On the Emergence of Attitudes towards Risk Some Simulation Results

Transcription

1 On the Emergence of Attitudes towards Risk Some Simulation Results Ste en Huck Wieland Müller Martin Strobel Humboldt University September 28, 1999 Abstract By means of computer simulations we investigate the emergence of preferences towards risk in the context of 2-by-2 games. Preferences are learnt in the following sense: Based on current preferences individuals play an equilibrium of the underlying game. This determines monetary success and preferences yielding higher monetary success are more likely to be copied by individuals than others. We nd that individuals learn to be risk averse though the degree of risk aversion which is developed varies with some parameters of the setup and may vary over time. 1. Introduction Most economic models of learning studyhow individuals adapt behavior over time. These models di er mainly in the rules according to which individuals determine to act and these rules are usually assumed to be stable. But, in general, learning is far more complicated than that since individuals also learnonthe level of rules. In the present study we introduce an approach to this kind of learning. We will focus on a rule by which individuals evaluate the consequences of their behavior, a rule specifying how to value outcomes of interaction. In economic terms such rules are usually called preferences and, in fact, we will focus on how preferences may be learned over time. For economists it has long been taboo to analyze (or even mention) the possibility of preference changes over time. Their point of view has been epitomized by Stigler and Becker s (1977) credo de gustibus non est disputandum. Today the Department of Economics, Institute for Economic Theory III, Spandauer Str. 1, Berlin, Germany, Fax , huck..., wmueller... or strobel@wiwi.hu berlin.de.

2 Latin phrase may still be true with respect to chatter at social events but no longer with respect to the discourse in social sciences: Endogenous preferences have become a serious and well established topic on the agenda of economic research with a variety of authors having contributed to the new eld which is not yet dominated by one methodology. 1 Preferences may change over time due to various processes: Biological evolution may select amongst types of individuals, cultural evolution may do the same. The notion of type selection usually implies that an individual has xed preferences but that the preference pro le of a population may change. Beyond that, preferences may also be adapted on the individual level, namely in a learning process where the object of learning is the rule with which outcomes are evaluated. Formally, all these processes can be modelled in a very coherent way by separately analyzing how individuals behave for given preferences and, based on this, how preferences are selected. 2 In this study we apply this methodology to the issue of preferences towards risk where we reduce the real world to a world in which only 2-by-2 games are played. The basic logic is like this: Based on their current preferences individuals decide upon their behavior, i.e. they play an equilibrium of the game. This equilibrium determines monetary success and when it comes to the adaptation of preferences those preferences yielding higher material success are more likely to be copied by individuals than those yielding lower success. This de nes a dynamic process selecting amongst preferences. We nd that individuals learn to be risk averse though the degree of risk aversion which is developed varies with some parameters of the setup and may vary over time. The remainder of the paper is organized as follows. Section 2 deals with the topic of endogenous preferences towards risk, introduces the general problem we want to study and shows that the problem can hardly be solved analytically. Section 3 will sketch two setups considered in this study while Section 4 introduces the basic concepts of non cooperative game theory which are needed for this study. In Sections 5 and 6 we describe the procedures used in the simulations in more detail and present the results. Finally, we discuss our ndings in Section 7. 1 See, for example, the recent survey by Bowles (1998), who, however, neglects the contributions of evolutionary game theory to the topic which are closely related to our approach. 2 Due to this two fold nature of analyzing endogenous preference Güth and collaborators (see e.g. Güth and Yaari, 1992; Güth and Kliemt, 1994; Güth and Huck, 1997; or Bester and Güth, 1997) call this approach indirect. 2

