Epistemic logic meets epistemic game theory: a comparison between multi-agent Kripke models and type spaces

Paolo Galeazzi,Emiliano Lorini

doi:10.1007/s11229-015-0834-x

Abstract

In the literature there are at least two main formal structures to deal with situations of interactive epistemology: Kripke models and type spaces. As shown in many papers (see Aumann and Brandenburger in Econometrica 36:1161–1180, 1995; Baltag et al. in Synthese 169:301–333, 2009; Battigalli and Bonanno in Res Econ 53(2):149–225, 1999; Battigalli and Siniscalchi in J Econ Theory 106:356–391, 2002; Klein and Pacuit in Stud Log 102:297–319, 2014; Lorini in J Philos Log 42(6):863–904, 2013), both these frameworks can be used to express epistemic conditions for solution concepts in game theory. The main result of this paper is a formal comparison between the two and a statement of semantic equivalence with respect to two different logical systems: a doxastic logic for belief and an epistemic–doxastic logic for belief and knowledge. Moreover, a sound and complete axiomatization of these logics with respect to the two equivalent Kripke semantics and type spaces semantics is provided. Finally, a probabilistic extension of the result is also presented. A further result of the paper is a study of the relationship between the epistemic–doxastic logic for belief and knowledge and the logic STIT (the logic of “seeing to it that”) by Belnap and colleagues (Facing the future: agents and choices in our indeterminist world, 2001).

Highlights

In recent years many game theorists have focused on the epistemic part of playing a game, taking explicitly into account the knowledge and beliefs of the players involved in strategic interactions
In the literature there are at least two main formal structures to deal with situations of interactive epistemology: Kripke models, mainly used in logic and computer science Fagin et al (1995), and type spaces, more common in economics and game theory (Harsanyi 1967–1968)
We show how to interpret the extended language on type spaces and subsequently we define the corresponding class of epistemic-doxastic game models

Summary

Introduction

In recent years many game theorists have focused on the epistemic part of playing a game, taking explicitly into account the knowledge and beliefs of the players involved in strategic interactions. There are few games (e.g., the Prisoners’ Dilemma) that do not need any strategic reasoning about the others for a rational player to choose an action. In most of the situations instead players have to take into consideration what they think about the others’ actions and beliefs in order to choose an action. We need to consider what a player thinks about the others in order to assess her rationality. Strategic thinking comes out when the players reason about the others’ actions, knowledge and beliefs, and epistemic game theory makes it explicit

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Discussion

Conclusion