The Centipede Game in game theory is a classic example of a game in extensive form that highlights the power and limitations of the concept of backward induction.
It was first introduced by Robert W. Rosenthal in 1981.
The Setup
The game is played between two players, Player 1 and Player 2, and involves a series of turns. The game starts with a small sum of money (e.g., $1). At each turn, a player has two choices:
- Take: The player takes a slightly larger share of the money and ends the game.
- Pass: The player passes the turn to the other player, and the total amount of money in the pot increases.
The game is structured so that the payoff for taking is always slightly better than the payoff the player would receive if they passed and the opponent then took the money. However, if both players continue to pass, the total payoff for both of them at the end of the game is significantly higher.
The Paradox of Backward Induction
The standard game theory solution for the Centipede Game is derived using a process called backward induction. This involves reasoning from the end of the game backward to the beginning:
- Final Turn: Let’s say the game has a final, 100th turn. Player 2 is on this turn. They have two choices: take a large sum or pass to Player 1 for the very last turn. Player 2’s payoff for taking is greater than their payoff for passing (which would result in Player 1 taking an even larger sum). Therefore, a rational Player 2 will always take the money on the last turn.
- Penultimate Turn: Now consider the 99th turn, where Player 1 is making a decision. Player 1 knows, from the previous step, that if they pass, Player 2 will take the money on the next turn. Player 1’s payoff from taking the money on the 99th turn is slightly greater than their payoff from passing and letting Player 2 take the money on the 100th turn. Therefore, a rational Player 1 will take the money on the 99th turn.
- And so on… This logic continues all the way back to the very first move. Player 2, at turn 2, knows that if they pass, Player 1 will take the money on turn 3. So, Player 2 takes the money on turn 2. Player 1, at the very beginning of the game, knows that if they pass, Player 2 will take the money. Therefore, a perfectly rational Player 1’s optimal strategy is to take the money on the very first turn and end the game.
The paradox lies in the outcome: the game ends immediately, with both players receiving a very small payoff, even though there was a potential for a much larger, mutually beneficial outcome if they had both cooperated and passed the pot many times.
The Discrepancy with Human Behavior
The most fascinating aspect of the Centipede Game is the stark contrast between the theoretical prediction and what is observed in real-world experiments. When real people play the game, they almost never take the money on the first turn. Instead, they typically pass for several rounds, leading to a much larger pot, before one player finally takes the money.
This discrepancy challenges the core assumptions of classical game theory, particularly that players are perfectly rational and act only to maximize their own immediate payoff. The reasons for this observed behavior are a subject of ongoing research, but they are often attributed to:
- Bounded Rationality: Players are not capable of performing the complex backward induction calculation perfectly, especially in a long game.
- Trust and Reciprocity: Players may believe that their opponent is also cooperative and will pass the pot, hoping to reach a higher collective payoff.
- Social Norms: In many situations, cooperation and fairness are valued, which can override a purely selfish, “rational” strategy.
- Altruism: A player might pass to benefit the other player, especially if the pot is still small.
In conclusion, the Centipede Game serves as a powerful illustration of a fundamental paradox in game theory. While backward induction provides a logically sound solution, it fails to predict how people actually behave. It highlights the importance of psychological, social, and emotional factors—such as trust and altruism—in understanding strategic interactions, which are often overlooked by purely rational models.