Centipede Game¶

A sequential finite-horizon game in which each player can take the larger share or pass to grow the pot, where backward induction prescribes taking on the first move — isolating common knowledge of rationality as the load-bearing assumption cooperation depends on.

Version: v2 · 2026-09-19 · History
Domain-specific #: 141
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomain: Finite Horizon Cooperation Unravelling → Economics & Finance

Core Idea¶

The Centipede Game (Rosenthal, 1981) is a sequential two-player game with perfect information and a known finite horizon in which cooperation unravels by backward induction despite paying more the longer it lasts. At each node a player takes (ending the game with the larger share) or passes (growing the pot and handing over the move). Backward induction prescribes taking on the first move, yet subjects routinely pass for many rounds (McKelvey and Palfrey, 1992). The gap isolates common knowledge of rationality as the load-bearing assumption.

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The Grab-or-Pass Game

Two kids take turns with a pile of candy. On your turn you can grab the bigger part and end the game, or pass, and the pile gets bigger. If they keep passing, both end up with more candy. But a super-strict puzzle-solver, thinking backward from the last turn, says the first kid should grab right away—yet real kids usually pass a few times.

The Growing Pot Game

The Centipede Game is a pretend game two players take turns in, and both know exactly how many turns there are. On your turn you can 'take', which ends the game and gives you the bigger part of the pot, or 'pass', which makes the pot bigger and lets the other player choose. If you work backward from the very last turn, each player should always take before the other can, so the first player should take immediately and both get very little. When real people play, they usually pass for several turns and share much bigger prizes. The gap shows that the backward reasoning depends on everyone trusting that everyone else will reason exactly that way.

Centipede Game: Backward-Induction Paradox

The Centipede Game, introduced by Robert Rosenthal in 1981, is a two-player game where moves alternate, everyone sees everything, and the number of turns is fixed and known. At each turn the mover can take (end the game and get the larger share of the current pot) or pass (grow the pot and hand the choice to the other player). Solving it by backward induction—starting from the last turn and reasoning back—leads to the prediction that the first player takes right away, giving both players the smallest payoffs. Experiments, first systematically by McKelvey and Palfrey in 1992, find people usually pass for several rounds and earn far more. The gap shows that the backward-induction prediction depends on common knowledge of rationality: not just that each player is rational, but that each knows the other is, knows the other knows, and so on. If that chain breaks—because players have limited reasoning, are unsure how deeply the other thinks, or care about the other's payoff—cooperation can last a while before it falls apart.

The Centipede Game, introduced by Robert Rosenthal in 1981, is a finite, sequential two-player game of perfect information in which each node offers the mover a choice to take—ending the game with the larger share of the current pot—or to pass, growing the pot (for example by doubling) and handing the same choice to the opponent. Because the pot grows with every pass, sustained passing is collectively beneficial. Yet the unique subgame-perfect equilibrium, found by backward induction, is to take at the first node: the last mover takes, so the penultimate mover pre-empts, and the argument unravels to the start, yielding the minimum payoff for both. Experiments beginning with McKelvey and Palfrey in 1992 show subjects routinely pass for multiple rounds, often reaching the end or near-end and splitting much larger pots. This gap isolates common knowledge of rationality as the load-bearing assumption, since the induction requires unbounded mutual knowledge of rationality. Bounded rationality, uncertainty about the opponent's depth of reasoning, or other-regarding preferences break that chain and allow several rounds of cooperation before unraveling. The game closely parallels Selten's 1978 chain-store paradox, where backward induction says an incumbent should never fight entry even though fighting is often observed and strategically defensible.

Scope of Application¶

It lives across game theory wherever a situation is genuinely centipede-shaped — a sequential, perfect-information, common-knowledge finite horizon with a growing joint payoff and a known last move.

Game theory proper — the canonical case of finite-horizon backward-induction failure.
Behavioral economics — the standard testbed where equilibrium diverges from play.
Bounded-rationality modeling — the proving ground for level-k and quantal-response models.
Reputation and trust experiments — designs letting reputation accumulate across rounds.
Finite-horizon negotiation — the unravelling of cooperation once the end-date is known.

Clarity¶

The game's clarifying force is that it isolates a single load-bearing assumption and makes its failure observable. Real cooperation breakdowns are overdetermined; the centipede strips away information asymmetry, coordination problems, and dominant-strategy temptation, leaving only the backward-induction chain, which rests on common knowledge of rationality. It converts an abstract worry into a replicable measurement of how far reasoning propagates, and it forces a specific reading of any observed deviation.

Manages Complexity¶

Cooperation breakdowns come as overdetermined messes. The centipede shuts off all but one force by design, collapsing the sprawl of candidate causes to a single isolated assumption. That imposes a clean three-branch structure on what a deviation can mean — bounded depth, belief-uncertainty, or other-regarding preferences — from which the modeller reads off the rest. The compression travels: recognizing a situation as centipede-shaped imports the whole apparatus without re-derivation.

Abstract Reasoning¶

It licenses backward-induction prediction folding the solution from the last node; isolation-by-design diagnosis that pins any deviation on the common-knowledge chain; three-branch attribution of an observed deviation to a testable model; and skeleton-recognition transfer that imports the apparatus to any finite-horizon cooperation and predicts that sustaining cooperation requires breaking the common-knowledge chain on purpose.

Knowledge Transfer¶

Within game theory the centipede transfers as model wherever a situation is genuinely centipede-shaped — the chain-store paradox, finite-horizon negotiation, finitely-repeated PD with a known endpoint — each a real instance, not a likeness. But this breadth is one substrate (reasoning agents) at many scales, not cross-domain reach; stripped of game-theoretic vocabulary it has no counterpart in chemistry or ecology. The portable content is one level up: the parent finite_horizon_unravelling, kin to backward_induction.

Relationships to Other Abstractions¶

Current abstraction Centipede Game Domain-specific

Parents (2) — outgoing typed relations

Centipede Game includes Subgame Perfect Equilibrium Domain-specific

The canonical centipede construction contains its subgame-perfect solution, whose every-node credibility requirement yields take on the first move.
Centipede Game includes Common Knowledge Prime

The centipede model contains common knowledge of rationality and the final move as the epistemic chain that propagates backward to immediate taking.

Hierarchy paths (7) — routes to 6 parentless roots

Centipede Game → includes Subgame Perfect Equilibrium → is a kind of Nash Equilibrium → is a kind of, typical Equilibrium → is a kind of, typical Fixed Point

Neighborhood in Abstraction Space¶

Centipede Game sits in a crowded region of the domain-specific corpus (25^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Game-Theoretic Models & Paradoxes (38 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25