Cooperative Benefit and Cost Games under Fairness Concerns


1 School of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran

2 School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran

3 LCFC, Arts et Métiers Paris Tech, Metz, France


Solution concepts in cooperative games are based on either cost games or benefit games. Although cost games and benefit games are strategically equivalent, that is not the case in general for solution concepts. Motivated by this important observation, a new property called invariance property with respect to benefit/cost allocation is introduced in this paper. Since such a property can be regarded as a fairness criterion in cooperative games when deciding on choosing the solution concepts in coordination contracts, it is crucially important for players to check if the solution concepts available in contract menu possesses this property. To this end, we showed that some solution concepts such as the Shapley value, and the  -value satisfy invariance property with respect to benefit/cost allocation but some others such as Equal Cost Saving Method (ECSM) and Master Problem variant I ( ), do not. Furthermore, a measure for fairness with respect to equitable payoffs and utility is defined and related to invariance property. To validate the proposed approach, a numerical example extracted from the existing literature in benefit/cost cooperative games is solved and analyzed. The results of this research can be generalized for all solution concepts in cooperative games and is applicable for n-person games.


1.     Ghosh, S. and Khanra, S., "Channel coordination among a manufacturer and a retailer in two-layer supply chain", International Journal of Applied and Computational Mathematics,  Vol. 4, No. 1, (2018), DOI:
2.     Hou, D., Sun, H., Sun, P. and Driessen, T., "A note on the shapley value for airport cost pooling game", Games and Economic Behavior,  Vol. 108, (2017), 162-169.
3.     Dror, M. and Hartman, B.C., "Survey of cooperative inventory games and extensions", Journal of the Operational Research Society,  Vol. 62, No. 4, (2011), 565-580.
4.     Göthe-Lundgren, M., Jörnsten, K. and Värbrand, P., "On the nucleolus of the basic vehicle routing game", Mathematical programming,  Vol. 72, No. 1, (1996), 83-100.
5.     Engevall, S., Göthe-Lundgren, M. and Värbrand, P., "The heterogeneous vehicle-routing game", Transportation Science,  Vol. 38, No. 1, (2004), 71-85.
6.     Zibaei, S., Hafezalkotob, A. and Ghashami, S.S., "Cooperative vehicle routing problem: An opportunity for cost saving", Journal of Industrial Engineering International,  Vol. 12, No. 3, (2016), 271-286.
7.     Abbasia, B., Niaki, S. and Hosseinifardc, S., "A closed-form formula for the fair allocation of gains in cooperative n-person games", International Journal of Engineering-Transactions C: Aspects,  Vol. 25, No. 4, (2012), 303-310.
8.     Li, J., Feng, H. and Zeng, Y., "Inventory games with permissible delay in payments", European Journal of Operational Research,  Vol. 234, No. 3, (2014), 694-700.
9.     Drechsel, J. and Kimms, A., "Computing core allocations in cooperative games with an application to cooperative procurement", International Journal of Production Economics,  Vol. 128, No. 1, (2010), 310-321.
10.   Krajewska, M.A., Kopfer, H., Laporte, G., Ropke, S. and Zaccour, G., "Horizontal cooperation among freight carriers: Request allocation and profit sharing", Journal of the Operational Research Society,  Vol. 59, No. 11, (2008), 1483-1491.
11.   Frisk, M., Göthe-Lundgren, M., Jörnsten, K. and Rönnqvist, M., "Cost allocation in collaborative forest transportation", European Journal of Operational Research,  Vol. 205, No. 2, (2010), 448-458.
12.   Lozano, S., Moreno, P., Adenso-Díaz, B. and Algaba, E., "Cooperative game theory approach to allocating benefits of horizontal cooperation", European Journal of Operational Research,  Vol. 229, No. 2, (2013), 444-452.
