On Nash Equilibria in Stochastic Games

Abstract. We study two-player timed games where the objectives of the two players are not opposite. We focus on the standard notion of Nash equilibrium and propose a series of transformations that builds two finite turn-based games out of a timed game, with a precise correspondence between Nash equilibria in the original and in final games. This provides us with an algorithm to compute Nash equilibria in two-player timed games for large classes of properties.

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“…There is a pseudo Nash equilibrium in with payoff ( 1 , 2 ) iff there are two runs 1 in 1 and 2 in 2 s.t. 9 …”

Section: Some General Decidability Resultsmentioning

confidence: 99%

Computing Equilibria in Two-Player Timed Games via Turn-Based Finite Games

Bouyer

Brenguier

Markey

2010

Lecture Notes in Computer Science

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“…Chatterjee & al. [4] showed that one can compute a Nash equilibrium of a multiplayer parity game in nondeterministic polynomial time. They also showed that it is NP-hard to determine whether a multiplayer game with reachability conditions has a Nash equilibrium where all players win.…”

Section: Related Workmentioning

confidence: 99%

“…Previous research on algorithms for finding Nash equilibria in infinite games has focused on computing some Nash equilibrium [4]. However, a game may have several Nash equilibria with different payoffs, so one might not be interested in any Nash equilibrium but in one that fulfils certain requirements.…”

Section: Computational Complexitymentioning

confidence: 99%

“…Therefore, traditionally, the research in this area has mostly looked at two-player games where each play is won by precisely one of the two players, so-called two-player zero-sum games. However, motivated by the modelling of distributed systems, interest in the general case has increased in recent years [3,4].…”

Section: Introductionmentioning

confidence: 99%

“…Chatterjee & al. [4] showed that any infinite multiplayer game with ω-regular winning conditions has a Nash equilibrium in pure strategies, and they also gave an algorithm for computing one. We argue that this is not satisfactory.…”

Section: Introductionmentioning

confidence: 99%

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The Complexity of Nash Equilibria in Infinite Multiplayer Games

Ummels

Foundations of Software Science and Computational Structures

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Abstract. We study the complexity of Nash equilibria in infinite (turnbased, qualitative) multiplayer games. Chatterjee & al. showed the existence of a Nash equilibrium in any such game with ω-regular winning conditions, and they devised an algorithm for computing one. We argue that in applications it is often insufficient to compute just some Nash equilibrium. Instead, we enrich the problem by allowing to put (qualitative) constraints on the payoff of the desired equilibrium. Our main result is that the resulting decision problem is NP-complete for games with co-Büchi, parity or Streett winning conditions but fixed-parameter tractable for many natural restricted classes of games with parity winning conditions. For games with Büchi winning conditions we show that the problem is, in fact, decidable in polynomial time.We also analyse the complexity of strategies realising a Nash equilibrium. In particular, we show that pure finite-state strategies as opposed to arbitrary mixed strategies suffice to realise any Nash equilibrium of a game with ω-regular winning conditions with a qualitative constraint on the payoff.

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Two-Player Nonzero-Sum ω-Regular Games

Chatterjee

2005

Lecture Notes in Computer Science

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On Nash Equilibria in Stochastic Games

Cited by 96 publications

References 25 publications

Computing Equilibria in Two-Player Timed Games via Turn-Based Finite Games

Computing Equilibria in Two-Player Timed Games via Turn-Based Finite Games

The Complexity of Nash Equilibria in Infinite Multiplayer Games

Two-Player Nonzero-Sum ω-Regular Games

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