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Cost sharing in production economies. (English) Zbl 1455.91126

Sotomayor, Marilda (ed.) et al., Complex social and behavioral systems. Game theory and agent-based models. New York, NY: Springer. Encycl. Complex. Syst. Sci. Ser., 421-462 (2020).
Summary: Throughout we will use a fixed set of agents \(N = \{1, 2, \dots, n\}\) where \(n\) is a given natural number. For subsets \(S, T\) of \( N\), we write \(S \subset T\) if each element of \(S\) is contained in \(T\); \(T\backslash S\) denotes the set of agents in \(T\) except those in \(S\). The power set of \(N\) is the set of all subsets of \(N\); each coalition \(S \subset N\) will be identified with the element \(1_S \in \{0, 1\}^N\), the vector with \(i\)-th coordinate 1 precisely when \(i \in S\). Fix a vector \(x \in \mathbb{R}^N\) and \(S \subset N\). The projection of \(x\) on \(\mathbb{R}^S\) is denoted \(x_S\), and \(x_{N\backslash S}\) is sometimes more conveniently denoted \(x_{-S}\). For any \(y \in \mathbb{R}^S\), \((x_{-S}, y)\) stands for the vector \(z \in \mathbb{R}^N\) such that \(z_i = x_i\) if \(i \in N\backslash S\) and \(z_i = y_i\) if \(i \in S\). We denote \(x(S)= \sum_{i\in S}x_i\). The vector in \(\mathbb{R}^S\) with all coordinates equal zero is denoted \(\mathbf{0}_S\). Other notation will be introduced when necessary.
This entry focuses on different approaches in the literature through a discussion of a couple of basic and illustrative models, each involving a single facility for the production of a finite set \(M\) of outputs, commonly shared by a fixed set \(N :=\{1, 2, \dots, n\}\) of agents. The feasible set of outputs for the technology is identified with a set \(X \subset \mathbb{R}^M_+\). It is assumed that the users of the technology may freely dispose over any desired quantity or level of the outputs; each agent \(i\) has some demand \(x_i \in X\) for output. Each profile of demands \(x \in X\!^N\) is associated with its cost \(c(x)\), i.e., the minimal amount of the idiosyncratic input commodity needed to fulfill the individual demands. This defines the cost function \(c : X\!^N \rightarrow \mathbb{R}_+\) for the technology, comprising all the production externalities. A cost sharing problem is an ordered pair \((x, c)\) of a demand profile \(x\) and a cost function \(c\). The interpretation is that \(x\) is produced, and the resulting cost \(c(x)\) has to be shared by the collective \(N\). Numerous practical applications fit this general description of a cost sharing problem.
In mathematical terms, a cost sharing problem is equivalent to a production sharing problem where output is shared based on the profile of inputs. However, although many concepts are just as meaningful as they are in the cost sharing context, results are not at all easily established using this mathematical duality. In this sense consider [J. Leroux, Games Econ. Behav. 62, No. 2, 558–572 (2008; Zbl 1137.91306)] as a warning to the reader, showing that the strategic analysis of cost sharing solutions is quite different from surplus sharing solutions. This monograph will center on cost sharing problems. For further reference on production sharing, see [L. D. Israelseni, “Collectives, communes, and incentives”, J. Comp. Econ. 4, No. 2, 99–124 (1980; doi:10.1016/0147-5967(80)90024-4); J. Leroux, Econ. Lett. 85, No. 3, 335–340 (2004; Zbl 1254.91149); 2008, loc. cit.; H. Moulin and S. Shenker, Econometrica 60, No. 5, 1009–1037 (1992; Zbl 0766.90013)].
For the entire collection see [Zbl 1457.91008].

MSC:

91B32 Resource and cost allocation (including fair division, apportionment, etc.)
