Iterated Admissibility Through Forcing in Strategic Belief Models
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Iterated Admissibility Through Forcing in Strategic Belief Models Fernando Tohmé1
· Gianluca Caterina2 · Jonathan Gangle2
© Springer Nature B.V. 2020
Abstract Iterated admissibility embodies a minimal criterion of rationality in interactions. The epistemic characterization of this solution has been actively investigated in recent times: it has been shown that strategies surviving m +1 rounds of iterated admissibility may be identified as those that are obtained under a condition called rationality and m assumption of rationality in complete lexicographic type structures. On the other hand, it has been shown that its limit condition, with an infinity assumption of rationality (R∞A R), might not be satisfied by any state in the epistemic structure, if the class of types is complete and the types are continuous. In this paper we analyze the problem in a different framework. We redefine the notion of type as well as the epistemic notion of assumption. These new definitions are sufficient for the characterization of iterated admissibility as the class of strategies that indeed satisfy R∞A R. One of the key methodological innovations in our approach involves defining a new notion of generic types and employing these in conjunction with Cohen’s technique of forcing. Keywords Iterated admissibility · Possibility models · Forcing
1 Introduction Non-Cooperative Game Theory is concerned with the interactions of self-interested agents in structured environments. While its main elements were introduced by von
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Fernando Tohmé [email protected] Gianluca Caterina [email protected] Jonathan Gangle [email protected]
1
IMABB-Conicet and Dept. of Economics-UNS, Avda. Alem 1253, Bahía Blanca, Argentina
2
Center for Diagrammatic and Computational Philosophy, Endicott College, 376 Hale Street, Beverly, MA 01915, USA
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Neumann and Morgenstern (1944) the seminal contributions of Nash (1950, 1951) shaped its current form. Its focus is on the characterization of equilibria as fixed points in the space of profiles of mixed strategies, such that no player can benefit by deviating alone. For decades, Game Theory studied these equilibria and their refinements, without a formal discussion of how the way in which players face the uncertainties of a game guides their choices towards those outcomes. Theorists were well aware that there was a need for mathematical models of the reasoning processes carried out by the players in games. Epistemic Game Theory arose as a response to those concerns. It addresses the question of what information use the players when choosing their actions. They can be uncertain about either the game itself, or (as in this paper) about the actions chosen by the other players. In either case, each agent will have to act based on her beliefs, the class of possibilities she deems possible. In certain cases, these beliefs can be strengthened to knowledge when a property is actually the case and is present in all the states a player considers possible. The final decision made by a player
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