On (Subgame Perfect) Secure Equilibrium in Quantitative Reachability Games

Abstract

We study turn-based quantitative multiplayer non zero-sum games played on finite graphs with reachability objectives. In such games, each player aims at reaching his own goal set of states as soon as possible. A previous work on this model showed that Nash equilibria (resp. secure equilibria) are guaranteed to exist in the multiplayer (resp. two-player) case. The existence of secure equilibria in the multiplayer case remained and is still an open problem. In this paper, we focus our study on the concept of subgame perfect equilibrium, a refinement of Nash equilibrium well-suited in the framework of games played on graphs. We also introduce the new concept of subgame perfect secure equilibrium. We prove the existence of subgame perfect equilibria (resp. subgame perfect secure equilibria) in multiplayer (resp. two-player) quantitative reachability games. Moreover, we provide an algorithm deciding the existence of secure equilibria in the multiplayer case. ACM CCS: [Theory of computation]:Theory and algorithms for application domains-Algorithmic game theory and mechanism design-Algorithmic game theory.

Subgame perfection in recursive perfect information games

“…A subgame perfect ε-equilibrium, for every ε > 0, was previously established in games with only nonnegative rewards (Flesch et al 2010a), in free transition games (Kuipers et al 2013), and in games where each player only controls one state (Kuipers et al 2016). In the literature, we can also find sufficient conditions for other classes of games, such as in the classical papers by Fudenberg and Levine (1983) and Harris (1985), and more recently, in the papers by Solan and Vieille (2003), Flesch et al (2010b), Purves and Sudderth (2011), Brihaye et al (2013), Roux and Pauly (2014), Flesch and Predtetchinski (2016), Roux (2016), Mashiah-Yaakovi (2014), Cingiz et al (2019) and Flesch et al (2019). We further refer to the recent book by Alós-Ferrer and Ritzberger (2016), and the surveys by Jaśkiewicz and Nowak (2016) and Bruyère (2017).…”

Existence of Secure Equilibrium in Multi-player Games with Perfect Information

“…Algorithmic result: Let us consider multi-player quantitative reachability games played on a finite graph, with deterministic transitions, where each payoff is determined by the number of moves it takes to get in a particular set of states. As a corollary of Theorem 1 and some results of [2] (the proof of Theorem 4.1, Proposition 4.5 and Remark 4.7), we derive an algorithm to obtain, in ExpSpace, a secure equilibrium such that finite payoffs are bounded by 2 · |N | · |S| in such games. We intend to further investigate algorithmic questions for other classes of objectives.…”

“…Subgame-perfect secure equilibrium: Brihaye et al [2] introduced the concept of subgame-perfect secure equilibrium, and showed its existence in twoplayer quantitative reachability games. We do not know if Theorem 1 can be extended to subgame-perfect secure equilibrium.…”

“…Remark 1 or [7, Example 2.2.34]). That is why we choose to call a strongly secure equilibrium an equilibrium according to [5,Definition 8] (see Remark 1), and we choose to call, as it has already been done in [1,2], a secure equilibrium an equilibrium according to the alternative characterization given in [5,Proposition 4], extended to the quantitative framework. Note that, with these definitions, every secure equilibrium is automatically strongly secure if there are only two players.…”

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

“…Flesch et al (2014) proposed and examined a refinement, called strong -SPE, which avoids this shortcoming. As another refinement, Brihaye et al (2013) investigated so-called secure SPE, which can be applied for assume-guarantee synthesis and model checking. Secure refers to a property that the players, in some sense, do not have to fear deviations when an opponent changes his strategy to another one which gives this opponent the same payoff.…”