91A10 Noncooperative games
91A12 Cooperative games
91A80 Applications of game theory
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References:

[1] Aadland D, Kolpin V (1998) Shared irrigation cost: an empirical and axiomatical analysis. Math Soc Sci 35:203-218 · Zbl 0927.90080
[2] Aadland D, Kolpin V (2004) Environmental determinants of cost sharing. J Econ Behav Organ 53:495-511
[3] Albizuri MJ, Zarzuelo JM (2007) The dual serial cost-sharing rule. Math Soc Sci 53:150-163
[4] Albizuri MJ, Santos JC, Zarzuelo JM (2003) On the serial cost sharing rule. Int J Game Theory 31:437-446 · Zbl 1083.91059
[5] An M (1998) Logconcavity versus logconvexity, a complete characterization. J Econ Theory 80:350-369 · Zbl 0911.90071
[6] Archer A, Feigenbaum J, Krishnamurthy A, Sami R, Shenker S (2004) Approximation and collusion in multicast costsharing. Games Econ Behav 47:36-71 · Zbl 1080.90014
[7] Arin J, Iñarra E (2001) Egalitarian solutions in the core. Int J Game Theory 30:187-193 · Zbl 1082.91011
[8] Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Financ 9:203-228 · Zbl 0980.91042
[9] Atkinson AB (1970) On the measurement of inequality. J Econ Theory 2:244-263
[10] Aumann RJ (1959) Acceptable points in general cooperative n-person games. In: Contributions to the theory of games, vol IV. Princeton University Press, Princeton · Zbl 0085.13005
[11] Aumann RJ, Maschler M (1985) Game theoretic analysis of a bankruptcy problem from the Talmud. J Econ Theory 36:195-213 · Zbl 0578.90100
[12] Aumann RJ, Shapley LS (1974) Values of non-atomic games. Princeton University Press, Princeton · Zbl 0311.90084
[13] Baumol W, Bradford D (1970) Optimal departure from marginal cost pricing. Am Econ Rev 60:265-283
[14] Baumol W, Panzar J, Willig R (1988) Contestable markets and the theory of industry structure. Harcourt College Pub, Revised Edition
[15] Bergantino A, Coppejans L (1997) A game theoretic approach to the allocation of joint costs in a maritime environment: a case study. Occasional papers 44. Department of Maritime Studies and International Transport, University of Wales, Cardiff
[16] Billera LJ, Heath DC (1982) Allocation of shared costs: a set of axioms yielding a unique procedure. Math Oper Res 7:32-39 · Zbl 0509.90009
[17] Billera LJ, Heath DC, Raanan J (1978) Internal telephone billing rates: a novel application of non-atomic game theory. Oper Res 26:956-965 · Zbl 0417.90059
[18] Binmore K (2007) Playing for real: a text on game theory. Oxford University Press, Oxford · Zbl 1127.91015
[19] Bird CG (1976) On cost allocation for a spanning tree: a game theoretic approach. Networks 6:335-350 · Zbl 0357.90083
[20] Bjorndal E, Hamers H, Koster M (2004) Cost allocation in a bank ATM network. Math Meth Oper Res 59:405-418 · Zbl 1148.91324
[21] Bochet O, Klaus B (2007) A note on Dasgupta, Hammond, and Maskin’s (1979) domain richness condition. Discussion paper RM/07/039, ME-TEOR, Maastricht
[22] Bondareva ON (1963) Some applications of linear programming to the theory of cooperative games. Probl Kybern 10:119-139. [in Russian] · Zbl 1013.91501
[23] Brânzei R, Ferrari G, Fragnelli V, Tijs S (2002) Two approaches to the problem of sharing delay costs in joint projects. Ann Oper Res 109:359-374 · Zbl 1005.91010
[24] Chen Y (2003) An experimental study of serial and average cost pricing mechanisms. J Public Econ 87:2305-2335
