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Estate division problem: In search of a fair mechanism

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  • Zak, F.

    (Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow, Russia)

Abstract

. In the Nash Divide-the-Dollar (DD) game, n agents simultaneously submit claims on an estate of size E. If the total claim does not exceed E, each agent receives the amount requested; otherwise, all agents receive zero. The set of Nash equilibria coincides with the simplex of n-tuples of non-negative numbers summing to E. However, it is intuitively clear - and confirmed by experiments - that an equal division of the estate among the participants is the most reasonable and fair outcome. A large body of literature is devoted to resolving this tension by considering various mathematical models. Owing to its simplicity, the DD game serves as a prototype for many problems in which the question arises whether the standard tools of mathematical economics - such as different notions of equilibrium - can be modified to explain empirical data re?ecting the idea of fairness. In this survey article we review various modifications of the DD game, including those based on non-selfish preferences and bankruptcy rules. We discuss Nash and Kant equilibria (multiplicative and additive), models with restricted demands and penalties, as well as utility functions re?ecting different degrees of morality and inequality aversion.

Suggested Citation

  • Zak, F., 2026. "Estate division problem: In search of a fair mechanism," Journal of the New Economic Association, New Economic Association, vol. 70(1), pages 12-46.
    • Handle: RePEc:nea:journl:y:2026:i:70:p:12-46
    DOI: 10.31737/22212264_2026_1_12-46
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    References listed on IDEAS

    as
    1. Ingela Alger & Jörgen W. Weibull, 2017. "Strategic Behavior of Moralists and Altruists," Games, MDPI, vol. 8(3), pages 1-21, September.
    2. Alexander W. Cappelen & Roland Iwan Luttens & Erik Ø. Sørensen & Bertil Tungodden, 2019. "Fairness in Bankruptcies: An Experimental Study," Management Science, INFORMS, vol. 67(6), pages 2832-2841, June.
    3. Robert J. Aumann, 2025. "Game-Theoretic Analysis of a Bankruptcy Problem from the Talmud," World Scientific Book Chapters, in: SELECTED CONTRIBUTIONS TO GAME THEORY, chapter 9, pages 219-242, World Scientific Publishing Co. Pte. Ltd..
    4. Ingela Alger & Jörgen W. Weibull, 2013. "Homo Moralis—Preference Evolution Under Incomplete Information and Assortative Matching," Econometrica, Econometric Society, vol. 81(6), pages 2269-2302, November.
    5. Ashlagi, Itai & Karagözoğlu, Emin & Klaus, Bettina, 2012. "A non-cooperative support for equal division in estate division problems," Mathematical Social Sciences, Elsevier, vol. 63(3), pages 228-233.
    Full references (including those not matched with items on IDEAS)

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    Keywords

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    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C78 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Bargaining Theory; Matching Theory
    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D62 - Microeconomics - - Welfare Economics - - - Externalities
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement

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