3 2. The basic problem Individuals often display a negative attitude towards risk, i.e. when confronted with a choice between various gambles they often take those which have a lower expected value than others but at the same time less variance. The rst important contribution to this matter has been Bernoulli s treatment of the so called St. Petersburg paradox. He resolves the paradox which essentially seems paradoxical because it illustrates that most people will refuse to pay even modest amounts of money in exchange for a gamble with an in nite expected value by arguing that the marginal value of wealth is decreasing. And if this is so, it may be perfectly rational to choose a nite amount of money that is certain rather than the gamble yielding an expected in nite amount of money. That people are often risk averse (i.e. that they have a concave utility function over money) is not only suggested bycasual empiricismbut also byhard scienti c data gathered in economic experiments. The study of Mosteller and Nogee (1951) gives early evidence, and the studies conducted thereafter are innumerable. For a survey see Camerer (1995). But if people are risk averse, why are they so? Obviously, this is a question about preference formation and a possible answer could be: Because individuals who are risk averse are more successful than others. And indeed, this is what our contribution suggests. There have been other studies dealing with endogenous riskpreferences, e.g. Robson s (1996a) paper in which he shows that male individuals may have a tendency to seek risks since this is an e cient way to get female mates and, therefore, to replicate. Further references can be found there or in Robson (1996b). All these studies share their focus on one person decision problems. There are two studies which also consider the e ect of interaction. Ely and Yilankaya (1997) show that under certain informational assumptions individuals will basically be risk neutral in the long run. To (1997) shows that in the context of a Knightian model of entrepreneurial risk and investment the population converges to a stationary distribution where both risk averse and risk loving types are present and where only the risk loving types invest. In the study at hand the approach is as follows. Taking the class of 2-by-2 games which has often been claimed as capturing the most basic features of strategic interaction we analyze what e ect playing these games has on attitudes towards risk. More speci cally, we study which preference pro les result when individuals of a population do nothing but play 2-by-2 games. Individuals are randomly matched and the games are drawn from a uniform distribution over all possible games. A pair of individuals plays the game according to its rational solution, i.e. there is no learning process on the level of 3

4 actual behavior. However, there is a selection process on the level of preferences which can be interpreted as learning on the rule level. The principle assumption for this dynamic selection process is that players whose preferences made them successful in monetary terms are more likely to replicate. This replication can be seen as biological but also as a kind of learning process based on the imitation of preferences. In general, such a process requires certain informational assumptions. Individuals must have knowledge about who was/is successful and about which attitude towards risk these successful individuals have. Depending on the available amount of information the resulting learning dynamics may slightly di er. However, in this contribution we do not discuss these intricate details. Instead, we rather illustrate the basic approach by applying simple and easy to implement routines. One could de ne the dynamics formally and try to obtain analytical results. However, the particular problemdealt with in this study proves to be unsolvable. This is mainly due to one aspect of the problem, namely the highly stochastic nature of the model: Each matching requires a randomly chosen game which means that there are eight chance moves picking all possible payo s for the two players. Even if one reduces the amount of randomness in the dynamics by assuming an in nite population which allows taking expected values, the problem remains analytically intractable since the payo functions include eight multiple integrals which cannot be solved. 3 Computational methods, however, can be easily applied. This is illustrated in the next section. 3. The setups A problem like the one described above can be modelled in many di erent ways and depending on assumptions about the population size, the matching scheme, the dynamics, etc. it is possible to obtain markedly di erent results. 4 Therefore, we study two di erent setups to check for the robustness of our results What the setups have in common In both setups interaction between individuals takes place by playing randomly generated 2-by-2 games. (A game-theoretical digression on 2-by-2 games follows after this section.) We assume that all 2-by-2 games are equally likely, i.e. each (monetary) payo is chosen from a uniform distribution over all possible payo s. 3 Engelmann (1998) derives an analytical result for a special class of 2-by-2 games. 4 An example of this phenomenon can be found in a recent paper by Vriend (1998). 4

5 Furthermore, all individuals are equally likely to be matched with each other to play a game. Figure 3.1: Illustration of the Indirect Evolutionary Approach. Individuals are endowed with cardinal utility functions over money. Given the utility functions of two randomly matched players, the game, originally de ned by monetary payo s, can be transformed in a standard game where the payo s represent cardinal utility levels. The standard game is then played according to its game-theoretic solution (which is sometimes obtained by applying an equilibrium selection theory). However, the dynamics selecting amongst preferences are based on monetarypayo s rather than on utilities, i.e. in both setups we assume that learning is a function of material success rather than of sheer happiness. This makes sense when looking at the real world, especially since true material success is usually observable while true happiness is rarely so. An illustration of this basic approach is given in Figure