[25] Clarke EH (1971) Multipart pricing of public goods. Public Choice 11:17-33
[26] Dasgupta P, Hammond P, Maskin E (1979) The implementation of social choice rules: some general results on incentive compatibility. Rev Econ Stud 46:185-216 · Zbl 0413.90007
[27] Davis M, Maschler M (1965) The kernel of a cooperative game. Nav Res Logist Q 12:223-259 · Zbl 0204.20202
[28] de Frutos MA (1998) Decreasing serial cost sharing under economies of scale. J Econ Theory 79:245-275 · Zbl 0911.90137
[29] Demers A, Keshav S, Shenker S (1990) Analysis and simulation of a fair queueing algorithm. J Internetworking 1:3-26
[30] Denault M (2001) Coherent allocation of risk capital. J Risk 4:1
[31] Dewan S, Mendelson H (1990) User delay costs and internal pricing for a service facility. Manag Sci 36:1502-1517 · Zbl 0717.90029
[32] Dutta B, Ray D (1989) A concept of egalitarianism under participation constraints. Econometrica 57:615-635 · Zbl 0703.90105
[33] Dutta B, Ray D (1991) Constrained egalitarian allocations. Games Econ Behav 3:403-422 · Zbl 0753.90076
[34] Flam SD, Jourani A (2003) Strategic behavior and partial cost sharing. Games Econ Behav 43:44-56 · Zbl 1045.91001
[35] Fleurbaey M, Sprumont Y (2009) Sharing the cost of a public good without subsidies. J Public Econ Theory 11:1-9
[36] Friedman E (2002) Strategic properties of heterogeneous serial cost sharing. Math Soc Sci 44:145-154 · Zbl 1027.91046
[37] Friedman E (2004) Paths and consistency in additive cost sharing. Int J Game Theory 32:501-518 · Zbl 1098.91012
[38] Friedman E, Moulin H (1999) Three methods to share joint costs or surplus. J Econ Theory 87:275-312 · Zbl 1016.91056
[39] Friedman E, Shenker S (1998) Learning and implementation on the Internet. Working paper 1998-21. Department of Economics, Rutgers University
[40] González-Rodríguez P, Herrero C (2004) Optimal sharing of surgical costs in the presence of queues. Math Methods Oper Res 59:435-446 · Zbl 1076.90014
[41] Granot D, Huberman G (1984) On the core and nucleolus of minimum cost spanning tree games. Math Program 29(1984):323-347 · Zbl 0541.90099
[42] Green J, Laffont JJ (1977) Characterization of satisfactory mechanisms for the revelation of preferences for public goods. Econometrica 45:427-438 · Zbl 0366.90021
[43] Groves T (1973) Incentives in teams. Econometrica 41:617-663 · Zbl 0311.90002
[44] Haimanko O (2000) Partially symmetric values. Math Oper Res 25:573-590 · Zbl 1073.91509
[45] Harsanyi J (1967) Games with incomplete information played by Bayesian players. Manag Sci 14:159-182 · Zbl 0207.51102
[46] Hart S, Mas-Colell A (1989) Potential, value, and consistency. Econometrica 57:589-614 · Zbl 0675.90103
[47] Haviv M (2001) The Aumann-Shapley price mechanism for allocating congestion costs. Oper Res Lett 29:211-215 · Zbl 0993.90027
[48] Henriet D, Moulin H (1996) Traffic-based cost allocation in a network. RAND J Econ 27:332-345
[49] Hougaard JL, Moulin H (2014) Sharing the cost of redundant projects. Games Econ Behav 87:339-352 · Zbl 1302.91117
[50] Hougaard JL, Moulin H (2017) Sharing the cost of risky projects. Economic Theory. https://doi.org/10.1007/s00199-017-1034-3 · Zbl 1402.91222
[51] Hougaard JL, Thorlund-Petersen L (2000) The stand-alone test and decreasing serial cost sharing. Economic Theory 16:355-362 · Zbl 0969.91019
[52] Hougaard JL, Thorlund-Petersen L (2001) Mixed serial cost sharing. Math Soc Sci 41:51-68 · Zbl 1152.91615
