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Dividing the Indivisible

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  • Steven J. Brams
  • Todd R. Kaplan

Abstract

Political parties in Northern Ireland recently used a divisor method of apportionment to choose, in sequence, ten cabinet ministries. If the parties have complete information about each other’s preferences, we show that it may not be rational for them to act sincerely by choosing their most-preferred ministry that is available. One consequence of acting sophisticatedly is that the resulting allocation may not be Pareto-optimal, making all the parties worse off. Another is non-monotonicity – choosing earlier may hurt rather than help a party. We introduce a mechanism, combining sequential choices with a structured form of trading, that results in sincere choices for two parties that avoids these problems. Although there are dif.culties in extending this mechanism to more than two parties, other approaches are explored, such as permitting parties to make consecutive choices not prescribed by an apportionment method. But certain problems, such as eliminating envy, remain.

Suggested Citation

  • Steven J. Brams & Todd R. Kaplan, 2004. "Dividing the Indivisible," Journal of Theoretical Politics, , vol. 16(2), pages 143-173, April.
    • Handle: RePEc:sae:jothpo:v:16:y:2004:i:2:p:143-173
    DOI: 10.1177/0951629804041118
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    References listed on IDEAS

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    1. Steven J. Brams & Peter C. Fishburn, 2000. "Fair division of indivisible items between two people with identical preferences: Envy-freeness, Pareto-optimality, and equity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 17(2), pages 247-267.
    2. Steven J. Brams & D. Marc Kilgour, 2001. "Competitive Fair Division," Journal of Political Economy, University of Chicago Press, vol. 109(2), pages 418-443, April.
    3. David A. Kohler & R. Chandrasekaran, 1971. "A Class of Sequential Games," Operations Research, INFORMS, vol. 19(2), pages 270-277, April.
    4. Warwick, Paul V. & Druckman, James N., 2001. "Portfolio Salience and the Proportionality of Payoffs in Coalition Governments," British Journal of Political Science, Cambridge University Press, vol. 31(4), pages 627-649, October.
    5. Edelman, Paul & Fishburn, Peter, 2001. "Fair division of indivisible items among people with similar preferences," Mathematical Social Sciences, Elsevier, vol. 41(3), pages 327-347, May.
    6. Steven J. Brams & Morton D. Davis, 1978. "Optimal Jury Selection: A Game-Theoretic Model for the Exercise of Peremptory Challenges," Operations Research, INFORMS, vol. 26(6), pages 966-991, December.
    7. Arthur Roth & Joseph B. Kadane & Morris H. Degroot, 1977. "Optimal Peremptory Challenges in Trials by Juries: A Bilateral Sequential Process," Operations Research, INFORMS, vol. 25(6), pages 901-919, December.
    8. Friedrich Pukelsheim & Albert W. Marshall & Ingram Olkin, 2002. "A majorization comparison of apportionment methods in proportional representation," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 19(4), pages 885-900.
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    Cited by:

    1. T. Clark Durant & Michael Weintraub, 2014. "How to make democracy self-enforcing after civil war: Enabling credible yet adaptable elite pacts," Conflict Management and Peace Science, Peace Science Society (International), vol. 31(5), pages 521-540, November.
    2. Steven Brams & D. Kilgour & Christian Klamler, 2012. "The undercut procedure: an algorithm for the envy-free division of indivisible items," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 615-631, July.
    3. Mithun Chakraborty & Ulrike Schmidt-Kraepelin & Warut Suksompong, 2021. "Picking Sequences and Monotonicity in Weighted Fair Division," Papers 2104.14347, arXiv.org, revised Aug 2021.
    4. Brams, Steven J. & Kilgour, D. Marc & Klamler, Christian, 2013. "Two-Person Fair Division of Indivisible Items: An Efficient, Envy-Free Algorithm," MPRA Paper 47400, University Library of Munich, Germany.
    5. Haris Aziz, 2016. "A generalization of the AL method for fair allocation of indivisible objects," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 4(2), pages 307-324, October.
    6. Steven J. Brams & Daniel L. King, 2005. "Efficient Fair Division," Rationality and Society, , vol. 17(4), pages 387-421, November.

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