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The nonatomic assignment model. (English) Zbl 0808.90021
Summary: We formulate a model with a continuum of individuals to be assigned to a continuum of different positions which is an extension of the finite housing market version due to Shapley and Shubik. We show that optimal solutions to such a model exist and have properties similar to those established for finite models, namely, an equivalence among the following: (i) optimal solutions to the linear programming problem (and its dual) associated with the assignment model; (ii) the core of the associated market game; (iii) the Walrasian equilibria of the associated market economy.

91B50 General equilibrium theory
90C05 Linear programming
91A40 Other game-theoretic models
49J52 Nonsmooth analysis
Full Text: DOI
[1] Aumann, R.J.: Markets with a continuum of traders. Econometrica32, 39-50 (1964) · Zbl 0137.39003 · doi:10.2307/1913732
[2] Aumann, R.J., Shapley, L.S.: Values of non-atomic games. Princeton: Princeton University Press 1974 · Zbl 0311.90084
[3] Debreu, G., Scarf, H.: A limit theorem on the core of an economy. Int. Econ. Rev.4, 236-246 (1963) · Zbl 0122.37702 · doi:10.2307/2525306
[4] Diestel, J., Uhl, J.J.: Vector measures. Providence: American Mathematical Society 1977 · Zbl 0369.46039
[5] Dunford, N., Schwartz, J.T.: Linear operators part I: General theory. New York: Interscience 1958 · Zbl 0084.10402
[6] Dunz, K.: PhD thesis. Berkeley: University of California 1983
[7] Edgeworth, F.Y.: Mathematical physics. London: Paul 1881 · Zbl 0005.17402
[8] Faden, A.M.: The abstract transportation problem. In: Zarley, A.M. (ed.) Papers in quantitative economics, vol. 2. The University Press of Kansas 1971
[9] Faden, A.M.: Economics of time and space: the measure-theoretic foundations of social sciences. Ames: Iowa State University Press 1977 · Zbl 0414.90003
[10] Gretsky, N.E., Ostroy, J.M.: Thick and thin market nonatomic exchange economies. Proceedings of the Conference on General Equilibrium Theory, held at Indiana, Purdue University of Indianapolis, USA, February 10-12,1984. In: Advances in Equilibrium theory, no. 214 (Lecture notes in economics and mathematical system). Berlin Heidelberg New York: Springer 1985
[11] Hammond, P.J., Kaneko, M., Wooders, M.H.: Continuum economies with finite coalitions: Core, equilibria and widespread externalities. J. Econ. Theory49, 113-134 (1989) · Zbl 0678.90020 · doi:10.1016/0022-0531(89)90070-7
[12] Hildenbrand, W.: Core and equilibria of a large economy, no. 5. In: Princeton studies in mathematical economics. Princeton: Princeton University Press 1975
[13] Kaneko, M., Wooders, M.H.: The core of a game with a continuum of players and finite coalitions; the model and some results. Math. Soc. Sci.12, 105-137 (1986) · Zbl 0616.90106 · doi:10.1016/0165-4896(86)90032-6
[14] Kantorovich, L.V.: On the translocation of masses. (Translated into English and reprinted in Manage. Sci.5, 1-4 (1958)) Dokla. Akad. Nauk SSSR37, 199-201 (1942)
[15] Kantorovich, L.V., Akilov, G.P.: Functional analysis, 2nd. edn. Oxford New York: Pergamon Press 1982 · Zbl 0484.46003
[16] Kluvanek, I.: The range of a vector-valued measure. Math. Syst. Theory7, 44-54 (1973) · Zbl 0256.28008 · doi:10.1007/BF01824806
[17] Krabs, W.: Optimization and approximation. (Translation of Optimierung und Approximation) New York: Wiley & Sons 1979 · Zbl 0409.90051
[18] Levin, V.L.: Some applications of duality for the problem of translocation of masses with lower semicontinuous cost function: closed perferences and Choquet theory. Sov. Math. Dokl.24, 262-266 (1981)
[19] Levin, V.L.: The problem of mass transfer in a topological space, and probability measures having given marginal measures on the product of two spaces. Sov. Math. Dokl.29, 638-643 (1984) · Zbl 0622.60004
[20] Levin, V.L., Milyutin, A.A.: The problem of mass transfer with a discontinuous cost function and a mass statement of the duality problem for convex extremal problems. Russ. Math. Surv.34, 1-78 (1978) · Zbl 0437.46064 · doi:10.1070/RM1979v034n03ABEH003996
[21] Lewis, A.: Extreme point methods for infinite linear programming. PhD thesis, Cambridge University (1986)
[22] Lindenstrauss, J.: A remark on extreme doubly stochastic measures. Am. Math. Mon.72, 379-382 (1965) · Zbl 0152.24404 · doi:10.2307/2313497
[23] Makowski, L.: A characterization of perfectly competitive economies with production. J. Econ. Theory22, 208-221 (1980) · Zbl 0431.90017 · doi:10.1016/0022-0531(80)90040-X
[24] Ostroy, J.M.: The no-surplus condition as a characterization of perfectly competitive equilibrium. J. Econ. Theory22, 183-207 (1980) · Zbl 0437.90022 · doi:10.1016/0022-0531(80)90039-3
[25] Ostroy, J.M.: A reformulation of the marginal productivity theory of distribution. Econometrica52, 599-630 (1984) · Zbl 0569.90011 · doi:10.2307/1913467
[26] Ostroy, J.M., Zame, W.R.: Nonatomic economies and the boundaries of perfect competition. UCLA Economics Department, Working Paper No. 502 · Zbl 0798.90011
[27] Shapley, L.S.: Markets as cooperative games. Technical Report P-B29, Rand Corporation, Santa Monica, CA. Unpublished lecture notes from 1953-54 Princeton seminar (1955)
[28] Shapley, L.S., Shubik, M.: The assignment game, i: the core. Int. J. Game Theory1, 111-130 (1972) · Zbl 0236.90078 · doi:10.1007/BF01753437
[29] Simon, L., Zame, W.R.: Discontinuous games and endogenous sharing rules. Econometrica58, 861-872 (1990) · Zbl 0729.90098 · doi:10.2307/2938353
[30] Tulcea, A.I., Tulcea, C.I.: Topics in the theory of lifting. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 48. New York: Springer 1969 · Zbl 0179.46303
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