Abstract
We define a family of rules for dividing m indivisible goods among agents, parameterized by a scoring vector and a social welfare aggregation function. We assume that agents’ preferences over sets of goods are additive, but that the input is ordinal: each agent reports her preferences simply by ranking single goods. Similarly to positional scoring rules in voting, a scoring vector \(s = (s_1, \ldots , s_m)\) consists of m nonincreasing, nonnegative weights, where \(s_i\) is the score of a good assigned to an agent who ranks it in position i. The global score of an allocation for an agent is the sum of the scores of the goods assigned to her. The social welfare of an allocation is the aggregation of the scores of all agents, for some aggregation function \(\star \) such as, typically, \(+\) or \(\min \). The rule associated with s and \(\star \) maps a profile to (one of) the allocation(s) maximizing social welfare. After defining this family of rules, and focusing on some key examples, we investigate some of the social-choice-theoretic properties of this family of rules, such as various kinds of monotonicity, and separability. Finally, we focus on the computation of winning allocations, and on their approximation: we show that for commonly used scoring vectors and aggregation functions this problem is NP-hard and we exhibit some tractable particular cases.
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Note that the Borda scoring vector in voting is usually defined as \((m-1,m-2,\dots ,1,0)\). Here, together with Brams et al. [14], we define the Borda scoring vector by fixing the score of the bottom-rank object to 1, meaning that getting it is better than getting nothing. For scoring voting rules, a translation of the scoring vector has obviously no impact on winner determination (see Observation 2.2 in the work of Hemaspaandra and Hemaspaandra [27]); for scoring allocation rules, however, it does.
This choice comes with a loss of generality, as there are tie-breaking mechanisms that are not defined this way (we thank a reviewer for this remark). Also, we rule out the possibility of randomly breaking ties.
This is the case for all properties expressing that an agent prefers a set of allocations to another set of allocations (and applies, e.g., to object monotonicity); for these properties there is not a unique way of generalizing the property, unlike in voting where this is well-known, e.g., for strategy-proofness. For a study of strategy-proofness for scoring allocation correspondences, we refer to the work of Nguyen et al. [34].
If the scoring vector s is part of the input then the problem \(F_{s, \star }\)-FOA, \(\star \in \{\min , {{\mathrm{leximin}}}\}\), is \(\mathrm {NP}\)-hard (though not strongly \(\mathrm {NP}\)-hard in the sense of Garey and Johnson [23, 24]), even for two agents having identical preferences, by a direct reduction from Partition.
Here and later, we slightly abuse notation, as X and \(C_i\) will refer both to the initial sets and their corresponding sets of goods.
References
Aziz, H., Gaspers, S., Mackenzie, S., & Walsh, T. (2015). Fair assignment of indivisible objects under ordinal preferences. Artificial Intelligence, 227, 71–92.
Aziz, H., Walsh, T., & Xia, L. (2015). Possible and necessary allocations via sequential mechanisms. In Proceedings of the 24th International Joint Conference on Artificial Intelligence (pp. 468–474). AAAI Press/IJCAI.
Bansal, N., & Sviridenko, M. (2006). The Santa Claus problem. In Proceedings of the 38th ACM Symposium on Theory of Computing (pp. 31–40). ACM Press
Baumeister, D., & Rothe, J. (2015). Preference aggregation by voting. In J. Rothe (Ed.), Economics and computation. An introduction to algorithmic game theory, computational social choice, and fair division, chap 4 (pp. 197–325). Berlin: Springer.
Baumeister, D., Bouveret, S., Lang, J., Nguyen, N., Nguyen, T., Rothe, J., et al. (2014). Axiomatic and computational aspects of scoring allocation rules for indivisible goods. In A. Procaccia & T. Walsh (Eds.), Proceedings of the 5th international workshop on computational social choice. Pittsburgh, PA: Carnegie Mellon University.
Baumeister, D., Bouveret, S., Lang, J., Nguyen, T., Nguyen, N., & Rothe, J. (2014). Scoring rules for the allocation of indivisible goods. In Proceedings of the 21st European conference on artificial intelligence (pp. 75–80). IOS Press
Bouveret, S., & Lang, J. (2011). A general elicitation-free protocol for allocating indivisible goods. In Proceedings of the 22nd international joint conference on artificial intelligence (pp 73–78). AAAI Press/IJCAI.
Bouveret, S., & Lemaître, M. (2016). Characterizing conflicts in fair division of indivisible goods using a scale of criteria. Journal of Autonomous Agents and Multi-Agent Systems, 30(2), 259–290.
Bouveret, S., Endriss, U., & Lang, J. (2010). Fair division under ordinal preferences: Computing envy-free allocations of indivisible goods. In Proceedings of the 19th European conference on artificial intelligence (pp. 387–392). IOS Press.
Bouveret, S., Chevaleyre, Y., & Maudet, N. (2016). Fair allocation of indivisible goods. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. Procaccia (Eds.), Handbook of computational social choice chap 12. Cambridge: Cambridge University Press.
Brams, S., & Fishburn, P. (2002). Voting procedures. In K. Arrow, A. Sen, & K. Suzumura (Eds.), Handbook of social choice and welfare chap 4 (pp. 173–236). Amsterdam: North-Holland.
Brams, S., & King, D. (2005). Efficient fair division—help the worst off or avoid envy? Rationality and Society, 17(4), 387–421.
Brams, S., & Taylor, A. (1996). Fair division: From cake-cutting to dispute resolution. Cambridge: Cambridge University Press.
