Equilibria for games with combined qualitative and quantitative objectives


The overall aim of our research is to develop techniques to reason about the equilibrium properties of multi-agent systems. We model multi-agent systems as concurrent games, in which each player is a process that is assumed to act independently and strategically in pursuit of personal preferences. In this article, we study these games in the context of finite-memory strategies, and we assume players’ preferences are defined by a qualitative and a quantitative objective, which are related by a lexicographic order: a player first prefers to satisfy its qualitative objective (given as a formula of linear temporal logic) and then prefers to minimise costs (given by a mean-payoff function). Our main result is that deciding the existence of a strict \(\epsilon \) Nash equilibrium in such games is 2ExpTime-complete (and hence decidable), even if players’ deviations are implemented as infinite-memory strategies.

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Fig. 1


  1. 1.

    We remark that strict Nash equilibria appear naturally in the study of evolutionary game-theory [41]. Indeed, every strict Nash equilibrium in a symmetric game is evolutionary stable.

  2. 2.

    The transition systems on which these games are played are sometimes called “concurrent game structures” and sometimes “arenas”. Note that “concurrent” in this context simply means that players move at the same time, i.e., synchronously.

  3. 3.

    Note that the arena is similar to the one of [45, Fig. 3 in Lemma 7] to prove that Multi mean-payoff games require in general infinite memory strategies to be played optimally.

  4. 4.

    This definition extends the one provided in [43] which considers mean-payoff games.

  5. 5.

    This is an abuse of notation, as the transition functions of the two strategies are defined on a different set of decisions. Here we mean that the transition functions of \(\mathbf {\upsigma }'\) simply copy the ones in \(\mathbf {\upsigma }\) by ignoring the components expressed by agents \(a_{1}\) and \(a_{2}\)

  6. 6.

    With another abuse of notation, we here mean the restriction of sequences in \({\text {A}\!\text {P}} \cup \{p\}\) to sequences in \({\text {A}\!\text {P}}\).


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The authors would like to thank the reviewers for their careful reading of preliminary versions of this manuscript. This allowed us to significantly improve the quality and solidity of our results.

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Correspondence to Giuseppe Perelli.

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We gratefully acknowledge the financial support of ERC Advanced Investigator Grant 291528 at Oxford (J. Gutierrez, G. Perelli, and M. Wooldridge), INdAM Research Project 2017 “Logica e Autonomi per il Model Checking” at Naples (A. Murano), Marie Curie Fellowship of the Istituto Nazionale di Alta Matematica (S. Rubin), the EPSRC Centre for Doctoral Training in Autonomous Intelligent Machines and Systems EP/L015897/1 and the Ian Palmer Memorial Scholarship (T. Steeples). A preliminary version of this work appeared in [27].

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Gutierrez, J., Murano, A., Perelli, G. et al. Equilibria for games with combined qualitative and quantitative objectives. Acta Informatica (2020). https://doi.org/10.1007/s00236-020-00385-4

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