency will also vary in time. 1.4.2 Present study In this thesis I will examine one very simple model, called the Variable Ac-tivity Evolutionary Minority Game ( V A E M G ) , of agent inactivity within the Minority Game framework. In keeping with the ultimate goal of the M G I will attempt to characterize the model as completely as possible and, in doing, will focus on two aspects: 1. how the performance of an arbitrary agent evolves when he is given a variable activity level; and 2. how the dynamics of the game are affected under these same circumstances. In analogy with the gene value of the E M G , each agent in the V A E M G will have his own activity parameter which is simply the probability that the agent is active on any given round. The structure of this inactivity mechanism is intentionally simplistic so as to maintain tractability in the model. At the moment I am simply concerned with how the presence of an inactive state affects the behaviour of the game; it will be left to a later study to examine more complex and realistic inactivity mechanisms. The play of the V A E M G is split into two steps. In the first, each agent "decides" whether to be active or not. If an agent chooses to participate in the round then he simply plays an EMG-type game with all of the other active agents in the second step. To motivate the agents to be active, I have included a small penalty to an agent's score for each round he is inactive. When an agent's score falls below some predetermined level that agent is killed-off and 20 replaced by one with new activity and gene values that are hopefully better able to compete within the group dynamic. In this thesis I will only examine the effect of this inactivity mechanism within the framework of the E M G model. This choice has been made since the E M G is ideal for examining the properties of an individual agent and I am interested in how this performance will be affected when the agent has the ability to choose whether or not to participate in the game. 1.5 Organization of thesis The remainder of this study will be as follows: in chapter 2, I shall develop the formal definition of the E M G model and present the pertinent results of numerical simulations of the model. In chapter 3, I shall further discuss the choice of the inactivity mechanism for the V A E M G and then formalize the model's structure. This chapter will conclude with my developing a mean-field theory of the V A E M G . In chapter 4 I present the results of numerical simulations of the V A E M G model and compare these with the results of the theory developed in chapter 3. These results will focus on the two areas of the model indicated above: 1. how the performance and behaviour of an agent is affected when he has the ability to control his activity level; and 2. how the dynamics of the E M G model change when this modification is added. Finally, in chapter 5,1 restate the pertinent results of the paper along with any possible implications they may have and then conclude by discussing the future work that the results suggest. 21 Chapter 2 The Evolutionary Minority Game In this chapter I will review the Evolutionary Minority Game (EMG) model and discuss its behaviour with the use of numerical simulations. The work in this chapter is based upon [70] and will serve as a base for the rest of the thesis. 2.1 The basic E M G model structure While the definition of the E M G given in chapter 1 might be enough to con-struct a numerical algorithm for the model, it is not adequate for any sort of an analytical treatment. In order to give a more precise description of the model and motivate a later analytical theory, a formal definition of the E M G is needed. Let N be the (odd) number of agents in the game and let i G { 1 , . . . , N} be the index that runs over this group. The game is played repeatedly for T rounds. On round t 6 { 1 , . . . , T}, each agent independently makes a binary decision, Xi,t £ {—!>+!}) about whether to buy (which will be arbitrarily 22 labeled +1) or to sell (-1) the asset for that round. Once all agents have made their decisions the number who chose, to buy and the number who chose to sell are calculated. The winning option, p,t € {0,1}, is defined as another binary variable whose value is 0 if J2i Xi,t > 0 (i e- sellers win) and +1 if Xi,t < 0 (buyers win). Now let the memory, M, be the number of past rounds that each agent can remember and = HtM®Ht-{M-\)®- • -®^t-2®Pt-\ be the game's history of length M for round t. Given the binary nature of the game and a memory of length M, it follows that there are 2M possible histories, h^1; each history (a binary string) will be labeled by its decimal equivalent, j € {0,. . . , 2 M — 1}. A n agent's trading decision, Xi,t, on round t is determined by the inter-action between the current history, hf1, and his set of strategies. A strategy, S, is a function that maps each of the possible 2M histories onto some decision (<S : /x M —> 8) that will be made given that history is the current history. Each strategy will be represented as a bit string of length 2M where the jth element, Sj, is the agent's trading decision given that the jth history is the current his-tory. For concreteness I have listed all of the possible histories for M = 3 along with a sample strategy in figure 2.1 below. If the history bit string labelled by j — 3 is encounterred during round t of the game (ie. h^1 = Oi l ) and an agent is playing this strategy, then that agent will choose to buy the asset (s3 = +1) on that round. Because of the binary nature of the decision process there are 2 2 M unique strategies for the 2M possible histories; in the E M G agents only have access to 2 of these strategies. Let be the trend strategy. The jth entry of this 23 j hi SJ 0 0 0 0 -1 1 0 0 1 +1 2 0 1 0 +1 3 0 1 1 +1 4 1 0 0 -1 5 1 0 1 -1 6 1 1 0 +1 7 1 1 1 -1 Figure 2.1: The 8 possible histories, , for M = 3 along with the 8 corre-sponding strategy elements, Sj, of a sample strategy, S. binary string holds the decision value, Xi,t, that would have put the agent in the minority room the last time the jth history was encounterred in the game. Since the result of the game fluctuates between "buy" and "sell", <S* is a dynamic strategy that will be modified after each round. While it might seem intuitive to play the trend strategy, if all agents took this view then they would all lose because of the minority rule. Heterogeniety amongst the agents is introduced by their ability to employ the anti-trend strategy, , which is the strategy anti-correlated with the trend strategy. Finally, associated with agent i is a gene value, pi G [0,1]; this the probability that agent i will play the trend strategy. At the beginning of the game each agent starts with a score of 0 and both the trend and anti-trend strategies are initialized. To seed the first round of the game an imaginary history, hf1 = /J.-M+I ® • • • ® /^ o, is created. Details of these initializations will be presented in the next section. During each round t of the game agents select their trading decisions based upon their gene value. With probaility pi: agent i chooses the option 24 dictated by and with probability l—pi he chooses the other option. Once the minority decision is determined all those agents who chose the minority option have 1 point added to their score; those who chose the majority option have 1 point deducted from their score. At the end of every round the strategies are updated with the result of the round just played. A n agent will play the game until his score falls below some critical level, Zc < 0, at which point he is "killed-off" and replaced with a new agent. This mechanism is the evolutionary feature of the E M G . If an agent's performance is poor then it is reasonable to assume that eventually he will lose all of his money and pull out of the market. The new agent is given an initial score of 0 and a new gene value which is a uniformly distributed random number with radius, R G [0,1] (reflective boundary conditions are employed to ensure p £ [0,1]) centerred around the pi value for the killed-off agent. In this way a poorly performing gene value is replaced by one that is hopefully better able to compete within the complex group dynamic. 2.2 Numerical results A numerical algorithm has been created and run over a wide range of parameter values simulating the behaviour of the E M G model. These simulations will serve two purposes: 1. to characterize the basic E M G system; and 2. by comparing the results with the original work on the E M G [70] I hope to confirm the accuracy of the algorithm so it can be used as a base upon which to construct the Variable Activity E M G algorithm. 