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Uniqueness of Equilibria

In: Game Theory and Its Applications

Author

Listed:
  • Akio Matsumoto

    (Chuo University)

  • Ferenc Szidarovszky

    (Corvinus University)

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.

Suggested Citation

  • Akio Matsumoto & Ferenc Szidarovszky, 2025. "Uniqueness of Equilibria," Springer Books, in: Game Theory and Its Applications, edition 0, chapter 0, pages 105-111, Springer.
    • Handle: RePEc:spr:sprchp:978-981-96-0590-3_8
    DOI: 10.1007/978-981-96-0590-3_8
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