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Constrained implementation. (English) Zbl 1422.91233
Summary: Consider a society with two sectors (issues or objects) that faces a design problem. Suppose that the sector-2 dimension of the design problem is fixed and represented by a mechanism \(\Gamma^2\), and that the designer operates under this constraint for institutional reasons. A sector-1 mechanism \(\Gamma^1\) constrained implements a social choice rule \(\varphi\) in Nash equilibrium if for each profile of agents’ preferences, the set of (pure) Nash equilibrium outcomes of the mechanism \(\Gamma^1\times\Gamma^2\) played by agents with those preferences always coincides with the recommendations made by \(\varphi\) for that profile. If this mechanism design exercise could be accomplished, \(\varphi\) would be constrained implementable. We show that constrained monotonicity, a strengthening of (Maskin) monotonicity, is a necessary condition for constrained implementation. When there are more than two agents, and when the designer can use the private information elicited from agents via \(\Gamma^2\) to make a socially optimal decision for sector 1, constrained monotonicity, combined with an auxiliary condition, is sufficient. This sufficiency result does not rule out any kind of complementarity between the two sectors.
MSC:
91B14 Social choice
91A40 Other game-theoretic models
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[1] Abreu, D.; Matsushima, H., Virtual implementation in iteratively undominated strategies: complete information, Econometrica, 60, 993-1008, (1992) · Zbl 0766.90002
[2] Diamantaras, D.; Wilkie, S., A generalization of Kaneko’s ratio equilibrium for economies with private and public goods, J. Econ. Theory, 62, 499-512, (1994) · Zbl 0801.90016
[3] Dutta, B.; Sen, A., A necessary and sufficient condition for two-person Nash implementation, Rev. Econ. Stud., 58, 121-128, (1991) · Zbl 0717.90005
[4] Hayashi, T.; Lombardi, M., Implementation in partial equilibrium, J. Econ. Theory, 169, 13-34, (2017) · Zbl 1400.91165
[5] Hayashi, T.; Sakai, T., Nash implementation of competitive equilibria in the job-matching market, Int. J. Game Theory, 38, 453-467, (2009) · Zbl 1211.91178
[6] Kaneko, M., The ratio equilibrium and a voting game in a public goods economy, J. Econ. Theory, 16, 123-136, (1977) · Zbl 0399.90019
[7] Jackson, M. O., Implementation in undominated strategies: a look at bounded mechanisms, Rev. Econ. Stud., 59, 757-775, (1992) · Zbl 0771.90004
[8] Jackson, M. O., A crash course in implementation theory, Soc. Choice Welf., 18, 655-708, (2001) · Zbl 1069.91557
[9] Lombardi, M.; Yoshihara, N., A full characterization of Nash implementation with strategy space reduction, Econ. Theory, 54, 131-151, (2013) · Zbl 1284.91023
[10] Maskin, E., Nash equilibrium and welfare optimality, Rev. Econ. Stud., 66, 23-38, (1999) · Zbl 0956.91034
[11] Maskin, E.; Sjöström, T., Implementation theory, (Arrow, K.; Sen, A. K.; Suzumura, K., Handbook of Social Choice and Welfare, (2002), Elsevier Science: Elsevier Science Amsterdam), 237-288
[12] Moore, J.; Repullo, R., Nash implementation: a full characterization, Econometrica, 58, 1083-1100, (1990) · Zbl 0731.90009
[13] Ollár, M.; Penta, A., Full implementation and belief restrictions, Am. Econ. Rev., 107, 2243-2277, (2017)
[14] Schmeidler, D., Walrasian analysis via strategic outcome functions, Econometrica, 48, 1585-1593, (1980) · Zbl 0457.90014
[15] Sjöström, T., On the necessary and sufficient conditions for Nash implementation, Soc. Choice Welf., 8, 333-340, (1991) · Zbl 0734.90007
[16] Svensson, L.-G., Nash implementation of competitive equilibria in a model with indivisible goods, Econometrica, 59, 869-877, (1991) · Zbl 0748.90004
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