Structure and backward induction
In Rosenthal's (1981) centipede game each player, on their turn, chooses take or pass. Every pass enlarges the pot, but whoever takes gets the bigger share. The player at the last node will obviously take, so the player before should take first, and reasoning back to the start ends the game on the very first move. This is the subgame perfect equilibrium formalized by Selten (1965), and the puzzle belongs to the same family as Selten's (1978) chain store paradox. As in the prisoner's dilemma, individual rationality yields an outcome that is worse for both.
The common knowledge of rationality debate
Binmore (1987) and Reny (1992) pointed out that once the first player passes, the assumption that the opponent is rational has already been contradicted, so the rest of the backward-induction argument cannot simply be applied. Aumann (1995), by contrast, proved that in perfect-information games common knowledge of rationality implies the backward-induction outcome. Kreps, Milgrom, Roberts and Wilson (1982) showed that even a small chance that an opponent is cooperatively inclined can make it an equilibrium for rational players to cooperate for a while, and this reputation logic is also used to explain cooperation in the centipede game.
Experimental evidence
In McKelvey and Palfrey's (1992) experiments, games almost never ended on the first move; most players passed several times before taking. They explained this with a model allowing for some altruistic players and errors. A constant-sum version, in which the total pot is fixed, produced results closer to theory (Fey et al., 1996), and Nagel and Tang (1998) found early stopping rare even in normal form. Palacios-Huerta and Volij (2009) reported that chess masters almost always stop at the first node, but Levitt, List and Sadoff (2011) pushed back, finding that most chess players do not play the theoretical prediction either.
FAQ
- What is the centipede game in simple terms?
- Two people take turns passing a growing pot of money, and either can grab it at any time. Both do better the longer it goes, but theory says a rational player should grab it right away.
- How do real people play it?
- In experiments, few people stop on the first move; most pass a few times before taking. Researchers explain this by trust and by the expectation that some players are cooperatively inclined.
Sources reviewed: 2026-10-26
References
- Rosenthal, R. W. (1981). Games of perfect information, predatory pricing and the chain-store paradox. Journal of Economic Theory, 25(1), 92–100.
- Selten, R. (1965). Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift für die gesamte Staatswissenschaft, 121, 301–324, 667–689.
- Selten, R. (1978). The chain store paradox. Theory and Decision, 9(2), 127–159.
- Binmore, K. (1987). Modeling rational players: Part I. Economics and Philosophy, 3(2), 179–214.
- Aumann, R. J. (1995). Backward induction and common knowledge of rationality. Games and Economic Behavior, 8(1), 6–19.
- Reny, P. J. (1992). Rationality in extensive-form games. Journal of Economic Perspectives, 6(4), 103–118.
- Kreps, D. M., Milgrom, P., Roberts, J., & Wilson, R. (1982). Rational cooperation in the finitely repeated prisoners' dilemma. Journal of Economic Theory, 27(2), 245–252.
- McKelvey, R. D., & Palfrey, T. R. (1992). An experimental study of the centipede game. Econometrica, 60(4), 803–836.
- Fey, M., McKelvey, R. D., & Palfrey, T. R. (1996). An experimental study of constant-sum centipede games. International Journal of Game Theory, 25(3), 269–287.
- Palacios-Huerta, I., & Volij, O. (2009). Field centipedes. American Economic Review, 99(4), 1619–1635.
- Levitt, S. D., List, J. A., & Sadoff, S. E. (2011). Checkmate: Exploring backward induction among chess players. American Economic Review, 101(2), 975–990.
- Nagel, R., & Tang, F. F. (1998). Experimental results on the centipede game in normal form. Journal of Mathematical Psychology, 42(2–3), 356–384.