[53] Hougaard JL, Tind J (2007) Cost allocation and convex data envelopment. Eur J Oper Res 194:939-947 · Zbl 1168.90507
[54] Iñarra E, Isategui JM (1993) The Shapley value and average convex games. Int J Game Theory 22:13-29 · Zbl 0776.90093
[55] Israelsen D (1980) Collectives, communes, and incentives. J Comp Econ 4:99-124
[56] Jackson MO (2001) A crash course in implementation theory. Soc Choice Welf 18:655-708 · Zbl 1069.91557
[57] Joskow PL (1976) Contributions of the theory of marginal cost pricing. Bell J Econ 7:197-206
[58] Kalai E (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons. Econometrica 45:1623-1630 · Zbl 0371.90135
[59] Kaminski M (2000) ‘Hydrolic’ rationing. Math Soc Sci 40:131-155 · Zbl 0967.90070
[60] Kamiya S, Zanjani G (2017) Egalitarian equivalent capital allocation. N Am Actuar J 21:382-396 · Zbl 1414.91206
[61] Kolpin V, Wilbur D (2005) Bayesian serial cost sharing. Math Soc Sci 49:201-220 · Zbl 1111.91008
[62] Koster M (2002) Concave and convex serial cost sharing. In: Borm P, Peters H (eds) Chapters in game theory. Kluwer, Dordrecht
[63] Koster M (2005) Sharing variable returns of cooperation. CeNDEF working paper 05-06. University of Amsterdam, Amsterdam
[64] Koster M (2006) Heterogeneous cost sharing, the directional serial rule. Math Methods Oper Res 64:429-444 · Zbl 1102.91068
[65] Koster M (2007) The Moulin-Shenker rule. Soc Choice Welf 29:271-293 · Zbl 1280.91098
[66] Koster M (2012) Consistent cost sharing, Math Meth Oper Res 75:1-28. https://doi.org/10.1007/s00186-011-0372-3 · Zbl 1260.91131
[67] Koster M, Boonen T (2019) Constrained stochastic cost allocation. Math Soc Sc 101:20-30 · Zbl 1426.91146
[68] Koster M, Tijs S, Borm P (1998) Serial cost sharing methods for multicommodity situations. Math Soc Sci 36:229-242 · Zbl 0936.91037
[69] Koster M, Molina E, Sprumont Y, Tijs ST (2002) Sharing the cost of a network: core and core allocations. Int J Game Theory 30:567-599 · Zbl 1083.91021
[70] Koster M, Reijnierse H, Voorneveld M (2003) Voluntary contributions to multiple public projects. J Public Econ Theory 5:25-50
[71] Koutsoupias E, Papadimitriou C (1999) Worst-case equilibria.In: 16th annual symposiumon theoretical aspects of computer science, Trier, pp 404-413 · Zbl 1099.91501
[72] Legros P (1986) Allocating joint costs by means of the nucleolus. Int J Game Theory 15:109-119 · Zbl 0593.90088
[73] Leroux J (2004) Strategy-proofness and efficiency are incompatible in production economies. Econ Lett 85:335-340 · Zbl 1254.91149
[74] Leroux J (2008) Profit sharing in unique Nash equilibrium: characterization in the two-agent case. Games Econ Behav 62(2):558-572 · Zbl 1137.91306
[75] Littlechild SC, Owen G (1973) A simple expression for the Shapley value in a simple case. Manag Sci 20:370-372 · Zbl 0307.90095
[76] Littlechild SC, Thompson GF (1977) Aircraft landing fees: agame theory approach. Bell J Econ 8:186-204
[77] Maniquet F, Sprumont Y (1999) Efficient strategy-proof allocation functions in linear production economies. Economic Theory 14:583-595 · Zbl 0965.91025
[78] Maniquet F, Sprumont Y (2004) Fair production and allocation of an excludable nonrival good. Econometrica 72:627-640 · Zbl 1130.91362
[79] Maschler M (1990) Consistency. In: Ichiishi T, Neyman A, Tauman Y (eds) Game theory and applications. Academic, New York, pp 183-186
