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The Calculus of Selfishness (Princeton Series in Theoretical and Computational Biology)

Sigmund, Karl
Princeton University Press: Princeton, NJ, 2010
ISBN 9780691142753 (pb)

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Reviewed by Herbert Gintis
Santa Fe Institute and Central European University

Cover of book I have noticed in recent years a tendency for agent-based modelers to consider their craft as an alternative to analytical model-building. This is lamentable for two reasons. First, an agent-based computer model is a perfectly well defined mathematical structure, that off a finite Markov process. Under quite general circumstances a Markov process has a stationary distribution that characterizes its long-term behavior. In most cases, however, the transition matrix associated with an agent-based model will be much too large to use for the analytical solution for the stationary distribution. The computer simulation is an empirical estimation of the properties of the Markov process. Second, agent-based programmers often pride themselves on being ignorant of dynamical systems theory, which they consider old-fashioned (see, for instance, Wolfram 2002 for a particularly egregious example of computer modeling euphoria). This is a rather questionable stance, because the solid research with which I am acquainted uses agent-based modeling as one tool among many in the study of complex dynamical systems.

Enter Karl Sigmund, an accomplished mathematician with a serious love affair for the evolutionary modeling of social life, as witnessed by his previous contribution "Game of Life" (see Sigmund 1995). Sigmund develops classical game theory as well as the evolutionary game theory that takes a classical game as its stage game, which it embeds in a population structure such that sets of agents meet in each period and play the stage game. Periodically, more successful agents reproduce and less successful agents die off, as in a Darwinian dynamic. The result analytically is called a monotonic dynamic, of which the replicator dynamic is the most famous and commonly used example. There are many elegant properties of evolutionary games of this type (see Gintis 2009b), but Sigmund sticks to models that can be modeled two-dimensionally, usually using barycentric coordinates, so generally the models allow agents to have three different pure strategies.

This book is well-suited for teaching or self-learning. While Sigmund touches upon a variety of games, most of the material is devoted to two-player social dilemma games, the Prisoners Dilemma (PD) game in some form being most intensively studied. The central point is that in such games selfish players will never cooperate, but if the social setting is properly structured, they may cooperate.

Sigmund devotes a chapter to direct reciprocity, which is called reciprocal altruism in the biological literature (Trivers 1971) and tit-for-tat in the economic literature (Axelrod 1984). This leads him to deal with repeated games, stressing the role of errors in compromising efficiency. Sigmund then devotes a chapter to indirect reciprocity, which was proposed by the biologist Richard Alexander (1987) as an obvious extension of direct reciprocity. In an indirect reciprocity model, individuals develop reputations for good behavior that is common knowledge to all players. Players then cooperate with each other in the PD game, provided both are in good standing. Moreover, a player who defects on a another player who is in good standing moves to being in bad standing. A high level of cooperation can be sustained in such a model, but it is rather unrealistic to assume in most situations that reputations will be accurate enough to render cooperation an individual best response, especially when the error rate is not very low, or the condition of cooperating is private information of the two players. There is little evidence of either direct or indirect reciprocity in non-human creatures (Stephens, McLinn and Stevens 2002), but considerable evidence in humans (Fehr and Gächter 1998, Brandt and Sigmund 2004, Brandt and Sigmund 2005, Panchanathan and Boyd 2003, Panchanathan and Boyd 2004, Engelmann and Fischbacher 2009, Rockenbach and Milinski 2006).

Sigmund then devotes a chapter to what is known as the Trust Game (Berg, Dickhaut and McCabe 1995). This is a sequential PD game, where the first player cooperates by giving money to the experimenter, who doubles it and sends the money to the second player. The second player can then send money back to the first player. This game differs from the standard PD game only in that one player's decision is conveyed to the second player before the latter has to decide what to do, and this rule is known to both players.

In his brief foray into many-player games, Sigmund studies the Public Goods (PG) game, stressing the research of himself and his colleagues (Hauert et al. 2002a, Hauert et al. 2002b, Hauert, Haiden, and Sigmund 2004). This work is very elegant mathematically, but is rather a waste of time from a scientific standpoint because the conclusion is that when cooperation is sustained in their model, there is no material payoff to cooperation - individuals who act alone have equal payoffs with actors who cooperate in groups.

