Abstract
We define a subgradient algorithm to compute the maxmin value of a completely divisible good in both competitive and cooperative strategic contexts. The algorithm relies on the construction of upper and lower bounds for the optimal value which are based on the convexity properties of the range of utility vectors associated to all possible divisions of the good. The upper bound always converges to the optimal value. Moreover, if two additional hypotheses hold: that the preferences of the players are mutually absolutely continuous, and that there always exists relative disagreement among the players, then also the lower bound converges, and the algorithm finds an approximately optimal allocation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barbanel, J. (1999). Partition ratios, Pareto optimal cake division, and related notions. Journal of Mathematical Economics, 32(4), 401–428.
Barbanel, J. (2005). The geometry of efficient fair division. Cambridge: Cambridge University Press.
Bertsekas, D. P. (1999). Nonlinear programming (2nd ed.). Belmont, MA: Athena Scientific.
Boyd, S., & Vanderberghe, L. (2003). Convex optimization. Cambridge: Cambridge University Press.
Brams, S. J., & Taylor, A. D. (1996). Fair division: From cake-cutting to dispute resolution. Cambridge: Cambridge University Press.
Dall’Aglio, M. (2001). The Dubins–Spanier optimization problem in fair division theory. Journal of Computational and Applied Mathematics, 130(1–2), 17–40.
Dall’Aglio, M., Branzei, R., & Tijs, S. H. (2009). Cooperation in dividing the cake. TOP, 17(2), 417–432.
Dall’Aglio, M., & Hill, T. P. (2003). Maximin share and minimax envy in fair-division problems. Journal of Mathematical Analysis and Applications, 281, 346–361.
Dubins, L. E., & Spanier, E. H. (1961). How to cut a cake fairly. American Mathematical Monthly, 68, 1–17.
Dvoretzky, A., Wald, A., & Wolfowitz, J. (1951). Relations among certain ranges of vector measures. Pacific Journal of Mathematics, 1, 59–74.
Elton, J., Hill, T. P., & Kertz, R. P. (1986). Optimal-partitioning Inequalities for nonatomic probability measures. Transactions of the American Mathematical Society, 296(2), 703–725.
Kalai, E. (1977). Proportional solutions to bargaining situations: Interpersonal utility comparisons. Econometrica, 45(77), 1623–1630.
Kalai, E., & Smorodinsky, M. (1975). Other solutions to Nash’s bargaining problem. Econometrica, 43(3), 513–518.
Legut, J. (1988). Inequalities for \(\alpha \)-optimal partitioning of a measurable space. Proceedings of the American Mathematical Society, 104(4), 1249–1251.
Legut, J. (1990). On totally balanced games arising from cooperation in fair division. Games and Economic Behavior, 2(1), 47–60.
Legut, J., Potters, J. A. M., & Tijs, S. H. (1994). Economies with land: A game theoretical approach. Games Econom. Behav, 6(3), 416–430.
Legut, J., & Wilczynski, M. (1988). Optimal partitioning of a measurable space. Proceedings of the American Mathematical Society. 104(1), 262–264.
Shor, N. Z. (1985). Minimization methods for non-differentiable functions. Berlin: Springer-Verlag.
Walras, L. (1889). Eléments d économie politique pure, ou Théorie de la richesse sociale (Definitive ed.). Paris: Pichon and Durand-Auzias.
Acknowledgments
The authors would like to thank Professor Farhad Huseynov, School of Business, ADA University, Baku, Azerbaijan, and Visiting Professor at Luiss, Rome, Italy, for several comments and suggestions that helped improving the paper. The authors keep full responsibility for possible errors.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dall’Aglio, M., Di Luca, C. Finding maxmin allocations in cooperative and competitive fair division. Ann Oper Res 223, 121–136 (2014). https://doi.org/10.1007/s10479-014-1611-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-014-1611-9