Geodesic
On the complexity of the Shapley-Scarf economy with several types of goods
Kybernetika, Tome 45 (2009) no. 5, pp. 689-700
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In the Shapley-Scarf economy each agent is endowed with one unit of an indivisible good (house) and wants to exchange it for another, possibly the most preferred one among the houses in the market. In this economy, core is always nonempty and a core allocation can be found by the famous Top Trading Cycles algorithm. Recently, a modification of this economy, containing Q >= 2 types of goods (say, houses and cars for Q=2) has been introduced. We show that if the number of agents is 2, a complete description of the core can be found efficiently. However, when the number of agents is not restricted, the problem to decide the nonemptyness of the core becomes NP-hard already in the case of two types of goods. We also show that even the problem to decide whether an allocation exists in which each agent strictly improves compared to his endowment, is NP-complete.
In the Shapley-Scarf economy each agent is endowed with one unit of an indivisible good (house) and wants to exchange it for another, possibly the most preferred one among the houses in the market. In this economy, core is always nonempty and a core allocation can be found by the famous Top Trading Cycles algorithm. Recently, a modification of this economy, containing Q >= 2 types of goods (say, houses and cars for Q=2) has been introduced. We show that if the number of agents is 2, a complete description of the core can be found efficiently. However, when the number of agents is not restricted, the problem to decide the nonemptyness of the core becomes NP-hard already in the case of two types of goods. We also show that even the problem to decide whether an allocation exists in which each agent strictly improves compared to his endowment, is NP-complete.
Classification : 68Q17, 68Q25, 91A06, 91A12
Keywords: Shapley–Scarf economy; core; algorithm; NP-completeness 
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Cechlárová, Katarína. On the complexity of the Shapley-Scarf economy with several types of goods. Kybernetika, Tome 45 (2009) no. 5, pp. 689-700. http://geodesic.mathdoc.fr/item/KYB_2009_45_5_a0/

[1] A. Abdulkadiroglu, P. Pathak, A. Roth, and T. Sönmez: The Boston public school match. Amer. Econom. Rev. 95 (2005), 2, 368–371.

[2] D. Abraham, K. Cechlárová, D. Manlove, and K. Mehlhorn: Pareto optimality in house allocation problems. In: Algorithms and Computation (R. Fleischer and G. Trippen, eds., Lecture Notes in Comput. Sci. 3827). Springer–Verlag, Berlin 2005, pp. 1163–1175. | MR

[3] P. Berman, M. Karpinski, and A. D. Scott: Approximation Hardness of Short Symmetric Instances of MAX-3SAT. Electronic Colloquiumon Computational Complexity, Report No. 49, 2003.

[4] S. Fekete, M. Skutella, and G. Woeginger: The complexity of economic equilibria for house allocation markets. Inform. Process. Lett. 88 (2003), 5, 219–223. | MR

[5] M. R. Garey and D. S. Johnson: Computers and Intractability. Freeman, San Francisco 1979. | MR

[6] H. Konishi, T. Quint, and J. Wako: On the Shapley–Scarf economy: the case of multiple types of indivisible goods. J. Math. Econom. 35 (2001), 1–15. | MR

[7] A. Roth and M. A. O. Sotomayor: Two-sided matching: a study in game-theoretic modeling and analysis. (Econometric Society Monographs 18.) Cambridge University Press, Cambridge 1990. | MR

[8] A. Roth, T. Sönmez, and U. Ünver: Kidney exchange. Quarterly J. Econom. 199 (2004), 457–488.

[9] H. Scarf: The core of an $N$-person game. Econometrica 35 (1967), 50–69. | MR | Zbl

[10] L. Shapley and H. Scarf: On cores and indivisibility. J. Math. Econom. 1 (1974), 23–37. | MR

[11] Y. Yuan: Residence exchange wanted: A stable residence exchange problem. European J. Oper. Res. 90 (1996), 536–546. | Zbl