Sigmund finishes off "The Calculus of Selfishness" with an elegant chapter on kinship selection models on grids. As it turns out, it is relatively easy to generate cooperation on a spatial grid even without kinship, so long as agents are forced to remain near one another for an extended period of time. Essentially, PD games on a grid are simple extensions of tit-for-tat, which is relatively powerful in fostering cooperation.

The reader might wonderwhat is behind the title of this book. Sigmund stresses "selfishness" because he is well aware that a huge volume of empirical results from behavioral game theory over the past two decades show that both in the laboratory and in the field, human subjects sustain cooperation not through being selfish, but rather through exhibiting other-regarding preferences and favoring such character virtues as honesty and trustworthiness (Gintis 2009a). Sigmund wants to make the point that much cooperation in PG and PD games can be sustained without the need for such prosocial preferences. He does show this for PD games when there is sufficient information accuracy and publicity, but not otherwise. Indeed, even in small groups, the PG game does not sustain cooperation under plausible conditions without other-regarding preferences. If humans were truly selfish, they would not have had anything close to the evolutionary success they have enjoyed (Bowles and Gintis 2011).

* References

ALEXANDER, R D (1987) The Biology of Moral Systems. New York: Aldine

AXELROD, R (1984) The Evolution of Cooperation. New York: Basic Books

BERG, J, Dickhaut,J and McCabe, K (1995) Trust, Reciprocity, and Social History. Games and Economic Behavior 10, pp. 122-142

BOWLES, S and Gintis, H (2011) A Cooperative Species: Human Reciprocity and its Evolution. Princeton: Princeton University Press

BRANDT, H and Sigmund, K (2004) The Logic of Reprobation: Assessment and Action Rules for Indirect Reciprocation. Journal of Theoretical Biology 231, pp. 475-486

BRANDT, H and Sigmund, K (2005) Indirect Reciprocity, Image Scoring, andMoral Hazard.Proceedings of the National Academy of Sciences 102,7, pp. 2666-2670

ENGELMANN, D and Fischbacher, U (2009) Indirect Reciprocity and Strategic Reputation Building in an Experimental Helping Game. Games and Economic Behavior 67, pp. 399-407

FEHR, E and Gächter, S (1998) 'How Effective Are Trust- and Reciprocity-Based Incentives?'. In Putterman, L and Ben-Ner, A (Eds.), Economics, Values and Organizations. New York: Cambridge University Press, pp. 337-363

GINTIS, H (2009a) The Bounds of Reason: Game Theory and the Unification of the Behavioral Sciences. Princeton: Princeton University Press

GINTIS, H (2009b) Game Theory Evolving. Second Edition. Princeton: Princeton University Press

HAUERT, C, Haiden, N and Sigmund, K (2004) The Dynamics of Public Goods. Discrete and Continuous Dynamical Systems B 4, pp. 575-585

HAUERT, C, Demonte, S, Hofbauer, J. and Sigmund, K (2002a) Replicator Dynamics for Optional Public Goods Game. Journal of Theoretical Biology 218, pp. 187-194

HAUERT, C, Demonte, S, Hofbauer, J. and Sigmund, K (2002b) Volunteering as Red Queen Mechanism for Cooperation in Public Goods Game. Science 296, pp. 1129-1132

PANCHANATHAN, K and Boyd, R (2003) A Tale of Two Defectors: The Importance of Standing for Evolution of Indirect Reciprocity. Journal of Theoretical Biology 224, pp. 115-126

PANCHANATHAN, K and Boyd, R (2004) Indirect Reciprocity Can Stabilize Cooperation without the Second- Order Free Rider Problem. Nature 432, pp. 499-502

ROCKENBACH, B and, Milinski, M (2006) The Efficient Interaction of Indirect Reciprocity and Costly Punishment. Nature 444, pp. 718-723

SIGMUND, K (1995) Games of Life. New York: Penguin

STEPHENS, W, McLinn, C M and Stevens, J R (2002) Discounting and Reciprocity in an Iterated Prisoner's Dilemma. Science 298, pp. 2216-2218

TRIVERS,R L (1971) The Evolution of Reciprocal Altruism. Quarterly Review of Biology 46, pp. 35-57

WOLFRAM, S (2002) A New Kind of Science. Wolfram Media, accessible at: http://www.wolframscience.com/nksonline/toc.html


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