25 Throughout this work the focus will primarily be on an agent's perfor-mance within the collective actions of the group. The crucial feature differen-tiating one agent from another is his gene value, p,. In the E M G , Pi indicates agent z's willingness to follow trends; a pi « 1 means that the agent is a trend follower, while Pi « 0 means the agent employs a contrarian trading strategy. By quantifying an agent's performance in terms of his gene value we will hope-fully be able to shed light on what type of strategy is best suited for this type of supply-and-demand governed collective. For any agent, i, the stochastic process, {Ri,t}, of his payoff is a super-martingale - ie. the E M G is a "less-than-fair" game. Because of the payoff function (+1 point per round for winning and -1 point for losing) and the minority rule structure of the game, the agent's expected payoff per round is negative and so it is expected that he will be killed-off within some finite time. Instead of his point total, which will eventually fall to Zc, I will measure an agent's performance by the number of rounds he plays before dying - ie. his lifetime. For this reason, the simulation results will focus on the distribution of average lifetimes, for the group of agents. To initialize the simulations, the current history, , the trend, , and anti-trend, <i>t strategies, and the agents' gene values, p^, were randomized from a uniform distribution of their appropriate realizations. Simulations were also performed using other initializations, but in all cases L(p) was found to be invariant under these changes. 26 2.2.1 Time evolution of the E M G In chapter 1 I discussed the intrinsic use of inductive reasoning in the E M G . Since this feature makes no assumption about the equilibrium of the system, it will be useful to investigate its dynamic properties and whether or not it does actually reach an equilibrium state. I will describe the state of the system by \n<i\, the magnitude of the difference between the sizes of the two positions. In figure 2.2 is plotted two time seriess of \nd\ for the E M G . We can see that the system exhibits a transient 3501 1 1 1 1 1 round number - 1 Figure 2.2: The time evolution of \nd\ with Pi = 0.25 (—) and pi randomly initialized (• • •) for N = 501, M = 3, Zc = -100, and R=l. response which rapidly decays away after approximately 200 rounds. The 27 length and shape of this transient response appears relatively insensitive to the initial conditions of the system with one exception; when each agent's gene value is initially randomized the transient response is not clearly present. When the gene values are initially very similar to one another, the size of the majority group is very large (and therefore so is \nd\); for this initial period most agents will lose a point on each round and so will die off in « \ZC\ rounds. Once this first round of agents has been killed-off and replaced by agents with randomly distributed pi values (since R = 1), \nd\ then rapidly decays to a time-averaged value that remains constant for the rest of the game. The state of the system in this "dynamic equilibrium" (in which the average of \nd\ over some time interval t ~ O(103) rounds) reveals a very interesting property of the model. If we examine \nd\, the time-averaged value of \nd\, we will be able to determine how well the agents are utilizing the system's resources. As \nd\ —>• 1 the size of the minority group becomes largest and the least wealth is lost from the system. In figure 2.3 I have plotted \nd\ for several values of N. To compare the efficiency of the E M G ' s equilibrium state with the "random" case, I have also plotted \nd\ for simulations in which the agents simply select their position at random (ie. independent of the game's history). On a log-log scale, both curves are very linear with exponents of 1/2 demonstrating that \nd\ oc \/N as we might expect. What is interesting from figure 2.3 is the fact that the value of \nd\ for the E M G is significantly less than the "random" \nd\. This result tells us that the agents within the E M G naturally evolve (over the transient time of the system) into a state in which the systems' resources are used much 28 30 01 i i i i i i 0 200 400 600 800 1000 1200 number of agents - N Figure 2.3: The equilibrium average size difference between the two positions for the E M G (o) and when the agents select their positions randomly (•). Other parameters for the simulations are Zc = —100, M = 3, and R = 1. more efficiently than if the agents simply made their decisions at random. This emergence of "co-operation" is most likely a result of the evolutionary feature of the model and fascinating due to the fact that, as will be mentioned below, the equilibrium properties of the model are independent the agent's memory, M , which one might expect to be a large source of the model's history dependence and therefore its emergent behaviour. 2.2.2 Equil ibrium properties of L(p) A sample L(p) distribution is shown in Figure 2.4. The distribution is sym-29 600 | 1 i 1 1 1 1 1 1 r 500 o 300 H 200 h 100h "0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 gene value - p Figure 2.4: The distribution of average lifetimes, L(jp), for N = 101, M = 3, T = 106 rounds, Zc = - 5 , and R = 2. metric about pi = 1/2 and is distinctly bimodal with sharp peaks at p = 0 and p = 1. Let p* be an agent's optimal strategy - one that allows him to survive the longest. Interestingly, Figure 2.4 shows that this optimal strategy is to either always follow (p* = 1) or go against (p* = 0) the prevailing trend; if he uses any mixture of these strategies his performance will drop. In the case of p = 1/2 when an agent is most unsure of which strategy to choose, we can see that his performance is worst. Figure 2.5 shows the dependence of L(p) on the number of agents, N. As the number of agents increases the average lifetime of each agent increases also; this result can be understood if we consider an agent's expected payoff 30 600 500 h 400 H S300H 200 100 Figure 2.5: L(p) for M = 3, T = 106, Zc = - 5 , and R = 2 with 7V=31 (—); N=61 (—); and 7Y=101 (—-). per round. From above we know that \rid\/N oc l / x / i V when the system is in equilibrium and so, as the number of agents increases, we should therefore expect the fractional difference between the size of the two options to go to zero. If this fractional difference decreases then so too will the expected payoff of the agents thereby increasing their average lifetimes. Figure 2.6 shows the dependence of L(p) on Zc. As Zc decreases it is intuitive that L(p) should increase. Specifically we can see that L(p) scales very well with the size of the cutoff score, \ZC\, or L(p) ~ \ZC\. In addition, the dependence of L(p) on M and R was checked. In all 31 450 400 350 300 —o250 N o. - 1 200 150 100 50 0 "O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P Figure 2.6: L{p)/\ZC\ vs. p for Zc = -5 (—), -10 (—), -25 (—-), and -100 (• • •)• N = 101, M = 3, T = 106, and R = 2. cases there was no statistically significant dependence of L(p) on either of these parameters. Analysis of results I will now give a simple explanation [73] for the cause of the bimodal and symmetric shape of L(p) seen in figure 2.4. Consider the case N = 3 and assume that each agent can only take on the gene values, p =0, 1/2, or 1. I will label the agents 1, 2, and 3 and focus on the actions of agent 3. Given the 3 possible values for p it follows that there are 9 possible gene value combinations, (p\,p2), for agents 1 and 2. Depending on the specific 32 combination of (pi,P2) and his own gene value, p 3 , agent 3 can expect a range of payoffs which are shown below in figure 2.7. The mean and variance for (Pl»P2) (0,0) (0,1) i1-l) (i1) (1,0) (M) (1,1) P 3 = 0 -1 -1 -1 - i 1 " ? 0 -1 0 + 1 Ps = k 0 1 2 -1 i '•?. 1 "2 i "2 -1 l "2 0 P3 = 1 +1 0 -1 0 I "2 -1 -1 -1 -1 Figure 2.7: A l l possible expected payoffs for agent 3 when N = 3 given that each agent can only access the states p =0, 1/2, and 1. each of the rows of figure 2.7 are shown in figure 2.8. P3 mean payoff variance 0 i "2 2 ? 1 2 1 "2 i 1 I 2 Figure 2.8: The mean payoff and corresponding variance for each of agent 3's gene values when N = 3 and each agent only has access to the states p =0, 1/2, and 1. From figure 2.8 we can see, as expected, that the mean payoff for agent 3 is the same for each of his gene values (ie. there is no a priori best gene value), but that the variance is greater for p3 = 0 and p 3 = 1 than for p 3 = 1/2. Eventhough there are more situations in which agent 3 is guaranteed to choose the majority position (and therefore be penalized 1 point) with either p 3 =0 or 1, the larger variances for these gene values indicate that the agent also has a greater probability of choosing the minority position by out-guessing the other two agents; he is therefore more likely to live for a greater number of rounds before he is killed-off. 33 2.3 Discussion of results The results of simulations of the E M G presented in this chapter have demon-strated some of the interesting properties of the model. First, it has been shown that agents within the E M G framework naturally organize themselves into a state in which the systems' resources are utilized more efficiently than if the agents simply chose their positions at random (ie. by nipping a coin). This self-organized "co-operation" was shown to be independent of the system's ini-tial state and, since its equilibrium state was found to be independent of the agent's memory, is most likely a result of the history-dependent evolutionary mechanism of the model. Also significant is the bimodal and symmetric nature of the distribution of average lifetimes, L(p). Within the type of competitive group dynamic formed by the E M G , it has been shown that an agent's optimal strategy is to either always or never follow the trend set by the group. This result has been explained using the simple arguement that the variance for an agent's payoff is a minimum for p = 1/2 and a maximum for p = 0 and p = 1; when an agent is in one of these extreme strategies he therefore