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

• f:

◮

c, d, e, f ∈ C c d e f

◮ 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(c1c2c3 . . .), where c1, c2, c3, . . . are colors along trajectory of the pebble.

Definition

A payoff 휙 is positional if in all 휙-games players can play

• ptimally 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 훽1, 훽2, 훽3, . . . ∈ C 휔 the following holds. Assume that for any i ∈ N we have that 훼 and 훽i coincide in the first i elements. Then 휙(훼) = lim

i→∞ 휙(훽i)

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(x훼) > 휙(x훽), 휙(y훼) < 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.

. . . . . .

x y β α

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.

Thank you!