There are several distinctive features in Raymond Boudon s pioneering

<#> RATIONAL CHOICE, EVOLUTION AND THE BEAUTY CONTEST Andreas Diekmann ETH Zurich 1. SOCIAL INTERACTION AND GAME THEORY There are several distinctive features in Raymond Boudon s pioneering work. While others merely retell the work of classical sociologists, Boudon exploits the ideas of Alexis de Tocqueville, Emile Durkheim and other classical sociologists in a very creative way. Building on the sociological tradition, he was able to demonstrate the fruitful synthesis of classical ideas and modern analytical tools such as mathematical models, simulation methods and game theory. Models of social interactions and their unintended consequences have a central place in Boudon s writings. A fine example is the Logic of Frustration, part of a collection of essays published as Effets pervers et ordre social in 1977 (the English edition appeared under the title The Unintended Consequences of Social Action in 1982). Studying the causes of the French revolution, Tocqueville put forward the hypothesis that increasing wealth is paralleled by rising dissatisfaction. The core of Tocqueville s argument is captured by a game-theoretic model. This model is not simply a more precise description of the structure of social interaction among the members of a social system, it allows the deduction of testable hypotheses and specifies the conditions for positive or negative correlations between an increase in opportunities and the level of satisfaction experienced by the members of the social system. The game model also explains the empirical findings reported by Stouffer (1949) in the American Soldier. Stouffer com- 1

RAYMOND BOUDON A LIFE IN SOCIOLOGY pared two military units observing that the unit with a higher promotion rate exhibited a lower level of satisfaction. Given certain values for the structural parameters, Boudon s model explains this paradoxical effect. Classical, non-cooperative game theory with the concepts of dominant strategies, mixed strategies and the Nash-equilibrium were sufficient to fruitfully analyse an important class of structures of social interaction. As Boudon suggests there are many possible ways to elaborate the basic model. Yet, the charm of the model is its simplicity. Just a few assumptions lead to deep insights and far-reaching consequences. Surprisingly, rational choice sociologists have often denied or downplayed the usefulness of game theory in explaining sociological problems. Boudon, in contrast, recognized early on the viability of game models for the analysis of the structure of social interaction and he was one of the pioneers in applying game theory to classical sociological problems. 2. LIMITS OF RATIONALITY Rational choice sociologists using game models assume that actors decide rationally about alternatives and are aware of the consequences of possible actions. There are many examples of successful explanations using the rational actor model as, for example, Boudon s work demonstrates. On the other hand, there are problems of strategic interaction which are not easily explainable by rational actor models alone. Often foresight and knowledge of the consequences of decisions are very limited. People learn by trial and error, imitate successful modes of behaviour and adapt new strategies. Customs, conventions, social norms, and institutions are usually not a result of a rational design, but the outcome of evolutionary forces with cultural mutations, variations and a selection mechanism. Rational choice theory lays too much emphasis on intelligent design by purpose-oriented actors, while in fact many social phenomena are generated by the process of cultural evolution. The study by Gürerk, Irlenbusch and Rockenbach (06) is an instructive example. This study builds on Fehr and Gächter s (02) public good experiment with altruistic sanctions. Fehr and Gächter were able to show that participants in a public good experiment refrained from free riding if actors had the opportunity to sanction their co-players. Note that sanctions were costly for both actors. The punished person 2

