Existence of equilibria in repeated games with long-run payoffs

Abstract

Significance Nash equilibrium, of central importance in strategic game theory, exists in all finite games. Here we prove that it exists also in all infinitely repeated games, with a finite or countably infinite set of players, in which the payoff function is bounded and measurable and the payoff depends only on what is played in the long run, i.e., not on what is played in any fixed finite number of stages. To this end we combine techniques from stochastic games with techniques from alternating-move games with Borel-measurable payoffs.

Equilibrium in two-player stochastic games with shift-invariant payoffs

“…Remark 3.4. If s is an absorbing state, then by Ashkenazi-Golan, Flesch, Predtetchinski, and Solan [1], there exists an ε-equilibrium for the initial state s, for every ε > 0. Hence, by Assumption 3.3, (i) a state s is absorbing if and only there exists an ε-equilibrium for the initial state s, for every ε > 0, and (ii) once the run reaches an absorbing state, the payoffs do not depend on the continuation of the run.…”

“…The function D δ i is in essence a strategy of player I in this auxiliary game such that this strategy is winning in each subgame. For a precise exposition, we refer to Ashkenazi-Golan, Flesch, Predtetchinski, and Solan [1]; properties (2)-( 4) directly follow from their lemma, whereas property (1) follows from their construction. The extension to a finite state space S is carried out in Flesch and Solan [19].…”

“…Tail-measurability of the payoffs, which is slightly weaker than shift-invariance (see also Section 4), was recently used in two related papers for repeated games, i.e., stochastic games with only one state. In Ashkenazi-Golan, Flesch, Predtetchinski, and Solan [1], it is shown that every repeated game with countably many players, finite action sets, and bounded, Borel-measurable, and tail-measurable payoffs, admits an ε-equilibrium for all ε > 0. In Ashkenazi-Golan, Flesch, Predtetchinski, and Solan [2], the regularity properties of the minmax and maxmin values are investigated in the context of multiplayer repeated games, and they are applied to show the existence of ε-equilibria in different classes of repeated games.…”

Repeated Games with Tail-Measurable Payoffs

“…Remark 5.4 The proof method above can be slightly adjusted to show that Theorem 2.3 holds even if the number of players is countably infinite, see Ashkenazi-Golan, Flesch, Predtetchinski, Solan (2022a). Since this involves more technical details on the measuretheoretic foundation of the model and the results, we only briefly discuss how the method can be applied.…”

“…When the winning set is not Borel-measurable, the game is not always determined. Blackwell (1969) studied analogous games where the two players move simultaneously: in every stage both players simultaneously choose actions, and player 1 wins if and only and Solan (2022aSolan ( , 2022b, the fourth proof is new.…”