6 Setup I Setup II Population in nite nite Preferences parameterized class piecewise linear approximation of the whole space Dynamics replicator dynamics monotone but stochastic Mutations no yes Table 3.1: Characteristics of the two setups 3.2. What the setups do not have in common Table 3.1 gives an overview of the main di erences between the two setups. In Setup I we assume that the population is in nite and that there is only a certain class of monotonic (parameterized) preferences which are available to individuals. This class of preferences is further narrowed down by allowing only a nite set of parameter values. On the other hand, in Setup II the population is assumed to be nite while virtually the whole space of (piecewise linear) utility functions is available. 5 The di erent assumptions about population sizes induce further di erences. The most important one relates to the dynamics. In Setup I it is possible to work with standard replicator dynamics. In Setup II this does not work. Having a nite population of xed size requires dynamics which are a bit more tricky if one wants to stick to the monotonicity assumption prescribing that more successful preferences should be more likely to spread than others. (All this will be discussed in further detail in following sections.) The last important di erence which we want to mention here is that given the very large set of possible utility functions in Setup II we have to assume that mutations are possible, while this is not necessary in Setup I where we simply assume that the initial state has full support. 4. A digression on two-player games We consider the evolution of utility functions in the context of two player games. Therefore, all needed concepts are only de ned for this special class of games. De nition 1. A 2 person normal form game G is a 4 tuple G = (S 1 ; S 2 ; p 1 ; p 2 ) which has the following components: 5 Of course, any computer simulation can only deal with nite spaces. Thus, in the end we work with a (large) nite set of utility functions. However, this set is not arbitrarily narrowed down but only through the nite grid induced by computer variables of type double. 6

7 Player 2 s 1 2 s 2 2 Player 1 s 1 1 a 11 ; b 11 a 12 ; b 12 s 2 1 a 21 ; b 21 a 22 ; b 22 Table 4.1: The representation of 2-by-2 game as a bi-matrix. ² S i (i = 1; 2) is a non empty set consisting of the pure strategies of player i. A single element of set S i is denoted by s i. The tuple s = (s 1 ;s 2 ) consisting of a pure strategy s i 2 S i for each player i is called pure strategy vector. We will write S = f(s 1 ;s 2 )js i 2 S i g = S 1 S 2 for the set of all pure strategy vectors. ² p i (i = 1; 2) is a mapping p i : S! R which is the payo function of player position i assigning the payo p i (s) of player i to each strategy vector s 2 S. In this paper we will be concerned with normal form games in which the strategy sets of the two players consist of only two elements. These games can easily be described by using a bi matrix as shown in Table 4.1. Here player 1 chooses a row and player 2 chooses a column. For each combination of actions the matrix shows the associated payo s. Players are allowed to use mixed strategies: De nition 2. A mixed strategy of player i in the normal form game G is a probability distribution ¾ i : S i! [0; 1] assigning to every pure strategy s i 2 S i a probability ¾ i (s i ) such that P s i 2S i ¾ i (s i ) = 1: The set of all mixed strategies of player i will be denoted by 4(S i ): The tuple ¾ = (¾ 1 ;¾ 2 ) consisting of a mixed strategy ¾ i 2 4(S i ) for each player i is called mixed strategy vector. We will write 4(S) = f(¾ 1 ;¾ 2 )j¾ i 2 4(S i )g = 4(S 1 ) 4(S 2 ) for the set of all mixed strategy vectors. If players use mixed strategies the expected payo P i of player i = 1; 2 is given by P i (¾) = P s2s (¾ 1(s 1 )¾ 2 (s 2 ))p i (s): The 4 tuple G M = (4(S 1 ); 4(S 2 ); P 1 ; P 2 ) is referred to as the mixed extension of the normal form game G: The fundamental solution concept of two agents playing a game is given in the following De nition 3. The mixed strategy vector ¾ = (¾ 1 ;¾ 2 ) of the mixed extension of the two player normal form game G is a Nash equilibrium if it holds that P 1 (¾ 1 ;¾ 2 ) P 1(¾ 1 ;¾ 2 ) for all ¾ 1 2 4(S 1 ) and P 2 (¾ 1 ;¾ 2 ) P 2(¾ 1 ;¾ 2) for all ¾ 2 2 4(S 2 ): 7