[80] Maschler M (1992) The bargaining set, kernel and nucleolus. In: Aumann RJ, Hart S (eds) Handbook of game theory with economic applications, vol I. North-Holland, Amsterdam · Zbl 0968.91502
[81] Maschler M, Reijnierse H, Potters J (1996) Monotonicity properties of the nucleolus of standard tree games. Int J Game Theory 39:89-104 · Zbl 1211.91083
[82] Maskin E, Sjöström T (2002) Implementation theory. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare, vol I. North-Holland, Amsterdam
[83] Matsubayashi N, Umezawa M, Masuda Y, Nishino H (2005) Cost allocation problem arising in hub-spoke network systems. Eur J Oper Res 160:821-838 · Zbl 1061.90015
[84] McLean RP, Pazgal A, Sharkey WW (2004) Potential, consistency, and cost allocation prices. Math Oper Res 29:602-623 · Zbl 1082.91055
[85] Mirman L, Tauman Y (1982) Demand compatible equitable cost sharing prices. Math Oper Res 7:40-56 · Zbl 0496.90016
[86] Monderer D, Shapley LS (1996) Potential games. Games Econ Behav 14:124-143 · Zbl 0862.90137
[87] Moulin H (1987) Equal or proportional division of a surplus, and other methods. Int J Game Theory 16:161-186 · Zbl 0631.90093
[88] Moulin H (1994) Serial cost-sharing of an excludable public good. Rev Econ Stud 61:305-325 · Zbl 0807.90040
[89] Moulin H (1995a) Cooperative microeconomics: agame-theoretic introduction. Prenctice Hall, London
[90] Moulin H (1995b) On additive methods to share joint costs. Jpn Econ Rev 46:303-332
[91] Moulin H (1996) Cost sharing under increasing returns: a comparison of simple mechanisms. Games Econ Behav 13:225-251 · Zbl 0851.90019
[92] Moulin H (1999) Incremental cost sharing: characterization by coalition strategy-proofness. Soc Choice Welf 16:279-320 · Zbl 1066.91502
[93] Moulin H (2000) Priority rules and other asymmetric rationing methods. Econometrica 68:643 · Zbl 1038.91536
[94] Moulin H (2002) Axiomatic cost and surplus-sharing. In: Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare. Handbooks in economics, vol 19. North-Holland Elsevier, Amsterdam, pp 289-357
[95] Moulin H (2008) The price of anarchy of serial, average and incremental cost sharing. Economic Theory 36:379-405 · Zbl 1146.91008
[96] Moulin H (2010) An efficient and almost budget balanced cost sharing method. Games Econ Behav 70:107-131 · Zbl 1206.91014
[97] Moulin H, Shenker S (1992) Serial cost sharing. Econometrica 60:1009-1037 · Zbl 0766.90013
[98] Moulin H, Shenker S (1994) Average cost pricing versus serial cost sharing; an axiomatic comparison. J Econ Theory 64:178-201 · Zbl 0811.90008
[99] Moulin H, Shenker S (2001) Strategy-proof sharing of submodular cost: budget balance versus efficiency. Economic Theory 18:511-533 · Zbl 1087.91509
[100] Moulin H, Sprumont Y (2005) On demand responsiveness in additive cost sharing. J Econ Theory 125:1-35 · Zbl 1122.91015
[101] Moulin H, Sprumont Y (2006) Responsibility and cross-subsidization in cost sharing. Games Econ Behav 55:152-188 · Zbl 1138.91327
[102] Moulin H, Vohra R (2003) Characterization of additive cost sharing methods. Econ Lett 80:399-407 · Zbl 1254.91025
[103] Moulin H, Watts A (1997) Two versions of the tragedy of the commons. Econ Des 2:399-421
[104] Mutuswami S (2004) Strategyproof cost sharing of a binary good and the egalitarian solution. Math Soc Sci 48:271-280 · Zbl 1080.91007
[105] Myers SC, Read JA (2001) Capital allocation for insurance companies. J Risk Insur 68:545-580
