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Interactive Purchasing Situations


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  • Groote Schaarsberg, M.
  • Borm, P.E.M.
  • Hamers, H.J.M.
  • Reijnierse, J.H.

    (Tilburg University, Center for Economic Research)


Abstract: This paper introduces a new class of interactive cooperative purchasing situations and provides an explicit alternative characterization of the nucleolus of cooperative games, which offers an alternative to Kohlberg (1971). In our cooperative purchasing situation, the unit price of a commodity depends on the largest order quantity within a cooperating group of players. Due to a decreasing unit price function, players can obtain cost savings by purchasing cooperatively. However, to establish fruitful cooperation a decision has to be made about an adequate allocation of the corresponding cost savings. We analyze Maximum Cooperative Purchasing (MCP)-situation from the perspective of allocation by defining corresponding cooperative MCP-games. It turns out that in an MCP-game all coalitional values can be determined from the values of two-player coalitions. Moreover, it is shown that a decreasing unit price function is a sufficient condition for a non-empty core: the Direct Price solution is both a core element and a marginal vector. It is seen that the nucleolus of an MCP-game can be derived in polynomial time from the Direct Price solution, using a socalled nucleolus determinant. To show this result, the explicit alternative characterization of the nucleolus is used. Using the decomposition of an MCP-game into unanimity games we find an explicit expression for the Shapley value. Interestingly, the Shapley value can be interpreted as a specific tax and subsidize system. Finally, the behavior of the three solution concepts is compared numerically.

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Bibliographic Info

Paper provided by Tilburg University, Center for Economic Research in its series Discussion Paper with number 2012-035.

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Date of creation: 2012
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Handle: RePEc:dgr:kubcen:2012035

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Keywords: cooperative purchasing; direct pricing; nucleolus; Shapley value;

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  1. Krishnan S. Anand & Ravi Aron, 2003. "Group Buying on the Web: A Comparison of Price-Discovery Mechanisms," Management Science, INFORMS, vol. 49(11), pages 1546-1562, November.
  2. A. van den Nouweland & P. Borm & W. van Golstein Brouwers & R. Groot Bruinderink & S. Tijs, 1996. "A Game Theoretic Approach to Problems in Telecommunication," Management Science, INFORMS, vol. 42(2), pages 294-303, February.
  3. Daniel Granot & Greys So\v{s}i\'{c}, 2005. "Formation of Alliances in Internet-Based Supply Exchanges," Management Science, INFORMS, vol. 51(1), pages 92-105, January.
  4. Sprumont, Yves, 1990. "Population monotonic allocation schemes for cooperative games with transferable utility," Games and Economic Behavior, Elsevier, vol. 2(4), pages 378-394, December.
  5. Cruijssen, Frans & Cools, Martine & Dullaert, Wout, 2007. "Horizontal cooperation in logistics: Opportunities and impediments," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 43(2), pages 129-142, March.
  6. Tella, Eija & Virolainen, Veli-Matti, 2005. "Motives behind purchasing consortia," International Journal of Production Economics, Elsevier, vol. 93(1), pages 161-168, January.
  7. Reijnierse, Hans & Potters, Jos, 1998. "The -Nucleolus of TU-Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 77-96, July.
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Cited by:
  1. Huijink, S. & Borm, P.E.M. & Reijnierse, J.H. & Kleppe, J., 2013. "Bankruptcy and the Per Capita Nucleolus," Discussion Paper 2013-059, Tilburg University, Center for Economic Research.


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