has the greatest probability of choosing the minority position and surviving the longest. While the unrealistic features of the M G have been discussed in chap-ter 1, there is one point specific to the E M G that I feel should be emphasized further. It has already been mentioned several times that the E M G is a super-Martingale and so the expected payoff for each agent is always negative. If the results of the model are discussed in financial terms then one immediately en-counters the problem If, on average, an agent expects to lose money, regardless 34 of which position he takes, then why would he trade in the first place? In reality, the E M G is actually more a realistic model of a group of agents either betting "red" or "black" on a roulette wheel than a group of agents trading a financial asset in a market. The only agents that would play the E M G are those that are risk-preferring - ie. agents would only participate for the thrill of the game and not with the realistic expectation of increasing their wealth. Since the lifetime is the only decent measure of such a risk-preferring agent, the bimodal shape of the distribution of average lifetimes is not unexpected; the strategies p = 0 and p = 1 are the riskiest strategies and so it is reasonable that they are also the best performing. In keeping with the theme of developing the E M G into a more realistic market model, an interesting (and essential) extension would be to incorporate a payoff function in which the expected payoff is positive and the agents there-fore have an incentive to participate. By defining a "wealth" as an agent's ac-cumulated score and incorporating an evolutionary mechanism which replaces the gene values of a certain percentage of the worst performing agents every z » 1 rounds of the game, one can then examine the group's performance with the final distribution of wealth and of gene values. Because of the sub-Martingale property of this proposed model it is likely that using either p = 0 or p = 1 will no longer be optimal - ie. they will not maximize an agent's wealth. In fact, it is quite possible that p = 1/2 will emerge as an agent's optimal gene value due to its lesser probability of creating a situation in which the agent is guaranteed to choose the minority room and lose (see figure 2.7). 35 Even though the current binary payoff function is financially unrealistic, I will use it in this work and leave the above modifications for a later study. 36 Chapter 3 The Variable Activity Evolutionary Minority Game Results of simulations of the E M G from chapter 2 have shown that the model is well-suited for demonstrating properties of an individual agent within the Minority Game framework. Specifically, it has been shown that an agent's optimal strategy (defined as the gene value which maximizes his expected lifetime) is to always play the same pure strategy - either always follow the previous winning option (p = 1), or always go against this trend (p = 0). Several questions arise if the agents are given the ability to abstain from playing. Firstly, will the symmetry of the E M G be broken? It has been shown in chapter 2 that the trend-following and contrarian pure strategies of the E M G are equally optimal. It is interesting to ask whether this situation will persist, or if the ability to be inactive will cause one of these strategies to become preferable over the other. Secondly, what will the optimal strategy be when inactivity is introduced; will this strategy be "pure" such as always play or never play, or will it be some mixture of the two; are there symmetries 37 of agent performance over the strategy space; and are there conditions under which an agent's optimal strategy can change? These are some of the questions that have motivated the development of the Variable Activity Evolutionary Minority Game ( V A E M G ) . In this chapter I will discuss and formalize the structure of the V A E M G model and specifically the structure of the mechanism used to generate agent inactivity within the standard E M G framework. With the V A E M G I have attempted to model this mechanism so as to capture some realistic features of a group of competitive and adaptive agents while keeping the model as simple as possible. The chapter will conclude using this formal definition to develop an analytical theory of the V A E M G which will later be used as a comparison with the results of numerical simulations of the model. 3.1 Choice of activity structure The first problem that must be addressed is the way in which to model agent inactivity within the framework of the E M G . The possibilities for this choice are endless. For instance, we could use: 1. a very simple "group" condition where specific agents are randomly se-lected to sit-out on each round, 2. inactivity spreading via a bond percolation mechanism like in the Cont-Bouchaud model [31], or 3. some more realistic mechanism of controlling inactivity at the single 38 agent level through feedback of the agent's past performance For this study it would be ideal if the inactivity mechanism generated emergent properties of the system, but also was very simple. With choice (1) above, while the global inactivity structure is very simple, it would unlikely produce any interesting results since it lacks any mechanism for generating correlations between the individual agents. With choices (2) and (3), mech-anisms exist to cause the necessary correlations, but in both cases they are overly complex and will likely produce results that are difficult to analyze. The E M G model provides an ideal starting point for this goal. As has already been noted, the evolutionary nature of the gene value, p in the standard E M G allows us to examine how an agent's performance depends on his strategy selection. Even though its structure is relatively simple, this evolution is important due to the feedback that introduces into the system. It is hoped that, if the agent's activity level is modeled in an analogous way, the modified model will produce interesting emergent behaviour (such as a relationship between agent performance and activity level) while remaining relatively easy to treat analytically. 3.2 Model structure The play of the V A E M G is the same as the E M G with several exceptions. In the V A E M G , each agent i € {1, . . . , AT}) has an activity parameter, ^ e [0,1], which is simply that agent's probability of being active during a round. In an analogous way with pi in the standard E M G , ai is an evolution-39 ary variable and therefore introduces feedback (memory dependence) into an agent's decision on his activity status. The game proceeds in two independent steps. In the first step, each agent "chooses" whether or not to be active based upon his activity parameter, cti. Once these decisions are made, the total number of active agents, Na < N, is determined. While not strictly necessary, we have chosen to exclude the possibility of having an even number of active agents (ie. Na G {1 ,3 , . . . , N}). In addition to keeping the model structure as close to the original E M G as possible and preventing us from having to deal with the case when there is no distinct minority group (rid — 0), the exclusion of A^-even, as will be shown in a later section, significantly reduces the complexity of the analytical theory. In the numerical algorithm the A^-even condition is enforced simply by requiring the agents to replay a round's first step until an odd number of active agents is obtained. In the second step of the round the active agents play a normal round of the E M G with N — Na. Each of the agents chooses either to "buy" or "sell" based upon the past history and their individual strategy preference, pi, and then is either rewarded or penalized based upon whether their choice puts them in the minority or majority group, respectively. As the V A E M G is currently structured, there is no incentive to partici-pate in the second step of the game. If an agent decides to be inactive for the round, then he simply sits out and waits for the next round. While he does have the possibility of choosing the minority option and raising his score, his expected payoff, given that he plays, is negative; he would therefore be better 40 off by choosing never to play. To prevent this "trivial" solution (all agents always inactive) I will in-troduce a global parameter, I £ [—1,0], to act as a penalty to an agent's score for being inactive during a round. More than just a mathematical convenience, / can be thought of as a parameter that models the effect of inflation on the system. To explain this point, imagine a group of agents within a financial market who own cash and a single type of asset. When an agent is confident (in his own mind) about the future movement of the asset he will "play" the market and either buy or sell some quantity of it and in turn change the quan-tity of his cash reserve. If, on the other hand, he is unsure of the asset's future movements he can choose to avoid the risk of playing the market and hold the quantities of his cash and asset constant. While this second option is less risky than the first, it will not entirely preserve his wealth. Because of inflation, the value of his cash, which could have been invested in the form of the asset, will be worth less in the future and so his net worth will effectively decrease. The game proceeds until agent i's score falls strictly below the cut-off score, Zc. As in the E M G the agent is then killed-off and replaced by a new agent with an initial score of zero and redistributed values of both OJJ and Pi. These redistributions are uncorrected random variables uniformly distributed around the old values of Ui and pi with radii Ra G [0,1] and Rp € [0,1], respectively. 