We study the strategic impact of players’ higher-order uncertainty over the observability of actions in general two-player games. More specifically, we consider the space of all belief hierarchies generated by the uncertainty over whether the game will be played as a static game or with perfect information. Over this space, we characterize the correspondence of a solution concept which captures the behavioral implications of Rationality and Common Belief in Rationality (RCBR), where ‘rationality’ is understood as sequential whenever the game is dynamic. We show that such a correspondence is generically single-valued, and that its structure supports a robust refinement of rationalizability, which often has very sharp implications. For instance: (i) in a class of games which includes both zero-sum games with a pure equilibrium and coordination games with a unique efficient equilibrium, RCBR generically ensures efficient equilibrium outcomes (eductive coordination); (ii) in a class of games which also includes other well-known families of coordination games, RCBR generically selects components of the Stackelberg profiles (Stackelberg selection); (iii) if it is commonly known that player 2’s action is not observable (e.g., because 1 is commonly known to move earlier, etc.), in a class of games which includes all of the above RCBR generically selects the equilibrium of the static game most favorable to player 1 (pervasiveness of first-mover advantage).
We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players' behavior deviates from rationality. Instead we assume that there exists a probability p such that all players play rationally with at least probability p, and all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information, and they can be applied to observed frequencies of play to derive a measure p that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.Keywords: strategic interaction, correlated equilibrium, robustness to bounded rationality, approximate knowledge, incomplete information, measure of rationality, experiments. JEL Classification: C72, D82, D83.
Predictions under common knowledge of payoffs may differ from those under arbitrarily, but finitely, many orders of mutual knowledge; Rubinstein's (1989) Email game is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for all types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. This paper studies how a wide class of departures from common belief in rationality impact Weinstein and Yildiz's discontinuity. We weaken ICR to ICR λ , where λ is a sequence whose term λ n is the probability players attach to (n − 1)th-order belief in rationality. We find that Weinstein and Yildiz's discontinuity remains when λ n is above an appropriate threshold for all n, but fails when λ n converges to 0. That is, if players' confidence in mutual rationality persists at high orders, the discontinuity persists, but if confidence vanishes at high orders, the discontinuity vanishes.
We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players' behavior deviates from rationality. Instead we assume that there exists a probability p such that all players play rationally with at least probability p, and all players believe, with at least probability p, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of p-rational outcomes, which we define and characterize for arbitrary probability p. We then show that this set varies continuously in p and converges to the set of correlated equilibria as p approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The p-rational outcomes are easy to compute, also for games of incomplete information, and they can be applied to observed frequencies of play to derive a measure p that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.Keywords: strategic interaction, correlated equilibrium, robustness to bounded rationality, approximate knowledge, incomplete information, measure of rationality, experiments. JEL Classification: C72, D82, D83.
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