Brain Teasers · The Growing Pot
Centipede Game Puzzle - Why Theory Says Take Now
The centipede game as a story: a pot that doubles on every pass, and logic that says take it immediately.
What is the centipede game?
The centipede game is a sequential bargaining game introduced by the economist Robert Rosenthal in 1981. Two players alternate moves over a fixed, known number of rounds. At each turn the current player can either take a pot of money, ending the game immediately with an unequal split favouring the taker, or pass, which grows the pot and hands the decision to the other player. If both players keep passing all the way to the final round, the total payoff is far larger than what either player could get by taking early.
Backward induction gives a sharp and famously unattractive answer. At the very last round, the player who must move there should take, since there is no future round left to protect by passing. Knowing this, the player one move earlier should also take rather than pass into a round where the other player will take from them anyway. That reasoning propagates backward through every round to the very first move - so a game that could be worth an enormous amount to both players if they cooperated is theoretically supposed to end on the very first turn.
This result is one of the most studied examples of the gap between game-theoretic prediction and observed human behaviour. In laboratory experiments, most people do not take at the first opportunity - they pass several rounds into the game before someone defects - which suggests that the strict, unlimited common knowledge of rationality the backward-induction argument requires rarely holds in practice, and that reputational or trust considerations shape play even in a game with no repeated interaction.
The centipede game is often used to probe the limits of backward induction and common knowledge of rationality as descriptive assumptions: it demonstrates that the same tool that cleanly solves a game like the pirates' vote can produce a theoretically airtight prediction that virtually no real player follows.
How it works
Four pounds sits on the table, 3 on Ann's side. She can take her 3 and end it, or pass - each pass doubles the pot and hands the choice across to Ben. Six rounds, reaching 256 if everyone passes, and whoever takes gets the larger share. Both see the whole ladder up front. The story asks how far the pot actually gets. You lock in a guess, then step through the answer and breakdown.
How scoring works
No grading - your guess is displayed back beside the answer. The story teasers share one lifetime unlock for free users.
The key insight
Start at the last move: the final mover takes, since there is no future to protect. The player before, knowing that, takes first - and the logic chains all the way back, so Ann takes the 3 on move one and the pot never grows. The collapse depends on common knowledge of rationality running the full six levels deep; in experiments even a little doubt keeps people passing well into the middle of the ladder.
The full step-by-step solution lives at /brain-teasers/centipede/solution.
A worked example
Round 6: Ben takes the larger share rather than let the game end paying him less. Round 5: Ann knows this, so passing hands Ben the take - she takes first. Each earlier node collapses identically, landing on Ann taking 3 at move one with 253 pounds of potential surplus never created.
Common mistakes
• Guessing 'they pass a few times then take' as the theoretical answer - that is the empirical result, not the equilibrium.
• Reasoning forward from move one instead of backward from the last node.
• Confusing this with the prisoner's dilemma - here nothing dominates; the unravelling comes from the known endpoint.
Why interviews test this
Interviewers want the unravelling argument stated confidently, then the acknowledgment that observed play differs and why - a test of treating models as tools. The written solution is at /brain-teasers/centipede/solution.
Play The Growing Pot on Outcry
The Growing Pot is free to play on Outcry, in the Brain Teasers section, and every run is timed and scored. Read the guide once, then play The Growing Pot and check your score against the strategy above. The guide book has a written guide for every other game, and the practice hub and mock assessments put the same skills under a real screen's clock.