Abstract
We study cooperative interval games. These are cooperative games where the value of a coalition is given by a closed real interval specifying a lower bound and an upper bound of the possible outcome. For interval cooperative games, several (interval) solution concepts have been introduced in the literature. We assume that each player has a different attitude towards uncertainty by means of the so-called Hurwicz coefficients. These coefficients specify the degree of optimism that each player has so that an interval becomes a specific payoff. We show that a classical cooperative game arises when applying the Hurwicz criterion to each interval game. On the other hand, the same Hurwicz criterion can also be applied to any interval solution of the interval cooperative game. Given this, we say that a solution concept is Hurwicz compatible if the two procedures provide the same final payoff allocation. When such compatibility is possible, we characterize the class of compatible solutions, which reduces to the egalitarian solution when symmetry is required. The Shapley value and the core solution cases are also discussed.
This is a preview of subscription content, access via your institution.
Notes
For example, assume that players form an oligopoly that plans to create a cartel. The cartel can then anticipate their benefit as a monopoly. However, if two or more players are not present, the remaining players can not anticipate their exact benefit, as it would depend on whether the other players merge or not.
Other applications of Hurwicz coefficients in interval games appear in Lardon (2017) and Li (2016), who also deduce a (classical) cooperative game by using a selection via degrees of optimism. However, these degrees are coalition-dependent, not individual. Hence, they cannot be identified as Hurwicz coefficients in the same way we do here.
This is always possible to do since interval cooperative games generalize classical ones.
References
Abe, T., & Nakada, S. (2019). The weighted-egalitarian Shapley values. Social Choice and Welfare, 52(2), 197–213.
Alparslan Gök, S., Brânzei, R., & Tijs, S. (2008). Cores and stable sets for interval-valued games. Discussion Paper number 2008-17 2008-17. Tilburg University, Center for Economic Research.
Alparslan Gök, S., Brânzei, R., & Tijs, S. (2009a). Airport interval games and their Shapley value. Operations Research and Decision, 2, 571–575.
Alparslan Gök, S., Brânzei, R., & Tijs, S. (2009b). Convex interval games. Journal of Applied Mathematics and Decision Sciences2009 (14) Article ID: 342089.
Alparslan Gök, S. Z., Brânzei, R., Fragnelli, V., & Tijs, S. (2013). Sequencing interval situations and related games. Central European Journal of Operations Research, 21, 225–236.
Alparslan Gök, S. Z., Brânzei, R., & Tijs, S. (2010). The interval Shapley value: An axiomatization. Central European Journal of Operations Research, 18(2), 131–140.
Alparslan Gök, S., Miquel, S., & Tijs, S. (2009c). Cooperation under interval uncertainty. Mathematical Methods of Operations Research, 69, 99–109.
Béal, S., Casajus, A., Huettner, F., Rémila, E., & Solal, P. (2016). Characterizations of weighted and equal division values. Theory and Decision, 80(4), 649–667.
Béal, S., Rémila, E., & Solal, P. (2019). Coalitional desirability and the equal division value. Theory and Decision, 86(1), 95–106.
Bergantiños, G., & Vidal-Puga, J. (2004). Additive rules in bankruptcy problems and other related problems. Mathematical Social Sciences, 47(1), 87–101.
Bergantiños, G., & Vidal-Puga, J. (2009). A value for PERT problems. International Game Theory Review, 4(11), 419–436.
Bolton, G. E., & Ockenfels, A. (2000). ERC: A theory of equity, reciprocity, and competition. American Economic Review, 90(1), 166–193.
Brânzei, R., & Alparslan Gök, S. Z. (2008). Bankruptcy problems with interval uncertainty. Economics Bulletin, 3(56), 1–10.
Brânzei, R., Branzei, O., Alparslan Gök, S. Z., & Tijs, S. (2010). Cooperative interval games: a survey. Central European Journal of Operations Research, 18(3), 397–411.