[106] Myerson RR (1980) Conference structures and fair allocation rules. Int J Game Theory 9:169-182 · Zbl 0441.90117
[107] Myerson RR (1991) Game theory: analysis of conflict. Harvard University Press, Cambridge, MA · Zbl 0729.90092
[108] Nash JF (1950) Equilibrium points in n-person games. Proc Natl Acad Sci 36:48-49 · Zbl 0036.01104
[109] O’Neill B (1982) A problem of rights arbitration from the Talmud. Math Soc Sci 2:345-371 · Zbl 0489.90090
[110] Osborne MJ (2004) An introduction to game theory. Oxford University Press, New York
[111] Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge · Zbl 1194.91003
[112] Peleg B, Sudhölter P (2004) Introduction to the theory of cooperative games. Series C: theory and decision library series. Springer-Verlag Berlin Heidelberg
[113] Pérez-Castrillo D, Wettstein D (2006) An ordinal Shapley value for economic environments. J Econ Theory 127:296-308 · Zbl 1126.91008
[114] Potters J, Sudhölter P (1999) Airport problems and consistent allocation rules. Math Soc Sci 38:83-102 · Zbl 1111.91310
[115] Razzolini L, Reksulak M, Dorsey R (2004) An experimental evaluation of the serial cost sharing rule. Theor Decis 63:283-314 · Zbl 1206.91018
[116] Ritzberger K (2002) Foundations of non-cooperative game theory. Oxford University Press, Oxford
[117] Rosenthal RW (1973) A class of games possessing pure-strategy Nash equilibria. J Econ Theory 2:65-67 · Zbl 0259.90059
[118] Roth AE (1988) The Shapley value, essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 307-319
[119] Samet D, Tauman Y, Zang I (1984) An application of the Aumann-Shapley prices for cost allocation in transportation problems. Math Oper Res 9:25-42 · Zbl 0531.90014
[120] Sánches SF (1997) Balanced contributions axiom in the solution of cooperative games. Games Econ Behav 20:161-168 · Zbl 0894.90183
[121] Sandsmark M (1999) Production games under uncertainty. Comput Econ 14:237-253 · Zbl 0943.91009
[122] Schmeidler D (1969) The nucleolus of a characteristic function game. SIAM J Appl Math 17:1163-1170 · Zbl 0191.49502
[123] Shapley LS (1953) A value for n-person games. Ann Math Study 28:307-317. Princeton University Press, Princeton · Zbl 0050.14404
[124] Shapley LS (1967) On balanced sets and cores. Nav Res Logist Q 14:453-460
[125] Shapley LS (1969) Utility comparison and the theory of games. In: La decision: Aggregation et dynamique des ordres de preference. Editions du Centre National de la Recherche Scientifique, Paris, pp 251-263. Also in Roth AE (ed) (1988) The Shapley value, essays in honor of Lloyd S. Shapley. Cambridge University Press, Cambridge, pp 307-319
[126] Shapley LS (1971) Cores of convex games. Int J Game Theory 1:1-26 · Zbl 0222.90054
[127] Sharkey W (1982) Suggestions for a game-theoretic approach to public utility pricing and cost allocation. Bell J Econ 13:57-68
[128] Sharkey W (1995) Network models in economics. In: Ball MO et al (eds) Network routing. Handbook in operations research and management science, vol 8. North-Holland, Amsterdam · Zbl 0861.90023
[129] Shubik M (1962) Incentives, decentralized control, the assignment of joint cost, and internal pricing. Manag Sci 8:325-343 · Zbl 0995.90578
[130] Skorin-Kapov D (2001) On cost allocation in hub-like networks. Ann Oper Res 106:63-78 · Zbl 1019.90010
[131] Skorin-Kapov D, Skorin-Kapov J (2005) Threshold based discounting network: the cost allocation provided by the nucleolus. Eur J Oper Res 166:154-159 · Zbl 1066.90015
