Abstract
The stable marriage problem is a well-known problem of matching men to women so that no man and woman who are not married to each other both prefer each other. Such a problem has a wide variety of practical applications, ranging from matching resident doctors, to hospitals to matching students to schools. A well-known algorithm to solve this problem is the Gale–Shapley algorithm, which runs in quadratic time in the number of men/women. It has been proven that stable marriage procedures can always be manipulated. Whilst the Gale–Shapley algorithm is computationally easy to manipulate, we prove that there exist stable marriage procedures which are NP-hard to manipulate. We also consider the relationship between voting theory and stable marriage procedures, showing that voting rules which are NP-hard to manipulate can be used to define stable marriage procedures which are themselves NP-hard to manipulate. Finally, we consider the issue that stable marriage procedures like Gale–Shapley favour one gender over the other, and we show how to use voting rules to make any stable marriage procedure gender neutral.
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References
Arrow K. J., Sen A. K., Suzumura K. (2002) Handbook of social choice and welfare. Elsevier, North Holland
Bartholdi J., Orlin J. (1991) Single transferable vote resists strategic voting. Social Choice and Welfare 8(4): 341–354
Bartholdi J. J., Tovey C. A., Trick M. A. (1989) The computational difficulty of manipulating an election. Social Choice and Welfare 6(3): 227–241
Conitzer, V., & Sandholm, T. (2003). Universal voting protocol tweaks to make manipulation hard. In Proceedings of the IJCAI’03, pp. 781–788.
Conitzer, V., & Sandholm, T. (2006). Nonexistence of voting rules that are usually hard to manipulate. In Proceedings of the AAAI’06, AAAI Press.
Demange G., Gale D., Sotomayor M. (1987) A further note on the stable matching problem. Discrete Applied Mathematics 16: 217–222
Dubins L., Freedman D. (1981) Machiavelli and the Gale–Shapley algorithm. American Mathematical Monthly 88: 485–494
Gale D., Shapley L. S. (1962) College admissions and the stability of marriage. American Mathematical Monthly 69: 9–14
Gale D., Sotomayor M. (1985) Some remarks on the stable matching problem. Discrete Applied Mathematics 11: 223–232
Gale D., Sotomayor M. (1985) Machiavelli and the stable matching problem. American Mathematical Monthly 92: 261–268
Gibbard A. (1973) Manipulation of voting schemes: A general result. Econometrica 41(3): 587–601
Gusfield D., Irving R. W. (1989) The stable marriage problem: Structure and algorithms. MIT Press, Boston Mass
Gusfield, D. (1987). Three fast algorithms for four problems in stable marriage. SIAM Journal of Computing, 16(1).
Huang, C.-C. (2006). Cheating by men in the Gale–Shapley stable matching algorithm. In Proceedings of the ESA’06, pp. 418-431, Springer-Verlag.
Irving R.W., Leather P., Gusfield D. (1987) An efficient algorithm for the “optimal” stable marriage. JACM 34(3): 532–543
Klaus B., Klijin F. (2006) Procedurally fair and stable matching. Economic Theory 27: 431–447
Knuth, D. E. (1976). Mariages stables et leurs relations avec d’autres problèmes combinatoires. (French) Introduction à l’analyse mathématique des algorithmes. Collection de la Chaire Aisenstadt. Les Presses de l’Université de Montréal, Montreal, Que.
Kobayashi, H., & Matsui, T. (2008). Successful manipulation in stable marriage model with complete preference lists. In Proceedings of the MATCH-UP workshop: A satellite workshop of ICALP’08, pp. 17–22.
Liebowitz J., Simien J. (2005) Computational efficiencies for multi-agents: A look at a multi-agent system for sailor assignment. Electronic Government: An International Journal 2(4): 384–402
Masarani F. Gokturk S.S. (1989) On the existence of fair matching algorithms. Theory and Decision 26: 305–322 Kluwer
Procaccia A. D., Rosenschein J. S. (2007) Junta distributions and the average-case complexity of manipulating elections. JAIR 28: 157–181
Roth A. (1982) The economics of matching: Stability and incentives. Mathematics of Operations Research 7: 617–628
Roth A. (1984) The evolution of the labor market for medical interns and residents: A case study in game theory. Journal of Political Economy 92: 991–1016
Roth A. (2008) Deferred acceptance algorithms: History, theory, practice, and open questions. International Journal of Game Theory, Special Issue in Honor of David Gale on his 85th birthday 85: 537–569
Roth A., Sotomayor M. (1990) Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press, Cambridge
Roth A., Vande Vate J. H. (1991) Incentives in two-sided matching with random stable mechanisms. In Economic Theory 1(1): 31–44
Satterthwaite M. A. (1975) Strategy-proofness and arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare function. Economic Theory 10(3): 187–217
Teo C.-P., Sethuraman J., Tan W.-P. (2001) Gale–Shapley stable marriage problem revisited: Strategic issues and applications. Management Science 47(9): 1252–1267
Walsh, T. (2009). Where are the really hard manipulation problems? The phase transition in manipulating the veto rule. In Proceedings of the IJCAI’09, pp. 324–329.
Xia, L., Conitzer, V. (2008). Generalized scoring rules and the frequency of coalitional manipulability. In Proceeding of the ACM conference on electronic commerce 2008, pp. 109-118, ACM Press.
Zuckerman M., Procaccia A. D., Rosenschein J. S. (2009) Algorithms for the coalitional manipulation problem. Artificial Intelligence 173(2): 392–412
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Pini, M.S., Rossi, F., Venable, K.B. et al. Manipulation complexity and gender neutrality in stable marriage procedures. Auton Agent Multi-Agent Syst 22, 183–199 (2011). https://doi.org/10.1007/s10458-010-9121-x
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DOI: https://doi.org/10.1007/s10458-010-9121-x