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How long to Pareto efficiency?

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Abstract

We consider uncoupled dynamics (each player knows only his own payoff function) that reach outcomes that are Pareto efficient and individually rational. We show that in the worst case the number of periods it takes to reach these outcomes must be exponential in the number of players and hence the same number of periods it takes to reach Nash equilibria. For social welfare maximizing outcomes we provide a tight bound on the minimal number of steps required for reaching such an outcome by uncoupled dynamics.

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Notes

  1. We say that the bound \(f(n)\le g(n) \le h(n)\) on \(g(n)\) is tight if \(\underset{n \rightarrow \infty }{\lim } \frac{f(n)}{h(n)}=1\).

  2. ”Reach” is synonymous for ”play” in game theory and for ”know” in communication complexity. We mean it in the latter sense (but, as we will see later, it does not really matter).

  3. Throughout the paper \(\delta \) will denote the Dirac measure.

  4. We assume, without loss of generality, that \(n\) is even.

  5. The ”first one” is well defined because the set of \(PIR\) distributions is closed.

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Acknowledgments

This work is part of the author’s Ph.D. thesis. The author wishes to thank his supervisor Sergiu Hart for his support and guidance and Noam Nisan for useful discussion. This research was partially supported by ERC Grant 0307950, and by ISF Grant 0397679.

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  1. Department of Computing and Mathematical Sciences, Center for the Mathematics of Information, California Institute of Technology, Pasadena, CA,  91125, USA

    Yakov Babichenko

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  1. Yakov Babichenko
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Correspondence to Yakov Babichenko.

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Babichenko, Y. How long to Pareto efficiency?. Int J Game Theory 43, 13–24 (2014). https://doi.org/10.1007/s00182-013-0365-y

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  • DOI: https://doi.org/10.1007/s00182-013-0365-y

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