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On the Structure of Weakly Acyclic Games

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Abstract

The class of weakly acyclic games, which includes potential games and dominance-solvable games, captures many practical application domains. In a weakly acyclic game, from any starting state, there is a sequence of better-response moves that leads to a pure Nash equilibrium; informally, these are games in which natural distributed dynamics, such as better-response dynamics, cannot enter inescapable oscillations. We establish a novel link between such games and the existence of pure Nash equilibria in subgames. Specifically, we show that the existence of a unique pure Nash equilibrium in every subgame implies the weak acyclicity of a game. In contrast, the possible existence of multiple pure Nash equilibria in every subgame is insufficient for weak acyclicity in general; here, we also systematically identify the special cases (in terms of the number of players and strategies) for which this is sufficient to guarantee weak acyclicity.

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Notes

  1. In some of the economics literature, the terms “weak finite-improvement path property” (weak FIP) and “weak finite best-response path property” (weak FBRP) are also used, for weak acyclicity under better- and best-response dynamics, respectively.

  2. For example, this might happen if the link between 2 and d temporarily fails. 2 would always choose to send traffic to 1 (if anywhere); 1 would eventually converge to sending traffic directly to d (with 2 sending its traffic to 1), and 3 would then be able to send its traffic along 321d. Once the failed link between 2 and d is restored, 2’s best response to the choices of the other nodes is to send its traffic directly to d, resulting in the first configuration of the cycle above.

  3. Yamamori and Takahashi use the terms quasi-acyclicity for weak acyclicity under best response, and Pure Nash Equilibrium Property (PNEP) for subgame stability.

  4. Each player has 2 or more strategies.

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Author notes

  1. Aaron D. Jaggard

    Present address: Formal Methods Section (Code 5543), U.S. Naval Research Laboratory, Washington, DC, USA

Authors and Affiliations

  1. Google Research, Mountain View, CA, USA

    Alex Fabrikant

  2. DIMACS Center, Rutgers University, Piscataway, NJ, USA

    Aaron D. Jaggard

  3. Department of Computer Science, Colgate University, Hamilton, NY, USA

    Aaron D. Jaggard

  4. School of Computer Science and Engineering, Hebrew University of Jerusalem, Jerusalem, Israel

    Michael Schapira

Authors
  1. Alex Fabrikant
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  2. Aaron D. Jaggard
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  3. Michael Schapira
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Corresponding author

Correspondence to Aaron D. Jaggard.

Additional information

The first author was supported by a Cisco URP grant and a Princeton University postdoctoral fellowship. The second author was partially supported by NSF grants 0751674, 0753492, and 1101690. The third author was supported by a grant from the Israel Science Foundation (ISF) and by the Marie Curie Career Integration Grant (CIG). This is a revised and expanded version of a paper that appeared in the Proceedings of SAGT 2010.

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Fabrikant, A., Jaggard, A.D. & Schapira, M. On the Structure of Weakly Acyclic Games. Theory Comput Syst 53, 107–122 (2013). https://doi.org/10.1007/s00224-013-9457-0

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