88,911 research outputs found

    Hierarchies of belief and interim rationalizability

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    In games with incomplete information, conventional hierarchies of belief are incomplete as descriptions of the players' information for the purposes of determining a player's behavior. We show by example that this is true for a variety of solution concepts. We then investigate what is essential about a player's information to identify behavior. We specialize to two player games and the solution concept of interim rationalizability. We construct the universal type space for rationalizability and characterize the types in terms of their beliefs. Infinite hierarchies of beliefs over conditional beliefs , which we call Delta-hierarchies, are what turn out to matter. We show that any two types in any two type spaces have the same rationalizable sets in all games if and only if they have the same Delta-hierarchies.Interim rationalizability, belief hierarchies

    HIERARCHIES OF BELIEF AND INTERIM RATIONALIZABILITY

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    In games with incomplete information, conventional hierarchies of belief are incomplete as descriptions of the players’ information for the purposes of determining a player’s behavior. We show by example that this is true for a variety of solution concepts. We then investigate what is essential about a player’s information to identify rationalizable behavior in any game. We do this by constructing the universal type space for rationalizability and characterizing the types in terms of their beliefs. Infinite hierarchies of beliefs over conditional beliefs, what we call delta-hierarchies, are what turn out to matter. We show that any two types in any two type spaces have the same rationalizable sets in all games if and only if they have the same delta-hierarchies.

    The Alternative to Equilibrium Existence

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    This paper establishes and interprets a necessary and sucient condition for existence of (countably additive) correlated equilibrium in n-person games, assuming only that utility functions are bounded, measurable. A sequence of deviation profiles is consistent if there exists a correlated strategy that makes every profile in the sequence unprofitable with respect to the sum of utilities. An equilibrium exists if and only if every sequence of deviation profiles has a consistent subsequence. This condition fails to characterize Nash equilibrium. As a direct corollary, existence of (communication) equilibrium is characterized in games with incomplete information on type spaces large enough to include the universal one. Exact conditions for existence of approximate correlated equilibrium are also obtained, as well as a value for two-person zero-sum games.correlated equilibrium, consistency, duality, discontinuous games.

    Incomplete Information

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    In interactive contexts such as games and economies, it is important to take account not only of what the players believe about substantive matters (such as payoffs), but also of what they believe about the beliefs of other players. Two different but equivalent ways of dealing with this matter, the semantic and the syntactic, are set forth. Canonical and universal semantic systems are then defined and constructed, and the concepts of common knowledge and common priors formulated and characterized. The last two sections discuss relations with Bayesian games of incomplete information and their applications, and with interactive epistemology -- the theory of multi-agent knowledge and belief as formulated in mathematical logic

    Finite state verifiers with constant randomness

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    We give a new characterization of NL\mathsf{NL} as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio

    Essays in economic theory

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    This dissertation consists of two essays in Game Theory with Incomplete Information. The common theme of the essays pertains to the interpretation and origin of incomplete information in strategic situations. In the first essay, I study bargaining between a buyer and a seller when the buyer can invest in generating outside options at a cost, and the seller cannot observe his investment choice. I model the negotiation phase as an incentive compatible and ex-post individually rational direct mechanism, which maximizes a weighted average of the buyer's and seller's expected payoff. When the weight on the buyer is larger, trade happens with certainty, the price equals the seller’s cost, and the buyer does not invest in generating outside options. When the weight on the seller is larger, the optimal mechanism is a posted price mechanism at the worst outside option that the buyer could have, provided that the cost function of the buyer is decreasing in first order stochastic dominance. The probability of trade is strictly below 1, even if it would be socially efficient to trade. In the second essay, I propose a notion of Rationalizability, called Incomplete Preference Rationalizability, for games with incomplete preferences. Under an appropriate topological condition, the incomplete preference rationalizable set is non-empty and compact. I argue that incomplete orderings can be used to model incomplete information in strategic settings. Drawing on this connection, I show that in games with private values the sets of incomplete preference rationalizable actions, of belief-free rationalizable actions (Battigalli et al., 2011, Bergemann and Morris, 2017), and of interim correlated rationalizable actions (Dekel et al., 2007) of the universal type space coincide
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