Chapter 6: Game Theory and Players’ Beliefs on the Play
Christian Schmidt 6.1 INTRODUCTION For over a decade, several game theorists, assisted by logicians and computer scientists, have been working to uncover the epistemic foundations of interactive situations between many agents. Their objective is to produce a general framework for investigating players’ knowledge assumptions. However, due to information conditions, players’ knowledge is not necessarily true. Therefore the term ‘knowledge’ has been replaced by ‘belief’. The result has been the construction of belief systems, derived from modal logics, which are associated with the available data of the game.1 The simplest way to capture the main features of a belief system is to start with individual situations of decision-making à la Savage. The beliefs of the individual decision-makers are pictured by a set of ﬁnite states of the world, deﬁned as a system of subjective partitions. In such a situation, the decisionmaker’s beliefs are independent of his chosen action. This is no longer the case in a game situation, where the states depend on the other players’ actions and on the player’s own actions. The consequence is an intricate system of degrees (beliefs, beliefs of beliefs, etc.) to formulate the system of players’ beliefs as a hierarchical order encoded in a consistent format. Supplementing a game by a system of players’ beliefs changes the traditional perspective of game theory or, at least, its way of modelling a game situation. Roughly speaking, in the classical approach developed by game theory, a game situation raises a problem to be solved by the game theorists...
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