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Refining Nash Implementation

  • Luis C. Corchón
Chapter
  • 37 Downloads

Abstract

In the two preceding chapters we have studied the Nash equilibrium approach to the problem of implementation. Various authors have put forward certain undesirable consequences of the property of monotonicity which, as you will remember, is a necessary condition for implementation in Nash equilibria. Firstly, monotonicity prohibits any type of consideration based on the cardinality of utility functions. Secondly, in some cases, distributional considerations may collide with monotonicity. The following example (taken from Moore and Repullo, 1988) will illustrate this point. We assume that there is a public good (which can take two values, 0 or 1), and a private good. The utility functions are quasi linear of the form u i = a i y + x i and the cost of 1 (resp. 0) is 1 (resp. 0). An allocation is a list (y, t 1 ,…, t n ) where y ∈ {0, 1} and t i is the tax paid by i. An economy u is a list (a i ,…, a n ) (the parameter a i is called the marginal propensity to pay). Consider an economy u for which the allocation (1, t1,…, t n ) is optimal. We now consider an economy u′ such that all the marginal propensities to pay, apart from that of the first individual, increase. Then, monotonicity implies that (1, t1,…, t n ) is also optimal for u′ no matter how much the marginal propensities to pay of all the other consumers have increased.

Keywords

Nash Equilibrium Social Choice Social Choice Function Strong Equilibrium Marginal Propensity 
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References

  1. The pioneering contribution in stage games is R. Selten (1975), ‘A Reexamination of the Perfection Concept for Equilibrium Points in Extensive Games’, International Journal of Game Theory, pp. 25–55.Google Scholar
  2. For a general introduction to subgame perfection see D. Fudenberg and J. Tirole (1991), Game Theory (MIT Press)Google Scholar
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  4. Implementation by means of subgame perfect Nash equilibrium was proposed in the following article: J. Moore and R. Repullo (1988), ‘Subgame Perfect Implementation’, Econometrica, vol. 56, no. 5 (September), pp. 1191–220.CrossRefGoogle Scholar
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  7. This paper offers references on the early work by Farquharson (1957) and Moulin (1979) that were forerunners of this approach. The pioneering paper on non-cooperative foundations of bargaining solutions by using Subgame Perfect Nash Equilibrium is: A. Rubinstein (1982), ‘Perfect Equilibrium in a Bargaining Model’, Econometrica, 50, pp. 97–109.CrossRefGoogle Scholar
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  16. The previous list does not exhaust all the possible refinements. For instance for implementation in trembling hand perfect Nash equilibria see T. Sjöström (1993), ‘Implementation in Perfect Equilibria’, Social Choice and Welfare, 10, pp. 97–106.CrossRefGoogle Scholar
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  18. Virtual Implementation is considered in the following papers: H. Matsushima (1988), ‘A New Approach to the Implementation Problem’, Journal of Economic Theory, 45, pp. 128–44CrossRefGoogle Scholar
  19. D. Abreu and A. Sen (1991), ‘Virtual Implementation in Nash Equilibrium’, Econometrica, vol. 59, pp. 997–1007CrossRefGoogle Scholar
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  21. The latter paper aroused an interchange of notes between Abreu and Matsushima (1992b) and Glazer and Rosenthal (1992) in Econometrica, 60, 6, pp. 1435–1441 (1992). Virtual Implementation in Economic Enviroments has been studied by J. Bergin and A. Sen. ‘Implementation in Generic Environments’, Mimeo, Queens University, 1995.Google Scholar

Copyright information

© Luis C. Corchón 1996

Authors and Affiliations

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

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