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The Power of Counting Steps in Quantitative Games

Authors Sougata Bose , Rasmus Ibsen-Jensen , David Purser , Patrick Totzke , Pierre Vandenhove

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Author Details

Sougata Bose
  • University of Liverpool, UK
Rasmus Ibsen-Jensen
  • University of Liverpool, UK
David Purser
  • University of Liverpool, UK
Patrick Totzke
  • University of Liverpool, UK
Pierre Vandenhove
  • LaBRI, Université de Bordeaux, France

Funding

P. Vandenhove is funded by ANR project G4S (ANR-21-CE48-0010-01). S. Bose and P. Totzke acknowledge support from the EPSRC (EP/V025848/1, EP/X042596/1). Collaboration initiated via Royal Society International Exchanges grant (IES{backslash}R1{backslash}201211).

Cite As Get BibTex

Sougata Bose, Rasmus Ibsen-Jensen, David Purser, Patrick Totzke, and Pierre Vandenhove. The Power of Counting Steps in Quantitative Games. In 35th International Conference on Concurrency Theory (CONCUR 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 311, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CONCUR.2024.13

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
Keywords
  • Games on graphs
  • Markov strategies
  • quantitative objectives
  • infinite-state systems

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