Centipede Game

The centipede game is a sequential game introduced by Robert Rosenthal in 1981 to illustrate the tension between backward induction and observed human behavior. Two players alternate in choosing whether to "take" a larger payoff (ending the game) or "pass" (continuing the game with increased payoffs for both). The game typically has a finite number of rounds — often depicted as a centipede-shaped tree, hence the name.

The paradox is stark. Backward induction predicts that the first player will take immediately, since at the final node the last player would take rather than pass, and working backward, every player should anticipate this and take at their turn. Yet in laboratory experiments, players routinely pass — sometimes through most or all of the game's rounds — leaving money on the table according to the theory.

The Game Structure

In the standard centipede game, Player 1 moves first and chooses between Take (receiving a payoff and ending the game) or Pass (giving Player 2 a turn with larger potential payoffs). Player 2 then faces the same choice, with payoffs growing at each round. If both players pass through all rounds, they receive the largest possible payoffs.

The payoffs are structured so that at every decision node, taking yields more than passing and having the opponent take on the next move, but less than passing and having the opponent pass. This creates a local incentive to pass (if you trust the opponent) and a global incentive to take (if you believe the opponent will take eventually).

Experimental Evidence

Since McKelvey and Palfrey's (1992) landmark experiments, the centipede game has been one of the most robust empirical rejections of standard game theory. Key findings include:

Players pass in the vast majority of early rounds, contrary to backward induction.
Pass rates decrease but remain significant even in later rounds.
When stakes are increased, pass rates decrease but do not collapse to zero.
Experience and repetition reduce passing but do not eliminate it entirely.

These results have been replicated across cultures, stake sizes, and experimental designs. The centipede game is not a fragile anomaly; it is a persistent empirical regularity that standard theory cannot explain.

Competing Explanations

Social preferences. Players may pass because they care about the opponent's payoff as well as their own. Altruism, fairness, and inequity aversion can sustain passing if players derive utility from mutual cooperation.

Bounded rationality. Players may not perform full backward induction, especially in games with many rounds. Level-k reasoning and cognitive hierarchy models predict that players reason only a few steps ahead, leading to partial passing.

Belief-dependent reasoning. The most subtle explanation is that passing is rational given beliefs. If Player 1 believes that Player 2 will pass, then passing is optimal. The question becomes: why would Player 1 hold this belief? One answer is that the belief is self-fulfilling: if both players believe the other will pass, both will pass, and both beliefs are confirmed.

Reputation and learning. In repeated interactions, players may pass to build a reputation for cooperation that pays off in future games. Even in one-shot experiments, subjects may import social norms from repeated-game contexts.

The Systems Interpretation

From a systems perspective, the centipede game reveals a fundamental mismatch between closed-form game-theoretic reasoning and open-ended social interaction. Backward induction requires that the game be fully specified, that all players know it is fully specified, that all players know that all players know this — an infinite regress of common knowledge that no real social system satisfies.

The empirical regularity of passing suggests that human agents do not reason from the end backward. They reason forward, adaptively, and contextually. They ask not "what would happen at the last node?" but "what is likely to happen next?" This is not irrational; it is a different rationality — one suited to open systems where the terminal node is not known, the payoffs are not fixed, and the other player's reasoning is opaque.

The centipede game is not a puzzle about irrationality. It is a demonstration that the wrong model, applied with perfect logic, produces the wrong prediction. Backward induction is not flawed mathematics; it is flawed metaphysics — a method that assumes the world is a closed game when the world is an open process. The players who pass are not making mistakes. They are playing a different game than the one the theorist wrote down. And in the game they are actually playing — a game of trust, reciprocity, and mutual expectation — passing is not merely rational. It is the only move that makes sense.