Abstract
One-sided matching mechanisms are fundamental for assigning a set of indivisible objects to a set of self-interested agents when monetary transfers are not allowed. Two widely-studied randomized mechanisms in multiagent settings are the Random Serial Dictatorship (RSD) and the Probabilistic Serial Rule (PS). Both mechanisms require only that agents specify ordinal preferences and have a number of desirable economic and computational properties. However, the induced outcomes of the mechanisms are often incomparable and thus there are challenges when it comes to deciding which mechanism to adopt in practice. In this paper, we first consider the space of general ordinal preferences and provide empirical results on the (in)comparability of RSD and PS. We analyze their respective economic properties under general and lexicographic preferences. We then instantiate utility functions with the goal of gaining insights on the manipulability, efficiency, and envyfreeness of the mechanisms under different risk-attitude models. Our results hold under various preference distribution models, which further confirm the broad use of RSD in most practical applications.
Similar content being viewed by others
Notes
This problem is sometimes called the assignment problem or house allocation problem in the literature.
Given utility functions for the agents, where \(u_i(j)\) is the utility agent i derives from being assigned object j, the (utilitarian) social welfare of an assignment A is \(\sum _i \sum _j A_{i,j}u_i(j)\).
A recent experimental study on the incentive properties of PS shows that human subjects are less likely to manipulate the mechanism when misreporting is a Nash equilibrium. However, subjects’ tendency for misreporting is still significant even when it does not improve their assignments [25].
References
Abdulkadiroğlu, A., Pathak, P. A., & Roth, A. E. (2009). Strategy-proofness versus efficiency in matching with indifferences: Redesigning the NYC high school match. The American Economic Review, 99(5), 1954–1978.
Abdulkadiroğlu, A., & Sönmez, T. (1998). Random serial dictatorship and the core from random endowments in house allocation problems. Econometrica, 66(3), 689–701.
Abdulkadiroğlu, A., & Sönmez, T. (1999). House allocation with existing tenants. Journal of Economic Theory, 88(2), 233–260.
Arrow, K. J. (1971). Essays in the theory of risk-bearing. Markham economics series. North-Holland. https://books.google.com/books?id=vivwAAAAMAAJ.
Ashlagi, I., Fischer, F., Kash, I. A., & Procaccia, A. D. (2015). Mix and match: A strategyproof mechanism for multi-hospital kidney exchange. Games and Economic Behavior, 91, 284–296.
Aziz, H., Brandt, F., & Brill, M. (2013). The computational complexity of random serial dictatorship. Economics Letters, 121(3), 341–345.
Aziz, H., Chen, J., Filos-Ratsikas, A., Mackenzie, S., & Mattei, N. (2015). Egalitarianism of random assignment mechanisms. arXiv preprint, arXiv:1507.06827.
Aziz, H., Gaspers, S., Mackenzie, S., Mattei, N., Narodytska, N., & Walsh, T. (2015). Equilibria under the probabilistic serial rule. In Proceedings of the 24th international conference on artificial intelligence, IJCAI 2015 (pp. 1105–1112). AAAI Press. http://dl.acm.org/citation.cfm?id=2832249.2832402.
Aziz, H., Gaspers, S., Mackenzie, S., Mattei, N., Narodytska, N., & Walsh, T. (2015). Manipulating the probabilistic serial rule. In Proceedings of the 14th international conference on autonomous agents and multiagent systems (AAMAS 2015) (pp. 1451–1459). International Foundation for Autonomous Agents and Multiagent Systems.
Aziz, H., Gaspers, S., Mackenzie, S., & Walsh, T. (2015). Fair assignment of indivisible objects under ordinal preferences. Artificial Intelligence, 227, 71–92.
Berg, S. (1985). Paradox of voting under an urn model: The effect of homogeneity. Public Choice, 47(2), 377–387.
Bhalgat, A., Chakrabarty, D., & Khanna, S. (2011). Social welfare in one-sided matching markets without money. In L. A. Goldberg, K. Jansen, R. Ravi, & J. D. P. Rolim (Eds.), Approximation, randomization, and combinatorial optimization. Algorithms and techniques (pp. 87–98). Berlin: Springer.
