Positionality and strategy improvement for continuous payoffs A. - PowerPoint PPT Presentation

Positionality and strategy improvement for continuous payoffs A. Kozachinskiy University of Warwick, Coventry, UK Highlights of L., G. and A. 2020

Infinite games on finite graphs Fix a function 휙 : C 휔 → R (called a payoff ). A 휙 -game consist of: e f c c, d, e, f ∈ C d ◮ ◮ Min and Max are shifting a pebble (Min in △ -nodes and Max in � -nodes) along the edges. Infinitely many shifts. ◮ Zero sum: Min pays Max a fine of size 휙 ( c 1 c 2 c 3 . . . ) , where c 1 , c 2 , c 3 , . . . are colors along trajectory of the pebble. Definition A payoff 휙 is positional if in all 휙 -games players can play optimally via a positional strategy.

Continuous payoffs Definition A payoff 휙 : C 휔 → R is continuous if for any 훼 ∈ C 휔 and for any infinite sequence 훽 1 , 훽 2 , 훽 3 , . . . ∈ C 휔 the following holds. Assume that for any i ∈ N we have that 훼 and 훽 i coincide in the first i elements. Then 휙 ( 훼 ) = lim i →∞ 휙 ( 훽 i ) Can be defined by the cylinder topology, which is compact. Examples: (multi)discounted payoff is continuous, Parity and Mean Payoff are not.

Characterizing positional payoffs A payoff 휙 : C 휔 → R is called prefix-monotone if there are no x , y ∈ C ∗ and 훼, 훽 ∈ C 휔 such that 휙 ( x 훼 ) > 휙 ( x 훽 ) , 휙 ( y 훼 ) < 휙 ( y 훽 ) . Theorem Let 휙 : C 휔 → R be a continuous payoff. Then 휙 is positional if and only if 휙 is prefix-monotone. . . . α x y . . . β Figure: The “only if” part.

What else can be said Generalizing some results for (multi)discounted payoffs. ◮ strategy improvement (all continuous positional payoffs) ◮ LP-type problems and subexponential randomized algorithms (all continuous positional payoffs). ◮ Strong bounds on strategy improvement (for generalized or non-linear discounted payoff). What about stochastic games? ◮ Continuous + stochastically positional = ⇒ (multi)discounted.

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