Legitimate equilibrium

Janos Flesch, Dries Vermeulen*, Anna Zseleva

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


We present a general existence result for a type of equilibrium in normal-form games, which extends the concept of Nash equilibrium. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a legitimate equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a legitimate equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from (Vasquez (2017)), but is more direct as it does not make use of countably additive representations of finitely additive measures by (Yosida and Hewitt (1952)).
Original languageEnglish
Pages (from-to)787-800
Number of pages14
JournalInternational Journal of Game Theory
Issue number4
Early online date12 Apr 2021
Publication statusPublished - Dec 2021


  • Nonzero-sum games
  • Games with infinite action spaces
  • Equilibrium existence
  • Finitely additive strategies

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