3 research outputs found
Quantitative games with interval objectives
Traditionally quantitative games such as mean-payoff games and discount sum
games have two players -- one trying to maximize the payoff, the other trying
to minimize it. The associated decision problem, "Can Eve (the maximizer)
achieve, for example, a positive payoff?" can be thought of as one player
trying to attain a payoff in the interval . In this paper we
consider the more general problem of determining if a player can attain a
payoff in a finite union of arbitrary intervals for various payoff functions
(liminf, mean-payoff, discount sum, total sum). In particular this includes the
interesting exact-value problem, "Can Eve achieve a payoff of exactly (e.g.)
0?"Comment: Full version of CONCUR submissio
Infinite Separation between General and Chromatic Memory
In this note, we answer a question from [Alexander Kozachinskiy. State
Complexity of Chromatic Memory in Infinite-Duration Games, arXiv:2201.09297].
Namely, we construct a winning condition over a finite set of colors such
that, first, every finite arena has a strategy with 2 states of general memory
which is optimal with respect to , and second, there exists no such that
every finite arena has a strategy with states of chromatic memory which is
optimal with respect to
Characterizing Omega-Regularity Through Finite-Memory Determinacy of Games on Infinite Graphs
We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of ?-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is ?-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be ?-regular. This provides a game-theoretic characterization of ?-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which ?-regularity depends on the value of some parameters