Rational choice and AGM belief revision

Abstract

We establish a correspondence between the rationalizability of choice studied in the revealed preference literature and the notion of minimal belief revision captured by the AGM postulates. A choice frame consists of a set of alternatives , a collection E of subsets of (representing possible choice sets) and a function f : E ! 2 (representing choices made). A choice frame is rationalizable if there exists a total pre-order R on such that, for every E 2 E, f (E) coincides with the best elements of E relative to R. We re-interpret choice structures in terms of belief revision. An interpretation is obtained by adding a valuation V that assigns to every atom p the subset of at which p is true. Associated with an interpretation is an initial belief set and a partial belief revision function. A choice frame is AGM-consistent if, for every interpretation of it, the associated partial belief revision function can be extended to a full-domain belief revision function that satis…es the AGM postulates. It is shown that a …nite choice frame is AGM-consistent if and only if it is rationalizable.

Belief Change in Branching Time: AGM-consistency and Iterated Revision

“…In view of the above example, a natural question to ask is whether there exists a property of branching-time belief revision frames that guarantees that the partial belief revision functions generated by models based on frames that satisfy that 10 This is a consequence of the following result, which is proved in the Appendix (Lemma 13). Let K be a consistent belief set and BK : !…”

“…The following proposition, which is proved in the Appendix, builds on results given in [10] and [21]. 13 Note that the Qualitative Bayes Rule (Property 4 of Denition 3) is necessary for the validity of Proposition 6.…”

“…6 = f (E) E. Choice functions are used in economics to represent the choices made by an individual when faced with possible menus of alternatives. In [21] a necessary and suf cient condition is given for the rationalizability of a choice function in terms of a preference relation and in [10] choice functions are shown to be interpretable in terms of one-shot belief revision. In the proof given in the Appendix more details are given on how results in [10] and [21] can be extended to branching-time belief revision frames to obtain Proposition 6.…”

“…The notion of AMG-consistency de ned below guarantees that no inconsistecies could be detected by contemplating hypothetical information in addition to the actual information. 13 Both [10] and [21] deal with choice functions f : E ! 2 , where E is a collection of subsets of , satisfying the property that if E 6 = ?…”

Reasoning About Strategies and Rational Play in Dynamic Games

“…If E i 2 Ω \∅ then Conditions 1-4 in Definition 4.1 are necessary but not sufficient for the existence of such a plausibility relation. The existence of a plausibility relation that rationalizes the function f i (ω, ·) : E i → 2 Ω is necessary and sufficient for the belief revision policy encoded in f i (ω, ·) to be compatible with the theory of belief revision introduced in Alchourrón et al (1985), known as the AGM theory (see Bonanno (2009)). …”