41 3.3 Theory of the V A E M G Like the standard E M G , the V A E M G is a super-martingale for all values of / < 0. Since each agent will eventually die (for \ZC\ < oo) that agent's performance is best measured by his lifetime. In the V A E M G , however, there are two evolutionary variables, p^ and OJJ, and since correlations between these variables may emerge, the study will focus on the generalized average lifetime distribution, L(ojj,pj). The goal of this section is to derive an analytical expression for L(a,p) (the subscript i has been omitted since the theory should be general for any agent within the group). The derivation will mirror that of Lo et al. [71] and will use a mean-field formulation. Because agents within the V A E M G only interact through its global history, the system should be ammenable to a such a mean-field treatment. 3.3.1 Derivation of an expression for the average life-time distribution, L(a,p) I will begin by defining the expected payoff to an agent in states a and p as (n a p ) . Since (n Q p ) < 0 it is expected that approximately l /| (n a p )| rounds will be required for the agent to lose 1 point. Because the agent is killed-off when his score, S, reaches Zc < 0 points, his average lifetime, L(a,p), should be given by L(a,p) = 42 The above equation is actually only valid for the standard E M G where an agent's score can only change by an integer amount and therefore (Sc(a,p)) = Zc is a constant (where (Sc(a,p)} is, on average, the agent's score when he is killed-off). In the V A E M G , if an agent is inactive during a round then his score will change by a non-integer amount, I E [—1,0] and therefore (Sc(a,p)) will most likely not equal Zc. The average number of rounds required to reach the cut-off score (ie. the average lifetime) will therefore be slightly modified in the V A E M G to During each round of the V A E M G , there are 3 possible moves for an agent in states a and p: 1. He can choose to be inactive. This choice occurs with probability 1 — a and results in a payoff of I points. 2. He can choose to participate in the round and then choose the minority option. This occurs with probability aPw(a,p) (where Pw(a,p) is the probability that an agent with activity level a and gene value p chooses the winning option given that he is active) and results in a payoff of +1. 3. He can choose to play and then choose the majority option. This occurs with probability a ( l — Pw(a,p)) and results in a payoff of -1 point. The expected payoff to the agent is therefore L(a,p) = {Sc(a,p)) (n Q p ) (3.1) (n Q p ) = a{2Pw(a,p) - 1) + 7(1 - a) (3.2) 43 Analytical form for the expected cutoff score, (Sc(a,p)) To derive an expression for (Sc(a,p)) I will examine the actions of only one agent. It will be assumed that this agent never chooses the minority option (ie. the agent is always inactive or active but chooses the majority position) thereby neglecting the effect of positive score changes. This assumption should cause no loss of generality since positive changes are always integer valued and therefore do not contribute to the variability in (Sc(a,p)). The expected payoff of this agent is therefore aPw(a,p) + (1 — a)I and so, on average, the number of rounds it will take for his score to drop to Zc will be Zc/(aPw(a,p) + (1 - at)I)- Since the agent will not be killed-off until S < Zc this number of rounds must be rounded-up to the nearest integer (a fractional number of rounds is not allowed). The expectation of the agent's score, {Sc(oi,p)), at this point will simply be this number of rounds multiplied by his expected payoff, or (Sc{a,p)) = (aPw(a,p) + (1 - a)I) x ceil where ceil(x) is defined as the smallest integer no smaller than x. 3.3.2 Derivation of an expression for Pw(a,p) The derivation of L(a,p) is now reduced to finding an expression for Pw(a,p). Towards this goal, I will focus on some round, t » 1, when the system has reached its equilibrium state. At this point I will consider the second step of the round in which there are Na active agents and, specifically, the actions of the ith agent in a pool of Na - 1 background agents. 44 aPw(a,p) + (1 - a)I_ (3.3) I define FNa (n) as the probability that n of the total iV a active agents choose the option (buy or sell) predicted by the trend strategy. Also let G1Na_l{n) be the probability that n agents, chosen from the background pool, choose the option predicted by the trend strategy. FNa(n) and G V 0 - i ( n ) a r e then related by FNa(n) = PiG*Na_1(n - 1) + (1 - Pl)GiNa_1(n) (3.4) which simply states that there are two ways in which n agents choose the trend strategey; either if agent i chooses this strategy along with n — 1 of the background agents, or if n of the background agents choose the strategy while agent i chooses the anti-trend strategy. Agent i will choose the winning option if one of the two following situ-ations occur: 1. Either agent i chooses the trend strategy and no more than (Na — 3)/2 of the background agents choose this same option, or 2. Agent i chooses the anti-trend strategy while at least (Na + l ) /2 of the background agents choose the trend strategy The probability that agent i chooses the minority (winning) option, pw(Na), given that he is both active and there are Na active agents is therefore Pw(Na) = P l GlNA-AN) + (1 - ft) E °lNA-M (3-5) Equation (3.5) is z-dependent. To remove this dependence and gener-alize the result for all agents, (3.4) must be rewritten as FNa(n) = G ^ i n ) - px ( ^ ( n ) - G ^ n - 1)) (3.6) 45 and note its boundary conditions FNa(0) = £ ~ PJGN.-M (3.7) FNa(Na) = PiG^iNa - 1) (3.8) Now summing over both sides of (3.6) from n = 1 to n = (Na — 3)/2 and cancelling terms gives (Na-3)/2 (iV„-3)/2 /V - 3 E F " » = E ^ . - i ( « ) + f t ^ . - i ( o ) - ^ . - i ( - V " ) n=l n=l Using the b.c. (3.7) the above equation reduces to (JVa-3)/2 (tf„-3)/2 „ _ o E G J V a _ 1 ( n ) = E (3-9) n=0 n=0 In an analagous manner, (3.6) can be summed from n = (Na + l ) /2 to n = N. After the appropriate cancellations and use of b.c. (3.8) we are left with E G J V a _ 1 ( n ) = E FNA^) - Pi^-d^Y1) (3-10) n=(Na + l)/2 n=(iV0 + l)/2 Substituting (3.9) and (3.10) into (3.5) and using the relation (from (3.4)) PiG^dNa - 3)/2) = FNa((Na - l)/2) - (1 - pi)GlNa_1((Na - l ) /2) gives ^ Na Pv,{Na)=PiY^FNa{n) + (l-Pi) E - 2 P i ( l - ^ G k - i ( ^ l i ) (3-11) 46 n=0 To remove all i-dependence from pw(Na) the third term on the RHS of (3.11) needs to be expressed in terms of F^ 0 (n). Again, rewriting (3.4) we have ftGVxfn - 1) = FNa(n) - (1 - Vi)G%Na^{n) (3.12) Applying (3.12) to itself then results in the relation PtGiNa-,(n - 1) = FNa(n) - ^ [F* B (n + 1) - (1 - Pi)GiNa_1(n + 1)} and after successive applications, we are left with P * G k - i ( " - 1) = - L-*FNa(n + l) + 1 - p ' P F N o ( n + 2) + . . . p - 1 P Na-n FNa(Na) or simply G i r . - i ( n - l ) = ^E ^ i=o P - 1 . P (3.13) Because of the factor [(p - l)/p]j, equation (3.13) is not numerically practical for p « 0 . To correct this problem, we write (3.4) as (1 - Pl)GlNa_l(n) = FNa(n) - plG*Na_l{n - 1) (3.14) After recursively applying Eq. (3.14) to itself as was done with Eq. (3.12) we get G 1 n y 3=0 V 1 3 P - 1 FNa(n-j) (3.15) which is much more usable in the region p « 0. 47 Together, equations (3.11), (3.13), and (3.15) give the probability, pw{Na), that any agent chooses the minority option given that he is both active and there are Na active agents during the round in question. But € {1,3, . . . , N} is variable. To find the total probability that the agent chooses the minority option, it is necessary to perform a weighted sum over all possible states, Na. Towards this goal I will now focus on the first step of the round and, specifically, on the actions of the ith agent within a pool of N — 1 background agents (all N agents are still "active" at this stage). Let QN(NO) be the proba-bility that ther are Na active agents, out of a possible N. Also let ^ ^ ( A ^ —1) be the probability that, excluding the ith agent, there are Na — 1 active back-ground agents during the round. Simply by definition, the probability that agent i selects the minority option given that he is active is then N Pw(a,p) = £ $iv-iW» - l)pw(Na) (3-16) NA=3 In a similar way to Eq (3.4), <dN(Na) and $ l w _ 1 ( N a - 1) are related by QNW = a&^iNa - 1) + (1 - ax)&N_x(Na) (3.17) which states that there can be active agents if agent i is active along with Na - 1 of the background agents, or if agent i is inactive while A^a background agents are active. The expression for Pw(a,p) is conditional on agent i being active; if he is inactive then Pw(a,p) = 0 by definition. The 2 n d term on the RHS of (3.17) assumes that agent i is inactive and it therefore makes no contribution to Pw(a,p). For our purposes then, (3.17) reduces to ^-l(Na-l) = ^eff(Na) (3.18) 48 and so the final, z-independent form of Pw(a,p) is N a 1 N Pv,{a,p) = - T ®N(Na)pw(Na) (3.19) Na=3 3.3.3 Analytical forms for QN(Na) and FNa(n) The final question that must be answerred is how to deal with QN(NQ) and F^a(n). Qualitatively, 0/v(Aa) and F^a(n) can be regarded as the sum of N and Na independent actions, respectively. At this time I will restrict N 3> 1 and assume that Na » 1 also. Since cti,Pi € [0,1] the Central Limit Theo-rem tells us that QN(NO) and FNa(n) should both be approximately gaussian shaped with means, jia and /j,p, and variances, o2a and ap, respectively, given by fia = N f f aP(a,p)dpda a\ = N f J a(l - a)P(a,p)dpda (3.20) J a J p J a J p PP = ^2eN(Na)Na / / Pp(a>P)dPda NA J A J J } °l = E QN{Na)Na f fp(l- p)P(a,p)dpda M JaJp (3.21) NA where P(a,p) is simply the joint frequency distribution of a and p and is given by P ( a , p ) = , r L T { ( a , P L . (3-22) Ja SpL{a,p)dadp when the system is in equilibrium (this will be experimentally verified in a later section). 