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Algorithms may not learn to play a unique Nash equilibrium

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Abstract

There is a widespread hope that, in the near future, algorithms become so sophisticated that “solutions” to most problems are found by machines. In this note, we throw some doubts on this expectation by showing the following impossibility result: given a set of finite-memory, finite-iteration algorithms, a continuum of games exist, whose unique and strict Nash equilibrium cannot be reached from a large set of initial states. A Nash equilibrium is a social solution to conflicts of interest, and hence finite algorithms should not be always relied upon for social problems. Our result also shows how to construct games to deceive a given set of algorithms to be trapped in a cycle without a Nash equilibrium.

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Notes

  1. It is well-known that Lemke–Howson algorithm [11] can find a Nash equilibrium of any finite game, although the time it takes can be exponential (Savani and Stengel [16]). Our motivation is completely different from this line of research.

  2. In order to always choose a pure-action best response, some tie-breaking rule must be added. This caveat applies to all behavior rules in the following, but our impossibility result is independent of the tie-breaking rules.

  3. The standard fictitious play rule uses the entire history to compute the “observed frequency” and thus requires unlimited memory. However, we can modify the definition of the “observed frequency” to allow bounded memory.

  4. The standard model of the level-k theory is for a single population model with a symmetric component game.

  5. Since we allow learning of how to choose a behavior rule, it is not a simple “learning process”.

  6. For an illustration of a rule-learning model, see Fig. 1 of Stahl [18].

  7. We can allow an S1-player to choose a mixed-action best response \(\sigma _i \in BR_i(a_j)\) for some \(a_j \in A_j \mid _h\). This, however, complicates the analysis without affecting our impossibility result.

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Acknowledgements

The authors are grateful to an anonymous referee for useful comments.

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Authors and Affiliations

  1. Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

    Takako Fujiwara-Greve

  2. Dipartimento di Economia e Finanza, Catholic University of Milan, Via Necchi 5, 20123, Milan, Italy

    Carsten Krabbe Nielsen

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  1. Takako Fujiwara-Greve
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  2. Carsten Krabbe Nielsen
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Correspondence to Takako Fujiwara-Greve.

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Fujiwara-Greve, T., Nielsen, C.K. Algorithms may not learn to play a unique Nash equilibrium. J Comput Soc Sc 4, 839–850 (2021). https://doi.org/10.1007/s42001-021-00109-9

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