Centipede Game

The setup and the unraveling

Rosenthal's 1981 centipede game has two players, Alice and Bob, and a sequence of decision nodes. The game starts with a small "pot" of money — say, $0.40 split as $0.30 for Alice, $0.10 for Bob. At each node:

• The active player can TAKE the current pot, ending the game with the current split.
• Or they can PASS, doubling the pot and giving the next move to the other player. The split now favours the other player slightly.

The classic six-node version has pots growing: $0.40, $0.80, $1.60, $3.20, $6.40, $12.80, with terminal $25.60 if both keep passing through node 6. Crucially, at every node the active player would get slightly more by TAKING now than they would after their opponent takes next round. So the immediate temptation always favours TAKE.

Why backward induction predicts immediate TAKE

Apply backward induction from the last node. At node 6 the active player faces a choice: take the full $25.60 (their preferred split) or pass for a hypothetical $51.20 that... ends the game with no further moves. With no future, the player at the terminal node simply takes. So at node 6, rational play is TAKE.

Move back to node 5. The player at node 5 knows that if they pass, the rival will TAKE at node 6. So passing earns them whatever node-6 TAKE gives them (the lower split), while TAKING at node 5 gives them the higher split. Rational play is TAKE at node 5.

Continue this reasoning. At node 4 the active player knows the rival will take at node 5, so they take at node 4. Iterate. The unraveling reaches node 1, where Alice should TAKE the smallest pot of $0.40 immediately. The unique subgame-perfect Nash equilibrium under common knowledge of rationality is: TAKE at the first opportunity.

Mutual cooperation through all six nodes would have yielded $25.60 — sixty-four times the equilibrium payoff. Both players know this. Yet rationality forbids it.

What real players do

McKelvey and Palfrey ran the canonical experiment in 1992. Subjects played the six-node version with the $0.40-to-$25.60 payoff ladder. Headline findings from 281 plays:

• Only 7% of first movers took at node 1.
• About 80% of pairs reached node 3 or deeper.
• About 35% reached node 5.
• Roughly 8% played through to node 6 without anyone taking — earning the full $25.60.

The pattern repeats almost identically across dozens of replications. Nagel and Tang (1998), Palacios-Huerta and Volij (2009), Levitt-List-Sadoff (2011), and many others have re-run the centipede with various subjects (undergraduates, chess masters, professional poker players, game-theory PhD students). Cooperation depth varies by population but no group reliably plays the backward-induction equilibrium.

A concrete six-node walkthrough

Use the canonical payoffs: pot at node n is $0.40 × 2ⁿ⁻¹. At each node the active player can take $0.30 × 2ⁿ⁻¹ for themselves and leave $0.10 × 2ⁿ⁻¹ for the rival.

Node	Active player	Pot	TAKE → active gets	TAKE → rival gets	If both PASS through, terminal split
1	Alice	$0.40	$0.30	$0.10	—
2	Bob	$0.80	$0.60	$0.20	—
3	Alice	$1.60	$1.20	$0.40	—
4	Bob	$3.20	$2.40	$0.80	—
5	Alice	$6.40	$4.80	$1.60	—
6	Bob	$12.80	$9.60	$3.20	Bob $19.20 / Alice $6.40

Backward induction: at node 6 Bob TAKES ($9.60 > if game ended he gets less). At node 5 Alice prefers to TAKE her $4.80 rather than wait for Bob's TAKE that leaves her $3.20. Continue: each player at each node prefers to take immediately rather than allow the opponent the next move. Equilibrium: Alice TAKES $0.30 at node 1.

Empirically — Alice almost never takes at node 1. The empirical mean stopping node is around 3.5, leaving total payoffs of roughly $3 in the pot — twentyfold the equilibrium prediction.

Centipede vs other rationality puzzles

	Centipede	Prisoner's Dilemma	Ultimatum Game	Dictator Game	Trust Game	Public Goods
Move structure	Sequential, alternating	Simultaneous	Sequential, 2 moves	One-sided	Sequential, 2 moves	Simultaneous
Rational prediction	TAKE at node 1	Both defect	Accept any ε > 0	Keep everything	Defect	Free ride
Typical observed play	Cooperate 3-4 nodes deep	40-50% cooperate one-shot	Reject offers < 30%	Give 20-30%	Trust 50-70%	40-60% contribute
Friction	Backward induction failure	Dominance failure	Fairness preferences	Altruism	Trust + reciprocity	Cooperation norm
Canonical citation	Rosenthal 1981; McKelvey-Palfrey 1992	Axelrod 1984	Güth et al. 1982	Forsythe et al. 1994	Berg-Dickhaut-McCabe 1995	Ledyard 1995
Theory response	Level-k, QRE	Repeated games	Inequity aversion	Altruism functions	Reciprocity	Reciprocal types

The centipede game is the deepest of these puzzles for game theorists because it directly tests backward induction — the strongest of all rationality requirements. If backward induction fails here, it must be qualified in every sequential game.