49 3.3.4 Solving for the analytical form of L(a,p) The derivation is now complete. Together, (3.1-3.3), (3.11), (3.13), (3.15), and (3.19-3.22) form a self-consistent set of equations for L(a,p). Unfortunately this system is implicit and so L(a,p) cannot be solved directly. To solve for L(a,p) I will use the following iterative routine: 1. Assume an initial form for L(a,p). 2. Assume an initial gaussian form (with p, — a2 = N/2) for QN{NO) for use in Eq. (3.22). 3. Calculate pa, pp, o\, and a2 using equations (3.20) and (3.21). 4. Using equations (3.13) and (3.15), calculate pW(NA) from (3.11) 5. Using (3.19) calculate Pw(a,p) 6. Using (3.3) calculate the cutoff score, (Sc(a,p)) 7. Calculate the expected payoff, (nQp), (3.2) and then find a corrected form for L(a,p) using (3.1) 8. Using the new L(a,p), repeat steps 3-6 until the difference between suc-cessive L(a,p)'s falls within some convergence criterion. 50 Chapter 4 Results and Discussion In this chapter I will examine the properties and behaviour of the V A E M G model developed in chapter 3. As with the E M G model of chapter 2, I will focus on results of numerical simulations of the model and then supplement and compare these with the theory developed in section 3.3. The results will focus on two aspects of the model: 1. the performance of an agent when he is able to control his activity level; and 2. how the properties of the E M G model react to a variable level of agent activity. In section 4.1 I will begin the chapter with a short discussion about the details of initializing the routines for generating both the numerical and theoretical results. In section 4.2 I will discuss the transient response of the V A E M G model. In section 4.3 I will then present and discuss results concerned with the activity-related properties of a single agent. In order to focus on this aspect of the model, I will use a version of the V A E M G in which the gene value, p, is non-evolutionary for all agents. As will be explained later, this formulation reduces the complexity of the model and allows us to focus more 51 easily on the activity dependence of an agent's behaviour. Finally, in section 4.4 I will use the full version of the V A E M G model developed in section 3.2 to examine how the addition of a variable activity level affects global properties of the E M G model. 4.1 Setup of numerical routines The data presented in this chapter have been generated from numerical rou-tines. The purpose here is to simply describe some aspects of the setup and any relevant initial conditions of these routines. 4.1.1 Experimental The "experimental" data (data from numerical simulations of the V A E M G model) have been produced using a routine built-up from that used to simulate the E M G in chapter 2. As with those simulations, the initial history bit string, , the trend and anti-trend strategies (5* and respectively), and an agent's activity level, a, € [0,1] were all randomly initialized. In addition, an agent's gene value, Pi E [0,1], was randomly initialized in section 4.4 where the full V A E M G model is simulated. In section 4.3 however, pi is non-evolutionary and so it has been set the same for all agents; the choice of this constant will be discussed further in section 4.3. Simulations of the V A E M G are computationally very large. Whereas the number of rounds, T, used in a simulation of the E M G in chapter 2 was 106, I will show in section 4.3 that T must be much larger (typically T ~ O(10 8 — 52 109)) for the V A E M G model. As a result of this increased computational size, I have used the ran2() random number generator found in Numerical Recipes in C [74]. This random number generator is of the L'Ecuyer type with a Bays-Durham shuffle and, most importantly, has a very long period (> 2 x 1018) which should prevent any serial correlations between rounds. 4.1.2 Theoretical The routine, given in section 3.3.4, for generating the theoretical distribution of average lifetimes has several points that must be discussed. First, the "con-tinuous" variables, a and p (where appropriate), have each been discretized into 100 evenly spaced points over the interval (0,1). Second, an initial average lifetime distribution was required to seed the numerical routine; for simplicity reasons a constant, normalized distribution was used. The sensitivity of the algorithm to this initial distribution was tested using various other forms and, in all cases it was found that the final distribution was independent of this choice. The precision of the final distribution is dependent on the convergence criterion used in the algorithm. Let Li(a,p) be the ith approximation to the actual theoretical distribution. We define to be a measure of the difference between the i — 1th and ith iterations of the routine (and the ^2ap is a sum over all possible, discrete combinations of a and p). When the convergence criterion is set to be 7; < 10~6 there is no visible 53 li = change in the distribution between steps i — 1 and i. For safety sake I will use li < 10~ 1 0 as the convergence criterion throughout this study. 4.2 Transient response of the V A E M G While I will mainly focus on the equilibrium properties of the V A E M G model, I would also like to characterize its approach to this state. With the E M G the approach to the dynamic equilibrium was quantified using \nd\, the magnitude of the difference between the number of agents who selected "buy" and who se-lected "sell". With the V A E M G there are three possible outcomes each round (buy, sell, and hold) and so two quantities are needed: the magnitude of the difference between the number of buyers and sellers, \na\; and the magnitude of the difference between the number of active and inactive agents, — N\. Time series for each of these quantities are shown in figure 4.1. The transient responses of both \nj\ and \2Na — N\ appear to have died out completely by t « 1000 rounds. There are two interesting points from figure 4.1. First, from figure 2.2 we know that, within the E M G framework, \nd\ will not display a rapidly decaying transient if pi is randomly initialized; figure 4.1 re-confirms this result for the V A E M G . What is more interesting, however, is the fact that \2Na — N\ does rapidly evolve out of this randomly initialized state with a characteristic time ~ 0(|ZC|). As will hopefully become obvious later, this spontaneous evolution of \2Na — N\ out of a state in which OJJ is randomly distributed (in comparison to the evolution of |n^| into a state which behaves similarly to one 54 180 round number - 1 Figure 4.1: Time series for \nd\ and |2JVa - N\ for N = 301, M = 3, and / = —0.5 with both pi and ctj evolutionary and randomly initialized. in which pi is randomly distributed) is a result of an agent's performance being asymmetrically dependent on OJJ. Second, once |27V0 — N\ has evolved into its dynamic equilibrium level, it appears to undergo much larger fluctuations (over a timescale of ~ 0(1000) rounds) than \rid\; at this time I have no good explanation for this behaviour. 55 4.3 Individual agent behaviour in the V A E M G 4.3.1 Introduction This section will focus on the behaviour of a single, arbitrary agent within the framework of the V A E M G model. I will examine three main properties of the model: 1. the performance of an agent with a variable level of activity; 2. an agent's optimal performance for a given set of model conditions; and 3. how this activity dependent behaviour varies with certain model parameters such as the number of agents, N, the inflation rate, / , and the bankruptcy level, Zc. How an agent's activity level affects his performance is, presently, the main concern. While the V A E M G is capable of, and, in fact, has been specifi-cally designed to demonstrate this effect, the model includes unnecessary com-plexity that will detract from the activity dependence that is the present focus. For this section I will use a simplified version of the V A E M G model that will provide a more direct window into the activity dependent properties of an individual agent. This model is identical to the full V A E M G with the exception that the gene value, Pi, is now the same for all agents (pi = p), is fixed at p = 1/2, and is non-evolutionary (ie. Rv = 0). The specific simplifications were chosen for two reasons. First, they reduce the complexity of the theory that was developed in section 3.3. While it might seem intuitive that eliminating the evolutionarity of p will simplify the associated theory, of almost equal importance in this simplification is the choice of p = 1/2; this point will be discussed further in section 4.3.3. Second, 56 they simplify the dynamics of the game. In the second step of a round, each agent's strategy preference has been made irrelevant since he now has a p = 1/2 probability of choosing either to buy or sell. This second step is now equivalent to Na agents each nipping an unbiased coin to determine their move for the round. By removing the evolutionarity of p we are eliminating the system's' dependence on its history so that we can more easily focus on the its a dependence. 