Casajus, A., & Huettner, F. (2013). Null players, solidarity, and the egalitarian Shapley values. Journal of Mathematical Economics, 49(1), 58–61.
Casajus, A., & Huettner, F. (2014a). Weakly monotonic solutions for cooperative games. Journal of Economic Theory, 154, 162–172.
Casajus, A., & Huettner, F. (2014b). On a class of solidarity values. European Journal of Operational Research, 236(2), 583–591.
Dutta, B., & Ray, D. (1989). A concept of egalitarianism under participation constraints. Econometrica, 57, 615–535.
Han, W., Sun, H., & Xu, G. (2012). A new approach of cooperative interval games: The interval core and Shapley value revisited. Operations Research Letters, 40(6), 462–468.
Hougaard, J. L., & Moulin, H. (2018). Sharing the cost of risky projects. Economic Theory, 65(3), 663–679.
Hurwicz, L. (1951). The generalized Bayes minimax principle: A criterion for decision making under uncertainty. Discussion paper 335, Cowles Commission.
Koster, M. & Boonen, T. J. (2019). Constrained stochastic cost allocation. Working paper at University of Amsterdam.
Lardon, A. (2017). Endogenous interval games in oligopolies and the cores. Annals of Operations Research, 248(1), 345–363.
Li, D.-F. (2016). Models and methods for interval-valued cooperative games in economic management. New York: Springer.
Montemanni, R. (2006). A Benders decomposition approach for the robust spanning tree problem with interval data. European Journal of Operational Research, 174(3), 1479–1490.
Moretti, S., Gök, S. Z. A., Brânzei, R., & Tijs, S. (2011). Connection situations under uncertainty and cost monotonic solutions. Computers and Operations Research, 38(11), 1638–1645.
Pereira, J., & Averbakh, I. (2011). Exact and heuristic algorithms for the interval data robust assignment problem. Computers and Operations Research, 38(8), 1153–1163.
Shapley, L. S. (1953). A value for n-person games. In H. Kuhn & A. Tucker (Eds.), Contributions to the theory of games, volume II of annals of mathematics studies (pp. 307–317). Princeton NJ: Princeton University Press.
Thrall, R., & Lucas, W. (1963). N-person games in partition function form. Naval Reseach Logistic Quarterly, 10, 281–298.
van den Brink, R. (2007). Null or nullifying players: The difference between the Shapley value and equal division solutions. Journal of Economic Theory, 136, 767–775.
van den Brink, R., Chun, Y., Funaki, Y., & Park, B. (2015). Consistency, population solidarity, and egalitarian solutions for TU-games. Theory and Decision, 81(3), 427–447.
van den Brink, R., Palanci, O., & Gök, S. Z. A. (2017). Interval solutions for TU-games. Ti 2017-094/ii, Tinbergen Institute.
Wu, W., Iori, M., Martello, S., & Yagiura, M. (2018). Exact and heuristic algorithms for the interval min-max regret generalized assignment problem. Computers and Industrial Engineering, 125, 98–110.
Xue, J. (2018). Fair division with uncertain needs. Social Choice and Welfare, 51(1), 105–136.
Yokote, K., Kongo, T., & Funaki, Y. (2018). The balanced contributions property for equal contributors. Special Issue in Honor of Lloyd Shapley: Seven topics in game theory. Games and Economic Behavior, 108, 113–124.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Financial support by the Spanish Ministerio de Economía y Competitividad (ECO2014-52616-R), Ministerio de Economía, Industria y Competitividad (ECO2017-82241-R) and Xunta de Galicia (GRC 2015/014) is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Mallozzi, L., Vidal-Puga, J. Uncertainty in cooperative interval games: how Hurwicz criterion compatibility leads to egalitarianism. Ann Oper Res 301, 143–159 (2021). https://doi.org/10.1007/s10479-019-03379-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-019-03379-9
Keywords
- Cooperative interval games
- Hurwicz criterion
- Hurwicz compatibility