Centipede game
A growing pot that rational theory says you should grab immediately — but people don't.
What it means
A sequential game in which two players alternately choose to 'take' a slowly growing pile or 'pass' it to the other, where taking ends the game. Backward induction implies the first player should take at the very first move, since the other would take at the last, and so on back to the start — yet experimental players routinely pass for many rounds, building far larger payoffs. The gap is a celebrated challenge to the assumption of common knowledge of rationality, explained by altruism, reciprocity, reputation, and uncertainty about the opponent's rationality. It matters because it dramatizes the divide between game-theoretic prediction and actual human cooperation in dynamic settings.
The unraveling argument
The classic four-move version starts with a small pot; at each node the mover can take, keeping the larger share, or pass, which grows the stakes and hands the choice to the other. Reason from the end: the last mover strictly prefers taking, since passing simply gives the larger share away. Knowing that, the second-to-last mover also takes; the logic zips back to the opening move, where the first player, foreseeing the whole chain, should take at once. Each step is individually airtight, yet the conclusion — grab the smallest pot on offer — feels absurd.
What the experiments show
Rosenthal posed the game as a thought experiment in 1981; McKelvey and Palfrey brought it into the lab in 1992. Across 662 plays of four- and six-move versions, only 37 ended with an immediate take at the first node and just 23 ran all the way to the last; the rest stopped somewhere in between. Taking grows far more likely as the final node nears, where the stakes and the temptation peak. With repetition players drift toward stopping earlier, a partial unraveling, but even experienced subjects rarely grab at the opening move.
Why players pass
Two explanations survive scrutiny. The first keeps rationality but drops certainty: if even a few opponents might be unconditional cooperators, a selfish player can profit by passing early to imitate them and harvest a bigger pot later, so cooperation becomes a rational gamble rather than an error. McKelvey and Palfrey fit their data needing only about five percent altruistic types. The second drops unlimited reasoning: people iterate the induction one or two steps, not all the way, as level-k and cognitive-hierarchy models assume. Aumann proved that common knowledge of rationality does force the take-first outcome, so the puzzle sits in that assumption.
Does expertise close the gap?
If the gap were a simple reasoning failure, experts in backward induction should avoid it. The evidence is contested. Palacios-Huerta and Volij (2009) reported that chess players stop far earlier than students, with every grandmaster taking at the first node. Levitt, List and Sadoff (2011) could not replicate this: none of their sixteen grandmasters took first, and although the same players backward-inducted cleanly in a "race to one hundred," their centipede choices bore no relation to that skill. Strategic reasoning appears context-bound — unwinding a pure logic puzzle does not transfer to a game where motives are in play.
Where it shows up
The structure recurs wherever mutual restraint compounds value but either side can cash out for a one-time gain that ends the relationship: informal trade credit, staged financing released in tranches, any tit-for-tat exchange without a contract. Real relationships routinely sustain cooperation the pure theory says should collapse, largely because the endpoint is uncertain and reputations carry forward. The warning still bites at the edges: when a horizon becomes visible and fixed — a retiring partner, a fund winding down, a final delivery — reasons to keep passing weaken and cooperation unravels backward from the end.
Examples
Players keep passing a doubling pot back and forth, each trusting the other not to grab, accumulating far more than the take-first logic allows.
Two firms trading on an unwritten arrangement could each grab the margin today and walk away. Neither does, year after year, and both end up far richer than take-first logic predicts.
A freelancer delivers before invoicing; the client pays before the next brief lands. Either could stop and pocket the difference at any handover, and backward induction says stop at the first one.
Two rival states de-escalate a border standoff by withdrawing troops in stages. Either could exploit the other's exposed flank and end the thaw, yet each keeps pulling back.
Two labs swap unpublished datasets one release at a time, each trusting the other not to scoop. Either could publish first and break the exchange, but the trades continue for years.
First described in Robert Rosenthal (1981); McKelvey & Palfrey (1992).
Key references
- Levitt, S. D., List, J. A., & Sadoff, S. E. (2011). Checkmate: Exploring backward induction among chess players. American Economic Review, 101(2), 975-990. doi.org/10.1257/aer.101.2.975
- Palacios-Huerta, I., & Volij, O. (2009). Field centipedes. American Economic Review, 99(4), 1619-1635. doi.org/10.1257/aer.99.4.1619
- Nagel, R., & Tang, F. F. (1998). Experimental results on the centipede game in normal form: An investigation on learning. Journal of Mathematical Psychology, 42(2), 356-384. doi.org/10.1006/jmps.1998.1213
- Aumann, R. J. (1998). On the centipede game. Games and Economic Behavior, 23(1), 97-105. doi.org/10.1006/game.1997.0605
- McKelvey, R. D., & Palfrey, T. R. (1992). An experimental study of the centipede game. Econometrica, 60(4), 803-836. doi.org/10.2307/2951567
- Rosenthal, R. W. (1981). Games of perfect information, predatory pricing and the chain-store paradox. Journal of Economic Theory, 25(1), 92-100. doi.org/10.1016/0022-0531(81)90018-1