ANDREAS DIEKMANN RATIONAL CHOICE, EVOLUTION AND THE BEAUTY CONTEST had to pay a fine and the person who imposed the sanction had to pay for punishment. A homo economicus would not punish and cooperation is predicted to break down. However, the experiment demonstrated that subjects punish norm violators at their own cost (i.e. altruistic punishment) thereby increasing the rate of cooperation. Altruistic punishment may be due to a tendency towards strong reciprocity either because of fairness considerations or because of the inclination to reciprocate the unkind actions of co-players (Charness and Rabin 02, Fehr and Gächter 02, McCabe, Rigdon and Smith 03). Now, let us imagine that actors have the choice between two different regimes or institutions. Regime one is a public good game with sanctioning opportunities ( sanctioning institution SI ) as described above, while under regime two sanctioning opportunities are absent ( sanction-free institution SFI ). In an experiment conducted by Gürerk et al, subjects played over 30 rounds and in any round they had the choice of changing to the other type of institution. All subjects were informed about the gains of other players in both institutions. At the beginning of the experiment most subjects exhibited a preference for the public good game without sanctions. Yet, a minority in the sanctioning regime was successful in enhancing and sustaining cooperation, while free riding became the mode of conduct if sanctions were absent. Payoffs of free riders in the SFI groups surpassed the earnings of cooperators in the SI groups in rounds one to three, but thereafter the gains of cooperation increased steadily. When learning of the success of the SI regime, more and more subjects changed the type of institution. And migrating individuals changed their behaviour too. Former free riders in a SFI group rapidly adapted to the norm of cooperation under the sanctioning regime. This process led to extreme polarization in the society. In the end, in round 30, the researchers observed a small number of SFI groups with low payoffs and a large majority of more than 90 percent of subjects in the SI groups showing almost full cooperation. Gürerk et al. comment that the findings demonstrate the competitive advantage of sanctioning institutions and exemplify the emergence and manifestation of social order driven by institutional selection. Full cooperation and the predominance of sanctioning institutions were not the result of a rational choice by purpose-oriented actors with a talent for foreseeing the consequences of their actions. On the contrary, most subjects were attracted by the sanction-free regime in the beginning. In 3

RAYMOND BOUDON A LIFE IN SOCIOLOGY an evolutionary process, actors adapted step-by-step to more successful strategies leading to the unintended or in Boudon s words paradoxical result observed in the experiment. One might argue that step-bystep adaptation or myopic behaviour does not necessarily contradict the doctrines of rational choice theory. Even if we concede that this argument might be justified, the Gürerk et al. experiment and many other examples of cultural evolution demonstrate that assumptions on globally optimising behaviour in rational choice models are often too restrictive. To achieve a better understanding, if evolutionary forces are present, these assumptions should be substituted by principles of learning, imitation, adaptation of successful strategies and selection by cultural evolution. 3. BEAUTY CONTEST In the Gürerk et al. experiment on the competition between two institutional regimes, rational subjects with foresight would join the group with a sanctioning institution early. The more actors are blessed with foresight, the faster is the evolutionary process. In a beauty-contest game even fully rational actors would not adopt the strategy which finally evolves at the beginning. The beauty-contest game is a metaphor for the behaviour of actors with different degrees of foresight on financial markets. In a beauty contest, N players choose a number in the interval [0, 0]. The winner is the player who chooses the number closest to pµ with mean µ and p 0. The value of p is known to the players, i.e. p is common knowledge (Nagel 1995). The prize is shared equally if there is more than one winner. In a more general version of the game, a player s strategy set is the interval [a, b], x i (i = 1, 2, N) is a number chosen by player i, θ is a function of the x i, and x i pθ = Min! is the winning number. For N = 2 and p < 1 a beauty-contest game with θ = µ has a special feature. In this case the lower number always wins and picking the number zero is a (weakly) dominating strategy 1. Zero is the unique Nash-equilibrium strategy and also an evolutionarily stable strategy (ESS, Maynard-Smith and Price 1974). For larger N a dominating strategy does not exist. The game is more interesting if N > 2. What is the best choice in a beauty contest when, for example, p = 2/3 and θ = µ and N > 2? Assume that other players will pick a 4

ANDREAS DIEKMANN RATIONAL CHOICE, EVOLUTION AND THE BEAUTY CONTEST number randomly. Therefore, the expected mean is 50 and a best choice is 2/3 of 50 or 33.33. Yet, other players are expected to think strategically as well. If one believes that co-players are about to pick 33.33, second degree reasoning leads to the choice of 22.22. Third degree reasoning results in choosing 14.81 and so on. Actually, experimental investigations show that the numbers 33, 22, are chosen more frequently (Nagel 1995, Selten and Nagel 1998). A boundedly rational strategy is to form a belief about the degree n of reasoning of the co-players and then to go one step further to n + 1 and to pick the respective number. We call this an n+1 strategy for short. 2 Stock market brokers may employ this strategy. Even if an investor is afraid of a stock market bubble and even if he is convinced that stocks are overpriced, he nevertheless may decide to buy, because his assumption of other actors expectations of rising prices. John Maynard Keynes describes this reasoning in The General Theory of Interest, Employment and Money. The following citation (Keynes 1936, p. 6) also makes clear why the game earned its name beauty contest : Or, to change the metaphor slightly, professional investment may be linked to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one s judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees (Keynes 1936: 6). Of course, in general an n + 1-strategy is not the rational choice in a strict sense. One strong premise of rationality theories is the symmetry principle. An actor has to assume that all other actors are guided by the 5

