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Implementation in Nash Equilibrium (I): General Results

  • Luis C. Corchón
Chapter
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Abstract

In this chapter we will study the implementation of social choice correspondences by means of Nash equilibria. As you will remember from the previous chapter, the motivation for studying this type of implementation is to obtain positive results by relaxing the concept of equilibrium: instead of requiring the equilibrium strategy for each agent to be a good response to any possible strategy of the other players, the Nash equilibrium requires it to be a good reply for those who constitute a good reply of the other players (see Definition 3, Chapter 2).

Keywords

Utility Function Nash Equilibrium Social Choice Optimal Decision Social Choice Function 
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References

  1. The following articles offer a general view of implementation in Nash equilibrium: A. Postlewaite (1985), ‘Implementation via Nash Equilibria in Economic Environments’, Chap. 7 in L. Hurwicz, D. Schmeidler and H. Sonnenschein (eds), Social Goals and Social Organization (Cambridge University Press)Google Scholar
  2. J. Moore (1992), ‘Implementation, Contracts and Renegotiation in Environments with Complete Information’ in J.J. Laffont (ed.), Advances in Economic Theory (vol. I), VI World Congress of the Econometric Society (Cambridge University Press). The latter paper should be consulted for matters concerned with two-person implementation, renegotiation and implementation of social choice functions (see also exercises 24 and 27).Google Scholar
  3. The characterization of social choice correspondences which are implementable in Nash equilibria was first studied in E. Maskin (1977), ‘Nash Equilibrium and Welfare Optimality’, mimeo, MIT.Google Scholar
  4. The following papers present proofs that the no veto power and monotonicity imply that the social choice correspondence is implementable in Nash equilibria when there are more than two agents. S. Williams (1984), ‘Sufficient Conditions for Nash Implementation’, IMA mimeo, Minneapolis.Google Scholar
  5. R. Repullo (1987), ‘A Simple Proof of Maskin’s Theorem on Nash Implementation’, Social Choice and Welfare, 4, pp. 39–41.CrossRefGoogle Scholar
  6. T. Saijo (1988), ‘Strategy Space Reduction in Maskin’s Theorem: Sufficient Conditions for Nash Implementation’, Econometrica, vol. 56, no. 3 (May), pp. 693–700.CrossRefGoogle Scholar
  7. R.D. McKelvey (1989), ‘Game Forms for Nash Implementation of General Social Choice Correspondences’, Social Choice and Welfare, 6, pp. 139–56.CrossRefGoogle Scholar
  8. One problem which has drawn attention from the theorists is that of finding a condition which at the same time is necessary and sufficient for implementing social choice correspondences in Nash equilibria. See J. Moore and R. Repullo (1990), ‘Nash Implementation: A Full Characterization’, Econometrica, vol. 58, no. 5 (September), pp. 1083–99CrossRefGoogle Scholar
  9. V. Danilov (1992), ‘Implementation Via Nash Equilibrium’, Econometrica, vol. 60, pp. 43–56CrossRefGoogle Scholar
  10. T. Sjöström (1991), ‘On the Necessary and Sufficient Conditions for Nash Implementation’, Social Choice and Welfare, vol. 8, pp. 333–40.CrossRefGoogle Scholar
  11. The following papers can be consulted on the revelation principle in Nash equilibrium: R. Repullo (1986), ‘On the Revelation Principle under Complete and Incomplete Information’ in K. Binmore and P. Dasgupta (eds), Economic Organizations as Games (Oxford: Basil Blackwell)Google Scholar
  12. D. Mookherjee and Reichelstein, S. (1990), ‘The Revelation Approach to Nash Implementation’, Mimeo Graduate School of Business, Stanford UniversityGoogle Scholar
  13. D. Mookherjee and S. Reichelstein (1990), ‘Implementation via Augmented Revelation Mechanisms’, Review of Economic Studies, July, pp. 453–75Google Scholar
  14. H. Matsushima (1988) ‘A New Approach to the Implementation Problem’, Journal of Economic Theory, 45, pp. 128–44.CrossRefGoogle Scholar
  15. The theorems relating the Nash and Lindahl equilibria were published in: L. Hurwicz (1979), ‘On Allocations Attainable Through Nash Equilibria’ in J.J. Laffont (ed.), Aggregation and Revelation of Preferences (Amsterdam: North-Holland), also published in the Journal of Economic Theory, vol. 21, pp. 40–65 (1979); W. Thomson ‘Comment’ in Laffont (ed.), Aggregation and Revelation of PreferencesGoogle Scholar
  16. D. Schmeidler (1982), ‘A Condition Guaranteeing that the Nash Allocation is Walrasian’, Journal of Economic Theory, vol. 28, pp. 376–8.CrossRefGoogle Scholar
  17. A similar result to that of proposition 6 can be found in L. Hurwicz, E. Maskin and A. Postlewaite (1984), ‘Feasible Implementation of Social Choice Correspondences by Nash Equilibria’, Mimeo, Department of Economics of Minnesota and Pennsylvania.Google Scholar
  18. A question not studied in this chapter is the characterization of the allocations generated by Nash equilibria of manipulating games. The original contribution is due to Hurwicz. The interested reader may consult the following paper and the references there: W. Thomson (1984), ‘The Manipulability of Resource Allocation Mechanism’, Review of Economic Studies, vol. 51, pp. 447–60.CrossRefGoogle Scholar
  19. The theory of implementation without commitment has been developed in: E. Maskin and J. Moore (1987), ‘Implementation with Renegotiation’, Mimeo, Harvard UniversityGoogle Scholar
  20. B. Chakravorty, L. Corchôn and S. Wilkie (1992), ‘Credible Implementation’, Working paper, Bellcore 1992Google Scholar
  21. S. Baliga, L. Corchön and T. Sjöström (1994), ‘The Theory of Implementation when the Planner is a Player’, Working Paper, Harvard University, 1994.Google Scholar
  22. On signaling games the reader may consult D. Fudenberg and J. Tirole (1991), Game Theory (Boston: MIT Press), pp. 446–59.Google Scholar

Copyright information

© Luis C. Corchón 1996

Authors and Affiliations

  • Luis C. Corchón
    • 1
  1. 1.Universidad de AlicanteAlicanteSpain

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