Common belief in future and restricted past rationality
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Common belief in future and restricted past rationality Rubén Becerril‑Borja1,2 · Andrés Perea3 Accepted: 3 March 2020 © The Author(s) 2020
Abstract We introduce the idea that a player believes at every stage of a dynamic game that his opponents will choose rationally in the future and have chosen rationally in a restricted way in the past. This is summarized by the concept of common belief in future and restricted past rationality, which is defined epistemically. Moreover, it is shown that every properly rationalizable strategy of the normal form of a dynamic game can be chosen in the dynamic game under common belief in future and restricted past rationality. We also present an algorithm that uses strict dominance, and show that it selects exactly those strategies that can be chosen under common belief in future and restricted past rationality. Keywords Epistemic game theory · Dynamic games · Proper rationalizability · Belief in future rationality · Belief in restricted past rationality JEL Classification C72 · C73
The authors would like to thank two editors and three anonymous referees for comments made on this paper that helped improve the presentation and consider some interesting queries. * Andrés Perea [email protected] Rubén Becerril‑Borja [email protected] 1
Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, 09340 Mexico City, Mexico
2
EpiCenter, School of Business and Economics, Maastricht University, P.O. Box 616, MD 6200 Maastricht, The Netherlands
3
EpiCenter and Department of Quantitative Economics, School of Business and Economics, Maastricht University, P.O. Box 616, MD 6200 Maastricht, The Netherlands
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R. Becerril‑Borja, A. Perea
1 Introduction Epistemic game theory deals with the reasoning processes of an individual about his opponents before he makes a decision. This requires a belief about the choices of his opponents, but also a belief about the opponents’ beliefs about their oppo‑ nents’ choices, and so on. Such reasoning processes have been studied thoroughly in the framework of static games, in various forms of the concept of common belief in rationality. However, the extension of these concepts to the framework of dynamic games is not entirely trivial. One possible way to extend the idea of common belief in rationality would require that the players believe their opponents make only rational choices, in particular that past choices have been rational. However, in many cases this is not possible, since there may be stages in the game where play‑ ers have to conclude that an opponent has chosen irrationally in the past. To solve this problem some alternative concepts have been proposed. Battigalli and Siniscalchi (2002) propose the concept of common strong belief in rational‑ ity, in which players, whenever possible, must believe that their opponents are implementing rational strategies. Perea (2014) proposed the concept of common belief in future rationality, in whic
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