Incomplete Information in Repeated Coordination Games 2 November 2007

Asymmetric information can help achieve a good equilibrium in repeated coordination games. If there is a small probability that one player can play only one of a continuum of moves, that player can pretend to be of the constrained type, and other players will coordinate with him. Dan R. and Catherine M. Dalton Professor, Department of Business Economics and Public Policy, Kelley School of Business, Indiana University. Visitor (2007-2008), Nuffield College, Oxford University. Office: 011-44-1865 554-163 or (01865) 554-163. Nuffield College, Room C3, New Road, Oxford, England, OX1 1NF. Erasmuse@indiana.edu. http://www.rasmusen. org. This paper: Http://www.rasmusen. org/ papers/goodtype-rasmusen.pdf. I thank the Indiana Business Economics Brown Bag Lunch and Michael Baye for helpful comments. 1.Introduction It is well known that coordination games have multiple equilibria, depending on player expectations, even if one equilibrium is pareto superior and players can communicate. This multiplicity is present even in the oneshot game, and just gets worse when the game is repeated. Few of the refinements of Nash equilibrium that have been suggested in the context of other kinds of games help with coordination games, and none has gained more than minimal acceptance. The problem of multiple equilibria in coordination games has attracted attention from various authors. One way to try to predict which equilibrium is played out is to use the behavioral idea of “focal points” from Thomas Schelling (1960): that a human’s attention is drawn to certain equilibria because they look “different”. Thus, if a game’s equilibria had payoffs of (1,1), (2,2), and (100,100), the focal point would be (100,100). This is a difficult notion to formalize, though: if the alternatives were (1,1), (99,99), and (100,100) would we predict that that the players would end up at (1,1) because it is the most distinctive? Clearly, the idea of the focal point is important. Philosopher David Lewis (1965) divides the idea of the salience of a choice into two parts. The choice has “primary salience” to a player if he believes it is salient to he himself; it has “secondary salience” if he believes it has salience to other players. Judith Mehta, Chris Starmer & Robert Sugden (1994, p. 661) add “Schelling salience” to primary and secondary salience, as a choice that “seems obvious or natural to people who are looking for ways of solving coordination problems. ” In their article they report results of experiments trying to distinguish between primary salience– the answers subjects gave to questions when there was no reward for coordination– and secondary or Schelling salience– the answers when subjects were rewarded for successful coordination. They found that subjects indeed were picking with an eye towards what other subjects would pick; for example when asked to write down any day of the year, only 6% of the first set of subjects answered December 25, but 44% did when they were rewarded for successful cooperation. A second approach tries to use derive the unique equilibrium from rationality. Gauthier (1975, p. 201) defines the “Principle of Coordination” as that “in a situation with one and only one outcome which is both optimal and a best equilibrium, if each person takes every other person to be rational and to share a common conception of the situation, it is rational for each person to perform that action which has the best equilibrium as one of the possible outcomes. ” Bacharach ( 1993) Harsanyi & Selten (1988) and Janssen (2000, 2001) have pursued this approach, trying to add axioms for rational behavior that require players to avoid dominated equilibria. Repeating the game does not reduce the number of equilibria, but it does introduce a new angle: finding the optimal way to play a game starting without a convention as to the equilibria. What is the optimal strategy for the two players if they must first grope their ways towards coordination by guessing what the other player will do before they end up at the same action and use it thereafter? That is the project in Crawford& Haller (1990), who find a learning procedure that converges in finite time.