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On the Performance of Mildly Greedy Players in k-Coloring Games

Authors Vittorio Bilò , Andrea D'Ascenzo , Mattia D'Emidio , Giuseppe F. Italiano

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Subject Classification

ACM Subject Classification
  • Theory of computation → Exact and approximate computation of equilibria
  • Mathematics of computing → Graph coloring
Keywords
  • Coloring games
  • (Approximate) Nash Equilibria
  • Price of Anarchy

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Abstract

We study the performance of mildly greedy players in k-coloring games, a relevant subclass of anti-coordination games. A mildly greedy player is a selfish agent who is willing to deviate from a certain strategy profile only if her payoff improves by a factor of more than ε, for some given ε ≥ 0. In presence of mildly greedy players, stability is captured by the concept of (1+ε)-approximate Nash equilibrium. In this paper, we first show that, for any k-coloring game, the (1+ε)-approximate price of anarchy, i.e., the price of anarchy of (1+ε)-approximate pure Nash equilibria, is at least (k-1)/((k-1)ε +k), and that this bound is tight for any ε ≥ 0. Then, we evaluate the approximation ratio of the solutions achieved after a (1 + ϵ)-approximate one-round walk starting from any initial strategy profile, where a (1 + ϵ)-approximate one-round walk is a sequence of (1 + ε)-approximate best-responses, one for each player. We provide a lower bound of min{(k-2)/k, (k-1)/((k-1)ε+k)}  on this ratio, for any ε ≥ 0 and k ≥ 5; for the cases of k = 3 and k = 4, we give finer bounds depending on ε. Our work generalizes the results known for cut games, the special case of k-coloring games restricted to k = 2.

Cite As Get BibTex

Vittorio Bilò, Andrea D'Ascenzo, Mattia D'Emidio, and Giuseppe F. Italiano. On the Performance of Mildly Greedy Players in k-Coloring Games. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 21:1-21:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.21

Author Details

Vittorio Bilò
  • University of Salento, Lecce, Italy
Andrea D'Ascenzo
  • Luiss University, Rome, Italy
Mattia D'Emidio
  • University of L'Aquila, Italy
Giuseppe F. Italiano
  • Luiss University, Rome, Italy

Funding

Andrea D'Ascenzo and Giuseppe F. Italiano: Partially supported by the Italian Ministry of University and Reseach under PRIN Project n. 2022TS4Y3N - EXPAND: scalable algorithms for EXPloratory Analyses of heterogeneous and dynamic Networked Data.
  • Bilò, Vittorio: Partially supported by GNCS-INdAM; the PNRR MIUR project FAIR - Future AI Research (PE00000013), Spoke 9 - Green-aware AI; MUR - PNRR IF Agro@intesa; the Project SERICS (PE00000014) under the NRRP MUR program funded by the EU – NGEU.
  • D'Emidio, Mattia: Partially supported by: National Group for Scientific Computation of Istituto Nazionale di Alta Matematica (GNCS-INdAM); European Union under the Italian National Recovery and Resilience Plan of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 - program RESTART), project MoVeOver/SCHEDULE ("Smart interseCtions witH connEcteD and aUtonomous vehicLEs", CUP J33C22002880001).

Acknowledgements

The authors would like to thank Prof. Gianpiero Monaco (University of Chieti-Pescara) for offering constructive comments on the methodological approach to address the problem that motivated this study.

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