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COMPUTING SCIENCE

New Dilemmas for the Prisoner

Brian Hayes

Darwinian Dilemmas

2013-11HayesF3.jpgClick to Enlarge ImageOne possible escape is to put the game in the larger context of evolutionary biology, where Prisoner’s Dilemma offers a framework for understanding how cooperation might arise in an environment that seems to reward only selfishness. The Darwinian mechanism “closes the loop” on the game: Each agent’s probability of success or failure depends on the composition of the entire pool of players, but the composition of that pool depends in turn on which players succeed and fail.

Soon after the Press-Dyson report appeared, Christoph Adami and Arend Hintze of Michigan State University tested various zero-determinant strategies in an evolutionary simulation. The coercive strategies did well against certain opponents, but eventually they were displaced by other players, most notably Pavlov. The reason is that a “nasty” player can become a victim of its own success. The reward for winning in an evolutionary game is to become more common in the population, with the result that you encounter more members of your own species. Dictators and extortionists do not thrive in that environment. Thus Adami and Hintze concluded that zero-determinant strategies are unlikely to evolve in the wild.

 But this is not the end of the story. It turns out that not all zero-determinant strategies are weapons wielded by brutes and bullies. Alexander J. Stewart and Joshua B. Plotkin of the University of Pennsylvania have identified a set of “generous” zero-determinant players that form a mirror image to the extortionate ones. An extorter tries to claim more than his or her fair share, but when this gambit fails must accept the low payoff for mutual defection. A generous player offers to accept less than a fair share of the average payoff as an inducement to mutual cooperation. In other words, the generous player is willing to be a patsy if that’s what it takes to secure cooperation.

Generous behavior might seem like a maladaptive invitation to abuse, but Stewart and Plotkin found otherwise. In a series of evolutionary experiments, the generous subset of zero-determinant strategies were the dominant species in all contests except those with a very small population (fewer than about 10 individuals). Stewart and Plotkin went on to prove that generosity is a “robust” strategy, able to establish itself and proliferate in a diverse population and then repel invasion attempts by others. Apparently it pays to put up with a little unfairness if that leads to greater opportunities for beneficial cooperation.

Is that the moral of the story? The players of these games are very simple and mechanistic; they are algorithms, not personalities. Nevertheless, it’s hard to resist giving them value-laden labels such as “extortionate” or “generous.” Axelrod’s analysis of tit-for-tat clearly echoes the fundamental principle of lex talionis: take an eye for an eye (but no more than that). The evolutionary results of Stewart and Plotkin hint at a new dispensation: Mercy is greater than justice.

©Brian Hayes

Bibliography

  • Adami, C., and A. Hintze. 2012. Evolutionary instability of zero-determinant strategies demonstrates that winning is not everything. Nature Communications 4:2193.
  • Axelrod, R. 1984, 2006. The Evolution of Cooperation. New York: Basic Books.
  • Boerlijst, M. C., M. A. Nowak, and K. Sigmund. 1997. Equal pay for all prisoners. American Mathematical Monthly 104:303–305.
  • Press, W. H., and F. J. Dyson. 2012. Iterated prisoner’s dilemma contains strategies that dominate any evolutionary opponent. Proceedings of the National Academy of Sciences of the U.S.A. 109:10409–10413.
  • Stewart, A. J., and J. B. Plotkin. 2013. From extortion to generosity, the evolution of zero-determinant strategies in the prisoner’s dilemma. Proceedings of the National Academy of Sciences of the U.S.A. 110:15348–15353.





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