[132] Sprumont Y (1998) Ordinal cost sharing. J Econ Theory 81:126-162 · Zbl 0910.90277
[133] Sprumont Y (2000) Coherent cost sharing. Games Econ Behav 33:126-144 · Zbl 0981.91050
[134] Sprumont Y (2005) On the discrete version of the Aumann-Shapley cost-sharing method. Econometrica 73:1693-1712 · Zbl 1151.91453
[135] Sprumont Y, Ambec S (2002) Sharing a river. J Econ Theory 107:453-462 · Zbl 1033.91503
[136] Sprumont Y, Moulin H (2007) Fair allocation of production externatlities: recent results, Revue d’Économie Politique 2007/1 (117)
[137] Sudhölter P (1998) Axiomatizations of game theoretical solutions for oneoutput cost sharing problems. Games Econ Behav 24:42-71
[138] Suijs J, Borm P, Hamers H, Koster M, Quant M (2005) Communication and cooperation in public network situations. Ann Oper Res 137:117-140 · Zbl 1138.91358
[139] Tauman Y (1988) The Aumann-Shapley prices: a survey. In: Roth A (ed) The Shapley value. Cambridge University Press, Cambridge, pp 279-304
[140] Thomas LC (1992) Dividing credit-card costs fairly. IMA J Math Appl Bus Ind 4:19-33 · Zbl 0825.90837
[141] Thomson W (1996) Consistent allocation rules. Mimeo, Economics Department, University of Rochester, Rochester
[142] Thomson W (2001) On the axiomatic method and its recent applications to game theory and resource allocation. Soc Choice Welf 18:327-386 · Zbl 1069.91512
[143] Tijs SH, Driessen TSH (1986) Game theory and cost allocation problems. Manag Sci 32:1015-1028 · Zbl 0595.90110
[144] Tijs SH, Koster M (1998) General aggregation of demand and cost sharing methods. Ann Oper Res 84:137-164 · Zbl 0916.90081
[145] Timmer J, Borm P, Tijs S (2003) On three Shapley-like solutions for cooperative games with random payoffs. Int J Game Theory 32:595-613 · Zbl 1098.91015
[146] van de Nouweland A, Tijs SH (1995) Cores and related solution concepts for multi-choice games. Math Methods Oper Res 41:289-311 · Zbl 0837.90133
[147] von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton · Zbl 0063.05930
[148] Watts A (2002) Uniqueness of equilibrium in cost sharing games. J Math Econ 37:47-70 · Zbl 1005.91006
[149] Weber RJ (1988) Probabilistic values for games. In: Roth AE (ed) The Shapley value. Cambridge University Press, Cambridge
[150] Yaari ME (1987) The dual theory of choice under risk. Econometrica 55:95-115 · Zbl 0616.90005
[151] Yeh CH (2008) Secured lower bound, composition up, and minimal rights first for bankruptcy problems. J Math Econ 44:925-932 · Zbl 1142.91574
[152] Young HP (1985a) Producer incentives in cost allocation. Econometrica 53:757-765 · Zbl 0564.90003
[153] Young HP (1985b) Monotonic solutions of cooperative games. Int J Game Theory 14:65-72 · Zbl 0569.90106
[154] Young HP (1985c) Cost allocation: methods, principles, applications. North-Holland, Amsterdam
[155] Young HP (1988) Distributive justice in taxation. J Econ Theory 44:321-335 · Zbl 0637.90027
[156] Young HP (1994) Cost allocation. In: Aumann RJ, Hart S (eds) Handbook of game theory, vol II. Elsevier, Amsterdam, pp 1193-1235 · Zbl 0925.90085
[157] Young HP (1998) Cost allocation, demand revelation, and core implementation. Math Soc Sci 36:213-229 · Zbl 0927.91012
[158] Zanjani G (2002) Pricing and capital allocation in catastrophe insurance. J Financ Econ 65:283-
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.