Interactive Purchasing Situations

CentER Discussion Paper Series No. 2012-035

30 Pages Posted: 26 Apr 2012

See all articles by Peter Borm

Peter Borm

Tilburg University - Center for Economic Research (CentER); Tilburg University - Department of Econometrics & Operations Research

Herbert Hamers

Tilburg University - Center for Economic Research (CentER); Tilburg University - Department of Econometrics & Operations Research

Hans Reijnierse

Tilburg University - Center for Economic Research (CentER); Radboud University Nijmegen - Department of Mathematics

Mirjam Groote Schaarsberg

Tilburg University - Center for Economic Research (CentER)

Date Written: April 23, 2012

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.

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

JEL Classification: C71

Suggested Citation

Borm, Peter E. M. and Hamers, Herbert and Reijnierse, Hans and Groote Schaarsberg, Mirjam, Interactive Purchasing Situations (April 23, 2012). CentER Discussion Paper Series No. 2012-035. Available at SSRN: https://ssrn.com/abstract=2044624 or http://dx.doi.org/10.2139/ssrn.2044624

Peter E. M. Borm

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

Tilburg University - Department of Econometrics & Operations Research

Tilburg, 5000 LE
Netherlands

Herbert Hamers

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands
+31 13 4666 2660 (Phone)

Tilburg University - Department of Econometrics & Operations Research

Tilburg, 5000 LE
Netherlands

Hans Reijnierse

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

Radboud University Nijmegen - Department of Mathematics ( email )

Nijmegen, 6500 GL
Netherlands

Mirjam Groote Schaarsberg (Contact Author)

Tilburg University - Center for Economic Research (CentER) ( email )

P.O. Box 90153
Tilburg, 5000 LE
Netherlands

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