Efficient Effort Equilibrium in Cooperation with Pairwise Cost Reduction

García-Martínez, Jose A. and Mayor-Serra, Antonio J. and Meca, Ana (2020): Efficient Effort Equilibrium in Cooperation with Pairwise Cost Reduction.

Abstract

There are multiple situations in which bilateral interaction between agents results in considerable cost reductions. Such interaction can occur in settings where agents are interested in sharing resources, knowledge or infrastructures. Their common purpose is to obtain individual advantages, e.g. by reducing their respective individual costs. Achieving this pairwise cooperation often requires the agents involved to make some level of effort. It is natural to think that the amount by which one agent could reduce the costs of the other may depend on how much effort the latter exerts. In the first stage, agents decide how much effort they are to exert, which has a direct impact on their pairwise cost reductions. We model this first stage as a non-cooperative game, in which agents determine the level of pairwise effort to reduce the cost of their partners. In the second stage, agents engage in a bilateral interaction between independent partners. We study this bilateral cooperation as a cooperative game in which agents reduce each other's costs as a result of cooperation, so that the total reduction in the cost of each agent in a coalition is the sum of the reductions generated by the rest of the members of that coalition. In the non-cooperative game that precedes cooperation with pairwise cost reduction, the agents anticipate the cost allocation that results from the cooperative game in the second stage by incorporating the effect of the effort exerted into their cost functions. Based on this model, we explore the costs, benefits, and challenges associated with setting up a pairwise effort network. We identify a family of cost allocations with weighted pairwise reduction which are always feasible in the cooperative game and contain the Shapley value. We show that there are always cost allocations with weighted pairwise reductions that generate an optimal level of efficient effort and provide a procedure for finding the efficient effort equilibrium.

• Item Type: MPRA Paper
• Language: English
• Keywords: Allocation, Cost models, Efficiency, Game Theory
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

[1] Amin, W., Huang, Q., Afzal, M., Khan, A. A., Umer, K., and Ahmed, S. A. (2020). A converg- ing non-cooperative & cooperative game theory approach for stabilizing peer-to-peer electricity trading. Electric Power Systems Research, 183, 106278.

[2] Bernstein F., Kok A.G., and Meca A. (2015). Cooperation in assembly systems: The role of knowledge sharing networks. European Journal of Operational Research, 240, 160-171.

[3] Bilbao, J. M., Fernández, J. R., Jim nez, N., and López, J. J. (2008). The Shapley value for bicooperative games. Annals of Operations Research, 158(1), 99-115.

[4] Brandenburger, A., and Stuart, H. (2007). Biform games. Management science, 53(4), 537-549.

[5] Cachon, G. P., and Netessine, S. (2006). Game theory in supply chain analysis. In Models, methods, and applications for innovative decision making (pp. 200-233). INFORMS.

[6] Curiel I., Pederzoli G., and Tijs S. (1989). Sequencing games, European Journal of Operational Research, 40, 344-351.

[7] Dewatripont, M., Jewitt, I. and and Tirole, J. (1999). The economics of career concerns, Part I: Comparing information structures. Review of Economic Studies 66(1), 183-198.

[8] Edhan, O. (2014). alue Solutions in Cooperative Games, Journal of Economic Literature 52(1), 211-249.

[9] Fiestras-Janeiro, M. G., García-Jurado, I., and Mosquera, M. A. (2011). Cooperative games and cost allocation problems. Top, 19(1), 1-22.

[10] González-Díaz, J., Fiestras-Janeiro, M. G., García-Jurado (2010). An introductory course on mathematical game theory. Graduate Studies in Mathematics, vol 115. American Mathematical Society.

[11] Guajardo, M., and Ronnqvist, M. (2016). A review on cost allocation methods in collaborative transportation. International transactions in operational research, 23(3), 371-392.

[12] Fang, D., and Wang, J., (2019). Horizontal capacity sharing between asymmetric competitors. Omega, in press, Article 102109.

[13] Holmstrom, B. (1982). Moral hazard in teams. Bell Journal of Economics 13(2), 324-40.

[14] Holmstrom, B. (1999). Managerial incentive problems: A dynamic perspective. Review of Eco- nomic Studies 66(1), 169-182.

[15] Kar, A., Mitra, M., and Mutuswami, S. (2009). On the coincidence of the prenucleolus with the Shapley value. Mathematical Social Sciences, 57(1):16-25.

[16] Kimms, A.,and Kozeletskyi, I. (2016). Shapley value-based cost allocation in the cooperative traveling salesman problem under rolling horizon planning. EURO Journal on Transportation and Logistics, 5(4), 371-392.

[17] Le, P. H., Nguyen, T. D., and Bekta s, T. (2018). Efficient computation of the Shapley value for large-scale linear production games. Annals of Operations Research, 1-21.

[18] Li, S., and Zhang, Q. (2009). A simplified expression of the Shapley function for fuzzy game. European Journal of Operational Research, 196(1), 234-245.

[19] Littlechild S.C. and Owen G., 1973, A simple expression for the Shapley value in a special case, Management Science, 20, 370-372.

[20] Lozano, S., Moreno, P., Adenso-Díaz, B., and Algaba, E. (2013). Cooperative game theory ap- proach to allocating benefits of horizontal cooperation. European Journal of Operational Re- search, 229(2), 444-452.

[21] Meca, A.,and Sosi'c, G. (2014). Benefactors and Beneficiaries: The Effects of Giving and Receiving on Cost-Coalitional Problems. Production and Operations Management, 23(9), 1549-1560.

[22] Meca, A., García-Martínez, J. A., and Mayor-Serra, A. J. (2019). The Shapley alue of Corpo- ration Tax Games with Dual Benefactors. Handbook of the Shapley alue, 313.

[23] Moretti, S., and Patrone, F. (2008). Transversality of the Shapley value. Top, 16(1), 1.

[24] Moulin, H. (1987). Equal or proportional division of a surplus, and other methods. International Journal of Game Theory, 16(3), 161-186.

[25] Moulin, H. (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press.

[26] Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the national academy of sciences, 36(1), 48-49.

[27] Nash, J. (1951). Non-cooperative games. Annals of mathematics, 286-295.

[28] Park, J., and an der Schaar, M. (2010). A game theoretic analysis of incentives in content production and sharing over peer-to-peer networks. IEEE Journal of Selected Topics in Signal Processing, 4(4), 704-717.

[29] Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on applied mathematics, 17(6), 1163-1170.

[30] Serrano, R. (2009), Cooperative Games: Core and Shapley alue, in R. Meyers (ed.), Encyclo- pedia of Complexity and Systems Science, Springer, New York.

[31] Shapley, L. S. (1953). A value for n-person games. Contributions to the Theory of Games, 2(28), 307-317.

[32] Shapley, L. S. (1971). Cores of convex games. International journal of game theory, 1(1), 11-26.

[33] Slikker, M., and Van den Nouweland, A. (2012). Social and economic networks in cooperative game theory (Vol. 27). Springer Science & Business Media.

[34] Thomson, W. (2010). Fair Allocation Rules. Chapter 21 in the Handbook of Social Choice and Welfare, vol 2, 393-492.

[35] Todeva, E., and Knoke, D. (2005). Strategic Alliances & Models of Collaboration. Management Decision, 43(1).