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One-step-ahead implementation. (English) Zbl 1417.91294
Summary: In many situations, agents are involved in an allocation problem that is followed by another allocation problem whose optimal solution depends on how the former problem has been solved. In this paper, we take this dynamic structure of allocation problems as an institutional constraint. By assuming a finite number of allocation problems, one for each period/stage, and by assuming that all agents in society are involved in each allocation problem, a dynamic mechanism is a period-by-period process. This process generates at any period-\(t\) history a period-\(t\) mechanism with observable actions and simultaneous moves. We also assume that the objectives that a planner wants to achieve are summarized in a social choice function (SCF), which maps each state (of the world) into a period-by-period outcome process. In each period \(t\), this process selects for each state a period-\(t\) socially optimal outcome conditional on the complete outcome history realized up to period \(t - 1\). Heuristically, the SCF is one-step-ahead implementable if there exists a dynamic mechanism such that for each state and each realized period-\(t\) history, each of its subgame perfect Nash equilibria generates a period-by-period outcome process that coincides with the period-by-period outcome process that the SCF generates at that state from period \(t\) onwards. We identify a necessary condition for SCFs to be one-step-ahead implemented, one-step-ahead Maskin monotonicity, and show that it is also sufficient under a variant of the condition of no veto-power when there are three or more agents. Finally, we provide an account of welfare implications of one-step-ahead implementability in the contexts of trading decisions and voting problems.

91B32 Resource and cost allocation (including fair division, apportionment, etc.)
91B14 Social choice
91A20 Multistage and repeated games
Full Text: DOI
[1] Abreu, D.; Sen, A., Subgame perfect implementation: a necessary and almost sufficient condition, J. Econ. Theory, 50, 285-299, (1990) · Zbl 0694.90010
[2] Abreu, D.; Sen, A., Virtual implementation in Nash equilibrium, Econometrica, 59, 997, (1991) · Zbl 0732.90007
[3] Aghion, P.; Fudenberg, D.; Holden, R.; Kunimoto, T.; Tercieux, O., Subgame-perfect implementation under information perturbations, Q. J. Econ., 1843-1881, (2012) · Zbl 1400.91047
[4] Arrow, K. J., The role of securities in the optimal allocation of risk-bearing, Rev. Econ. Stud., 31, 91-96, (1964)
[5] Arrow, K. J.; Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, 265-290, (1954) · Zbl 0055.38007
[6] Athey, S.; Segal, I., An efficient dynamic mechanism, Econometrica, 81, 2463-2485, (2013) · Zbl 1304.91080
[7] Bergemann, D.; Välimäki, J., The dynamic pivotal mechanism, Econometrica, 78, 771-789, (2010) · Zbl 1229.91206
[8] De Donder, P.; Le Breton, M.; Peluso, E., Majority voting in multidimensional policy spaces: Kramer-Shepsle versus Staskelberg, J. Publ. Econ. Theory, 14, 879-909, (2012)
[9] Eső, P.; Szentes, B., Dynamic contracting: An irrelevance theorem, Theor. Econ., 12, 109-139, (2017) · Zbl 1396.91365
[10] Hayashi, T.; Lombardi, M., Implementation in partial equilibrium, J. Econ. Theory, 169, 13-34, (2017) · Zbl 1400.91165
[11] Herrero, M. J.; Srivastava, S., Implementation via backward induction, J. Econ. Theory, 56, 70-88, (1992) · Zbl 0764.90012
[12] Hurwicz, L.; Maskin, E.; Postlewaite, A., Feasible nash implementation of social choice rules when the designer does not know endowment or production sets, (Ledyard, J. O., The Economics of Informational Decentralization. Complexity, Efficiency and Stability, (1995), Kluwer: Kluwer Amsterdam), 367-433
[13] Jackson, M. O., Implementation in undominated strategies: A look at bounded mechanisms, Rev. Econ. Stud., 59, 757-775, (1992) · Zbl 0771.90004
[14] Kalai, E.; Ledyard, J. O., Repeated implementation, J. Econ. Theory, 83, 308-317, (1998) · Zbl 0915.90005
[15] Krusell, P.; Quadrini, V.; Rios-Rull, J.-V., Politico-economic equilibrium and economic growth, J. Econom. Dynam. Control, 21, 243-272, (1997) · Zbl 0875.90154
[16] Lee, J.; Sabourian, H., Efficient repeated implementation, Econometrica, 79, 1967-1994, (2011) · Zbl 1241.91046
[17] Lombardi, M.; Yoshihara, N., A full characterization of nash implementation with strategy space reduction, Econ. Theory, 54, 131-151, (2013) · Zbl 1284.91023
[18] Lombardi, M.; Yoshihara, N., Natural implementation with semi-responsible agents in pure exchange economies, Int. J. Game Theory, 46, 1015-1036, (2017) · Zbl 1411.91373
[19] Maskin, E., Nash equilibrium and welfare optimality, Rev. Econ. Stud., 66, 23-38, (1999) · Zbl 0956.91034
[20] Matsushima, H., A new approach to the implementation problem, J. Econ. Theory, 45, 128-144, (1988) · Zbl 0642.90011
[21] McKenzie, L. W., On equilibrium in Graham’s model of world trade and other competitive systems, Econometrica, 22, 147-161, (1954) · Zbl 0055.13702
[22] Mezzetti, C.; Renou, L., Repeated nash implementation, Theor. Econ., 12, 249-285, (2017) · Zbl 1396.91149
[23] Moore, J.; Repullo, R., Subgame perfect implementation, Econometrica, 56, 1191-1220, (1988) · Zbl 0657.90005
[24] Moore, J.; Repullo, R., Nash implementation: A full characterization, Econometrica, 58, 1083-1100, (1990) · Zbl 0731.90009
[25] Ollár, M.; Penta, A., Full implementation and belief restrictions, Am. Econ. Rev., 107, 2243-2277, (2017)
[26] Osborne, M.; Rubinstein, A., A course in Game Theory, (1994), MIT Press · Zbl 1194.91003
[27] Palfrey, T.; Srivastava, S., Nash-implementation using undominated strategies, Econometrica, 59, 479-501, (1991) · Zbl 0734.90004
[28] Pavan, A.; Segal, I.; Toikka, J., Dynamic mechanism design: A myersonian approach, Econometrica, 82, 601-653, (2014) · Zbl 1419.91147
[29] Penta, A., Robust dynamic implementation, J. Econ. Theory, 160, 280-316, (2015) · Zbl 1369.91056
[30] Persson, T.; Tabellini, G., Political Economics, (2002), The MIT Press
[31] Prescott, E. C.; Mehra, R., Recursive competitive equilibrium: The case of homogeneous households, Econometrica, 48, 1365-1379, (1980) · Zbl 0464.90006
[32] Radner, R., Existence of equilibrium of plans, prices, and price expectations in a sequence of markets, Econometrica, 40, 289-304, (1972) · Zbl 0245.90006
[33] Radner, R., Equilibrium under uncertainty, (Arrow, K. J.; Intriligator, M. D., Handbook of Mathemical Economics, Vol II, (1982), North-Holland: North-Holland Amsterdam), 923-1006, (Chapter 20)
[34] Vartiainen, H., Subgame perfect implementation: a full characterization, J. Econ. Theory, 133, 111-126, (2007) · Zbl 1280.91061
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