Abstract
In cooperative games, players have a possibility to form different coalitions. This leads to the questions about ways to motivate all players to collaborate, i.e. to motivate the players to form the so-called grand coalition. One of such ways is captured by the concept of nucleolus, which dates back to Babylonian Talmud. Weighted voting games form a class of cooperative games, that are often used to model decision making processes in parliaments. In this paper, we provide an algorithm for computing the nucleolus for an instance of a weighted voting game in pseudo-polynomial time. This resolves an open question posed by Elkind et al. (Ann Math Artif Intell 56(2), 109–131, 2007).
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References
Agrawal, M., Kayal, N., Saxena, N.: Errata: PRIMES is in P. Ann. Math. (2) 189(1), 317–318 (2019)
Aumann, R., Maschler, M.: Game theoretic analysis of a bankruptcy problem from the talmud. J. Econ. Theory 36(2), 195–213 (1985)
Baï ou, M., Barahona, F.: On the nucleolus of shortest path games. In: Algorithmic Game Theory, volume 10504 of Lecture Notes in Comput. Sci., pp. 55–66. Springer, Cham (2017)
Deng, X., Fang, Q., Sun, X.: Finding nucleolus of flow game. J. Comb. Optim. 18(1), 64–86 (2009)
Elkind, E., Goldberg, L.A., Goldberg, P.W., Wooldridge, M.: On the computational complexity of weighted voting games. Ann. Math. Artif. Intell. 56(2), 109–131 (2009)
Elkind, E., Pasechnik, D.: Computing the nucleolus of weighted voting games. In: Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 327–335. SIAM, Philadelphia (2009)
Faigle, U., Kern, W., Kuiper, J.: Computing the nucleolus of min-cost spanning tree games is NP-hard. Int. J. Game Theory 27(3), 443–450 (1998)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics: Study and Research Texts, vol. 2. Springer, Berlin (1988)
Hamers, H., Klijn, F., Solymosi, T., Tijs, S., Vermeulen, D.: On the nucleolus of neighbor games. Eur. J. Oper. Res. 146(1), 1–18 (2003)
Kern, W., Paulusma, D.: Matching games: the least core and the nucleolus. Math. Oper. Res. 28(2), 294–308 (2003)
Koenemann, J., Pashkovich, K., Toth, J.: Computing the nucleolus of weighted cooperative matching games in polynomial time. ArXiv e-prints (2018)
Maschler, M., Peleg, B., Shapley, L.S.: Geometric properties of the kernel, nucleolus, and related solution concepts. Math. Oper. Res. 4(4), 303–338 (1979)
Núñez, M.: A note on the nucleolus and the kernel of the assignment game. Int. J. Game Theory 33(1), 55–65 (2004)
Paulusma, D.: Complexity aspects of cooperative games. ProQuest LLC, Ann Arbor, MI (2001). Thesis (Dr.)–Universiteit Twente (The Netherlands)
Schmeidler, D.: The nucleolus of a characteristic function game. SIAM J. Appl. Math. 17, 1163–1170 (1969)
Solymosi, T.: Review for the paper “Computing the nucleolus of weighted voting games” by Edith Elkind and Dmitrii Pasechnik, (Mathematical Reviews)
Acknowledgements
We would like to thank Dmitrii Pasechnik for pointing us to the problem of computing the nucleolus of weighted voting games in pseudo-polynomial time. We are also grateful to Jochen Koenemann and Justin Toth for helpful discussions. We would also like to thank two reviewers for their helpful and constructive comments which helped to make the paper better.
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Pashkovich, K. Computing the nucleolus of weighted voting games in pseudo-polynomial time. Math. Program. 195, 1123–1133 (2022). https://doi.org/10.1007/s10107-021-01693-4
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DOI: https://doi.org/10.1007/s10107-021-01693-4