On the centipede game with a social norm☆
Introduction
The centipede game, first introduced in Rosenthal (1981), is now a staple of game theory texts in illustrating the concept of backward induction. At each point in a discrete time, finite-period setting, two players take turns in deciding whether to take all that is on the table and stop the game; or to continue and take the risk of obtaining a smaller amount in the next period in the hope and expectation that the other player will do the same on her turn, and thereby furnish an opportunity to the initial mover of obtaining a larger amount by stopping the game in the period after next. Given the way the situation is formalized, the perverse magic of backward induction makes the lack of co-operation inevitable, and the unique subgame perfect equilibrium of the game, through the grammar of rationality, mandates that the initial player forego the possibility of a higher payoff in the future, and accept what is on offer in the very first turn. The fact that there is a last period, and that therefore there is no period subsequent to it, there is simply no option to the first mover but to stop the game as soon as she is able to do so. In extrapolating a particular form of rationality in regard to his or her “other”, a player is bound by necessity to act likewise.
Yet, experimental tests show human subjects rarely follow all that this stylized situation leads one to predict.2 As its twin, the prisoner’s dilemma game, the centipede game presents a conflict between self-interest and mutual benefit. If it could be enforced, both players would prefer to cooperate throughout the entire game, but a player’s rational self-interest, combined with his or her distrust, creates a situation whereby cooperation is seen from the inside as irrational and blind. Since the payoffs, even after cooperation for only a few periods, are so much larger than those obtained through immediate non-continuance and defection, the “rational” solution given by backward induction is seen as somewhat paradoxical, and has prompted debate over the relevance of the grammar of rationality, and the usefulness of the idealization.3
However, the whole point of the grammar4 is to stick to the parameters of the game-theoretic situation as it has been framed, and not change it by appealing to more “realistic” considerations of language and communication, or to the boundedness of rationality. But if one is to avoid thick formalizations of inter-subjectivity, one cannot abstain from it altogether. The only relationship between the two players in the centipede game is what the objectivity of the situation forces on them, and as such it is more akin to a competitive model rather than a game-theoretic one.5 In this paper, we keep to the basic parameters of the situation–rationality as subgame perfection, full information and backward induction–but introduce agent interdependence in the form of a random cost that each player would pay for ending the game, one that depends on some parameter embodying how closely related the players are to each other, and one that is an increasing function, in the sense of first-order stochastic dominance, of the number of periods the game is played. We then have an optimal stopping game, one in which the length of the game is explicitly brought into play. This leads to results suggesting that different levels of cooperation can then be achieved as “rational” outcomes.6
We surely do not argue for these considerations as replacing other explanations but simply as a complementary consideration in the direction of a player’s payoff depending not only on his action but on some aspect of the situation and the behavior of others who give it definition. It injects into the situation some point of reference (say, a quid-pro-quo norm) that dictates, again consistent with the grammar of the linguistic register, each player to incorporate how his or her other follows this reference. How an alternative perception of what is reasonable and rational leads to outcomes that explain away the puzzles.7 After a self-contained description of the centipede game in Section 2, our reformulation of it in Section 3, and the implications of our results in Section 4, we briefly connect the social-norm intuition introduced in this paper to the literature on the golden-rule and to the nature of the investigation that we have pursued.
Section snippets
A classical centipede game
Consider a -legged centipede game with two players, and with () the set of all odd (even) numbers less than or equal to . Let denote the set of stages of the game and denote a generic index chosen from . Player 1 starts first, and at the decision node , player has the opportunity to choose either (stop) or (continue) if no player previously chooses . The game ends once is chosen, or alternatively, all legs are passed. The payoff structure in terms of a
A reformulation with a social norm
As mentioned in the introduction, the SPNE prediction fails both on intuitive grounds and in experimental results. We now modify the centipede game by introducing inter-subjectivity into the situation by virtue of some social norm that can be seen as a formalization of the golden rule. When making a decision, players trade off private benefit and the potential cost of violating this norm, where the cost depends on both the past cooperation and the (subjective) proximity of the players. In
Discussion
Unlike the classical centipede game where the unique SPNE outcome is the most inefficient outcome, Theorem 1, Theorem 2 bring out the relevance of how the particular (objective) situation is modeled. The analysis of backward induction relies only on the expected value of the random cost function. Note that deterministic costs are just a special case of random costs since a constant random variable is itself a random variable. Indeed, as a direct consequence of the assumption that is
Conclusion
In this note, we have formalized inter-subjectivity in the centipede game by introducing random costs to allow some room for reciprocity considerations. One could also view the costs we have introduced in terms of players’ traits into their payoffs, as formalized in recent work of Khan et al. (2013) that builds on identity considerations introduced in Akerlof and Kranton (2000). Such considerations give yet another lever to investigate the possibility of cooperation in situations when
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The authors are grateful to Hülya Eraslan for her May 7, 2014 day-long workshop on “bargaining as a sequential game”, and to her and Omar Khan for stimulating discussion. We also thank the referee for his/her careful reading and for a variety of expository suggestions. Errors are solely the authors’.