Essays on equilibrium refinements

Iryna Topolyan, Purdue University

Abstract

In the first chapter we present some proofs of the existence of the minimax point of a strategic game that do not seem to be available in the literature. The existence of the minimax point plays an important role in the theory of games. In the second chapter we provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. For this purpose we formulate and prove a generalization of Eilenberg-Montgomery fixed-point theorem, which is a generalization of Kakutani's theorem. The third chapter provides a sufficient condition for a Nash equilibrium point to be strictly perfect in terms of the primitive characteristics of the game (payoffs and strategies). We show that continuity of the best response correspondence (which can be stated in terms of the primitives of the game) implies strict perfectness. We prove several other useful theorems regarding the structure of best response correspondence in normal form games. In the fourth chapter we propose a refinement of perfect equilibrium for the finite normal form games and extensive games with perfect recall. It relaxes the restrictive assumption made by Selten in his seminal work (23) regarding the common knowledge of the parameters characterizing the rationality breakdown for each player in the trembling-hand framework. The proposed solution concept in certain cases (for example, in the presence of very undesirable outcomes) eliminates unreasonable equilibria that, to the best of our knowledge, none of the known equilibrium refinements could rule out. The fifth chapter generalizes the well-known backward induction procedure to the case of extensive games with perfect recall having certain information structure (called simple information structure). We prove that the backward induction assessments are precisely sequential equilibria, and show that the set of backward induction assessments for any sequential game of simple information structure is nonempty. This provides a direct proof of the existence of sequential equilibria in such games. The described procedure suggests a direct method of computing sequential equilibria for such games.

Degree

Ph.D.

Advisors

Novshek, Purdue University.

Subject Area

Economics|Economic theory

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