Subgame-perfection in free transition games

Abstract

We prove the existence of a subgame-perfect ε-equilibrium, for every ε > 0, in a class of multi-player games with perfect information, which we call free transition games. The novelty is that a non-trivial class of perfect information games is solved for subgameperfection, with multiple non-terminating actions, in which the payoff structure is generally not semi-continuous. Due to the lack of semi-continuity, there is no general rule of comparison between the payoffs that a player can obtain by deviating a large but finite number of times or, respectively, infinitely many times. We introduce new techniques to overcome this difficulty.Our construction relies on an iterative scheme which is independent of ε and terminates in polynomial time with the following output: for all possible histories h, a pure action a

A Characterization of Subgame-Perfect Equilibrium Plays in Borel Games of Perfect Information

“…Thus, Lemma 6.4 implies that P F ξ (h t ) ⊇ P F ξ (h t+1 ). By unraveling the inclusions we obtain P F ξ (h t k +1 ) ⊇ P F ξ (h t k+1 ), which is exactly (12). Define B {q ∈ P: u i (q) ≥ β * }.…”

“…As u i (q) ≥ β * ≥ β ξ (g k ), and since ι(g k ) i is not an element of F, we have q ∈ P F ξ (g k ), as desired. Combining (12) and (13), we obtain the inclusions…”

“…In contrast, the "full" auxiliary zero-sum game employed here allows the deviator to deviate finitely or even infinitely many times. Unlike the algorithms of Harris and Flesch et al, the algorithm proposed here does not require the game to satisfy the one-shot deviation principle: indeed, the security payoffs capture the possibility of infinite as well as finite deviations.Flesch et al [12] consider the class of free transition games and proves the existence of subgame-perfect -equilibrium in mixed strategies using an algorithm of iterative elimination of plays. This algorithm operates…”

“…Flesch et al [12] consider the class of free transition games and proves the existence of subgame-perfect -equilibrium in mixed strategies using an algorithm of iterative elimination of plays. This algorithm operates…”

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

“…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.…”