Brams, S., Edelman, P., & Fishburn, P. (2004). Fair division of indivisible items. Theory and Decision, 5(2), 147–180.
Budish, E. (2011). The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. Journal of Political Economy, 119(6), 1061–1103.
Budish, E., & Cantillon, E. (2012). The multi-unit assignment problem: Theory and evidence from course allocation at Harvard. The American Economic Review, 102(5), 2237–2271.
Caragiannis, I., & Procaccia, A. (2011). Voting almost maximizes social welfare despite limited communication. Artificial Intelligence, 175(9–10), 1655–1671.
Caragiannis, I., Kurokawa, D., Moulin, H., Procaccia, A., Shah, N., & Wang, J. (2016). The unreasonable fairness of maximum Nash welfare. In Proceedings of the 17th ACM conference on economics and computation.
Darmann, A., & Schauer, J. (2015). Maximizing Nash product social welfare in allocating indivisible goods. European Journal of Operational Research, 247, 548–559.
Elkind, E., Faliszewski, P., Skowron, P., & Slinko, A. (2014). Properties of multiwinner voting rules. In Proceedings of the 13th international conference on autonomous agents and multiagent systems (pp. 53–60). IFAAMAS.
Faliszewski, P., & Hemaspaandra, L. (2009). The complexity of power-index comparison. Theoretical Computer Science, 410(1), 101–107.
Gardenfors, P. (1973). Assignment problem based on ordinal preferences. Management Science, 20(3), 331–340.
Garey, M., & Johnson, D. (1978). “Strong” NP-completeness results: Motivation, examples, and implications. Journal of ACM, 25(3), 499–508.
Garey, M., & Johnson, D. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco: W. H. Freeman and Company.
Garg, N., Kavitha, T., Kumar, A., Mehlhorn, K., & Mestre, J. (2010). Assigning papers to referees. Algorithmica, 58(1), 119–136.
Golovin, D. (2005). Max-min fair allocation of indivisible goods. Tech. Rep. CMU-CS-05-144, School of Computer Science. Carnegie Mellon University.
Hemaspaandra, E., & Hemaspaandra, L. (2007). Dichotomy for voting systems. Journal of Computer and System Sciences, 73(1), 73–83.
Herreiner, D., & Puppe, C. (2002). A simple procedure for finding equitable allocations of indivisible goods. Social Choice and Welfare, 19(2), 415–430.
Kohler, D., & Chandrasekaran, R. (1971). A class of sequential games. Operations Research, 19(2), 270–277.
Lang, J., & Rothe, J. (2015). Fair division of indivisible goods. In J. Rothe (Ed.), Economics and computation. An introduction to algorithmic game theory, computational social choice, and fair division chap 8 (pp. 493–550). Berlin: Springer.
Manlove, D. (2013). Algorithmics of matching under preferences. Series on theoretical computer science. Singapore: World Scientific Publishing.
Moulin, H. (1995). Cooperative microeconomics: A game-theoretic introduction. Upper Saddle River, NJ: Prentice Hall.
Moulin, H. (2004). Fair division and collective welfare. Cambridge, MA: MIT Press.
Nguyen, N., Baumeister, D., & Rothe, J. (2015). Strategy-proofness of scoring allocation correspondences for indivisible goods. In Proceedings of the 24th international joint conference on artificial intelligence (pp. 1127–1133). AAAI Press/IJCAI.
Pruhs, K., & Woeginger, G. (2012). Divorcing made easy. In: Proceedings of the 6th international conference on fun with algorithms (pp. 305–314). Springer.
Roth, A., & Sotomayor, M. (1990). Two-sided matching: A study in game-theoretic modeling and analysis. Econometric society monographs. Cambridge: Cambridge University Press.
Skowron, P., Faliszewski, P., & Slinko, A. (2013). Fully proportional representation as resource allocation: Approximability results. In Proceedings of the 23rd International Joint Conference on Artificial Intelligence (pp. 353–359). AAAI Press/IJCAI.
Thomson, W. (2011). Consistency and its converse: An introduction. Review of Economic Design, 15(4), 257–291.
Wilson, L. (1977). Assignment using choice lists. Operational Research Quarterly, 28, 569–578.
Zwicker, W. (2016). Introduction to the theory of voting. In F. Brandt, V. Conitzer, U. Endriss, J. Lang, & A. Procaccia (Eds.), Handbook of computational social choice chap 2. Cambridge: Cambridge University Press.
Acknowledgments
We are grateful to the anonymous ECAI’14 and COMSOC’14 reviewers for their helpful comments. In particular, we thank the reviewer who pointed out a proof sketch of Theorem 6 for her or his consent to include the result and its proof. This work was supported in part by Deutsche Forschungsgemeinschaft under grants RO 1202/14-1, RO 1202/14-2, and RO 1202/15-1, by a project of the DAAD-PPP / PHC PROCOPE program entitled “Fair Division of Indivisible Goods: Incomplete Preferences, Communication Protocols and Computational Resistance to Strategic Behavior,” by COST Action IC1205 on Computational Social Choice, by ANR project CoCoRICo-CoDec, by a grant for gender-sensitive universities funded by the NRW Ministry for Innovation, Science, and Research, and by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED Project No. 102.01-2015.33).
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Baumeister, D., Bouveret, S., Lang, J. et al. Positional scoring-based allocation of indivisible goods. Auton Agent Multi-Agent Syst 31, 628–655 (2017). https://doi.org/10.1007/s10458-016-9340-x
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DOI: https://doi.org/10.1007/s10458-016-9340-x