Payoff Information and Self-Confirming Equilibrium
In a self-confirming equilibrium, each player correctly forecasts the actions that opponents will take along the equilibrium path, but may be mistaken about the way that opponents would respond to deviations. Intuitively, this equilibrium concept models the possible steady states of a learning process in which, each time the game is played, players observe only the actions played by their opponents (as opposed to the complete specification of the opponents' strategies) so that they need never receive evidence that their forecasts of off-path play are incorrect. 3 Because self- confirming equilibrium (henceforth "SCE") allows beliefs about off-path play to be completely arbitrary, it (like Nash equilibrium) corresponds to a situation in which players have no prior information about the payoff fimctions of their opponents.4 This may be a good approximation of some real-world situations; it is also the obvious way to model play in game theory experiments in which subjects are given no itiormation about opponents' payoffs. In other cases, both in the real world and in the laboratory, it seems plausible that players do have some prior information about their opponents' payoffs. The goal of this paper is to develop a more restrictive version of SCE that incorporates the effects of such prior information. In carrying out this program, a key issue is what sort of prior itiormation about payoffs should be considered. It is well known that predictions based on common certainty of payoffs are not robust to even a small amount of uncertainty. Following Fudenberg, Kreps and Levine (1987), we are interested in the strongest possible assumption that is robust to payoff uncertainty. Past work suggests that this assumption should be that payoffs are almost common certainty in the sense of Monderer and Samet (1989).5 Therefore we start by developing a preliminary concept -- rationalizability at reachable nodes -- that is robust and incorporates almost common certainty of the payoffs. In particular, we suppose that players believe that their opponents' actions will maximize their presumed payoff fbnctions so long as the opponents have not been observed to deviate from anticipated play. However, players do not use the prior payoff iniiormation to restrict their beliefs about the play of opponents who have already been observed to deviate from expected play. Intuitively, this corresponds to players supposing that such deviations are signals that the deviator's payoff finction is different than had been expected. (This assumes, in addition to almost common certainty of the payoffs, that the payoffs are determined independently, so that the signal refers only to the deviator's payoffs. We discuss the issue of independence in section 6.) We verify in section 3 that this intuition is correct and that this concept is robust; in section 4 we show that it does correspond to assuming almost common certainty of payoffs and independence. To capture the idea of SCE, namely that play corresponds to the steady state of a learning process in which the path is observed each time the game is played, we add the assumption that the path of play is public information. This is in the spirit of but stronger than, the assumption underlying self- confirming equilibrium, which is that each player knows the path of play. For simplicity, we also impose the assumption that players' beliefs concerning their opponents' play correspond to independent randomizations. (As mentioned earlier, independence is discussed further in section 6.) Combining these assumptions lead to rationalizable self-confirming equilibrium, or "RSCE." If we think of equilibrium as describing the steady state of a random- matching process of the sort used in most game theory experiments, then there are many agents, who are allocated into groups, one for each player role in the game. The observations of different agents in the same player role may differ depending on the actions they take. If itiormation about the aggregate distribution of outcomes is not available to the subjects, as it is not in most experiments, and players observe only the outcomes in their own matches, then the appropriate notion of SCE is Fudenberg and Levine's (1993a) notion of a heterogeneous SCE.G The corresponding notion with prior payoff information is a heterogeneous RSCE. Perhaps the best motivation and illustration of our ideas is a pair of experiments by Prasnikar and Roth  on the "best-shot" game. in which two players sequentially decide how much to contribute to a public good. The backwards-induction solution to this game is for player 1 to contribute nothing and player 2 to contribute 4; this is also the only RSCE. There is also an imperfect Nash equilibrium in which player 1 contributes 4 and player 2 contributes nothing. Prasnikar and Roth ran two treatments of this game. In the first one, players were informed of the fimction determining opponents' monetary payoffs. Here, by the last few rounds of the experiment the first movers had stopped contributing, which is the prediction made by RSCE. In the second treatment, subjects were not given any irdlormation about the payoffs of their opponents. In this treatment even in the later rounds of the experiment many first movers contributed to the public good. This is not consistent with RSCE, but it is consistent with an (approximate, heterogeneous) SCE (Fudenberg and Levine  7 ). Thus these experiments provide evidence that information about other players' payoffs makes a difference, and that this difference corresponds to the distinction between SCE and RSCE.8 Papers by Rubinstein and Wolinksy  and Greenberg , like ours, are based on the idea that players form their forecasts of opponents play using prior information both about the opponents' payoffs and about the realized outcomes when the game is played. Both these papers, unlike ours, consider common certainty of rationality, while our desire for robust predictions leads us to consider almost common certainty instead. Like us, Greenberg works in an extensive-form context, with players observing terminal nodes but not intended off-path play; Rubinstein and Wolinksy consider strategic-form games and general "signal fictions". We should make clear from the outset that, although this paper is motivated by the learning-theoretic approach to equilibrium in games, we do not here provide an explicit learning-theoretic foundation for our concepts. We are confident that such foundations can be constructed by, for example, incorporating restrictions on the priors into the steady-state learning model of Fudenberg and Levine [1993 b], but we have not checked the details.
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