Pure subgame-perfect equilibria in free transition games

Abstract

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Stochastic Multiplayer Games : Theory and Algorithms

“…The question arises whether we can also guarantee the existence of a positional Nash equilibrium in such games. Kuipers et al (2009) proved that this is not the case.…”

Subgame-perfection in free transition games

“…It is shown in Kuipers et al [6] that this game has no stationary subgame-perfect 0-equilibrium, and similar arguments can be used to prove the non-existence of stationary subgame-perfect ε-equilibrium for small ε > 0.…”

“…Indeed, this property implies that a finite number of deviations cannot be profitable either, and because the termination payoffs are nonnegative, no player can profit from deviating infinitely many times (that can only happen if play never terminates). By making use of this one-deviation principle, Kuipers et al [6] showed the existence of a pure subgame-perfect 0-equilibrium.…”

“…Iterative methods have been proven useful in a number of subclasses. Kuipers et al [6] and Flesch et al [3] applied such methods, but in both cases it is crucial to assume that the termination payoffs are non-negative. These results have been generalized by Flesch et al [4], who proved the existence of subgame-perfect ε-equilibria, provided that the transitions are deterministic and that the payoff functions of the players, defined as a function of the whole play, are bounded and lower-semicontinuous.…”

“…Kuipers et al [6]: if with respect to a strategy profile, no player can improve his payoff in any subgame by deviating exactly once, then this strategy profile is a subgame-perfect 0-equilibrium. Indeed, this property implies that a finite number of deviations cannot be profitable either, and because the termination payoffs are nonnegative, no player can profit from deviating infinitely many times (that can only happen if play never terminates).…”