: We provide a model for reasoning about knowledge and probability together. We allow explicit mention of probabilities in formulas, so that our language has formulas that essentially say "according to agent i, formula ' holds with probability at least b." The language is powerful enough to allow reasoning about higher-order probabilities, as well as allowing explicit comparisons of the probabilities an agent places on distinct events. We present a general framework for interpreting such formulas, and consider various properties that might hold of the interrelationship between agents' probability assignments at different states. We provide a complete axiomatization for reasoning about knowledge and probability, prove a small model property, and obtain decision procedures. We then consider the effects of adding common knowledge and a probabilistic variant of common knowledge to the language. A preliminary version of this paper appeared in the Proceedings of the Second Conference on T...

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Understanding knowledge is a fundamental issue in many disciplines. In computer science, knowledge arises not only in the obvious contexts (such as knowledgebased systems), but also in distributed systems (where the goal is to have each processor "know" something, as in agreement protocols). A general semantic model of knowledge is introduced, to allow reasoning about statements such as "He knows that I know whether or not she knows whether or not it is raining." This approach more naturally models a state of knowledge than previous proposals (including Kripke structures). Using this notion of model, a model theory for knowledge is developed. This theory enables one to interpret the notion of a "finite amount of information". A preliminary version of this paper appeared in Proc. 25th IEEE Symp. on Foundations of Computer Science, 1984, pp. 268--278. This version is essentially identical to the version that appears in Journal of the ACM 38:2, 1991, pp. 382--428. y Part of th...

In practice, scoring rules elicit good probability estimates from individuals, while betting markets elicit good consensus estimates from groups. Market scoring rules combine these features, eliciting estimates from individuals or groups, with groups costing no more than individuals.

In their seminal paper, Mertens and Zamir (1985) proved the existence of a universal Harsanyi type space which consists of all possible types. Their method of proof depends crucially on topological assumptions. Whether such assumptions are essential to the existence of a universal space remained an open problem. We answer it here by proving that a universal type space does exist even when spaces are defined in pure measure theoretic terms. Heifetz and Samet (1996) showed that coherent hierarchies of beliefs, in the measure theoretic case, do not necessarily describe types. Therefore, the universal space here differs from all previously studied ones, in that it does not necessarily consist of all We study here the foundations of that part of the theory of games with incomplete information that deals with players ’ beliefs. We study it in the broadest and most natural setup, that of probability (or measure) theory without any topological notions, which have always been used for this purpose until now. We show that even under this general setup

We study a coordination game with randomly changing payoffs and small frictions in changing actions. Using only backwards induction, we find that players must coordinate on the risk dominant equilibrium. More precisely, a continuum of fully rational players are randomly matched to play a symmetric 2 \Theta 2 game. The payoff matrix changes over time according to some Brownian motion. Players observe these payoffs and the population distribution of actions as they evolve. The game has frictions: opportunities to change strategies arrive from independent random processes, so that the players are locked into their actions for some time. We solve the game using only backwards induction. As the frictions disappear, each player ignores what the others are doing and switches at her first opportunity to the risk dominant equilibrium. History dependence emerges in some cases when frictions remain positive. As an application we show how frictions and aggregate cost shocks can lead to the selecti...

Each player in an infinite population interacts strategically with a finite subset of that population. Suppose each player's binary choice in each period is a best response to the population choices of the previous period. When can behaviour that is initially played by only a finite set of players spread to the whole population? This paper characterizes when such contagion is possible for arbitrary local interaction systems (represented by general undirected graphs). Maximal contagion occurs when local interaction is sufficiently uniform and there is low neighbour growth, i.e., the number of players who can be reached in k steps does not grow exponentially in k.

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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. This paper develops a refinement of self-confirming equilibrium in which players use information about opponents' payoffs in forming beliefs about the way that opponents play off of the equilibrium path. We show that this concept is robust to payoff uncertainty. We also discuss its relationship to other concepts, and show that it is closely related to assuming almost common certainty of payoffs in an epistemic model with independent beliefs. Journal of Economic Literature Classification Numbers C72, D84. 2 1.