Abstract
Due to the lack of coordination, it is unlikely that the selfish players of a strategic game reach a socially good state. Using Stackelberg strategies is a popular way to improve the system’s performance. Stackelberg strategies consist of controlling the action of a fraction α of the players. However compelling an agent can be costly, unpopular or just hard to implement. It is then natural to ask for the least costly way to reach a desired state. This paper deals with a simple strategic game which has a high price of anarchy: the nodes of a simple graph are independent agents who try to form pairs. We analyse the optimization problem where the action of a minimum number of players shall be fixed and any possible equilibrium of the modified game must be a social optimum (a maximum matching).
For this problem, deciding whether a solution is feasible or not is not straitforward, but we prove that it can be done in polynomial time. In addition the problem is shown to be APX-hard, since its restriction to graphs admitting a vertex cover is equivalent, from the approximability point of view, to vertex cover in general graphs.
This work is supported by ANR, project COCA, ANR-09-JCJC-0066.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Gale, D., Shapley, L.S.: College admissions and the stability of marriage. The American Mathematical Monthly 69(1), 9–15 (1962)
Shapley, L.S., Shubik, M.: The assignment game i: The core. International Journal of Game Theory 1, 111–130 (1972)
Kaporis, A.C., Spirakis, P.G.: Stackelberg games: The price of optimum. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms. Springer, Heidelberg (2008)
Korilis, Y.A., Lazar, A.A., Orda, A.: Achieving network optima using stackelberg routing strategies. IEEE/ACM Trans. Netw. 5(1), 161–173 (1997)
Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)
Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. Computer Science Review 3(2), 65–69 (2009)
Papadimitriou, C.H.: Algorithms, games, and the internet. In: STOC, pp. 749–753 (2001)
Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. Theor. Comput. Sci. 410, 3327–3336 (2009)
Roughgarden, T.: Stackelberg scheduling strategies. SIAM J. Comput. 33(2), 332–350 (2004)
Kumar, V.S.A., Marathe, M.V.: Improved results for stackelberg scheduling strategies. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 776–787. Springer, Heidelberg (2002)
Swamy, C.: The effectiveness of stackelberg strategies and tolls for network congestion games. In: SODA, pp. 1133–1142. SIAM, Philadelphia (2007)
Fotakis, D.: Stackelberg strategies for atomic congestion games. Theory Comput. Syst. 47(1), 218–249 (2010)
Sharma, Y., Williamson, D.P.: Stackelberg thresholds in network routing games or the value of altruism. Games and Economic Behavior 67, 174–190 (2009)
Bonifaci, V., Harks, T., Schäfer, G.: Stackelberg routing in arbitrary networks. Math. Oper. Res. 35(2), 330–346 (2010)
Manoussakis, Y.: Alternating paths in edge-colored complete graphs. Discrete Applied Mathematics 56(2-3), 297–309 (1995)
Hochbaum, D.S.: Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston (1996)
Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162(1), 439–485 (2005)
Escoffier, B., Gourvès, L., Monnot, J.: Minimum regulation of uncoordinated matchings. CoRR abs/1012.3889 (2010)
Andelman, N., Feldman, M., Mansour, Y.: Strong price of anarchy. Games and Economic Behavior 65, 289–317 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Escoffier, B., Gourvès, L., Monnot, J. (2011). The Price of Optimum in a Matching Game. In: Persiano, G. (eds) Algorithmic Game Theory. SAGT 2011. Lecture Notes in Computer Science, vol 6982. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24829-0_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-24829-0_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24828-3
Online ISBN: 978-3-642-24829-0
eBook Packages: Computer ScienceComputer Science (R0)