Subgame-perfect $$\epsilon $$ ϵ -equilibria in perfect information games with sigma-discrete discontinuities

Abstract

Multi-player perfect information games are known to admit a subgameperfect -equilibrium, for every > 0, under the condition that every player's payoff function is bounded and continuous on the whole set of plays. In this paper, we address the question on which subsets of plays the condition of payoff continuity can be dropped without losing existence. Our main result is that if payoff continuity only fails on a sigma-discrete set (a countable union of discrete sets) of plays, then a subgame-perfect -equilibrium, for every > 0, still exists. For a partial converse, given any subset of plays that is not sigma-discrete, we construct a game in which the payoff functions are continuous outside this set but the game admits no subgameperfect -equilibrium for small > 0.

Perfect information games where each player acts only once

“…These are games in which the payoffs are continuous when restricted to a sufficiently large part of the domain. This somewhat technical condition is related to the work of Flesch and Predtetchinski (2016b). We show that this class of games admits a subgame perfect -equilibrium for every positive value of .…”

“…The condition that a game is continuous outside a countable set is weaker than the condition used inFlesch and Predtetchinski (2016b).4 We have added one to the payoffs inPeleg (1969) and have relabeled action 0 as action 1 and action 1 as action 0. Clearly, this is inconsequential for the analysis of the example.…”

Perfect Information Games Where Each Player Acts Only Once

“…, n}. The idea 5 The condition that a game is continuous outside a countable set is weaker than the condition used in Flesch and Predtetchinski (2016b).…”

Introduction to the symposium on discontinuous games

“…By the Kakutani-Fan-Glicksberg fixed point theorem, any such correspondence F must be somewhere empty valued. When the domain, X say, is the set of strategy profiles in a game, and, for any x ∈ X , the value F(x) is the set of strategy profiles whose i-th coordinate is a strictly profitable unilateral deviation for player i against x, then any x for which F(x) is empty is an equilibrium of the game (see, e.g., Debreu 1952;Shafer and Sonnenschein 1975). Prokopovych observes that it suffices to construct via this patching technique, a correspondence G that majorizes F, i.e., G(x) contains F(x) for every x, and he provides conditions under which such a majorization G must be somewhere empty valued.…”