What explains the discrepancy?

• Level-k thinking (Stahl 1993, Nagel 1995). Players are bounded: level-0 plays randomly, level-1 best-responds to level-0, level-2 best-responds to level-1. Most empirical subjects are level-1 or level-2, generating cooperation 2-4 nodes deep before someone defects.
• Quantal response equilibrium (McKelvey-Palfrey 1995). Players make noisy best responses with logit choice probabilities. Even small noise unravels the backward-induction prediction: PASS becomes positive-probability at every node, sustaining cooperation in equilibrium.
• Altruism / inequity aversion. Fehr-Schmidt (1999) and Bolton-Ockenfels (2000) preferences include a term for the rival's payoff. With altruistic preferences, cooperation through several nodes can be a true equilibrium.
• Reputation / KMRW. Even if a single player is fully rational, if they assign small positive probability to the rival being a "commitment-passer" type, they may cooperate to build reputation. McKelvey-Palfrey (1992) found their data consistent with about 5% commitment types.
• Confusion / non-strategic play. Some subjects simply do not work through the backward induction; they treat the game as "do I want a bigger or smaller pot?" and pass naively. Levitt-List-Sadoff (2011) found that experimenter explanation matters substantially.
• Risk aversion. If players are risk-averse, the certain stake from passing once may dominate the uncertain gain from a longer game.

The empirical literature consistently finds combinations of these mechanisms operate, with weights depending on the subject pool and exact experimental design.

A brief history

Robert Rosenthal (an economist at Boston University, 1944-2002) introduced the centipede game in a short 1981 paper "Games of Perfect Information, Predatory Pricing and the Chain-Store Paradox" in the Journal of Economic Theory. The game was intended as a teaching example illustrating that backward induction in extensive-form games can produce counterintuitive predictions. The paper attracted little immediate attention.

The empirical literature began with Richard McKelvey and Thomas Palfrey's 1992 Econometrica paper, which ran the experiment and found massive deviations from the rational prediction. The paper sparked the now-standard set of behavioural explanations: level-k thinking (Stahl-Wilson 1995, Nagel 1995), quantal response equilibrium (McKelvey-Palfrey 1995), and inequity aversion (Fehr-Schmidt 1999).

Palacios-Huerta and Volij's 2009 paper with chess players sparked another debate. They found chess players took earlier than students, supporting the rationality interpretation. Levitt, List, and Sadoff (2011) replicated with different subject pools and found the effect did not survive. The exchange clarified that the centipede game is a sensitive instrument for distinguishing among models of bounded rationality — not a single decisive test.

Today the centipede is standard third-week material in behavioural-economics courses and a benchmark used to calibrate cognitive-hierarchy and QRE models. The game has been played online with thousands of subjects through MTurk and Prolific; the qualitative patterns persist.

Variants and extensions

• Three-player centipede (Aumann 1995). Adding a third active player at alternating nodes preserves the backward-induction unraveling but with richer empirical deviations.
• Long centipede. Increase nodes to 10, 20, or 100. The backward-induction prediction never changes; empirical cooperation depth grows roughly with log(number of nodes).
• Equal-payoff centipede. Modify payoffs so the active player only barely gains by taking. Backward induction still predicts immediate take; empirical cooperation deepens substantially.
• Repeated centipede. Multi-game tournaments. Cooperation decays slowly across rounds as players learn but never reaches the backward-induction prediction.
• One-sided centipede. Only one player chooses at each node; the other passively waits. Different theoretical structure, similar empirical cooperation.
• Lab versus online. Cooperation is roughly similar in lab and well-incentivised online settings, though online subjects show slightly more variance.

Common pitfalls

• Treating the centipede as proof that rationality is wrong. The result is that the joint assumption of rationality plus common knowledge of rationality plus standard preferences fails to predict observed play. Each assumption can be relaxed individually.
• Confusing equilibrium with prediction. The backward-induction outcome is a unique subgame-perfect equilibrium, but humans rarely play SPE in dynamic games. SPE is normative, not always descriptive.
• Assuming all subjects play the same strategy. Heterogeneity is the rule: 7% take immediately, 8% cooperate to the end, the rest somewhere in between.
• Forgetting payoff scale matters. Centipede experiments with stakes of $25 produce different cooperation depth than with stakes of $25,000. High-stakes versions (Palacios-Huerta-Volij used roughly $50) still show cooperation but less.
• Reading "cooperation" as kindness. Passing in the centipede is a profit-maximising bet under bounded-rationality assumptions, not necessarily an altruistic act. Behavioural mechanisms differ.
• Equating with prisoner's dilemma. The centipede tests backward induction; the PD tests dominance. The failures are structurally different and have different theoretical responses.
• Assuming game-theory training cures the puzzle. Palacios-Huerta found chess players still cooperated; theoretical sophistication shrinks but does not eliminate the gap.