Bogomolnaia, A., & Heo, E. J. (2012). Probabilistic assignment of objects: Characterizing the serial rule. Journal of Economic Theory, 147(5), 2072–2082.
Bogomolnaia, A., & Moulin, H. (2001). A new solution to the random assignment problem. Journal of Economic Theory, 100(2), 295–328.
Bouveret, S., & Lang, J. (2014). Manipulating picking sequences. In Proceedings of the 21st European conference on artificial intelligence (ECAI14) (pp. 141–146). Prague: IOS Press. http://recherche.noiraudes.net/resources/papers/ECAI14.pdf.
Budish, E., & Cantillon, E. (2012). The multi-unit assignment problem: Theory and evidence from course allocation at Harvard. The American Economic Review, 102(5), 2237–71.
Che, Y. K., & Kojima, F. (2010). Asymptotic equivalence of probabilistic serial and random priority mechanisms. Econometrica, 78(5), 1625–1672.
Christodoulou, G., Filos-Ratsikas, A., Frederiksen, S. K. S., Goldberg, P. W., Zhang, J., & Zhang, J.(2016). Social welfare in one-sided matching mechanisms. In International conference on autonomous agents and multiagent systems (pp. 30–50). Springer.
Domshlak, C., Hüllermeier, E., Kaci, S., & Prade, H. (2011). Preferences in AI: An overview. Artificial Intelligence, 175(7), 1037–1052.
Ekici, Ö., & Kesten, O. (2015). An equilibrium analysis of the probabilistic serial mechanism. International Journal of Game Theory, 45, 1–20. https://doi.org/10.1007/s00182-015-0475-9.
Filos-Ratsikas, A., Frederiksen, S. K. S., & Zhang, J. (2014). Social welfare in one-sided matchings: Random priority and beyond. In R. Lavi (Ed.), Algorithmic game theory (pp. 1–12). Berlin: Springer.
Fishburn, P. C. (1974). Lexicographic orders, utilities and decision rules: A survey. Management Science, 20(11), 1442–1471.
Hadar, J., & Russell, W. R. (1969). Rules for ordering uncertain prospects. The American Economic Review, 59, 25–34.
Hosseini, H., & Larson, K. (2015). Strategyproof quota mechanisms for multiple assignment problems. arXiv preprint, arXiv:1507.07064.
Hugh-Jones, D., Kurino, M., & Vanberg, C. (2013). An experimental study on the incentives of the probabilistic serial mechanism. Technical report, Discussion Paper, Social Science Research Center Berlin (WZB), Research Area Markets and Politics, Research Unit Market Behavior.
Kendall, M. G. (1938). A new measure of rank correlation. Biometrika, 30(1/2), 81–93.
Kendall, M. G. (1948). Rank correlation methods. London: Charles Griffin & Co. Ltd.
Kojima, F. (2009). Random assignment of multiple indivisible objects. Mathematical Social Sciences, 57(1), 134–142.
Kojima, F., & Manea, M. (2010). Incentives in the probabilistic serial mechanism. Journal of Economic Theory, 145(1), 106–123.
Liu, Q., & Pycia, M. (2013). Ordinal efficiency, fairness, and incentives in large markets. (Unpublished mimeo).
Lu, T., & Boutilier, C. (2011). Learning mallows models with pairwise preferences. In Proceedings of the 28th international conference on machine learning (ICML-11) (pp. 145–152).
Lu, T., & Boutilier, C. (2011). Robust approximation and incremental elicitation in voting protocols. In Proceedings of the 22nd international joint conference on artificial intelligence, IJCAI 2011 (Vol. 1, pp. 287–293).
Mallows, C. L. (1957). Non-null ranking models. Biometrika, 44(1/2), 114–130.
Manea, M. (2009). Asymptotic ordinal inefficiency of random serial dictatorship. Theoretical Economics, 4(2), 165–197.