RAYMOND BOUDON A LIFE IN SOCIOLOGY same principles of rationality. However, an n+1 strategy violates the symmetry principle because an actor expects co-players strategies to be of degree n, while he or she tries to outsmart the co-players by reasoning one step ahead. Moreover, an n+1 strategy is not an equilibrium strategy. With p = 2/3 and θ the mean, there is only one strict sub-game perfect Nash-equilibrium strategy: the choice of zero. If all actors decide on zero, there is no incentive to deviate from the equilibrium strategy unilaterally. Also, the Nash-equilibrium strategy is Pareto-optimal and in equilibrium all players of the symmetric game receive an equal share of the prize. In a beauty contest, no coordination problem arises as with a chicken- or battle-of-sex-type dilemma and there is no efficiency problem, i.e. a discrepancy between the Nash-equilibrium payoff and the Pareto-optimal payoff as in a prisoner s dilemma type game. All desirable requirements of a rational strategy are met. There is no doubt that rationality theories (i.e. Harsanyi and Selten 1988) would recommend picking the number 0 as the unique solution of the game. However, is it rational to choose the rational strategy if one is sure to lose? Assume you have full knowledge of the game and you are fully aware of the Nash-equilibrium strategy, would you really choose it? You know for sure that most players won t employ the rational strategy maybe for the same reasons that you won t. In this case, you are probably going to form an expectation of other players irrational behaviour and then you will try to adapt your strategy to your expectations. In many situations of strategic interaction, individuals do not employ the rational strategy because they have not enough insight into the subtleties of the strategic situation. One example of this is the Gürerk et al. experiment regarding competition between different institutional regimes discussed above. However, in a beauty contest, actors do not employ the rational strategy even if they know the rational choice! Experiments show that only a small minority chooses zero in a beauty contest (Nagel 1995, Selten and Nagel 1998). A student project organized by my chair explored the behaviour of students enrolled at the ETH Zurich with an online experiment using a repeated beauty contest game. 3 In accordance with previous studies (e.g. Nagel 1995), the online experiment runs over five rounds. With a 3 2 design, group size and information feedback were varied. The group size factor had three categories: small, medium and large groups. In the limited information treatment, participants were informed about the mean and the target 6

ANDREAS DIEKMANN RATIONAL CHOICE, EVOLUTION AND THE BEAUTY CONTEST Table 1: Results of an online beauty-contest game: Arithmetic means of chosen numbers Group 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group size in round 1 16 61 61 2 222 Group size in round 5 14 13 50 53 188 178 Information full limited full limited full limited Round 1 35.3 27.4 35.7 32.6 30.0 31.5 Round 2 21.5 22.8.1.6 21.9 23.2 Round 3 17.1 16.9.9.7.7 16.1 Round 4 14.0.3 12.2 17.1 13.6 9.6 Round 5 13.4 14.8 12.9 11.4.8 6.9 Figures are from the data set of Kamm and Dahinden. See also Kamm and Dahinden (08) value (2/3 of the mean) of the previous round. In the full information condition, subjects also received a graph of the distribution of numbers. The winning number in each round and treatment earned a reward of CHF (about 13 ). The table contains the means for all rounds and conditions. There are no consistent differences in means of group size categories and information feedback. However, in accordance with former studies, a pattern of change is apparent in the distributions over the five rounds in all six treatments. In the first round, with no feedback subjects, frequent choices are about 50, 33, 22 resulting in an average in the range of 29.7 to 35.5 as shown in the table. With increasing round numbers the distributions and the mean moves further to the «left». The Figure shows the distributions of rounds 1 to 5 in the largest group. Subjects adapt step-by-step to the information feedback of the previous round. Moreover, in the first round the heterogeneity of decisions is rather large and the heterogeneity or variance of the distribution declines with the round number. The Nash-equilibrium strategy was chosen very rarely. All in all, 602 subjects made 2,703 decisions in five rounds. A «zero» was delivered in only seven out of 2,703 choices. 497 subjects participated in all five rounds. 7

