Weihrauch Degrees of Finding Equilibria in Sequential Games

2019

Dyn Games Appl

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We consider a dynamical approach to game in extensive forms. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic (finite) time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies.For infinite games in extensive form we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are ∆ 0 2 -sets.Observation 9. If s ։ s ′ , there is some strategy profile s ′′ such that s ⇀ s ′′ and v(s ′ ) = v(s ′′ ).Proof. By definition, lazy convertibility does not restrict the choice of the new induced play, merely the ability to alter the strategy off the new induced play.Corollary 10. The Nash equilibria of a game are exactly the terminal profiles of the lazy improvement ⇀.

“…That this indeed defines the difference hierarchy was observed by Motto-Ros [26, Section 7] extending previous work by Andretta and Martin [1]. A direct proof can be found in [20].…”

Section: Transfinite Lazy Improvement In the Difference Hierarchysupporting

confidence: 80%

Section: Transfinite Lazy Improvement In the Difference Hierarchymentioning

confidence: 92%

A Semi-Potential for Finite and Infinite Games in Extensive Form

2019

Dyn Games Appl

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“…The high-level proof idea follows earlier work by the authors on equilibria in infinite sequential games, using Borel determinacy as a blackbox [15] 1 -unlike the constructions there (cf. [16]), the present ones however are constructive and thus give rise to algorithms computing the equilibria in the multi-player multi-outcome games given suitable winning strategies in the two-player win/lose versions.…”

Section: Introductionmentioning

confidence: 98%

Extending Finite Memory Determinacy to Multiplayer Games

Electron. Proc. Theor. Comput. Sci.

2016

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We show that under some general conditions the finite memory determinacy of a class of two-player win/lose games played on finite graphs implies the existence of a Nash equilibrium built from finite memory strategies for the corresponding class of multi-player multi-outcome games. This generalizes a previous result by Brihaye, De Pril and Schewe. For most of our conditions we provide counterexamples showing that they cannot be dispensed with.Our proofs are generally constructive, that is, provide upper bounds for the memory required, as well as algorithms to compute the relevant winning strategies.

“…Regarding a potential extension of Corollary 28 to winning sets beyond ∆ 0 2 we shall make a tangential remark: The computational task of finding a Nash equilibrium in a two-player sequential game with ∆ 0 2 winning sets is just as hard as iterating the task of finding an accumulation point of a sequence over some countable ordinal. This follows from results in [18,17,21,4]. Finding a Nash equilibrium of a game with Σ 0 2 winning sets is strictly more complicated.…”

Section: Convergence In Infinite Gamesmentioning

confidence: 92%

A Semi-Potential for Finite and Infinite Sequential Games (Extended Abstract)

Electron. Proc. Theor. Comput. Sci.

2016

Self Cite

We consider a dynamical approach to sequential games. By restricting the convertibility relation over strategy profiles, we obtain a semi-potential (in the sense of Kukushkin), and we show that in finite games the corresponding restriction of better-response dynamics will converge to a Nash equilibrium in quadratic time. Convergence happens on a per-player basis, and even in the presence of players with cyclic preferences, the players with acyclic preferences will stabilize. Thus, we obtain a candidate notion for rationality in the presence of irrational agents. Moreover, the restriction of convertibility can be justified by a conservative updating of beliefs about the other players strategies.For infinite sequential games we can retain convergence to a Nash equilibrium (in some sense), if the preferences are given by continuous payoff functions; or obtain a transfinite convergence if the outcome sets of the game are ∆ 0 2 -sets.