Manlove, D. (2013). Algorithmics of matching under preferences. Singapore: World Scientific Publishing.
Marden, J. I. (1996). Analyzing and modeling rank data., CRC monographs on statistics & applied probability London: Chapman & Hall.
Mattei, N., & Walsh, T. (2013). PrefLib: A library of preference data http://preflib.org. In Proceedings of the 3rd international conference on algorithmic decision theory (ADT 2013). Lecture notes in artificial intelligence. Springer.
McLennan, A. (2002). Ordinal efficiency and the polyhedral separating hyperplane theorem. Journal of Economic Theory, 105(2), 435–449.
Mennle, T., & Seuken, S. (2013). Hybrid mechanisms: Trading off efficiency and strategyproofness in one-sided matching. arXiv preprint.
Mennle, T., & Seuken, S. (2013). Hybrid mechanisms: Trading off strategyproofness and efficiency of random assignment mechanisms. arXiv preprint, arXiv:1303.2558.
Mennle, T., & Seuken, S. (2015). Partial strategyproofness: An axiomatic approach to relaxing strategyproofness for assignment mechanisms. Technical report, Working paper.
Mennle, T., Weiss, M., Philipp, B., & Seuken, S. (2015). The power of local manipulation strategies in assignment mechanisms. In Proceedings of the 24th international conference on artificial intelligence, IJCAI’15 (pp. 82–89). AAAI Press. http://dl.acm.org/citation.cfm?id=2832249.2832261.
Pápai, S. (2000). Strategyproof multiple assignment using quotas. Review of Economic Design, 5(1), 91–105.
Pathak, P. A. (2006). Lotteries in student assignment. Harvard University. (Unpublished mimeo).
Pathak, P. A., & Sethuraman, J. (2011). Lotteries in student assignment: An equivalence result. Theoretical Economics, 6(1), 1–17.
Roth, A. E., Sönmez, T., & Ünver, M. U. (2004). Kidney exchange. The Quarterly Journal of Economics, 119(2), 457–488.
Saban, D., & Sethuraman, J. (2014). A note on object allocation under lexicographic preferences. Journal of Mathematical Economics, 50, 283–289.
Saban, D., & Sethuraman, J. (2015). The complexity of computing the random priority allocation matrix. Mathematics of Operations Research, 40(4), 1005–1014.
Schulman, L. J., & Vazirani, V. V. (2012). Allocation of divisible goods under lexicographic preferences. arXiv preprint, arXiv:1206.4366.
Sönmez, T., & Ünver, M. U. (2010). Course bidding at business schools. International Economic Review, 51(1), 99–123.
Svensson, L. G. (1999). Strategy-proof allocation of indivisible goods. Social Choice and Welfare, 16(4), 557–567.
Von Neumann, J. (1953). A certain zero-sum two-person game equivalent to the optimal assignment problem. Contributions to the Theory of Games, 2, 5–12.
Acknowledgements
We are grateful for the valuable feedback we received from the anonymous reviewers. This work was partially supported by NSERC.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix
1.1 Appendix A: Numerical results
Table 7 shows the results of comparing RSD and PS under ordinal preferences for various combinations of agents and objects. Note that in most instances, RSD and PS do not induce the same random assignment.
Appendix B: Descriptive statistics
1.1 Appendix B.1: Descriptive statistics
1.2 Appendix B.2: Descriptive statistics of the mechanisms grouped by risk attitudes
See Table 10.
1.3 Appendix B.3: Descriptive statistics of the mechanisms grouped by market type
See Table 11.
1.4 Appendix B.4: Pairwise comparison with both factors, market type and risk attitudes
Rights and permissions
About this article
Cite this article
Hosseini, H., Larson, K. & Cohen, R. Investigating the characteristics of one-sided matching mechanisms under various preferences and risk attitudes. Auton Agent Multi-Agent Syst 32, 534–567 (2018). https://doi.org/10.1007/s10458-018-9387-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10458-018-9387-y