Stability in barter exchange markets
The notion of stability is the foundation of several classic problems in economics and computer science that arise in a wide-variety of real-world situations, including Stable Marriage, Stable Roommate, Hospital Resident and Group Activity Selection. We study this notion in the context of barter exchange markets. The input of our problem of interest consists of a set of people offering goods/services, with each person subjectively assigning values to a subset of goods/services offered by other people. The goal is to find a stable transaction, a set of cycles that is stable in the following sense: there does not exist a cycle such that every person participating in that cycle prefers to his current “status”. For example, consider a market where families are seeking vacation rentals and offering their own homes for the same. Each family wishes to acquire a vacation home in exchange of its own home without any monetary exchange. We study such a market by analyzing a stable transaction of houses involving cycles of fixed length. The underlying rationale is that an entire trade/exchange fails if any of the participating agents cancels the agreement; as a result, shorter (trading) cycles are desirable. We show that given a transaction, it can be verified whether or not it is stable in polynomial time, and that the problem of finding a stable transaction is NP-hard even if each person desires only a small number of other goods/services. Having established these results, we study the problem of finding a stable transaction in the framework of parameterized algorithms.
KeywordsAlgorithm design Stability Barter exchange FPT, W-hard
We thank anonymous reviewers for their valuable comments and suggestions which helped to improve the paper.
- 1.Abraham, D. J., Blum, A., & Sandholm, T. (2007). Clearing algorithms for barter exchange markets: Enabling nationwide kidney exchanges. In Proceedings of the 8th ACM conference on electronic commerce (EC) (pp. 295–304).Google Scholar
- 2.Anderson, R., Ashlagi, I., Gamarnik, D., & Roth, A. E. (2015). Finding long chains in kidney exchange using the traveling salesman problem. In Proceedings of the National Academy of Sciences (Vol. 112, pp. 663–668).Google Scholar
- 3.Biró, P. (2007). Stable exchange of indivisible goods with restrictions. In Proceedings of the 5th Japanese-Hungarian Symposium (pp. 97–105). Citeseer.Google Scholar
- 4.Brandt, F., Conitzer, V., Endrisss, U., Lang, J., & Procaccia, A. D. (Eds.). (2016). Handbook of computational social choice. Cambridge: Cambridge University Press.Google Scholar
- 7.Dickerson, J., Manlove, D., Plaut, B., Sandholm, T., & Trimble, J. (2016). Position-indexed formulations for kidney exchange. In Proceedings of EC.Google Scholar
- 10.Drummond, J., Perrault, A., & Bacchus, F. (2015). Sat is an effective and complete method for solving stable matching problems with couples. In Proceedings of IJCAI.Google Scholar
- 15.Igarashi, A., Bredereck, R., & Elkind, E. (2017). On parameterized complexity of group activity selection problems on social networks. In Proceedings of AAMAS’17.Google Scholar
- 16.Igarashi, A., & Elkind, E. (2016). Hedonic games with graph-restricted communication. In Proceedings of AAMAS’16 (pp. 242–250).Google Scholar
- 17.Igarashi, A., Elkind, E., & Peters, D. (2017). Group activity selection on social network. In Proceedings of AAAI’17.Google Scholar
- 18.Intervac: www.intervac-homeexchange.com.
- 22.Knuth, D. E. (1997). Stable marriage and its relation to other combinatorial problems : An introduction to the mathematical analysis of algorithms. In CRM proceedings & lecture notes. Providence, RI: American Mathematical Society.Google Scholar
- 24.McBride, I. (2015). Complexity results and integer programming models for hospitals/residents problem variants. Ph.D. thesis, University of Glasgow.Google Scholar
- 25.NationalOddShoeExchange: http://www.oddshoe.org/.
- 26.ReadItSwapIt: http://www.readitswapit.co.uk/thelibrary.aspx.