Centipede Game

The centipede game is an extensive-form game in game theory in which two players alternately get a chance to take the larger share of a slowly increasing money stash. It is arranged so that if a player passes the stash to his opponent who then takes the stash, the player receives a smaller amount than if he had taken the pot. The centipede game concludes as soon as a player takes the stash, with that player getting the larger portion and the other player getting the smaller portion. The game has a pre-defined total number of rounds, which are known to each player in advance.

While not as well-known as the famed Prisoner’s Dilemma, the centipede game also highlights the conflict between self-interest and mutual benefit with which people have to grapple. It was first introduced by psychologist Robert Rosenthal in 1982. “Centipede game” is so-called because its original version consisted of a 100-move sequence.

As an example, consider the following version of the centipede game involving two players, Jack and Jill. The game starts with a total $2 payoff. Jack goes first, and has to decide if he should "take" the payoff or "pass." If he takes, then he gets $2 and Jill gets $0, but if he passes, the decision to “take or pass” now must be made by Jill. The payoff is now increased by $2 to $4; if Jill takes, she gets $3 and Jack gets $1, but if she passes, Jack gets to decide whether to take or pass. If she passes, the payoff is increased by $2 to $6; if Jack takes, he would get $4, and Jill would get $2. If he passes and Jill takes, the payoff increases by $2 to $8, and Jack would get $3 while Jill got $5. The game continues in this vein for a total of 100 rounds. If both players always choose to pass, they each receive a payoff of $50 at the end of the game. Note that the money is contributed by a third party and not by either player.

What does game theory predict? Using backward induction – which is the process of reasoning backward from the end of a problem – game theory predicts that Jack (or the first player) will choose to take on the very first move and both players will receive a $1 payoff.

In experimental studies, however, only a very small percentage of subjects chose to take on the very first move. This discrepancy could have several explanations. One reason is that some people are altruistic, and would prefer to cooperate with the other player by always passing, rather than taking down the pot. Another reason is that people may simply be incapable of making the deductive reasoning necessary to make the rational choice predicted by the Nash equilibrium. The fact that few people take the stash on the very first move is not too surprising, given the small size of the starting payoff when compared with the increasing payoffs as the game progresses.