zbMATH — the first resource for mathematics

Stable cost sharing in production allocation games. (English) Zbl 1422.91384
Summary: Suppose that a group of agents have demands for some good. Every agent owns a technology which allows them to produce the good, with these technologies varying in their effectiveness. If all technologies exhibit increasing returns to scale (IRS) then it is always efficient to centralize production of the good, whereas efficiency in the case of decreasing returns to scale (DRS) typically requires to spread production. We search for stable cost allocations while differentiating allocations with homogeneous prices, in which all units produced are traded at the same price, from allocations with heterogeneous prices. For the respective cases of IRS or DRS, it is shown that there always exist stable cost sharing rules with homogeneous prices. Finally, in the general framework (under which there may exist no stable allocation at all) we provide a sufficient condition for the existence of stable allocations with homogeneous prices. This condition is shown to be both necessary and sufficient in problems with unitary demands.

91B32 Resource and cost allocation (including fair division, apportionment, etc.)
91B38 Production theory, theory of the firm
91A12 Cooperative games
Full Text: DOI
[1] Anderson, RM; Aumann, RJ (ed.); Hart, S. (ed.), The core in perfectly competitive economies, No. 1, 413-457, (1992), New York · Zbl 0968.91521
[2] Anshelevich, E.; Dasgupta, A.; Kleinberg, J.; Tardos, E.; Wexler, T.; Roughgarden, T., The price of stability for network design with fair cost allocation, SIAM J Comput, 38, 1602-1623, (2008) · Zbl 1173.91321
[3] Bahel, E.; Trudeau, C., A discrete cost sharing model with technological cooperation, Int J Game Theory, 42, 439-460, (2013) · Zbl 1269.91051
[4] Bahel, E.; Trudeau, C., Stable lexicographic rules for shortest path games, Econ Lett, 125, 266-269, (2014) · Zbl 1311.91054
[5] Bergantinos, G.; Vidal-Puga, J., A fair rule in minimum cost spanning tree problems, J Econ Theory, 137, 326-352, (2007) · Zbl 1132.91366
[6] Bird, CJ, On cost allocation for a spanning tree: a game theoretic approach, Networks, 6, 335-350, (1976) · Zbl 0357.90083
[7] Camiña, E., A generalized assignment game, Math Soc Sci, 52, 152-161, (2006) · Zbl 1142.91323
[8] Crawford, VP; Knoer, EM, Job matching with heterogeneous firms and workers, Econometrica, 49, 437-450, (1981) · Zbl 1202.91141
[9] Gillies DB (1953) Some theorems on n-person games. Ph.D. Thesis, Department of Mathematics, Princeton University
[10] Jaume, D.; Massó, J.; Neme, A., The multiple-partners assignment game with heterogeneous sells and multi-unit demands: competitive equilibria, Polar Biol, 39, 2189-2205, (2016)
[11] Kaneko, M., On the core and competitive equilibria of a market with indivisible goods, Naval Res Logist, 21, 321-337, (1976) · Zbl 0366.90015
[12] Moulin, H., Cost sharing in networks: some open questions, Int Game Theory Rev, 15, 1340001, (2013) · Zbl 1274.91051
[13] Moulin, H.; Sprumont, Y., Fair allocation of production externalities: recent results, Rev d’Econ Politique, 117, 7-36, (2007)
[14] Núñez, M.; Rafels, C., The assignment game: the \(\tau \)-value, Int J Game Theory, 31, 411-422, (2002) · Zbl 1083.91027
[15] Núñez, M.; Rafels, C., A survey on assignment markets, J Dyn Games, 2, 227-256, (2017) · Zbl 1391.91020
[16] Quant, M.; Borm, P.; Reijnierse, H., Congestion network problems and related games, Eur J Oper Res, 172, 919-930, (2006) · Zbl 1111.90019
[17] Quinzii, M., Core and competitive equilibria with indivisibilities, Int J Game Theory, 13, 41-60, (1984) · Zbl 0531.90012
[18] Rosenthal, EC, Shortest path games, Eur J Oper Res, 224, 132-140, (2013) · Zbl 1292.91024
[19] Sanchez-Soriano, J.; Lopez, MA; Garcia-Jurado, I., On the core of transportation games, Math Soc Sci, 41, 215-225, (2001) · Zbl 0973.91005
[20] Shapley, LS, Cores of convex games, Int J Game Theory, 1, 11-26, (1971) · Zbl 0222.90054
[21] Shapley, LS; Shubik, M., The assignment game I: the core, Int J Game Theory, 1, 9-25, (1971)
[22] Sharkey, WW; Ball, MO (ed.); Magnanti, TL (ed.); Nonma, CL (ed.); Nemhauser, GL (ed.), Network models in economics, No. 8, 713-765, (1995), New York
[23] Sotomayor, M.; Majumdar, M. (ed.), The multiple partners game, 269-283, (1992), Berlin
[24] Sotomayor, M., A labor market with heterogeneous firms and workers, Int J Game Theory, 31, 269-283, (2002) · Zbl 1082.91005
[25] Sotomayor, M., Connecting the cooperative and competitive structures of the multiplepartners assignment game, J Econ Theory, 134, 155-174, (2007) · Zbl 1157.91411
[26] Sprumont, Y., On the discrete version of the Aumann-Shapley cost-sharing method, Econometrica, 73, 1693-1712, (2005) · Zbl 1151.91453
[27] Thompson, GL; Aumann, R. (ed.), Auctions and market games, (1981), Princeton
[28] Trudeau, C., Cost sharing with multiple technologies, Games Econ Behav, 67, 695-707, (2009) · Zbl 1190.91024
[29] Trudeau, C., Network flow problems and permutationally concave games, Math Soc Sci, 58, 121-131, (2009) · Zbl 1176.91009
[30] Trudeau, C., A new stable and more responsive cost sharing solution for minimum cost spanning tree problems, Games Econ Behav, 75, 402-412, (2012) · Zbl 1280.91099
[31] Yokote, K., Core and competitive equilibria: an approach from discrete convex analysis, J Math Econ, 66, 1-13, (2016) · Zbl 1368.91111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.