Checkmate: Exploring Backward Induction among Chess Players

It is difficult to overstate the profound impact that game theory has had on the economic approach and on the sciences more generally. For that reason, understanding how closely the assumptions that underpin game theoretic analysis conform to actual human decision-making is a question of first-order importance to economists. In this spirit, backward induction represents one of the most basic concepts in game theory. Backward induction played a prominent role in Reinhard Selten’s (1965) development of perfect equilibrium, and it has helped to shape the modern refinement literature. Although backward induction is a cornerstone of game theory, existing empirical evidence suggests that economic agents engage in backward induction less frequently than theorists might hope. Backward induction has fared especially poorly in the centipede game, which was introduced by Robert W. Rosenthal (1981) and has since been extensively analyzed (Kenneth G. Binmore 1987, Robert J. Aumann 1988, Philip J. Reny 1988, David M. Kreps 1990, Geir B. Asheim and Martin Dufwenberg 2003). The original centipede game is a two-player, finite-move game in which the subjects alternate choosing whether to end the game or to pass to the other player. The subject’s payoff to ending the game at a particular node is greater than the payoff he receives if the other player ends the game at the next node, but less than the payoff earned if the other player elects not to end the game. The player making the final choice gets paid more from stopping than from passing, and thus would be expected to stop. If the opponent will stop at the last node, then conditional on reaching the penultimate node, the player maximizes his earnings by stopping at that node. Following this logic further, backward induction leads to the unique subgame perfect equilibrium: the game is stopped at the first node. As pointed out in prior research (Rosenthal 1981, Robert J. Aumann 1992, Richard D. McKelvey and Thomas R. Palfrey 1992, Mark Fey, Richard D. McKelvey and Thomas R. Palfrey 1996, Klaus G. Zauner 1999), there are many reasons why players might take actions in the centipede game that diverge from that prescribed by backward induction. Players may face an aversion to the loss of a potential surplus. They may have social preferences for fairness, altruism, or cooperation; or, they may believe that enough other players in the population have these preferences that continuing the game becomes the optimal rational strategy (Robert J. Aumann 1995). Similarly, there may be enough