A Model of Minimal Probabilistic Belief Revision

A probabilistic belief revision function assigns to every initial probabilistic belief and every observable event some revised probabilistic belief that only attaches positive probability to states in this event. We propose three axioms for belief revision functions: (1) linearity, meaning that if the decision maker observes that the true state is in {a,b}, and hence state c is impossible, then the proportions of c''s initial probability that are shifted to a and b, respectively, should be independent of c''s initial probability; (2) transitivity, stating that if the decision maker deems belief β equally similar to states a and b, and deems β equally similar to states b and c, then he should deem β equally similar to states a and c; (3) information-order independence, stating that the way in which information is received should not matter for the eventual revised belief. We show that a belief revision function satisfies the three axioms above if and only if there is some linear one-to-one function ϕ, transforming the belief simplex into a polytope that is closed under orthogonal projections, such that the belief revision function satisfies minimal belief revision with respect to ϕ. By the latter, we mean that the decision maker, when having initial belief β₁ and observing the event E, always chooses the revised belief β₂ that attaches positive probability only to states in E and for which ϕ(β₂) has minimal Euclidean distance to ϕ(β₁).microeconomics ;

Epistemic Foundations for Backward Induction: An Overview

In this survey we analyze, and compare, various sufficient epistemic conditions for backward induction that have been proposed in the literature. To this purpose we present a simple epistemic base model for games with perfect information, and translate the different models into the language of this base model. As such, we formulate the various sufficient conditions for backward induction in a uniform language, which enables us to explictly analyze their differences and similarities.mathematical economics;

Proper Rationalizability and Belief Revision in Dynamic Games

In this paper we develop an epistemic model for dynamic games in which players may revise their beliefs about the opponents'' preferences (including the opponents'' utility functions) as the game proceeds. Within this framework, we propose a rationalizability concept that is based upon the following three principles: (1) at every instance of the game, a player should believe that his opponents are carrying out optimal strategies, (2) a player should only revise his belief about an opponent''s relative ranking of two strategies if he is certain that the opponent has decided not to choose one of these strategies, and (3) the players'' initial beliefs about the opponents'' utility functions should agree on a given profile u of utility functions. Common belief about these events leads to the concept of persistent rationalizability for the profile u of utility functions. It is shown that for a given profile u of utility functions, every properly rationalizable strategy for ``types with non-increasing type supports'''' is a persistently rationalizable strategy for u. This result implies that persistently rationalizable strategies always exist for all game trees and all profiles of utility functions.mathematical economics;

Rationalizability and Minimal Complexity in Dynamic Games

This paper presents a formal epistemic framework for dynamic games in which players, during the course of the game, may revise their beliefs about the opponents'' utility functions. We impose three key conditions upon the players'' beliefs: (a) throughout the game, every move by the opponent should be interpreted as being part of a rational strategy, (b) the belief about the opponents'' relative ranking of two strategies should not be revised unless one is certain that the opponent has decided not to choose one of these strategies, and (c) the players'' initial beliefs about the opponents'' utility functions should agree on a given profile u of utility functions. Types that, throughout the game, respect common belief about these three events, are called persistently rationalizable for the profile u of utility functions. It is shown that persistent rationalizability implies the backward induction procedure in generic games with perfect information. We next focus on persistently rationalizable types for u that hold a theory about the opponents of ``minimal complexity'''', resulting in the concept of minimal rationalizability. For two-player simultaneous move games, minimal rationalizability is equivalent to the concept of Nash equilibrium strategy. In every outside option game, as defined by van Damme (1989), minimal rationalizability uniquely selects the forward induction outcome.microeconomics ;

Nash Equilibrium as an Expression of Self-Referential Reasoning

Within a formal epistemic model for simultaneous-move games, we present the following conditions: (1) belief in the opponents'' rationality (BOR), stating that a player should believe that every opponent chooses an optimal strategy, (2) self-referential beliefs (SRB), stating that a player believes that his opponents hold correct beliefs about his own beliefs, (3) projective beliefs (PB), stating that i believes that j''s belief about k''s choice is the same as i''s belief about k''s choice, and (4) conditionally independent beliefs (CIB), stating that a player believes that opponents'' types choose their strategies independently. We show that, if a player satisfies BOR, SRB and CIB, and believes that every opponent satisfies BOR, SRB, PB and CIB, then he will choose a Nash equilibrium strategy (that is, a strategy that is optimal in some Nash equilibrium). We thus provide a set of sufficient conditions for Nash equilibrium strategy choice. We also show that none of these seven conditions can be dropped.mathematical economics;