Mallozzi, Lina and Vidal-Puga, Juan
(2019):
*Uncertainty in cooperative interval games: How Hurwicz criterion compatibility leads to egalitarianism.*

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## 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 be also 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.

Item Type: | MPRA Paper |
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Original Title: | Uncertainty in cooperative interval games: How Hurwicz criterion compatibility leads to egalitarianism |

Language: | English |

Keywords: | Cooperative interval games; Hurwicz criterion; Hurwicz compatibility |

Subjects: | C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games |

Item ID: | 92730 |

Depositing User: | Juan Vidal-Puga |

Date Deposited: | 13 Mar 2019 14:39 |

Last Modified: | 27 Sep 2019 10:04 |

References: | Abe, T. and Nakada, S. (2019). The weighted-egalitarian shapley values. Social Choice and Welfare, 52(2):197–213. Alparslan-Gok, S., Brânzei, R., and 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., Branzei, R., and Tijs, S. (2009a). Airport interval games and their Shapley value. Operations Research and Decisions, 2:571–575. Alparslan Gök, S., Branzei, R., and Tijs, S. (2009b). Convex interval games. Journal of Applied Mathematics and Decision Sciences, 2009:14–. Article ID 342089. Alparslan Gök, S., Miquel, S., and Tijs, S. (2009c). Cooperation under interval uncertainty. Mathematical Methods of Operations Research, 69:99–109. Alparslan-Gök, S. Z., Branzei, R., Fragnelli, V., and Tijs, S. (2013). Sequencing interval situations and related games. Central European Journal of Operations Research, 21(1):225–236. Alparslan Gök, S. Z., Branzei, R., and Tijs, S. (2010). The interval Shapley value: an axiomatization. Central European Journal of Operations Research, 18(2):131–140. Béal, Casajus, Huetter, Rémila, and Solal (2016). Characterizations of weighted and equal division values. Theory and Decision, 80(4):649–667. Béal, S., Rémila, E., and Solal, P. (2019). Coalitional desirability and the equal division value. Theory and Decision, 86(1):95–106. Bergantiños, G. and Vidal-Puga, J. (2004). Additive rules in bankruptcy problems and other related problems. Mathematical Social Sciences, 47(1):87–101. Bolton, G. E. and Ockenfels, A. (2000). Erc: A theory of equity, reciprocity, and competition. American Economic Review, 90(1):166–193. Branzei, R. and Alparslan Gök, S. Z. (2008). Bankruptcy problems with interval uncertainty. Economics Bulletin, 3(56):1–10. Branzei, R., Branzei, O., Alparslan Gök, S. Z., and Tijs, S. (2010). Cooperative interval games: a survey. Central European Journal of Operations Research, 18(3):397–411. Casajus, A. and Huettner, F. (2013). Null players, solidarity, and the egalitarian Shapley values. Journal of Mathematical Economics, 49(1):58–61. Casajus, A. and Huettner, F. (2014a). On a class of solidarity values. European Journal of Operational Research, 236(2):583–591. Casajus, A. and Huettner, F. (2014b). Weakly monotonic solutions for cooperative games. Journal of Economic Theory, 154:162–172. Dutta, B. and Ray, D. (1989). A concept of egalitarianism under participation constraints. Econometrica, 57(3):615–636. Han, W., Sun, H., and 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. and 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. and Boonen, T. J. (2019). Constrained stochastic cost allocation. Technical report, 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. 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., Branzei, R., and Tijs, S. (2011). Connection situations under uncertainty and cost monotonic solutions. Computers and Operations Research, 38(11):1638–1645. Pereira, J. and 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 Kuhn, H. and Tucker, A., editors, Contributions to the theory of games, volume II of Annals of Mathematics Studies, pages 307–317. Princeton University Press, Princeton NJ. Thrall, R. and 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(1):767–775. van den Brink, R., Chun, Y., Funaki, Y., and 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., and Gok, S. Z. A. (2017). Interval solutions for TU-games. Ti 2017-094/ii, Tinbergen Institute. Wu, W., Iori, M., Martello, S., and 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., and Funaki, Y. (2018). The balanced contributions property for equal contributors. Games and Economic Behavior, 108:113–124. Special Issue in Honor of Lloyd Shapley: Seven Topics in Game Theory. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/92730 |