What is the Centipede Game in Game Theory?

The Centipede Game is a classic extensive-form game in behavioral game theory that highlights the paradox between individual self-interest and collective cooperation. While standard game theory mathematically predicts that rational players will end the game on the very first move, real-world behavioral experiments show that human players frequently cooperate to secure much higher payouts. This article explains the rules of the Centipede Game, the mathematical paradox of backward induction, and what actual player behavior teaches us about human psychology and trust.

How the Centipede Game is Played

The Centipede Game is played by two players who take turns deciding whether to “take” a larger share of a growing pot of money, or “pass” the choice to the other player.

The game begins with a small pool of money. If a player chooses to “pass,” the total amount of money increases, and the other player gets the next turn. However, if a player chooses to “take,” they receive the larger portion of the current pot, the other player receives a smaller portion, and the game immediately ends. The game has a pre-determined, finite number of rounds, which visually resembles a centipede when drawn out as a decision tree.

The Paradox of Backward Induction

According to classical game theory, players are perfectly rational, self-interested, and assume the other player is the same. Under these assumptions, the game is solved using a method called backward induction—reasoning backward from the end of the game to the beginning:

1. The Final Round: If the game reaches the very last turn, the player making the choice will choose “take” because they get a higher payoff than if they “passed.”
2. The Second-to-Last Round: Knowing that the final player will inevitably choose “take” on the next turn, the player in the second-to-last round will choose “take” to secure a better payoff now rather than get less on the next turn.
3. The Cascade: This logical deduction cascades backward through every single turn of the game.

Ultimately, backward induction leads to a Nash Equilibrium where Player 1 chooses “take” on the very first turn of the game. Both players walk away with the smallest possible payout, missing out on the substantial rewards of mutual cooperation.

Behavioral Game Theory and Real Human Behavior

When behavioral economists test the Centipede Game in laboratory settings with real people, the mathematical prediction of immediate defection rarely occurs. Instead, researchers observe the following behavioral patterns:

• Altruism and Trust: Players regularly choose to “pass” for several rounds, demonstrating a level of trust that the other player will not immediately betray them.
• Reciprocity: Players reward cooperation with cooperation. As long as one player continues to pass, the other is likely to do the same, allowing the pot to grow.
• Late Defection: While players do cooperate, they rarely play until the absolute end of the game. Cooperation typically breaks down, and one player chooses to “take” a few steps before the final round.

The Centipede Game is a cornerstone of behavioral game theory because it proves that human decision-making is not guided solely by cold, mathematical rationality. Factors such as trust, social norms, reciprocity, and a desire for fairness heavily influence economic choices, often leading to better financial outcomes than pure self-interest would dictate.