Perfect-Information Games with Lower-Semicontinuous Payoffs

Abstract

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

“…This follows from more general results in Flesch et al (2010) and Purves and Sudderth (2011). Carmona (2005) shows that for every > 0, an -SPE exists under the assumption that the payoff functions are bounded and continuous at infinity.…”

“…As a consequence of Flesch et al (2010) and Purves and Sudderth (2011), these games admit a subgame-perfect -equilibrium, -SPE for brevity, for every > 0, provided that every player's payoff function is bounded and continuous on the whole set of plays. Here, continuity is meant with respect to the product topology on the set of plays, with the set of actions given its discrete topology.…”

On refinements of subgame perfect $$\epsilon $$ ϵ -equilibrium

“…Nevertheless, semicontinuity proved to be a useful condition. Flesch et al (2010) proved that an -SPE exists for every > 0 when the payoffs are bounded and lsc, whereas Purves and Sudderth (2011) proved the same when the payoffs are bounded and usc. Very general topological conditions for existence of SPE are given in Alós-Ferrer and Ritzberger (2013).…”

“…Following, among others, Martin (1975), Fudenberg and Levine (1983), Flesch et al (2010), Purves and Sudderth (2011), we endow the set A with the discrete topology and A N with the product topology. The topology on A N is completely metrizable, 3 and a basis of this topology is formed by the cylinder sets O(h) = {p ∈ A N : h < p} for h ∈ H , where for a history h ∈ H and a play p ∈ A N we write h < p if h is the initial segment of p. Thus a sequence of plays ( p n ) n∈N converges to a play p precisely when for every k ∈ N there exists an N k ∈ N such that p n coincides with p on the first k coordinates for every n ≥ N k .…”

“…in Carmona (2005) for games with bounded continuous at infinity payoffs, in Flesch et al (2010) for games with bounded lower semicontinuous payoffs, in Purves and Sudderth (2011) for games with bounded upper semicontinuous payoffs, in Flesch et al (2013) for free transition games, in Solan andVieille (2001, 2003), Solan (2005) and Mashiah-Yaakovi (2009) for quitting games. Laraki et al (2013) consider zerosum games with semicontinuous payoffs, and prove that both players have -optimal strategies and one of the players even has a subgame perfect -optimal strategy.…”