4.3.2 Experimental results Distribution of average lifetimes, L(a) Here I will focus on the 1-D distribution of average lifetimes, L(a), over the phase space of the V A E M G model. Figure 4.2 shows L(a) for several values of the penalty, I. We can see from the figure that the system displays the proper behaviour at a = 0, namely that L(0) = Zc/I. Also, the distributions are all monotonic; for small values of |7|, ^ < 0, while for larger values of |7|, ^ > 0. When inflation is small an agent will do better by playing less frequently, but, eventually, as inflation is increased the agent will be better off by playing more. This dependence of L(a) on I will be discussed much further in later sections. A n immediate question that arises is whether or not the V A E M G reaches an equilibrium state and produces distributions that are stable with T. The model has been simulated for T — 106 -» 109 with various I e [-1,0) and, ignoring fluctuations, in all cases L(a) is found to be stable in time (agreeing 57 2000 v i l l i 1800 -1600 -1400 -~t 1200 -CD E terage lifeti 00 O o o o o _ < 600 . — 400 200 0 1 1 1 1 1 1 1 1 1 — 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a Figure 4.2: Average lifetime distributions for I = -0.05 (—), -0.10 (---), -0.25 (— - ) , and -0.75 (• • •) with N = 101, Zc = -100, and Ra = 1. with the conclusions from above). With 1 = 0 this is not the case. Figure 4.3 shows L(a) at I = 0 normalized by LQ(Q) (defined to be L(a) for T = 106) for several values of T. As T increases, the distribution becomes more heavily weighted around a — 0. The reason for this result is that the state a = 0 acts as a sink in the model. When an agent dies and is replaced by one with a = 0, this new agent will live forever. As T increases, the probability of an agent attaining the a — 0 state will increase; eventually (if T is large enough) all agents will become "trapped" in this sink and L(a) —> 6(a). Figure 4.4 shows the variation in the distribution of average lifetimes with the number of agents, N. For all L(a) increases with increasing 58 1.5 1H 0 51 i i i i i i i i i i ' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a Figure 4.3: Normalized average lifetime distributions, L(a)/L0(a), for T = 2 x 107,1 x 108 and 1 x 109 when the penalty, 7 — 0. Other parameters are N = 101, Zc = -100, and Ra = 1. N, or, in other words, as the number of agents within the group increases, so too does the performance of each agent. To explain this result, time series of the number of active agents, Na, have been plotted for several values of N in Figure 4.5. The second step of a round within the V A E M G is identical to the E M G with Na agents. We know from chapter 2 that an agent's performance in the E M G increases with the number of agents. Since A ^ increases with N (from figure 4.5), it is then clear why the performance also increases with N in the V A E M G . In chapter 2 we saw that L(p) ~ \ZC\. Since the the average point loss 59 2200 2000 1800 1600 1400 1200 1000k^ 800 600 400 Figure 4.4: Average lifetime distributions for ^=31 (---), 101 (—), 301 ( ), and 501 (• • •) when the penalty, / is -0.1. Other parameters are Zc — —100 and Ra = 1. per round should be independent of the cutoff score, Zc, it is reasonable to expect the same behaviour in the V A E M G . Figure 4.6 shows the dependence of average lifetime distribution on this "bankruptcy level", Zc. While the scaling relation, L(a) ~ \ZC\, holds for large \ZC\, it breaks down for \ZC\ < 100. Specifically, for a ^ 0, an agent's average lifetime is larger than we expect in this small \ZC\ region. • The reason for this behaviour is due to the agent's inability to reach an equilibrium state. On average, we expect the ratio of number of active/inactive rounds to be approximately a/(I — a) for each agent. When \ZC\ is large, an agent's lifetime is large and this ratio is allowed to 60 time (t) Figure 4.5: Time evolution of Na for N = 31,51, and 101 for I = -0 .1 . The time series have been averaged over a window of 10000 rounds. approach its equilibrium value, but when \ZC\ is small this will not usually happen. In this case, the rounds when an agent is inactive (and therefore loses only a small amount) become statistically more significant than they otherwise would be. Since the equilibrium behaviour of the V A E M G is the present focus I have used Zc = —100 for all the simulations in this chapter. Finally, simulations of the V A E M G have also been performed for various Ra € (0,1]. For all values of this redistribution radius no variation was found in L(a). 61 10.4 10 .2h Figure 4.6: Average lifetime distributions for Zc=-10 (—), -25 (---), -100 ( ), and -200 (• • •) with I = -0.1 and N = 101. Optimal performance characteristics In chapter 2, it was shown that an agent's optimal strategy is to either always or never follow the trend strategy, if the agent follows this rule then his average point loss per round will be minimized and he will survive the longest. The reason that the inactivity mechanism of the V A E M G has been modeled after the strategy selection mechanism of the E M G is so that we can determine an agent's optimal behaviour as a function of his activity level. To facilitate this study I will define a* to be the activity level that results in an agent's optimal performance - ie. his maximal lifetime. For example, from figure 4.2, 62 a. -0.05 0 and so being inactive on every round will maximize the agent's lifetime. Motivated by the results of figure 4.2, I have run simulations of the V A E M G with N = 101 for a range of I. From each distribution, L(a), the optimal activity level, a*, has been determined and the results have been plotted vs. I in figure 4.7 below. Although not shown, a* = 0 V|J| < 0.1 and 0.125 Figure 4.7: Dependence of the optimal activity level, a*, on the inflation rate, \I\ for N = 101. a* = 1 V|/| > 0.125. These results show that there is a sharp transition from a pure strategy of optimal inactivity to the other pure strategy of optimal activity at some critical value of |/*| « 0.1125. For |/| < |/*| it is optimal for an agent to remain inactive for every round of the game. If the agent participates in a 63 round then there is a possibility that he will choose the minority option and increase his score, but, on average, his expected payoff will be less than if he had chosen not to play in the first place. As the rate of inflation is increased beyond |/*| the average loss being active will become smaller than / and so always playing the game will become the agent's optimal strategy. Taking a* to be the order parameter of the system suggests, due to the discontinuity in a*, that the active/inactive transition is first order. To explore this point further, figure 4.8 shows several distributions, L(a), for |/| fa Ignoring fluctuations, L(a>) appears to be very linear near |/*| which 89301 1 1 1 1 1 1 1 1 r a Figure 4.8: Average lifetime distributions, L(a), near the active/inactive phase transition, |/*|, for N = 101. confirms the first order nature of the transition. Furthermore, the linearity of 64 L(a) for \I\ « |/*| suggests that there is no preferred level of activity at the active/inactive transition; on average, all values of a will result in an equal lifetime at |J| = |J*|. The distributions in figure 4.8 were run with T = 109. This was the minimal number of rounds required to reduce the size of the fluctuations in L(a) to the level where its maximum is distinguishable for AI = 0.0005; this distinguishability is the reason why T must be so much larger for the V A E M G than for the E M G . A note should be made about the nature of the optimality of a*. For \I\ < \I*\, a* is a true global optimum of this simplified (p = 1/2) version of the V A E M G . Regardless of the actions of the other N — 1 agents in the group, when |/| < |/*| an agent's best strategy is to always remain inactive. When |/| > 1 h o w e v e r , the strategy of a = 1 (always play) is only optimal if the other agents play as they normally would. If a group of the background agents collude and decide to not play for a round then they can change the expected payoff of playing so that it is no longer the optimal strategy. The simplest example of this is the extreme case when \I\ > |/*| but where all of the other N — 1 agents "decide" not to play. Here the "optimal" strategy for the single agent under consideration is still to always play, but if he follows this strategy his expected payoff is —1 (a minimum) since he is guaranteed to always be in the majority group. This optimal strategy of activity is therefore not global since it is inherently dependent on the decisions of the other agents. It will be useful to characterize the active/inactive phase transition in terms of the number of agents, N. Figure 4.9 shows plots of a* vs. |/*| for 65 various N. It is clear that the position of the transition, \I*\, decreases with 11 . . | . . . . 