# Refining Nash Implementation

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## 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, *t*_{1},…, *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, *t*_{1},…, *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## Preview

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## References

- 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 - For a general introduction to subgame perfection see D. Fudenberg and J. Tirole (1991),
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*A Course in Microeconomic Theory*(Princeton University Press).Google Scholar - 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 - A refinement of the results obtained in the paper above was obtained by: D. Abreu and A. Sen (1990), ‘Subgame Perfect Implementation: A Necessary and Almost Sufficient Condition’,
*Journal of Economic Theory*, 50, pp. 285–299.CrossRefGoogle Scholar - The literature on Subgame Perfect Nash implementation is reviewed in the survey by J. Moore (1992), ‘Implementation, Contracts, and Renegotiation in Environments with Complete Information’ in J.J. Laffort (ed.),
*Advances in Economic Theory*(Cambridge University Press).Google Scholar - 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 - See also R. Serrano, ‘A Market to Implement the Core’.
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- Virtual Implementation is considered in the following papers: H. Matsushima (1988), ‘A New Approach to the Implementation Problem’,
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