Working Papers

Abstract. In this chapter we focus on the epistemic concept of common belief in future rationality (Perea (2011a)), which describes a backward induction type of reasoning for general dynamic games. It states that a player always believes that his opponents will choose rationally now and in the future, always believes that his opponents always believe that their opponents choose rationally now and in the future, and so on, ad infinitum. It thus involves infinitely many conditions, which might suggest that this concept is too demanding for real players in a game. In this chapter we show, however, that this is not true. For finite dynamic games we present some finite reasoning procedures that a player can use to reason his way towards common belief in future rationality.

Belief in the Opponents' Future Rationality (2011)

Abstract: For dynamic games we consider the idea that a player, at every stage of the game, believes that his opponents will choose rationally in the future. Not only this, we also assume that players, throughout the game, believe that their opponents always believe that their opponents will choose rationally in the future, and so on. This leads to the concept of common belief in future rationality, which we formalize within an epistemic model. Our main contribution is to present an iterative elimination procedure, backwards dominance, that selects exactly those strategies that can rationally be chosen under common belief in future rationality. The algorithm proceeds by successively eliminating strategies at every information set of the game. More specifically, in round k of the procedure we eliminate at a given information set h those strategies for player i that are strictly dominated at some player i information set h′ weakly following h, given the opponents' strategies that have survived at h′ until round k.

A Simple Bargaining Procedure for the Myerson Value (2010)
with Noemí Navarro

Abstract: We consider situations where the cooperation and negotiation possibilities between pairs of agents are given by an undirected graph. Every connected component of agents has a value, which is the total surplus the agents can generate by working together. We present a simple, sequential, bilateral bargaining procedure, in which at every stage the two agents in a link (i,j) bargain about their share from cooperation in the connected component they are part of. We show that, if the marginal value of a link is increasing in the number of links in the connected component it belongs to, then this procedure yields exactly the Myerson value payoff (Myerson, 1977) for every player.

Agreeing to Disagree with Lexicographic Prior Beliefs (2010)
with Christian W. Bach

Abstract: The robustness of Aumann’s seminal agreement theorem with respect to the common prior assumption is considered. More precisely, we show by means of an example that two Bayesian agents with almost identical prior beliefs can agree to completely disagree on their posterior beliefs. Secondly, a more detailed agent model is introduced where posterior beliefs are formed on the basis of lexicographic prior beliefs. We then generalize Aumann’s agreement theorem to lexicographic prior beliefs and show that only a slight perturbation of the common lexicographic prior assumption at some – even arbitrarily deep – level is already compatible with common knowledge of completely opposed posterior beliefs. Hence, agents can actually agree to disagree even if there is only a slight deviation from the common prior assumption.

A Foundation for Proper Rationalizability from an Incomplete Information Perspective (2010)
with Souvik Roy

Abstract: Proper rationalizability (Schuhmacher (1999), Asheim (2001)) is a concept that is based on two assumptions: (1) every player is cautious, i.e., does not exclude any opponent's strategy from consideration, and (2) every player respects the opponent's preferences, i.e., deems one opponent strategy to be infinitely more likely than another whenever the opponent prefers the one to the other. In this paper we provide a new foundation for proper rationalizability, by assuming that players have incomplete information about the opponent's utilities. We show that, if the uncertainty of each player about the opponent's utilities vanishes gradually in some regular manner, then the choices he can make under common belief in rationality are all properly rationalizable in the original game with no uncertainty about the opponent's utilities.

Commitment in Alternating Offers Bargaining (2009)
with Topi Miettinen

Abstract: We extend the Ståhl-Rubinstein alternating-offer bargaining procedure to allow players, prior to each bargaining round, to simultaneously and visibly commit to some share of the pie. If commitment costs are small but increasing in the committed share, then the unique outcome consistent with common belief in future rationality (Perea, 2010), or more restrictively subgame perfect Nash equilibrium, exhibits a second mover advantage. In particular, as the smallest share of the pie approaches zero, the horizon approaches infinity, and commitment costs approach zero, the unique bargaining outcome corresponds to the reversed Rubinstein outcome (δ/(1+δ),1/(1+δ)).

Algorithms for Cautious Reasoning in Games (2009)
with Geir Asheim

Abstract: We provide comparable algorithms for the Dekel-Fudenberg procedure, iterated admissibility and proper rationalizability by means of the concepts of preference restrictions and likelihood orderings. We apply the algorithms for comparing iterated admissibility and proper rationalizability, and provide a sufficient condition under which iterated admissibility does not rule out properly rationalizable strategies. Finally, we use the algorithms to examine an economically relevant strategic situation, namely a bilateral commitment bargaining game.