Iterated Expectations with Common Beliefs

This paper generalizes a result by Samet concerning iterated expecta- tions and common priors. When a player in some state of the world is allowed to ascribe probability zero to that state, something not allowed in Samet s framework, iterated expectations may not converge, and when they do, common knowledge of their limit may not characterize a common prior. It is shown here that replacing common knowledge with common belief, convergence is still lost in general, but when it obtains, the full characterization is restored.

Uniform Topologies on Types

We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet (1989)). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris (2006)). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.Rationalizability, Incomplete information, Higher-order beliefs, Strategic topology, Electronic mail game

Uniform topologies on types

We study the robustness of interim correlated rationalizability to perturbations of higher-order beliefs. We introduce a new metric topology on the universal type space, called uniform weak topology, under which two types are close if they have similar first-order beliefs, attach similar probabilities to other players having similar first-order beliefs, and so on, where the degree of similarity is uniform over the levels of the belief hierarchy. This topology generalizes the now classic notion of proximity to common knowledge based on common p-beliefs (Monderer and Samet 1989). We show that convergence in the uniform weak topology implies convergence in the uniform strategic topology (Dekel, Fudenberg, and Morris 2006). Moreover, when the limit is a finite type, uniform-weak convergence is also a necessary condition for convergence in the strategic topology. Finally, we show that the set of finite types is nowhere dense under the uniform strategic topology. Thus, our results shed light on the connection between similarity of beliefs and similarity of behaviors in games.Rationalizability, incomplete information, higher-order beliefs, strategic topology, electronic mail game

Characterizing the Strategic Impact of Misspecified Beliefs

Previous research has established that the predictions of game theory are quite sensitive to the assumptions made about the players’ beliefs. We evaluate the severity of this robustness problem by characterizing conditions on the primitives of the model—the players’ beliefs and higher-order beliefs about the payoff-relevant parameters—for the behavior of a given Harsanyi type to be approximated by the behavior of (a sequence of) perturbed types. This amounts to providing belief-based characterizations of the strategic topologies of Dekel, Fudenberg, and Morris (2006). We apply our characterizations to a variety of questions concerning robustness to perturbations of higher-order beliefs, including genericity of types that are consistent with a common prior, and we investigate the connections between our notions of robustness and the notion of common p-belief of Monderer and Samet (1989)

Subjective beliefs and coordination in games

An individual facing a problem of choice under uncertainty behaves optimally given beliefs about the relevant unknown factors. When these include other agents\u27 choices, some beliefs are more reasonable than others. The standard equilibrium notions in game theory require that, at a minimum, a player\u27s beliefs agree with other players\u27 prescribed behavior. In equilibrium, then, individual behavior is both optimal and coordinated. Repeated interaction enables coordination based on observation of past moves. In one-shot situations, coordination arises from exchange of information before actual choices are made. The first chapter of this dissertation is concerned with the question of whether coordination in infinitely repeated games is affected by the natural assumption that observation of past moves is imperfect and players choose not to recall far past events. We show that in many cases this assumption has the strong consequence of completely undoing the possibility of coordination. In the second chapter, we consider one-shot situations where players can exchange information through an external mediator before choosing, under the assumption that communication is restricted in that the mediator can only make public recommendations. The result is that the possibility of coordination is completely insensitive to this restriction, provided rich sets of messages are available. In both of the first chapters it is assumed that each player attaches a utility number to every outcome and then ranks strategies by the mathematical expectations of their utility. It is further assumed informally that each player believes other players similarly assess the desirability of their strategies. The third chapter deals with the foundational problem of what these assumptions mean in terms of the individual\u27s personal attitude towards uncertain prospects. The main results—appropriate extensions of classical results in one-agent theory—establish rules on subjective preference which fully characterize the assumption that each player is an expected utility maximizer, believes every other player is, believes every other player believes every other player is, and so on

Subjective expected utility in games

This paper extends the scope of Savage’s subjective approach from decision problems under exogenous uncertainty to choice in strategic environments. In these environments the decision maker understands the uncertainty she is facing is affected by other decision makers in a similar situation. This contrast with classical decision making complicates the appropriate specification of the state space: as it is not exogenous, the uncertainty concerns not only the other decision makers ’ choices but also the behavioral rationale behind them. First, this problem is solved, constructing the state space explicitly — using hierarchies of preference relations — and then showing that this space indeed contains every relevant aspect of the decision maker’s uncertainty. Since no restriction on preferences is imposed a priori, these results enable the analysis of behavior in games under any axiomatic structure. Second, conditions on preferences are characterized which imply that the decision maker behaves as if she is sure each other agent has preferences satisfying certain axioms, is himself sure each other agent’s preferences satisfy certain axioms, and so on. Third, such characterization is provided for Savage’s axioms. It is shown that a sequence of preference relations uniquely identifies the decision maker’s utilities and beliefs, and also tells whether according to these beliefs each other agent is an expected utility maximizer, believes each other agent is, and so on

A Robustness Result for Rationalizable Implementation

This paper investigates the extent to which rationalizable implementation depends on the implicit common knowledge restrictions of traditional models. Under general conditions such restrictions are without loss of generality: If a mechanism M implements a social choice function f on a type space X, then M implements f assuming only that agents’ k-order beliefs are among those described by an open set B containing X, while higher-order beliefs are unrestricted. The result is applied to direct implementation on payoff type spaces, and related to incentive compatible implementation

Subjective beliefs and coordination in games

An individual facing a problem of choice under uncertainty behaves optimally given beliefs about the relevant unknown factors. When these include other agents\u27 choices, some beliefs are more reasonable than others. The standard equilibrium notions in game theory require that, at a minimum, a player\u27s beliefs agree with other players\u27 prescribed behavior. In equilibrium, then, individual behavior is both optimal and coordinated. Repeated interaction enables coordination based on observation of past moves. In one-shot situations, coordination arises from exchange of information before actual choices are made. The first chapter of this dissertation is concerned with the question of whether coordination in infinitely repeated games is affected by the natural assumption that observation of past moves is imperfect and players choose not to recall far past events. We show that in many cases this assumption has the strong consequence of completely undoing the possibility of coordination. In the second chapter, we consider one-shot situations where players can exchange information through an external mediator before choosing, under the assumption that communication is restricted in that the mediator can only make public recommendations. The result is that the possibility of coordination is completely insensitive to this restriction, provided rich sets of messages are available. In both of the first chapters it is assumed that each player attaches a utility number to every outcome and then ranks strategies by the mathematical expectations of their utility. It is further assumed informally that each player believes other players similarly assess the desirability of their strategies. The third chapter deals with the foundational problem of what these assumptions mean in terms of the individual\u27s personal attitude towards uncertain prospects. The main results—appropriate extensions of classical results in one-agent theory—establish rules on subjective preference which fully characterize the assumption that each player is an expected utility maximizer, believes every other player is, believes every other player believes every other player is, and so on