RAYMOND BOUDON A LIFE IN SOCIOLOGY Frequencies 18 16 18 18 14 16 16 12 14 14 12 12 8 8 6 6 4 4 2 0 Round Arithmetic Round mean 1 30. 05 Arithmetic 2/3 of Round mean mean 1.03 = 30. 05 Arithmetic 2/3 of mean mean 2=.03 = 30. 05 2/3 of n mean = 2=.03 n = 2 0 5 30 30 35 35 40 40 45 45 50 50 55 55 60 60 65 65 70 70 75 75 80 80 85 85 90 90 95 95 0 0 0 5 30 35 40 Chosen 45 50 number 55 60 65 70 75 80 85 90 95 0 Round Arithmetic Round mean 2 21.87 Arithmetic 2/3 of mean mean 14.58 = 21.87 2/3 of mean 213= 14.58 n = 213 Frequencies 5 0 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 Round Arithmetic Round mean 3.66 Arithmetic 2/3 of mean mean.44 =.66 2/3 of mean 2=.44 n = 2 Frequencies 45 40 45 35 40 30 35 30 5 0 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 8

ANDREAS DIEKMANN RATIONAL CHOICE, EVOLUTION AND THE BEAUTY CONTEST Round 4 Arithmetic Mean = 13.61 2/3 of mean = 9.07 n = 194 Frequencies 35 30 5 0 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 Round 5 Arithmetic mean =.77 2/3 of mean = 7.18 n = 188 Frequencies 60 50 40 30 0 0 5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 0 Figure 1: Distribution of chosen numbers in rounds one to five (largest group #5) Computed from the data set of Dahinden and Kamm. See also Dahinden and Kamm (08) 4. RATIONAL CHOICE BY EVOLUTION Binmore (1992) argues that three requirements have to be fulfilled in order to predict subjects behaviour in game-theoretical experiments: (1) The structure of the game has to be simple, (2) actors should have experi- 9

RAYMOND BOUDON A LIFE IN SOCIOLOGY ence of the game, and (3) incentives have to be high enough to motivate serious decision making. A beauty-contest game is simple and, in experiments, payoffs are probably high enough to motivate subjects. There are good reasons to assume that experienced subjects who frequently participate in the game nevertheless have a low probability of choosing the Nash-equilibrium strategy in the first round of a beauty contest. 4 On the other hand, learning and adaptation to the strategies of co-players in successive rounds at least approaches the Nash-equilibrium. And one may hypothesize that an increase in the number of rounds to, for example, 0 will result in the mutual choice of the equilibrium strategy. The evolutionary path may be described by the equations of replicator dynamics or other formal models of adaptation and learning (Maynard-Smith 1982, Young 1998). For example, players are myopic, begin with µ 0 = 50 and simply imitate the successful strategies of round n in round n + 1. Thus, they choose x n + 1 = (2/3) µ n. By this process, eventually the Nash equilibrium will be approached. The rational choice comes about among myopic players by means of a simple adaptation mechanism. This does not mean that an evolutionary process always results in an optimal state. As is well known, evolutionary processes often do not lead to a global optimum. Path dependence and lock in are the common terms to describe the sub-optimal results of evolutionary processes. 5 Local interactions may feed the evolutionary process and, dependent on initial conditions, one of many Nash-equilibria will be approached in the end. The historical process of the evolution of left or right driving rules is a good example, analysed extensively by Young (1998). Also, actors rational choices in situations of strategic interaction do not necessarily imply an optimal outcome in the aggregate and the evolutionarily stable strategy or ESS does not necessarily lead to an optimal outcome on the group level (Maynard-Smith 1982). Moreover, there is no guarantee that an evolutionary process approaches equilibrium if environmental conditions change rapidly (Rohde 1995). Rational design and cultural evolution by learning and adaptation is not a contradiction. Technological change may serve as an example of the interplay between rationality and evolution. An engineer who designs new machinery usually builds on existing machinery developed by his forerunners. For technical devices with a history, such as steam engines or automobiles, there is a long sequence of technological evolution ana-