8 In our simulations we include only generic 2 by 2 games, i.e. 2 by 2 games with the property a 1i 6= a 2i and b i1 6= b i2 for i = 1; 2 (4.1) in order to exclude the possibility of continua of mixed strategy equilibria. 6 Two-by-two games with the property (4.1) can be classi ed according to the number and the nature of their equilibrium points: They either have a unique equilibrium in pure strategies or a unique equilibrium in mixed strategies or two equilibria in pure and one equilibrium in mixed strategies. The latter case poses an equilibrium selection problem. In order to resolve this problem Harsanyi and Selten (1988) suggest the concepts of payo dominance and risk dominance. De nition 4. Let s 1 and s 2 be two Nash equilibria of the 2-by-2 game G. Then s 1 payo -dominates s 2 if for every i = 1; 2 it holds that p i (s 1 ) > p i (s 2 ): De nition 5. Assume that s 1 = (s 1 1 ;s1 2 ) and s2 = (s 2 1 ;s2 2 ) are the two (pure) Nash equilibria of the 2-by-2 game G: De ne the player s deviation losses by r 1 := a 11 a 21 ; r 2 := b 11 b 12 ; t 1 := a 22 a 12 ; t 2 := b 22 b 21 : Then s 1 riskdominates s 2 if r 1 r 2 > t 1 t 2 ; and s 2 risk-dominates s 1 if t 1 t 2 > r 1 r 2 : While the concept of payo dominance incorporates collective rationality the concept of risk dominance exclusively relies on a very re ned notion of individual rationality. It is possible that one of two strict Nash equilibria of a 2-by-2 game payo -dominates the other whereas the latter risk-dominates the rst (see Harsanyi and Selten (1988)), i.e. the two concepts may give opposite recommendations as to which equilibrium should be considered the unique solution of a game. The theory as described in Harsanyi and Selten (1988) gives priority to the payo -dominant equilibrium. In theories published later priority is given to the risk- dominant equilibrium (e.g. Harsanyi (1995) or Selten (1995)). It is not the purpose of this study to enter into the discussion as to which concept is to be given precedence. Therefore, the approach followed here is two fold: For each of the two setups two simulations will be run, one giving priority to payo dominance, the other to risk dominance. 6 Note, that a game not having the regularity property is drawn with probability 0. Hence we reduce the overall class of 2 by 2 games only by a sub class of Lebesgue-measure 0. 8

9 5. Setup I 5.1. Procedure Population: We consider a large (in fact in nite) population of individuals where each individual is characterized by its strictly monotonic increasing utility function u : R +! R over non negative amounts of resources. We justi ably assume that there is a positive upper bound on available resources. Furthermore, cardinal utility functions are unique up to positive a ne transformations. Thus, we can con ne ourselves to consider strictly monotonic increasing utility functions of the kind: u : [0; 1]! [0; 1] with u(0) = 0; u(1) = 1: We assume that each of the individuals is endowed with one of n 2 N possible types of utility functions. We denote by x t = (x t 1 ;:::;xt n) the state of the population at time t where x t i is the population share endowed with utility function of type i = 1; 2;:::;n. Of course, it holds that P n i=1 xt i = 1 for all t: Matching and behavior: Pairs of individuals of the population are randomly matched to play a game which is determined by drawing eight monetary payo s from a uniform distribution over the unit interval. The behavior of individuals in this money game is guided by their preferences over these resources. In order to determine their behavior players do not consider the resource game but what we call the utility game which is derived from the resource game by replacing the resource payo s with the according utility payo s. More precisely, let the game shown in Table 4.1 be the resource game G R in which the numbers a ij ;b ij (i;j = 1; 2) are resource payo s with 0 a ij ;b ij 1. If the utility functions over resources of player 1 and 2 are u and v; respectively, then the utility game G U is given by the matrix Player 2 s 1 2 s 2 2 Player 1 s 1 1 u(a 11 );v(b 11 ) u(a 12 );v(b 12 ) s 2 1 u(a 21 );v(b 21 ) u(a 22 );v(b 22 ) Individuals choose an action according to a (selected) Nash equilibrium in the utility game G U : Games and the Equilibrium Selection Problem: In our simulations we only included 2-by-2 games having the regularity property (4.1). Hence we avoid the problem of continua of mixed equilibria. However, it is still possible that a game has more than one equilibrium. In this case one has to decide which equilibrium is to be considered the unique solution of the game. For reasons discussed above in 9

