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**11 - 15**of**15**### Justified Belief and Rationality

, 2011

"... Brandenburger and Dekel have shown that common belief of rationality (CBR) characterizes rationalizable strategies, which are also characterized by a refinement of subjective correlated equilibrium called a posteriori equilibrium. It is possible that players ’ beliefs might be incompatible, in the s ..."

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Brandenburger and Dekel have shown that common belief of rationality (CBR) characterizes rationalizable strategies, which are also characterized by a refinement of subjective correlated equilibrium called a posteriori equilibrium. It is possible that players ’ beliefs might be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require players beliefs to be justified, in the sense that all players must ascribe the actual world positive probability. Aumann has shown that, under the common prior assumption (CPA), common belief of rationality characterizes strategies in the support of an objective correlated equilibrium. We can show that, if the CPA holds, then we can assume without loss of generality that players ’ beliefs are justified. We then consider the consequences of common justified belief of rationality (CJBR), without the common prior assumption. We show that CJBR characterize strategies in the support of a subjective correlated equilibrium where all players ’ beliefs have common support. In the Bayesian view of the world, each player has a subjective probability distribution (describing her

### A Model of Monopoly Pricing under Incomplete Information

, 2004

"... Nash equilibrium defines a solution to strategic interactions among rational players. Deductive equilibrium selection studies players who have all the relevant information in the game and who reason the outcome of the game. But how do the players come about to possess the relevant information and ho ..."

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Nash equilibrium defines a solution to strategic interactions among rational players. Deductive equilibrium selection studies players who have all the relevant information in the game and who reason the outcome of the game. But how do the players come about to possess the relevant information and how do they select among possibly many equilibria? One solution to the incomplete information is given by Bayesian Nash equilibrium, where the players do not exactly know the opponents’ preferences but the preferences are chosen probabilistically among some alternatives. On the contrary, inductive equilibrium selection interprets equilibrium as a result of a dynamic adaptive process. Actually, this is one way to motivate the traditional equilibrium theory. We introduce the celebrated learning models: reinforcement, belief-based, and evolutionary-based learning. We develop an adjustment process that extracts information and results in Bayesian Nash equilibrium. This thesis studies incomplete information in a situation, where a monopoly sells a good to a population of buyers. The seller discriminates the buyers by offering a tariff, which specifies a price for each amount of good to be sold. The seller’s task is to design the optimal tariff that

### THE REVIEW OF SYMBOLIC LOGIC, Page 1 of 35 PEOPLE WITH COMMON PRIORS CAN AGREE TO DISAGREE

"... Abstract. Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteri-ors are equal ” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s p ..."

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Abstract. Robert Aumann presents his Agreement Theorem as the key conditional: “if two people have the same priors and their posteriors for an event A are common knowledge, then these posteri-ors are equal ” (Aumann, 1976, p. 1236). This paper focuses on four assumptions which are used in Aumann’s proof but are not explicit in the key conditional: (1) that agents commonly know, of some prior μ, that it is the common prior; (2) that agents commonly know that each of them updates on the prior by conditionalization; (3) that agents commonly know that if an agent knows a proposition, she knows that she knows that proposition (the “K K ” principle); (4) that agents commonly know that they each update only on true propositions. It is shown that natural weakenings of any one of these strong assumptions can lead to countermodels to Aumann’s key conditional. Examples are given in which agents who have a common prior and commonly know what probabil-ity they each assign to a proposition nevertheless assign that proposition unequal probabilities. To alter Aumann’s famous slogan: people can “agree to disagree”, even if they share a common prior. The epistemological significance of these examples is presented in terms of their role in a defense of the Uniqueness Thesis: If an agent whose total evidence is E is fully rational in taking doxastic attitude D to P, then necessarily, any subject with total evidence E who takes a different attitude to

### Proceedings of the Thirteenth International Conference on Principles of Knowledge Representation and Reasoning Ambiguous Language and Differences in Beliefs

"... Standard models of multi-agent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents ’ bel ..."

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Standard models of multi-agent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents ’ beliefs regarding whether or not there is ambiguity. We consider the impact of ambiguity on a seminal result in economics: Aumann’s result saying that agents with a common prior cannot agree to disagree. This result is known not to hold if agents do not have a common prior; we show that it also does not hold in the presence of ambiguity. We then consider the tradeoff between assuming a common interpretation (i.e., no ambiguity) and a common prior (i.e., shared initial beliefs). 1