ANDREAS DIEKMANN RATIONAL CHOICE, EVOLUTION AND THE BEAUTY CONTEST logical to the sequence of changing organisms in biological evolution. Another example is the emergence of, and change in, institutions. The various institutions in western democracies developed over centuries. And there are different types, such as the Presidential system, the British Westminster model or the federal democracy found in Switzerland. John Lock, Charles de Montesquieu and Thomas Jefferson were institutional engineers and rational designers of democratic institutions. Yet, rational designs were not developed from scratch but represent steps in a long-lasting evolutionary process. Evolutionary models have an important advantage when explaining social patterns and institutions. No unrealistic assumptions of foresight or knowledge of the consequences of actions are necessary. In the extreme, there is no purposive action at all and rational choice is merely a result of the process, not a prerequisite. Of course, if actors are less myopic or more rational, the speed of the process is often increased. From this perspective, rationality is nothing more and nothing less than a shortcut in the evolutionary process. The cultural evolution of social patterns, technologies, laws and institutions is accelerated by rational and sometimes ingenious design and invention. NOTES 1. For an experimental study of the two-person game see Grosskopf and Nagel 08. 2. Called iteration step n by Nagel. A person is strategic of degree n if he chooses the number 50pn (Nagel 1995:13). 3. The author initiated this project which was conducted by Dominik Kamm and Stefan Dahinden. The project was supervised by Stefan Wehrli. For a description of results, see Kamm and Dahinden (08). 4. In the online experiments, subjects were asked concerning their knowledge of the beauty-contest game, game theory, statistics and probability theory and knowledge of stock markets. With means of a regression analysis for the first round, we did not find significant differences compared to naïve subjects. However, there is a tendency to choose slightly lower numbers if subjects had (self-reported) knowledge of game theory or statistics and probability theory. About 3 % of the subjects answered yes to a question on knowledge of the beauty-contest game. In average, chosen numbers in this group were.7 lower in round one although the difference failed to reach significance (two-sided test of the regression coefficient, α = 0.05, p = 0.068). Because the questionnaire was presented in round five, only subjects were included who participated in all five rounds. 11

RAYMOND BOUDON A LIFE IN SOCIOLOGY 5. Many examples of path dependence, lock-in situations and unintended consequences are found in the history of technological inventions, particularly the evolution of railway networks. For example, railways in Europe, the US and many other countries use a track gauge of 1435 mm. The standard gauge goes back to a decision of George and Robert Stephenson, the engineers and inventors who constructed the first railroads in Britain. The decision for 1435 mm is probably not the optimal track gauge for today s railways after so much technological change and almost two hundred years later. Yet there is no chance of switching to an optimal solution because the costs of changing existing networks are prohibitive. REFERENCES Boudon, Raymond (1977) Effets pervers et ordre social. Paris, Presses Universitaire de France. (1982) The Unintended Consequences of Social Action. New York, St. Martin s Press. Charness, Gary and Rabin, Matthew (02) Understanding social preferences with simple tests. The Quarterly Journal of Economics, 817 869. Fehr, Ernst and Simon Gächter (02) Altruistic punishment in humans. Nature 4: 137-140. Grosskopf, Brit and Nagel, Rosemarie (08) The two-person beauty contest. Games and Economic Behavior, 62: 93 99. Gürerk, Özgür, Irlenbusch, Bernd, Rockenbach, Bettina (06) The competitive advantage of sanctioning institutions. Science 312: 8-111. Harsanyi, John C. and Selten, Reinhard (1988) A General Theory of Equilibrium Selection in Games. Cambridge, MA: MIT Press. Kamm, Dominik and Dahinden, Stefan (08) P-Beauty Contest mit unterschiedlich grossen Teilnahmefeldern. MTU-Gruppenarbeit, ETH-Zurich. Keynes, John M. (1936) The General Theory of Employment, Interest and Money. London, Macmillan. Maynard Smith, John (1982) Evolution and the Theory of Games. Cambridge, Cambridge University Press. Maynard Smith, John and Price, George. R. (1973) The logic of animal conflict. Nature 246: 18. McCabe, Kevin A., Rigdon Mary L., Smith, Vernon L. (03) Positive reciprocity and intentions in trust games. Journal of Economic Behavior and Organization 52: 267 275. Nagel, Rosemarie C. (1995) Unraveling in guessing games: an experimental study. American Economic Review 85: 1313 1326. Rohde, Klaus (1995) Nonequilibrium Ecology. Cambridge, Cambridge University Press Selten. Reinhard and Nagel, Rosemarie C. (1998) Das Zahlenwahlspiel. Hintergründe und Ergebnisse. Spektrum der Wissenschaft 2: 16 22. Stouffer, S.A. (1949) The American Soldier. Princeton, Princeton University Press. Young, H. Peyton (1998) Individual Strategy and Social Structure. An Evolutionary Theory of Institutions. Princeton, NJ: Princeton University Press. 12