Fixed Point Theorems with Applications to Economics and Game Theory

TLDR

Economists focus on ensuring consistency in economic models, and fixed point theorems provide the mathematical conditions for existence of solutions. The book investigates the conditions under which market equilibrium can exist by applying fixed point theorems to economic and game‑theoretic models. It presents fixed point techniques, emphasizing applications to economics and games even when agents’ preferences are non‑transitive. The book clarifies existing fixed point results, introduces new theorems on the core of games without transitivity, and serves as a useful reference for mathematical economists and advanced students.

Abstract

One of the problems in economics that economists have devoted a considerable amount of attention in prevalent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it is asked under what circumstances such equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems: the conditions under which sets of assumptions have a solution. This book gives the reader access to the mathematical techniques involved and goes on to apply fixed point theorems to proving the existence of equilibria for economics and for co-operative and noncooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. The author presents topical proofs of old results in order to further clarify the results. He also proposes fresh results, notably in the last chapter, that refer to the core of a game without transitivity. This book will be useful as a text or reference work for mathematical economists and graduate and advanced undergraduate students.