10 one run we give priority to the payo -dominant equilibrium (if it exists) whereas priority is given to the risk-dominant equilibrium in another run. Between two replications (see below) the population played m 2 N randomly generated games. To each of these m resource games G R j (j = 1;:::;m) in step t of the simulation we can assign the according population-game matrix A t j (j = 1;:::;m). The rows and columns of matrix A t j are the types of utility functions individuals of the population are endowed with. In order to make each game a symmetric one matched individuals are assumed to play once in each of the two player positions. Thus each entry (a t j ) lk (l;k = 1;:::;n) of matrix A t j is the sum (resulting from the two possible position assignments) of the expected resource payo s of an individual of type l playing against an individual of typek: Replication: We assume that preferences yielding higher monetary success are more likely to be copied by others than preferences yielding lower success. Imitation of this kind can give rise to replicator dynamics if it follows certain rules (Björnerstedt and Weibull, 1993). Instead of discussing this here in detail, we directlyassume that the learning process can be modelled with replicator dynamics. Recall that the population state at time t is given by x t = (x t 1 ;:::;xt n) where x t i is the population share endowed with the type of utility function i at time t: For the j-th of the m games an individual of type i receives the expected success e T i At j xt ; where e T i is the transpose of the i-th unit vector. The average success of the population per game is given by x t T A t j x t : The resulting material payo for playing m games per replication step is simply given by the sum of these payo s, i.e. P m j=1 et i At j xt is the success of type i and P m j=1 x t T A t j x t is the average success of the population after playing m games. The dynamics of the population shares is now governed by the usual discrete-time version of the replicator dynamics (see for example Weibull (1995)) x t+1 i = P m j=1 et i At j xt P m j=1 (xt ) T A t j xtxt i; i.e. the growth rate x t+1 i =x t i of type i is the success of type i divided by the average success of the population. Note that the share of type i grows (shrinks) with regard to this dynamics if it earns more (less) than average during the play of the m games. 7 7 In this formulation of the replicator dynamics it must hold that the denominator never vanishes. Note that sincea ij;b ij 0 (i;j = 1;2) this happens if and only if all entries of them basic bi-matrices games are zero a case excluded in the simulations. 10

11 5.2. Results In this subsection we present the results of simulation runs for Setup I. In the simulations reported each individual of the population was endowed with one of nine types of the utility functions u(x) = x where = 1:5 and 2 f 4; 3; 2; 1; 0; 1; 2; 3; 4g as illustrated in Figure 5.1: Note that for < 0, = 0; > 0; respectively, individuals are risk averse, risk neutral, or risk loving. 8 Thus, the utility function with = 4 exhibits the highest degree of risk aversion whereas the utility function with = 4 exhibits the highest degree of risk loving. Between two replications individuals were matched to play one game (m = 1). As we will see in the following subsections we distinguished between dif u(x) x Figure 5.1: The types of utility functions used in the simulations: u(x) = x = 1:5 and 2 f 4; 3; 2; 1; 0; 1; 2; 3; 4g with ferent equilibrium selection concepts and between di erent initial shares (equal, random) at the beginning of the simulation. For each combination we performed 20 runs in order to test for stability of the results. 8 The functionu(x) =x exhibits constant relative risk aversion (or decreasing absolute risk aversion). To see this note thatr R(x;u) := xu 00 (x)=u 0 (x) = 1 ; i.e. the coe cient of relative risk aversion atx;r R(x;u); equals the constant 1 : 11

12 Payo Dominance If in the presence of multiple equilibria payo dominance has priority over risk dominance, then the result is clear cut: Independent of initial conditions all individuals develop maximum risk aversion. Figures 5.2 and 5.3 are typical pictures emerging in the simulations. The result is always the same no matter whether the population initially consists of equal shares of all types (Figure 5.2) or whether initial shares are randomly drawn (Figure 5.3). In Figure 5.2 the type = 4 wins the race right from the start. Moreover, in all 20 runs with equal initial conditions the share of the type = 4 is always greater than :999 after 15; 400 replications. Figure 5.3 shows the result of a run with randomly drawn initial conditions. Here in the beginning only 2:2% of all individuals are of type = 4 whereas 25% of all individuals are of type = 3: As one can see the share of individuals of type = 3 propagate very fast during the rst 2; 000 replications. However, although decreasing from time to time the share of individuals being most risk averse grows quickly, too. And when the lines indicating the population shares of the two most risk averse types cross all other types have virtually disappeared. Therefore, from this point in time one of these lines is the mirror image of the other line, i.e. what one share loses the other share gains. Finally, after 20; 000 replications the population essentially consists only of individuals being of type = 4: Note furthermore that in all 20 runs with randomly drawn initial conditions the share of individuals being of type = 4 is greater than :999 after 27; 425 replications Risk dominance If the risk-dominant equilibrium is given priority in case multiple equilibria exist the following can be said: In all runs (20 runs with equal initial shares, 20 runs with random initial shares) individuals being risk neutral or risk loving disappear very quickly, i.e. all individuals learn to become risk averse. But there is no single type that nally wins the race. Figures 5.4, 5.5, and 5.6 show typical pictures emerging in the course of 100; 000 replications. In Figure 5.4 showing a run starting with equal initial shares, only utility functions with = 4 or = 3; respectively, survive in the long run. But the respective shares go up and down all the time. Since the shares of all other types quickly converge to zero, again the respective trajectories are mirror images of each other. Even after 100; 000 replications there is still a lot of volatility and these two shares 9 All the runs lasted for 50,000 replications, but we report only the rst 25,000 since absolutely nothing changed afterwards. 12

