Uniqueness of Equilibria

Akio Matsumoto () and Ferenc Szidarovszky ()
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Akio Matsumoto: Chuo University
Ferenc Szidarovszky: Corvinus University

Chapter Chapter 8 in Game Theory and Its Applications, 2025, pp 105-111 from Springer

Abstract: Abstract In examining the existence conditions for equilibria in N-person games either the Banach or the Kakutani fixed point theoremKakutani fixed point theorem was used. The Banach fixed point theoremBanach fixed point theorem guaranteed the existence of the unique equilibrium and an iteration algorithm was also suggested to compute the equilibrium. However the existence theorems based on the Kakutani fixed point theoremKakutani fixed point theorem (Theorems 5.3 and 5.4) do not guarantee uniqueness. For example, by selecting constant payoff functions all strategies provide equilibria, and constant functions are continuous as well as concave. So the conditions of the Nikaido-Isoda theoremNikaido-Isoda theorem are satisfied if the strategy sets are nonempty, convex, closed and bounded. It is well known from optimization theory that strictly concave functions cannot have multiple maximum points.

Date: 2025
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DOI: 10.1007/978-981-96-0590-3_8

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