# Congestion Games with Capacitated Resources

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## Abstract

The players of a congestion game interact by allocating bundles of resources from a common pool. This type of games leads to well studied models for analyzing strategic situations, including networks operated by uncoordinated selfish users. Congestion games constitute a subclass of potential games, meaning that a pure Nash equilibrium emerges from a myopic process where the players iteratively react by switching to a strategy that diminishes their individual cost. With the aim of covering more applications, for instance in communication networks, we extend congestion games to the setting where every resource is endowed with a capacity which possibly limits its number of users. From the negative side, we show that a pure Nash equilibrium is not guaranteed to exist in any case and we prove that deciding whether a game possesses a pure Nash equilibrium is **NP**-complete. Our positive results state that congestion games with capacities are potential games in the well studied singleton case. Polynomial algorithms that compute these equilibria are also provided.

## Keywords

Congestion games Pure nash equilibrium Potential function Algorithms## Notes

### Acknowledgments

We thank the anonymous referees for their comments which lead to significant improvements of the manuscript.

## References

- 1.Ackermann, H., Goldberg, P., Mirrokni, V., Röglin, H., Vöcking, B.: A unified approach to congestion games and two-sided markets. Internet Math.
**5**(4), 439–457 (2008)zbMATHMathSciNetCrossRefGoogle Scholar - 2.Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. J. ACM
**55**(6) (2008)Google Scholar - 3.Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. SIAM J. Comput.
**38**(4), 1602–1623 (2008)zbMATHMathSciNetCrossRefGoogle Scholar - 4.Bauer, S., Clark, D., Lehr, W.: The evolution of internet congestion In: TPRC 2009, 37th Research Conference on Communication, Information and Internet Policy (2009)Google Scholar
- 5.Bilò, V., Fanelli, A., Flammini, M., Moscardelli, L.: Graphical congestion games. Algorithmica
**61**(2), 274–297 (2011)zbMATHMathSciNetCrossRefGoogle Scholar - 6.Campbell, A., Aurrecoechea, C., Hauw, L.: A review of QoS architectures. ACM Multimedia Systems J.
**6**, 138–151 (1996)Google Scholar - 7.Chien, S., Sinclair, A.: Convergence to approximate nash equilibria in congestion games In: SODA, pp 169–178 (2007)Google Scholar
- 8.Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games In: STOC, pp 67–73 (2005)Google Scholar
- 9.Correa, J., Schulz, A., Stier-Moses, N.: Selfish routing in capacitated networks. Math. Oper. Res.
**29**(4), 961–976 (2004)zbMATHMathSciNetCrossRefGoogle Scholar - 10.Czerny, A.I., Mitusch, K., Tanner, A.: Priority rules versus scarcity premiums in rail markets. WHU Otto Beisheim School of Management - Working Paper Series in Economics
**10**(3) (2010)Google Scholar - 11.Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to nash equilibrium in load balancing. ACM Trans. Algoritm.
**3**(3) (2007)Google Scholar - 12.Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure nash equilibria In: STOC, pp 604–612 (2004)Google Scholar
- 13.Farzad, B., Olver, N., Vetta, A.: A priority-based model of routing. Chic. J. Theor. Comput. Sci., 1 (2008)Google Scholar
- 14.Feldman, M., Ron, T.: Capacitated network design games. In: Serna, M. (ed.) Algorithmic Game Theory - 5th International Symposium, SAGT 2012, Barcelona, Spain, 22–23 October 2012. Proceedings of Lecture Notes in Computer Science, vol. 7615, pp. 132–143. Springer (2012)Google Scholar
- 15.Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of nash equilibria for a selfish routing game. Theor. Comput. Sci.