13 1 β= β= β= Replication Figure 5.2: Setup I, run 1 out of 20, equilibrium selection: payo dominance, equal initial shares. may continue to oscillate forever without settling down to the favor of one of the two types. The same seems to be true in Figure 5.5 showing a run with randomly drawn initial conditions. But whereas in these two latter runs all but the two most risk averse utility functions become extinct it also happens that individuals being only slightly riskaverse, i.e. individuals of type = 1 dominate for a long period (see Figure 5.6). Here everything seems to be decided in favor of the type showing only slightly risk aversion: The population share of this type is essentially 1 from replication 22; 400 to replication 72; 000 with a small interlude of = 3 around replication 60,000. But then within about 7,000 replications (from replication 80,000 to replication 87,000) the picture changes drastically: Suddenly the share of individuals of type = 1 (having been on top for quite a long time) goes essentially down to zero such that the population is then dominated by individuals of type = 3: However, as before one cannot be con dent that this is the nal result. Table 5.1 reports the frequencies of having the highest share after 100,000 replications. = 4 is the most frequent outcome. But this can not be seen as the nal result since in many runs heavy and persistent uctuations are observed. All in all, the results are less clear when risk dominance has priority. Although in all runs individuals become risk averse, the degree of risk aversion is undetermined. 13

14 1 β= β= 1 β= 3 β= Replication Figure 5.3: Setup I, run 9 out of 20, equilibrium selection: payo dominance, random initial shares. Initial shares = 4 = 3 = 2 = 1 other equal random Table 5.1: Frequencies of highest share after 100,000 replications when risk dominance is used as the equilibrium selection concept. 6. Setup II 6.1. Procedure Population: In this part we consider a nite population. Each of the n agents is endowed with a randomly generated utility function. For this purpose we take the partition [ 0 ; 1 ;:::; k ] with j = j k (j = 0; 1;:::;k) of the unit interval [0; 1]; i.e. we partition the unit interval into k equidistant subintervals of size 1=k. Next we take one individual and assign to each k + 1 points j a function value f( j ) by a random draw from a uniform distribution over the interval [0; 1]: The points are joined by a polygon. This is done for all individuals who are nally endowed with a utility function f : [0; 1]! [0; 1] over money. But this time we neither 14

15 1 0.8 β= β= 3 β= Replication Figure 5.4: Setup I, run 20 out of 20, equilibrium selection: risk dominance, equal initial shares. assume utility functions to be strictly increasing nor do we impose the normalization f(0) = 0 and f(1) = 1. Matching and Behavior: A pair of individuals is drawn to play a randomly generated game. After that individuals are taken back into the population and the next pair is drawn. Between two replications of the population m 2 N games are played. Since individuals are taken back into the population after being drawn it is possible that one individual plays more than once or never at all. When two individuals are chosen to play a game they are randomly assigned a player position without letting them switch roles. Again individuals decide rationally according to a (selected) Nash equilibrium given their preferences. The individuals enter the replication stage with the average monetary payo received during the games to which they were called upon to act Of course, in the case an individual is never drawn to play a game this individual s average monetary payo is zero. However, due to the actual values of m used in the simulations this is a very unlikely event. 15

16 β= β= Replication Figure 5.5: Setup I, run 16 out of 20, equilibrium selection: risk dominance, random initial shares. Games and the Equilibrium Selection Problem: Again we only include 2- by-2 games having the regularity property (4.1). As in Setup I there is a run in which we give priority to the payo -dominant equilibrium (if it exists) whereas priority is given to the risk-dominant equilibrium in another run. Replication: For the replication we implement a payo -monotone dynamic, i.e. a dynamic that ensures that more successful agents replicate at a higher rate than less successful agents (see for example Weibull, 1995). At the same time we hold the population size xed. More precisely, we implement the following replication procedure: For the replication agents are ranked according to their average monetary success with the individual having the highest average payo getting rank one and so on. In case several agents earn the same payo during the play of m games the respective ranks are assigned randomly. After this the number C i which indicates how often the agent having rank i is replicated is 16

