The Price of Optimum: Complexity and Approximation for a Matching Game
- 224 Downloads
This paper deals with a matching game in which the nodes of a simple graph are independent agents who try to form pairs. If we let the agents make their decision without any central control then a possible outcome is a Nash equilibrium, that is a situation in which no unmatched player can change his strategy and find a partner. However, there can be a big difference between two possible outcomes of the same instance, in terms of number of matched nodes. A possible solution is to force all the nodes to follow a centrally computed maximum matching but it can be difficult to implement this approach. This article proposes a tradeoff between the total absence and the full presence of a central control. Concretely, we study the optimization problem where the action of a minimum number of agents is centrally fixed and any possible equilibrium of the modified game must be a maximum matching. In algorithmic game theory, this approach is known as the price of optimum of a game. For the price of optimum of the matching game, deciding whether a solution is feasible is not straightforward, but we prove that it can be done in polynomial time. In addition, the problem is shown APX-hard, since its restriction to graphs admitting a perfect matching is equivalent, from the approximability point of view, to vertex cover. Finally we prove that this problem admits a polynomial 6-approximation algorithm in general graphs.
KeywordsApproximation algorithm Srategic game Complexity Price of Anarchy Price of optimum Stackelberg strategy
We would like to thank the anonymous reviewers for their valuable comments and suggestions.
- 3.Papadimitriou, C.H.: Algorithms, games, and the internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, STOC ‘01, pp. 749–753. ACM, New York, NY, USA (2001). doi: 10.1145/380752.380883
- 5.Kaporis, A.C., Spirakis, P.G.: Stackelberg games: the price of optimum. In: Kao, M.Y. (ed.) Encyclopedia of Algorithms. Springer, Berlin (2008)Google Scholar
- 8.Kumar, V.S.A., Marathe, M.V.: Improved results for stackelberg scheduling strategies. In: Widmayer, P., Ruiz, F.T., Bueno, R.M., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) Lecture Notes in Computer Science. Springer, Berlin (2002)Google Scholar
- 9.Swamy, C.: The effectiveness of stackelberg strategies and tolls for network congestion games. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 1133–1142. SIAM, Philadelphia (2007)Google Scholar