The Centipede Game
The centipede game was first introduced by Rosenthal (1982) and has subsequently been studied by Binmore(1987), Kreps (1990) Reny (1988) and many others in different modified forms. The original version of the game consisted of a sequence of a hundred moves (hence the name centipede) with linearly increasing payoffs.
The centipede game is an extensive-form game in which two players alternately get a chance to take the larger portion of a contiually increasing pile of money. As soon as a player takes, the game ends with that player getting the larger portion of the pile while the other player gets the smaller portion. Passing strictly decreases a player’s payoff if the opponent takes on the next move. If the opponent also passes, the two players are faced with the same choice situation with reversed roles and increased payoffs. The game has a finite number of moves which is known in advance to both players.
In the above diagram, a 1 at a black circle ("decision node") denotes a decision opportunity for player 1. A 2 at a decision node tells us that person 2 can make a decision here. The top number at the end of each vertical line is a payoff for player 1 and the bottom number is a payoff for player 2.Player 1 has the first move: if she chooses D, both players get 1; if she chooses A, the opportunity to make a decision passes to player 2. Player 2 has the second move: if he chooses D, player 1 gets payoff of D and he gets 3; if he chooses A, the opportunity to make a decision passes to player 1. And so on to the end of the game tree. If both players always choose A, they both receive payoff of 100 at the end of the game tree.
We have just observed that both players receive payoff of 100 if both players always choose A rather than D. Note also that both players receive payoff of 1 if player 1 chooses D on his first move.
What does game theory predict will happen?
Game Theory predicts that Player 1 will choose D in his first move and thus both players will receive payoff of 1!
How can that be the case? Click here to learn how using backward induction to solve a game in extensive form leads to the prediction player 1 will choose D on his first move. Furthermore,since all Nash equilibria make the same outcomes prediction, any usual refinements of Nash equilibrium also make the same prediction. So we have an unambiguous theoretical prediction.
Typical Experimental Results
Studying actual behaviour in different versions ( a four move, six move, and high payoff versions) of the centipede game, McKelvey and Palfrey (1992) found that subjects rarely followed the theoretical predictions. In fact in only 7% of the four-move games, 1% of the six-move games, and 15% of the high payoff games did the first player choose to take on the first move. Similar results were reported by Nagel and Tang (1998).
Possible Explanations of "Irrational" Behavior
There are two types of explanation to account for the divergence. The first assumes that the subject pool contains a certain proportion of altruists who place a positive weight in their utililty function on the payoff of their opponent. Also to the extent that selfish players believe that there is some probability that other players are altruistics, they have an incentive to mimic altruistic behaviour by passing. The second explanation considers the possibility of action errors. Errors in action, or ‘noisy’ play, may result from subjects experimenting with different strategies. Or simply from subjects pressing the wrong key.
More Discussions on Variants of the Centipede Game
Binmore,K. & McCarthy,J. & Ponti,G. & ..., (1999). "A backward induction experiment," Working papers 34, University of Wisconsin Madison - Social Systems
R.McKelvey and T.Palfrey (1992) “ An experimental study of the centipede game,” Econometrica 60