17 1 β= β=0 β= 3 β= Replication Figure 5.6: Setup I, run 17 out of 20, equilibrium selection: risk dominance, random initial shares. determined for every i 2 f1; 2;:::;ng by the following procedure: < i 1 C i = rand@1; min : C X = i 1;n C j A ; where rand(a;b) is a function yielding a random integer in [a;b] if a b and 0 otherwise. C 0 with 2 C 0 n is a parameter denoting the upper bound of copies an agent can have. This procedure ensures a-priori a strictly payo -monotone dynamic while keeping the population size constant. Mutation: In order to search the whole preference space we introduced mutations which can occur during replications. We assume that in case of a mutation at least one of the function values at the grid points j of a utility function is altered. More precisely, let " be the mutation rate, i.e. the probability that a utility function is not exactly imitated. Mutations occur according to the following rule. When copied all function values are altered with a probability. In case of an alteration a new value is randomly drawn from a uniform distribution over the interval 17 j=1

18 [0; 1] of real numbers. For a given mutation rate " the probability (i.e. the probability of a change occurring at a grid point j ) is implicitly given by the equation 1 " = (1 ) k+1 : The left hand side of this equation is the probability that a utility function remains unchanged while being copied, whereas (1 ) is the probability that a single point f( j ) remains unchanged. Thus (1 ) k+1 ; i.e. the probability that none of the function values f( j ) gets changed must be equal to 1 ": 6.2. Results In this section we illustrate the results of simulation runs for Setup II by showing average utility functions f over time. The average function f is given by f( j ) := 1 P n n i=1 f i( j ) where f i is the utility function of individual i = 1;:::;n and j = j k (j = 0; 1;:::;k) are the grid points. In order to provide a measure of the heterogeneity (or homogeneity) of the population we also calculate the average mean standard deviation ¾(f) of the function values f( j ) at a given replication step. Of course, a small value of ¾(f) either means that the shapes of individuals utility functions are similar or (and this more likely) that only a small number of di erent types still exists. Furthermore, for each picture showing the average utility function after a certain replication step we also plot a regression curve using a quadratic speci cation. When running the simulations for Setup II we relied on the following parameters: number of individuals n = 100; grid points j = j k (j = 0; 1;:::;k = 10); upper bound for the number of individuals replicated C 0 = 10; mutation rate " = :01; number of games played between two consecutive replications m = 32; 000: Payo Dominance First we present results for the case in which the payo -dominant equilibrium is given priority. Figure 6.1 shows the population s average utility function after several replications steps. Since the function values are randomly assigned according to independent draws from a uniform distribution over the interval [0; 1] the initial average utility function looks essentially like a straight horizontal line. But already after 100 replications the average utility function of the population is almost monotonically increasing and after 1; 000 replications there is also a slight tendency towards a concave curvature. This e ect becomes stronger with higher numbers of replications (as can be seen by looking at the leading coe cient of the regression result) and is clearly visible to the naked eye after t = 20; 000: 11 This implies that one particular individual played on average 640 games between two replications. 18

19 Figure 6.1: Average utility of the population at time t and with average mean standard deviation ¾ if payo dominance is used as equilibrium selection concept. The vertical axes denote utility, the horizontal axes denote resources. 19

20 In summary, also bearing in mind the other runs conducted for this subsection, it seems fair to say that the dynamic forces at work in this setup again favor individuals who are risk averse Risk Dominance When risk dominance is given priority, our ndings are (as in Setup I) less clear. Figure 6.2 shows the population s average utility function after several replications steps in a run with the same parameters as above. Rather soon the average utility function becomes monotonic, but the curvature is very close to being linear and changes over time. After 1,000 replications there seems to be a slight trend to risk loving utility functions and this trend peaks after 5,000 replications. However, during the next 15,000 replications the average utility function becomes over a rst interval convex, over a second concave and over a third convex. This pattern stabilizes over last 30,000 replication steps in so far as di erent curvatures and in ection points remain present. However the overall shape tends to be more convex than concave. 7. Discussion Playing 2-by-2 games seems to promote risk aversion. This is the central message of our simulations. However, the degree of risk aversion which is achieved over time crucially depends on the equilibrium selection rule in case of games with multiple equilibria. If payo dominance is given priority the degree of endogenous risk aversion is in all settings considerably higher than in the case of risk dominance as the exclusive selection rule. Of course, there are many parameters in both our settings whose in uence we do not have systematically explored. This is due to the fact that the simulations are extremely time consuming. 12 Especially, in Setup II with priority for risk dominance many more simulation runs would be desirable. The main question is whether the changing curvature of the average utility function would remain stable. If so, a process likethatcould add to the understanding of utility functions with de ection points as theoretically suggested by Friedman and Savage (1948). Furthermore, this would challenge our other results pointing to the endogenous emergence of global risk aversion. Therefore, a more thorough analysis remains desirable In Setup II one run over 50,000 replication steps takes roughly 48 hours with a Pentium 233 and 64 MB RAM. 13 For future work it also remains to identify further the forces driving the results when applying the two equilibrium selection criteria. For that purpose it would be useful to include only special 20

