Game theory

Remarkably simple models are often believed to capture features of what might seem like sophisticated decision making by humans, animals and human organizations. A particular case on which many studies have been done is the so-called iterated Prisoner's Dilemma, in which two players make a sequence of choices a and b to "cooperate" (1) or "defect" (2), each trying to maximize their score m〚a, b〛 with m = {{1, -1}, {2, 0}}. At a single step, standard static game theory from the 1940s implies that a player should always defect, but in the 1960s a folk theorem emerged that if a whole sequence of steps is considered then a possible strategy for perfectly rational players is always to cooperate—in apparent agreement with some observations on human and animal behavior. In 1979 Robert Axelrod tried setting up computer programs as players and found that in tournaments between them the winner was often a simple "tit-for-tat" program that cooperates on the first step, then on subsequent steps just does whatever its opponent did on the previous step. The same winner was also often obtained by natural selection—a fact widely taken to explain cooperation phenomena in evolutionary biology and the social sciences. In the late 1980s similar studies were done on processes such as auctions (cf page 1015), and in the late 1990s on games such as Rock, Paper, Scissors (RoShamBo) (with m = {{0, -1, 1}, {1, 0, -1}, {-1, 1, 0}}). (A simpler game—certainly played since antiquity—is Penny Matching or Evens and Odds, with m = {{1, -1}, {-1, 1}}.) But even though they seemed to capture or better actual human behavior, the programs considered in all these cases typically just used standard statistical or Markov model methods, or matching of specific sequences—making them far too weak to make predictions about the kinds of complex behavior shown in this book. (Note that a program can always win the games above if it can in effect successfully predict each move its opponent will make. In a game between two arbitrary programs it can be undecidable which will win more often over the course of an infinite number of moves.)