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.
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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.
For a general introduction to subgame perfection see D. Fudenberg and J. Tirole (1991), Game Theory (MIT Press)
D. Kreps (1990), A Course in Microeconomic Theory (Princeton University Press).
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.
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.
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).
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.
See also R. Serrano, ‘A Market to Implement the Core’. Journal of Economic Theory, 67, 1995, pp. 285–94.
Subgame Perfect implementation of fair division solutions have been considered by V. P. Crawford, ‘A Self-Administer Solution to the Bargaining Problem’, Review of Economic Studies, 47 (1980) pp. 385–92.
by G. Demange, ‘Implementing Efficient Egalitarian Equivalent Allocations’, Econometrica, 52 (1984) pp. 1167–78.
The following paper first proposed implementation in undominated Nash equilibrium: T.R. Palfrey and S. Srivastava (1991), ‘Nash Implementation using Undominated Strategies’, Econometrica vol. 59 no. 2 pp. 479–501 (March).
Some of the constructions used in this paper were criticized by: M. Jackson (1992), ‘Implementation in Undominated Strategies: A Look at Bounded Mechanisms’, Review of Economic Studies, 59, pp. 757–75.
The following papers offer implementing mechanisms that are free from the above objections: M. Jackson, T. Palfrey and S. Srivastava (1994), ‘Undominated Nash Implementation in Bounded Mechanisms’, Games and Economic Behavior, 6, pp. 474–501
T. Sjöström (1994), ‘Implementation in Undominated Nash Equilibria without Integer Games’, Games and Economic Behavior, 6, pp. 502–511.
Double implementation in Nash and undominated Nash equilibria is considered by T. Yamato (1995), ‘Nash Implementation and Double Implementation: Equivalence Theorems’, Journal of Mathematical Economics
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.
Implementation in iteratively undominated strategies is considered in D. Abreu and H. Matsushima (1992a), ‘Exact Implementation’, Mimeo, Princeton University.
Virtual Implementation is considered in the following papers: H. Matsushima (1988), ‘A New Approach to the Implementation Problem’, Journal of Economic Theory, 45, pp. 128–44
D. Abreu and A. Sen (1991), ‘Virtual Implementation in Nash Equilibrium’, Econometrica, vol. 59, pp. 997–1007
D. Abreu and H. Matsushima (1992b), ‘Virtual Implementation in Iteratively Undominated Strategies: Complete Information’, Econometrica, 60, no. 5, pp. 993–1008 (September).
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.
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© 1996 Luis C. Corchón
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Corchón, L.C. (1996). Refining Nash Implementation. In: The Theory of Implementation of Socially Optimal Decisions in Economics. Palgrave Macmillan, London. https://doi.org/10.1057/9780230372832_6
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DOI: https://doi.org/10.1057/9780230372832_6
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