13.   Li, J., Cai, X. and Zeng, Y., "Cost allocation for less-than-truckload collaboration among perishable product retailers", OR spectrum,  Vol. 38, No. 1, (2016), 81-117.
14.   Audy, J.-F., D’Amours, S. and Rousseau, L.-M., "Cost allocation in the establishment of a collaborative transportation agreement—an application in the furniture industry", Journal of the Operational Research Society,  Vol. 62, No. 6, (2011), 960-970.
15.   Liu, W., Song, S. and Wu, C., "Impact of loss aversion on the newsvendor game with product substitution", International Journal of Production Economics,  Vol. 141, No. 1, (2013), 352-359.
16.   Sadjadi, S. and Bayati, M.F., "Two-tier supplier base efficiency evaluation via network dea: A game theory approach", International Journal of Engineering-Transactions A: Basics,  Vol. 29, No. 7, (2016), 931-938.
17.   Flisberg, P., Frisk, M., Rönnqvist, M. and Guajardo, M., "Potential savings and cost allocations for forest fuel transportation in sweden: A country-wide study", Energy,  Vol. 85, (2015), 353-365.
18.   Wu, Q., Ren, H., Gao, W. and Ren, J., "Benefit allocation for distributed energy network participants applying game theory based solutions", Energy,  Vol. 119, (2017), 384-391.
19.   Alaei, S. and Setak, M., "Designing of supply chain coordination mechanism with leadership considering (research note)", International Journal of Engineering-Transactions C: Aspects,  Vol. 27, No. 12, (2014), 1888-1896.
20.   Borm, P., Hamers, H. and Hendrickx, R., "Operations research games: A survey", Top,  Vol. 9, No. 2, (2001), 139.
21.   Kogan, K. and Tapiero, C.S., "Supply chain games: Operations management and risk valuation, Springer Science & Business Media,  Vol. 113,  (2007).
22.   Tijs, S.H. and Driessen, T.S., "Game theory and cost allocation problems", Management Science,  Vol. 32, No. 8, (1986), 1015-1028.
23.   Nguyen, T.-D., "The fairest core in cooperative games with transferable utilities", Operations Research Letters,  Vol. 43, No. 1, (2015), 34-39.
24.   Roger, B.M., "Game theory: Analysis of conflict", The President and Fellows of Harvard College, USA,  (1991).
25.   Peleg, B. and Sudhölter, P., "Introduction to the theory of cooperative games, Springer Science & Business Media,  Vol. 34,  (2007).
26.   Owen, G., Game theory, 3rd. 1995, Academic Press.
27.   Kahan, J.P. and Rapoport, A., "Theories of coalition formation, Psychology Press,  (2014).
28.   Branzei, R., Dimitrov, D. and Tijs, S., "Models in cooperative game theory, Springer Science & Business Media,  Vol. 556, (2008).
29.   Drechsel, J., Cooperation in supply chains, in Cooperative lot sizing games in supply chains. 2010, Springer.55-61.
30.   Young, H.P., Okada, N. and Hashimoto, T., "Cost allocation in water resources development", Water resources research,  Vol. 18, No. 3, (1982), 463-475.
31.   Lemaire, J., "An application of game theory: Cost allocation", ASTIN Bulletin: The Journal of the IAA,  Vol. 14, No. 1, (1984), 61-81.
32.   Elomri, A., Jemai, Z., Ghaffari, A. and Dallery, Y., Stability of hedonic coalition structures: Application to a supply chain game, in Applications of multi-criteria and game theory approaches. 2014, Springer.337-363.
33.   Maschler, M., Solan, E. and Zamir, S., Game theory (translated from the hebrew by ziv hellman and edited by mike borns). 2013, Cambridge University Press, New York.
34.   Tijs, S., "Bounds for the core and the τ–value, In game theory and mathematical economics, Amsterdam, North-Holland Publishing Company,  (1981).
35.   Tijs, S.a.O., G.J., , "Compromise values in cooperative game theory", Top,  Vol. 1, No. 11, (1993), 1-36.
36.   González-Dıaz, J., Garcıa-Jurado, I. and Fiestras-Janeiro, M.G.,, "An introductory course on mathematical game theory", Graduate Studies in Mathematics,  (2010).