Skip to main content

How long to Pareto efficiency?

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.

This is a preview of subscription content, access via your institution.

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.

References

  • Arieli I, Babichenko Y (2011) Average testing and the efficient boundary. Discussion Paper 567, Center for the Study of Rationality, Hebrew University.

  • Conitzer V, Sandholm T (2004) Communication complexity as a lower bound for learning in games. In: Brody CE (ed.) Machine Learning, Proceedings of the Twenty-First International Conference (ICML 2004), Banff, July 2004, ACM, New York, pp 185-192.

  • Hart S, Mas Colell A (2003) Uncoupled dynamics do not lead to Nash equilibrium. Am Econ Rev 93:1830–1836

    Article  Google Scholar 

  • Hart S, Mas Colell A (2006) Stochastic uncoupled dynamics and Nash equilibrium. Games Econ Behav 57:286–303

    Article  Google Scholar 

  • Hart S, Mansour Y (2010) How long to equilibrium? The communication complexity of uncoupled equilibrium procedures. Games Econ Behav 69:107–126

    Article  Google Scholar 

  • Kushilevitz E, Nisan N (1997) Communication complexity. Cambridge University Press, Cambridge

    Google Scholar 

  • Marden JR, Young HP, Pao LY (2011) Achieving Pareto optimality through distributed learning, Economics series working papers, 557. University of Oxford, Oxford

    Google Scholar 

  • Pradelski B, Young HP (2010) Efficiency and equilibrium in trial and error learning, Economics series working papers, 480. University of Oxford, Oxford

    Google Scholar 

  • Saari DG, Simon CP (1978) Effective price mechanisms. Econometrica 46:1097–1125

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yakov Babichenko.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

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

Keywords

  • Nash Equilibrium
  • Payoff Function
  • Communication Complexity
  • Pareto Improvement
  • Pareto Efficiency