21 Figure 6.2: Average utility of the population at time t and with average mean standard deviation ¾ if risk dominance is used as equilibrium selection concept. The vertical axes denote utility, the horizontal axes denote resources. 21

22 On a more general level, this study may serve as an illustration of learning on the rule level in contrast to learning on the behavioral level. Of course, our implementations are only two possibilities among many others to model this kind of learning. In fact, our dynamics are borrowed from evolutionary game theory. Sometimes evolutionary processes capture learning quite well sometimes they do not (see, for example, Brenner 1996). It was not our aim to clarify the many intricate details which arise when translating actual human learning into a formal dynamic model. Clearly, this would have worn out our paper. Instead, we chose dynamic processes which are well understood in itself, which are easy to implement, and which bear some plausibility with respect to learning. Finally, we like to stress that we learned from this study how useful computational methods can be. Our investigation into the principle matter of endogenous attitudes towards risk started as an analytical one. But, as pointed out earlier, the analytical approach did not prove fruitful. So, for two of us, this study turned into our rst computational one. This has been exciting but, more importantly, it has also been convincing. Clearly, the question of how well one does when interacting with others is of importance for the understanding of risk preferences and the computational analysis has shown what we expected, namely that being risk averse might be very useful when one has to deal with others. References [1] Bester, H. and Güth, W. (1998): Is altruism evolutionarily stable?, Journal of Economic Behavior and Organisation, 34 (2), [2] Björnerstedt, J. and Weibull, J. (1993): Nash equilibrium and Evolution by Imitation, in: Rationality in Economics, (eds.: K. Arrow and E. Colombatto), MacMillan, New York [3] Bowles, S. (1998): Endogenous preferences: The cultural consequences of markets and other economic institutions, Journal of Economic Literature, 36, [4] Brenner, T. (1998): Can Evolutionary Algorithms Describe Learning Processes?, Journal of Evolutionary Economics, 8, [5] Camerer, C. (1995): Individual decision making, in: Handbook of Experimental Economics, eds. J.H. Kagel and A.E. Roth, classes of 2 by 2 games into the simulations. 22

23 [6] Ely, J. and Yilankaya, O. (1997): Nash equilibrium and the evolution of preferences, CMS-EMS Discussion Paper 1191, Northwestern University. [7] Engelmann, D. (1998): Risk aversion pays in the class of 2x2 games with no pure equilibrium, Mimeo. [8] Friedman, M. and Savage, L.J. (1948): The utility analysis of choices involving risk, Journal of Political Economy, 56, [9] Güth, W. and Kliemt, H. (1994): Competition or co-operation On the evolutionary economics of trust, exploitation and moral attitudes, Metroeconomica, 45, [10] Güth, W. and Huck, S. (1997): A new justi cation of monopolistic competition, Economic Letters, 57, [11] Güth, W. and Yaari, M. (1992): An evolutionary approach to explain reciprocal behavior in a simple strategic game, in: Explaining Process and Change - Approaches to Evolutionary Economics, (ed.: U. Witt), The University of Michigan Press, Ann Arbor, [12] Harsanyi, J. C. (1995): A new theory of equilibrium selection for games with complete information, Games and Economic Behavior, 8, [13] Harsanyi, J. C. and Selten, R. (1988): A General Theory of Equilibrium Selection in Games, Cambridge, MA: MIT Press. [14] Mosteller, F. and Nogee, P. (1951): An experimental measurement of utility, Journal of Political Economy, 59, [15] Robson, A. J. (1996a): The evolution of attitudes to risk: Lottery tickets and relative wealth, Games and Economic Behavior; 14, [16] Robson, A. J. (1996b): A biological basis for expected and non expected utility, Journal of Economic Theory, 68, [17] Selten, R. (1995): An axiomatic theory of a risk dominance measure for bipolar games with linear incentives, Games and Economic Behavior, 8, [18] Stigler, G. J. and Becker, G. S. (1977): De Gustibus Non Est Disputandum, American Economic Review 67, [19] To, T. (in press): Risk and evolution, Economic Theory, forthcoming. 23

24 [20] Vriend, N. J. (1998): An illustration of the essential di erence between individual and social learning, and its consequences for computational analyses, Journal of Economic Dynamics and Control, (forthcoming). [21] Weibull, J. W. (1995): Evolutionary Game Theory, MIT Press 24