1 — . _ . _ i 1 — i - 1 1 1 1 j 0.9 - i • i 0.8 - • ' 0.7 - I 0.6 - 1 " 8 0.5 - • , 0.4 - ' l 0-3 - ' 0.2 - l 0.1 - 1 0 ' ' 1 •+ • -• ' - ' 1 1 fc= <=* 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 HI Figure 4.9: Active/inactive phase transitions for N = 31(—), 101(—), 301 (— - ) , and 1001(- • •)• increasing N. This result indicates that as the number of agents increases, the inflation rate that should "force" an agent into a state of activity decreases, or, in other words, an agent's optimal strategy is to take more risk as the size of the group increases. 4.3.3 Theoretical results It was stated above that the structure of the simplified V A E M G used in this section reduces the complexity of the associated theory. Specifically, there are 66 two main simplifications that result from the choice of pi = p = 1/2. F irst , al l functions of both a and p (L(a,p), (Sc(a,p)), and Pw(a,p)) now reduce to functions of a alone. A s a result, the integration over p in E q . (3.20) and (3.22) disappears and E q . (3.21) reduces to To discuss the second simplification, I w i l l restate E q . (3.11) here. ^ q - 1 N 2 "a PW(Na)=PiJ2FN^ + ( l - P J E ^ ( ^ - 2 ^ ( 1 - ^ ) ^ - 1 n=0 n - A r « + 1 i V a - 1 2 (4.2) Recall that FNa (n) is the probability that n agents follow the trend strategy. Because of the choice of px = 1/2 and the fact that 2^^=o-^a(n) = 1> E q . (4.2) reduces to Pw(Na) = \ - { ^ Y 1 ) (4-3) Since there is now no history dependence of the agents' strategy preferences, GlNa_l(n) also takes on a simplified form. The probability that n background agents choose the trend strategy is now simply the sum of Na independent (resulting from the lack of a history dependence on p) decisions where n agents choose one of two possible states and Na — n agents choose the other. This probability can therefore be written as QI - i j v a — ( N a \ (4-4) 67 and so Eq. (4.3) becomes pw(Na) = \- 1 2 2N°+1\?^F1 (4.5) The first term of Eq. (4.5) is simply an agent's probability of obtaining either of the two options when flipping a coin while the second is a finite size effect that reduces this probability given the games minority rule. The remainder of the theory is the same as was described in chapter 3. Figure 4.10 shows a comparison of the experimental results of figure 4.2 with the theory of the simplified V A E M G . Agreement between the theory 2000 0.016 (a) (b) Figure 4.10: (a) Average lifetime distribution, L(a), for / = -0.05 (o), -0.1 (A) , -0.25 (•), and -0.75 (o) with N = 101. The solid lines are the respec-tive theoretical distributions, (b) The relative error, 5(a) = (Lsimuiation(ct) -Ltheory(oi))lLsimulation(u), of the theoretical distributions in (a). and simulations is very good. Defining the relative error, 5(a), of the the theoretical distribution as L simulation^) ~~ Ltheory(a) 5(a) L simulation (4.6) 68 it can be seen from figure 4.10 (b) that 5(a) < 2% and much less than this in the majority of cases. Ltheory(a) have been generated over the model's entire phase space (defined by N, I, and Zc) and the results are in similar agreement as those above. Figure 4.10.b does indicate that 5(a) is, in general, an increasing function of a and so is largest at a = 1. Since the transition of a* is most dependent on the relative height of L(a) at its endpoints, it is worth examining how well the theory predicts the position of this transition. Figure 4.11 shows a comparison between the "experimental" active/ inactive phase transitions from section 4.3.2 and the theory. Again, the theory 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 Figure 4.11: Comparison of "experimental" phase transition plots (—) (from figure 4.9) with theory (—) for N = 31, 101, 301, and 1001. is in good agreement with the simulations and, moreover, becomes exact in 69 the thermodynamic limit. The discrepancy between theory and simulation is puzzling. Because of the simplified version of the model used in this section I am confident that Eq. (4.5) is exact. I am therefore left with the Gaussian assumption for G A r (A r a ) as the only remaining source of error. To test this assumption QN(NO) has been measured from the simulated data and the results are shown in figure 4.12 below. The parabola-shaped curves in figure 4.12 demonstrate that, in (a) N=31 (b) N=101 Figure 4.12: eN(NA) distributions for N — 31 (a) and N = 101 (b) for I = -0.01(o), -0.02 (A) , -0.05 (•), -0.1 (©), and -0.2 (y)-most cases, the Gaussian assumption for Gjv(iV0) appears justified. The only occasion when the assumption fails is for large N and small |ij where the large NA tail decays more slowly than a Gaussian. Since the disagreement between theory and simulation is most significant for small N (where the position of the transition, |/*|, is largest) this failure in the assumption of a Gaussian ©;v(-Na) is not able to account for the discrepancy. This discrepancy, however, raises an interesting point about possible 70 emergent behaviour within the V A E M G model. It was mentioned above that an agent's optimal level of risk increases as N increases; in figure 4.11 |/| can therefore be thought of as a measure of this risk. What is interesting is that, for a given number of agents, N, an agent within the simulation acts as though he were less risk averse than a "theoretical" agent (\I?heory\ < \I*simuiation\)• l t should be noted that not only is a* the optimal activity level for an agent, but it is also the agent's most likely activity level when the game is in equilibrium. Since the theory of the V A E M G has been developed using a purely mean-field approach, the more risk-preferring behaviour of an agent has possibly emerged through some more subtle mechanism generating correlations between agents other than the simple "mean-field" global history of the system. 4.4 Global behaviour of the V A E M G In this section I will focus on the general V A E M G and concentrate on two results: 1. the behaviour of an agent within this generalized model; and 2. how the properties of the E M G are affected when the agents are given access to a third possible state. Due to the added complexity introduced by evolutionarity in p I will not attempt to characterize the model over its entire phase space. The purpose here will be to simply present a few selected results of the full V A E M G model that demonstrate the model's behaviour with respect to the above two categories. Figure 4.13 shows the distribution of average lifetimes, L(a,p), gen-erated by numerical simulation and the relative error of the corresponding 71 theoretical distribution. The distribution demonstrates results that are con-(a) (b) Figure 4.13: (a) The distribution of average lifetimes, L(a,p), for N = 101, I = -0.05, M = 3, Zc = -100, and Ra = Rp = 2. (b) Relative error of the corresponding theoretical distribution. sistent with those that have already been shown. At a = 1, when an agent is always active, the distribution properly reduces to the bimodal distribution seen in figure 2.4. Also, although it is not easily seen in figure 4.13, at p = 1/2 the same convex, increasing nature of L(a, 1/2) (for |/| < |/*|) that has been shown in section 4.3.2 is seen. Theoretical distributions have been generated for a wide range of the model's phase space and agreement with the simulated data has been found to be very good in all cases (an example of which we have shown in figure 4.13 (b)). Because of this agreement, the discussion of the theoretical results in section 4.3.3 holds here as well and so no further mention will be made of the results of the theory. A noticeable feature of figure 4.13 (a) is the apparent symmetry of L(a, p) with respect to p. To quantify this symmetry I will define a symmetry 72 measure, r)a, for a fixed activity level, a, as £ p < l / 2 L(a'P) - E p > l / 2 L(a>p) (4.7) A plot of r]a for / = -0.05 is shown in figure 4.14. In chapter 2 we found that x 10 -| r - i r T r Figure 4.14: Measure of the p-symmetry in L(a,p) for the distribution shown in figure 4.13 (a). an agent's optimal strategy is symmetric about p = 1/2 in the E M G . Ignoring fluctuations, this symmetry has been preserved in the V A E M G . With the exception of a = 0 in which an agent's strategy preference, p, is irrelevant, for all a, p = 0 and p = 1 are optimal strategies. Furthermore, tossing a coin (p = 1/2) to determine whether to buy or sell is the worst possible strategy 73 for any activity level, a > 0. Another noticeable feature of figure 4.13 is the optimal state for an agent in the full V A E M G . When evolutionarity in p is permitted, we can see that the system is able to access states that greatly outperform those when p = 1/2. To demonstrate the /-dependence of the state (a,p), figure 4.15 shows L(a,p) for several values of / . Together with figure 4.13 (a) these distributions (a) (b) Figure 4.15: The distribution of average lifetimes, L(a,p), for I = 0.0(a) and I = -0.013(b). Other parameters were N = 101, ZC = - 1 0 0 , M = 3, and RQ — ^ ~^p — ^ • demonstrate three interesting characteristics of the V A E M G ' s opt imal state (a* ,p*) . 1. |/*| < |ip* = 1/ 2 l- When an agent's gene value is able to evolve in response to the history of the game he should take more risk. For N = 101 the position of the phase transition has been determined to be |7*| = 0.0141 ± 0 . 0 0 0 1 . 2. For |/| < |/*| (see figure 4.13(a)) both (a*,p*) = (1,0) and (1,1) are 74 equally optimal, while for |7| > |7*| (see figure 4.15(a)) all values of p are equally optimal (ignoring fluctuations). 