**410**(36), 3305–3326 (2009)zbMATHMathSciNetCrossRefGoogle Scholar - 16.Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: Computing nash equilibria for scheduling on restricted parallel linksIn: STOC, pp 613–622. ACM (2004)Google Scholar
- 17.Gale, D., Shapley, L.S.: College admissions and the stability of marriage. Am. Math. Mon.
**69**(1), 9–15 (1962)zbMATHMathSciNetCrossRefGoogle Scholar - 18.Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
- 19.Gourvès, L., Monnot, J., Moretti, S., Nguyen Kim, T.: Congestion games with capacitated resources. In: Serna, M. (ed.) Algorithmic Game Theory - 5th International Symposium, SAGT 2012, Barcelona, Spain, 22–23 October 2012. Proceedings of Lecture Notes in Computer Science, vol. 7615, pp.204–215. Springer (2012)Google Scholar
- 20.Gourvès, L., Moretti, S.: Progress in Combinatorial Optimization. Combinatorial optimization problems arising from interactive congestion situations. ISTE Ltd, Wiley (2011)Google Scholar
- 21.Ieong, S., McGrew, R., Nudelman, E., Shoham, Y., Sun, Q.: Fast and compact: a simple class of congestion games In: AAAI, pp 489–494 (2005)Google Scholar
- 22.Konishi, H., Mun, S.: Carpooling and congestion pricing: Hov and hot lanes. Reg. Sci. Urban Econ.
**40**(4), 173–186 (2010)CrossRefGoogle Scholar - 23.Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: 16th Annual Symposium on Theoretical Aspects of Computer Science, pp. 404–413 (1999)Google Scholar
- 24.Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. Comput. Sci. Rev.
**3**(2), 65–69 (2009)zbMATHCrossRefGoogle Scholar - 25.Lee, K.D., Leung, V.C.M.: Utility-based rate-controlled parallel wireless transmission of multimedia streams with multiple importance levels. IEEE Trans. Mob. Comput., 81–92 (2009)Google Scholar
- 26.Leyton-Brown, K., Tennenholtz, M.: Local-effect games. In: Lehmann, D.J., Muller, R., Sandholm, T. (eds.) Computing and Markets, volume 05011 of Dagstuhl Seminar Proceedings. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany IBFI, Schloss Dagstuhl, Germany (2005)Google Scholar
- 27.Lulli, G., Pietropaoli, U., Ricciardi, N.: Service network design for freight railway transportation: the italian case. J. Oper. Res. Soc.
**62**(12), 2107–2119 (2011)CrossRefGoogle Scholar - 28.Mangold, S., Choi, S., May, P., Klein, O., Hiertz, G., Stibor, L.: IEEE 802.11e Wireless LAN for quality of serviceProceedings of European Wireless, vol. 18, pp 32–39 (2002)Google Scholar
- 29.Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior
**13**(1), 111–124 (1996)zbMATHMathSciNetCrossRefGoogle Scholar - 30.Monderer, D., Shapley, L.: Potential games. Games and Economic Behavior
**14**, 124–143 (1996)zbMATHMathSciNetCrossRefGoogle Scholar - 31.Nash, J.: Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America
**36**(1), 48–49 (1950)zbMATHMathSciNetCrossRefGoogle Scholar - 32.Rosenthal, R.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory
**2**, 65–67 (1973)zbMATHCrossRefGoogle Scholar - 33.Roth, A.E.: The college admissions problem is not equivalent to the marriage problem. J. Econ. Theory
**36**(2), 277–288 (1985)zbMATHCrossRefGoogle Scholar - 34.Serna, M. (ed.) Algorithmic Game Theory - 5th International Symposium, SAGT 2012, Barcelona, Spain, 22–23 October 2012. Proceedings of Lecture Notes in Computer Science, vol. 7615. Springer (2012)Google Scholar
- 35.Su, X., De Veciana, G.: Predictive routing to enhance qos for stream-based flows sharing excess bandwidth. Comput. Netw.
**42**(1), 65–80 (2003)zbMATHCrossRefGoogle Scholar - 36.Vöcking, B.: Congestion games: optimization in competition. In: ACiD Workshop, Text in Algorithmics, vol. 7, pp. 9–20 (2006)Google Scholar
- 37.Voorneveld, M.: Potential Games and Interactive Decisions with Multiple Criteria. PhD thesis, Tilburg University (1999)Google Scholar
- 38.Zhao, Z., Willman, B., Weber, S., de Oliveira, J.C.: Performance analysis of a parallel link network with preemption In: 40th Annual Conference on Information Sciences and Systems, pp 271–276 (2006)Google Scholar