The Power of Counting Steps in Quantitative Games

TL;DR

This paper investigates the strategy complexity in infinite-duration deterministic graph games with quantitative objectives, focusing on the sufficiency of step-counter strategies and establishing bounds for various payoff conditions.

Contribution

It provides new bounds on the use of step-counter strategies for mean-payoff and total-payoff objectives in infinite graphs, clarifying their limitations and capabilities.

Findings

Step-counter strategies suffice for certain limsup objectives on finitely branching graphs.

Strategies with step counters and finite memory are insufficient for some liminf total-payoff objectives.

The results settle the case of classical quantitative objectives up to the second Borel hierarchy level.

Abstract

We study deterministic games of infinite duration played on graphs and focus on the strategy complexity of quantitative objectives. Such games are known to admit optimal memoryless strategies over finite graphs, but require infinite-memory strategies in general over infinite graphs. We provide new lower and upper bounds for the strategy complexity of mean-payoff and total-payoff objectives over infinite graphs, focusing on whether step-counter strategies (sometimes called Markov strategies) suffice to implement winning strategies. In particular, we show that over finitely branching arenas, three variants of limsup mean-payoff and total-payoff objectives admit winning strategies that are based either on a step counter or on a step counter and an additional bit of memory. Conversely, we show that for certain liminf total-payoff objectives, strategies resorting to a step counter and finite memory are not sufficient. For step-counter strategies, this settles the case of all classical quantitative objectives up to the second level of the Borel hierarchy.