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Finding maxmin allocations in cooperative and competitive fair division

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  • Marco Dall’Aglio
  • Camilla Luca

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. Copyright Springer Science+Business Media New York 2014

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  • Marco Dall’Aglio & Camilla Luca, 2014. "Finding maxmin allocations in cooperative and competitive fair division," Annals of Operations Research, Springer, vol. 223(1), pages 121-136, December.
    • Handle: RePEc:spr:annopr:v:223:y:2014:i:1:p:121-136:10.1007/s10479-014-1611-9
    DOI: 10.1007/s10479-014-1611-9
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    References listed on IDEAS

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    1. Marco Dall’Aglio & Rodica Branzei & Stef Tijs, 2009. "Cooperation in dividing the cake," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(2), pages 417-432, December.
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    4. Kalai, Ehud, 1977. "Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons," Econometrica, Econometric Society, vol. 45(7), pages 1623-1630, October.
    5. Barbanel, Julius, 1999. "Partition ratios, Pareto optimal cake division, and related notions," Journal of Mathematical Economics, Elsevier, vol. 32(4), pages 401-428, December.
    6. Legut, Jerzy, 1990. "On totally balanced games arising from cooperation in fair division," Games and Economic Behavior, Elsevier, vol. 2(1), pages 47-60, March.
    7. Barbanel,Julius B. Introduction by-Name:Taylor,Alan D., 2005. "The Geometry of Efficient Fair Division," Cambridge Books, Cambridge University Press, number 9780521842488, September.
    8. Legut, J. & Potters, J.A.M. & Tijs, S.H., 1994. "Economies with land : A game theoretical approach," Other publications TiSEM 37ff121d-d79c-4e41-a06a-9, Tilburg University, School of Economics and Management.
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    Cited by:

    1. Jerzy Legut, 2020. "How to obtain an equitable optimal fair division," Annals of Operations Research, Springer, vol. 284(1), pages 323-332, January.
    2. Natalia I. Naumova, 2022. "Some solutions for generalized games with restricted cooperation," Annals of Operations Research, Springer, vol. 318(2), pages 1077-1093, November.
    3. Erel Segal-Halevi & Shmuel Nitzan, 2019. "Fair cake-cutting among families," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 709-740, December.
    4. Sophie Bade & Erel Segal-Halevi, 2018. "Fairness for Multi-Self Agents," Papers 1811.06684, arXiv.org, revised Apr 2022.
    5. Erel Segal-Halevi & Balázs R. Sziklai, 2019. "Monotonicity and competitive equilibrium in cake-cutting," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 68(2), pages 363-401, September.

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