3. For |7| = |7*| (see figure 4.15(b)), L(a,p) is a strictly convex function and so only the activities, a = 0 and a = 1 are optimal at this rate of inflation. This is in contrast to the p = 1/2 case where the distribution is a-independent at the transition and so all values of a are equally optimal. There is another subtle, yet interesting property of the V A E M G that can be seen in the distributions of figure 4.15. Examining the optimal performance, L(l,p*), of an agent in the a = l state shows that L(l,p*) is dependent on 7. A more detailed plot of this relationship is shown in figure 4.16. Because the penalty, 7, has no direct effect on the performance of an agent who is always active, to first approximation this result seems counterintuitive. But even though this agent is in the state a = 1 there will always be other agents with a < 1 and so, on average, Na < N. Since an agent's performance increases with the number of agents playing the game (from figure 2.5 in chapter 2) and since 7Va increases with |7| it is therefore reasonable to expect that L(l,p*) to increase with |7|. Figure 4.16 shows this property for 0 < |7| < 0.1, but for |7| > 0.1 the agent's optimal performance decreases with |7|. This large |7| behaviour is contrary to the above reasoning and may be the result of some more subtle correlation between a and p. 75 90001 1 1 1 1 1 1 r Q 5000 E § 4 0 0 0 30001 2000 -n 1000' ' 1 1 1 1 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 III Figure 4.16: Optimal performance, L(l,p*), of an purely active agent vs. penalty, I. Other parameters are N = 101, Zc = —100, M = 3, and Ra = Rp = 1/2. 76 Chapter 5 Conclusions 5.1 Summary of results In this paper I have developed an extension of the Evolutionary Minor i ty Game ( E M G ) , called the Variable A c t i v i t y Evolutionary Minor i ty Game ( V A E M G ) , in which agents have the ability to sit-out on various rounds. The goal has been to characterize the model in terms of its parameters and determine how agent activity affects affects the behaviour of the game. The original E M G model was designed to investigate how a group of agents compete for a l imited resource when governed by supply-and-demand and a minority rule, both of which are believed to play important roles in the functioning of a financial market. Agent inactivity was the focus of this work for two main reasons. First , "forced" activity is one of the most unrealistic assumptions of the Minor i ty Game. Inclusion of a variable activity level w i l l serve to answer the question of how inactivity affects the specific structure of the E M G and w i l l move us one step closer to the development of a more realistic 77 market model. Second, from a purely academic standpoint, it is interesting to wonder how the inclusion of an asymmetric third state affects the dynamics of the self-organizing complex system formed by the E M G . The structure of the inactivity mechanism in the V A E M G is intention-ally simplistic. Work on the E M G has demonstrated how an agent's perfor-mance depends on his strategy preference, p; this preference, or gene value, is simply the probability that the agent will follow the trend in the game. In order to determine a similar relationship between the agent's performance and his activity level, the inactivity mechanism has been modeled in analogy with the gene value of the E M G . Finally, in order to retain the super-martingale nature of the E M G (ie. to prevent the trivial situation where an agent never participates, but lives forever), a penalty, which can be thought of as a form of inflation, has also been included in the model. In order to focus on the activity dependence of an agent's performance, I first concentrated on a simplified version of the V A E M G . In this case each agent's gene value was fixed at p = 1/2 which was demonstrated to remove the system's history dependence on p to more easily focus on its activity-level dependence. Many of the results of this simplified V A E M G can be understood di-rectly in terms of the E M G . First, it has been found that the dynamics of the game are independent of an agent's memory size, M, and the redistribu-tion radius, Ra, of his activity level. Also, with the exception of the special case, 1 = 0, where the a — 0 state acts as a sink to the distribution and L(a) —> 8(a) as T —> oo, an agent's behaviour has been found to be indepen-78 dent of the length of the game, T. However, it has been found that L(a) does not scale with the bankruptcy-level, Zc, for all Zc as with the E M G . For small \ZC\ (\ZC\ <~ 100) an agent tends to lose less per round than when \ZC\ is large. This counter-intuitive result is caused by the non-integer nature of the penalty, J , and has been ex-plained in terms of the system's inability to reach a fully equilibrated state when \ZC\ is small. The most significant result of the simplified V A E M G was that an agent is found to perform better (live longer) by playing more often as the inflation (measured by \I\) increases. Specifically, a first order transition from a phase of optimal inactivity to a phase of optimal activity has been found as \I\ is increased above some critical value, |/*|. Finally, the position of this transition has been found to be strongly dependent on the number of agents within the group; specifically, —» 0 in the thermodynamic limit. Next, I studied the general formulation of the V A E M G in which both an agent's activity level and gene value were allowed to evolve in response to the history of the game; the purpose being to examine the global properties of the model and how the dynamics of the E M G are affected when the agents are allowed access to a third state. Several interesting results have been observed with this model. First, it has been found that the V A E M G retains the symmetry in the gene value, p, as well as the characteristic bimodal dependence of an agent's performance on p. This result tells us that, for any given activity level, a, p = 0 and p = 1 are always an agent's equally optimal strategies. Furthermore, the optimal states 79 (a*,p*) = (1,0) and (1,1) tend to dominate over all other a ^ 1 states of the system even for values of the penalty much lower than the transition, |/*= 1/2I> of the simplified V A E M G . This result, that \I*eneral\ < l^i^l) means that an agent's optimal strategy is to be much less risk averse when he is allowed to modify his strategy preference based on the history of the game. Finally, it has been found that the optimal performance of an agent who is always active, L ( l , p * ) , is dependent on the value of / . This somewhat counterintuitive result has been explained in the region of small |/| where L(l,p*) increases with |/| using the empirical facts that the number of active agents, Na, increases with |/| and an agent's performance increases with Na. For larger however, it has been found that L{l,p*) decreases with |/| and it is believed that this is a result of some subtle correlation between an agent's activity level, a, and his gene value, p. Finally, I have also developed an analytical theory of the V A E M G model based upon a mean-field approach. For both versions of the model (simplified and full) and for a large region of the system's phase space, the theory has been found to approximate the simulations very well and appears to become exact in the thermodynamic limit. Furthermore, the discrepancy at finite N has been explained in terms of an agent within the simulation taking on more risk than what is predicted by theory. Since the theory has been based on mean-field assumptions, this discrepancy indicates a possible emergence of subtle inter-agent correlations within the model for finite N. 80 5.2 Future directions From its inception, the Minority Game was designed with a very simple struc-ture to provide a solid base upon which to build a tractable model of a financial market. This paper has extended the basic E M G , but it is only one small step and there is still a lot of work left to be done. Within the context of this paper, future work on the Minority Game will focus on two main areas. First, the structure of the inactivity mechanism is admittedly very simplistic in the V A E M G and, most likely, too simplistic to be used in a real-world model. It will be interesting study the V A E M G using inactivity rules that more closely resemble those found in a real market. For instance, while an agent's activity level, a, in the present model is dependent on the history of his performance, this dependence is only very weak; a much more realistic mechanism would incorporate specific rules based on past ex-perience in the agent's decision making process. It will be interesting to see whether a more realistic mechanism is necessary to generate some of the styl-ized facts of a real market and, if so, what effect this will have on the global properties of the model. Finally, as was stated above, many simplifying assumptions of the E M G are unrealistic and need modification before there is hope that the model can be used to make real-world predictions. Along with agent inactivity, work also needs to be done in developing realistic mechanisms for such things as the payoff structure, market maker, and diversity of the agent population. 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