Abstract
We consider clustering games in which the players are embedded in a network and want to coordinate (or anti-coordinate) their choices with their neighbors. Recent studies show that even very basic variants of these games exhibit a large Price of Anarchy. Our main goal is to understand how structural properties of the network topology impact the inefficiency of these games. We derive topological bounds on the Price of Anarchy for different classes of clustering games. These topological bounds provide a more informative assessment of the inefficiency of these games than the corresponding (worst-case) Price of Anarchy bounds. As one of our main results, we derive (tight) bounds on the Price of Anarchy for clustering games on Erdős-Rényi random graphs, which, depending on the graph density, stand in stark contrast to the known Price of Anarchy bounds.
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
Notes
- 1.
We note that Valiant and Roughgarden [23] study Braess’ paradox in large random graphs (see Related Work).
- 2.
The game is called a coordination game if all edges are coordination edges and an anti-coordination game (or cut game) if all edges are anti-coordination edges.
- 3.
In this paper, we use [k] to denote the set \(\{1, \dots , k\}\) for a given integer \(k \ge 1\).
- 4.
In the full version, we extend our ideas and provide a characterization of the existence of pure Nash equilibria in symmetric coordination games, complementing a result by Anshelevich and Sekar [3].
- 5.
Our results do not seem to extend to clustering games on directed graphs. One could model a directed edge \(e = (i,j)\) by setting \(\alpha _{ij} = 0\) and \(\alpha _{ji} > 0\). E.g., Theorem 1 does not apply then as \(\bar{\alpha } = \infty \) in this case.
- 6.
Although this model was first introduced by Gilbert, it is often referred to as the Erdős-Rényi random graph model.
- 7.
Some of our results naturally extend to more general distribution rules, but we omit the (technical) details here because they do not provide additional insights.
- 8.
One may focus on any set of \(\lceil c_n/4\rceil \) nodes. The important thing to note is that we need a set of nodes with many edges on its induced subgraph and a perfect matching (it is not sufficient to find two different sets each satisfying one of these properties). Moreover, if \(c_n \ge 4n\), we consider \(W_n = \{1,\dots ,n\}\) and then the same argument works.
- 9.
Note that here we implicitly use that the intersection of two probabilistic events which occur with high probability also occurs with high probability.
- 10.
In general, this is not true if \(c \ge 3\). For example, consider a cycle of length three with only anti-coordination edges.
- 11.
A strategy profile s is an \((\epsilon ,k)\)-equilibrium with \(\epsilon \ge 1\) and \(k \in [n]\) if for every set of players \(K \subseteq V\) with \(|K| \le k\) and every deviation \(s'_K = (s'_i)_{i \in K}\), there is at least one player \(j \in K\) such that \(\epsilon \cdot u_j(s) \ge u_j(s_{-K},s'_K)\). We turn to \((\epsilon , k)\)-equilibria because pure Nash equilibria are not guaranteed to exist in asymmetric coordination games (see, e.g., [4]).
- 12.
Note that in the deterministic setting the Price of Anarchy does not improve if all players have a color in common (see [21]).
References
Amiet, B., Collevecchio, A., Scarsini, M.: Pure nash equilibria and best-response dynamics in random games. CoRR abs/1905.10758 (2019)
Anantharam, V., Salez, J.: The densest subgraph problem in sparse random graphs. Ann. Appl. Probab. 26(1), 305–327 (2016)
Anshelevich, E., Sekar, S.: Approximate equilibrium and incentivizing social coordination. In: Proceedings of the 28th AAAI Conference on Artificial Intelligence, pp. 508–514 (2014)
Apt, K.R., de Keijzer, B., Rahn, M., Schäfer, G., Simon, S.: Coordination games on graphs. Int. J. Game Theory 46(3), 851–877 (2017)
Bárány, I., Vempala, S., Vetta, A.: Nash equilibria in random games. Random Struct. Algor. 31(4), 391–405 (2007)
Bilò, V., Fanelli, A., Flammini, M., Monaco, G., Moscardelli, L.: Nash stable outcomes in fractional hedonic games: existence, efficiency and computation. J. Artif. Intell. Res. 62, 315–371 (2018)
Carosi, R., Flammini, M., Monaco, G.: Computing approximate pure nash equilibria in digraph k-coloring games. In: Proceedings of the 16th Conference on Autonomous Agents and Multi Agent Systems, pp. 911–919 (2017)
Carosi, R., Monaco, G.: Generalized graph k-coloring games. In: Wang, L., Zhu, D. (eds.) COCOON 2018. LNCS, vol. 10976, pp. 268–279. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94776-1_23
Chen, H.L., Roughgarden, T., Valiant, G.: Designing network protocols for good equilibria. SIAM J. Comput. 39(5), 1799–1832 (2010)
Drèze, J.H., Greenberg, J.: Hedonic coalitions: optimality and stability. Econometrica 48(4), 987–1003 (1980)
Feldman, M., Friedler, O.: A unified framework for strong price of anarchy in clustering games. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 601–613. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-47666-6_48
Frieze, A., Karoński, M.: Introduction to Random Graphs. Cambridge University Press, Cambridge (2015)
Gilbert, E.N.: Random graphs. Ann. Math. Stat. 30(4), 1141–1144 (1959)
Gopalakrishnan, R., Marden, J.R., Wierman, A.: Potential games are necessary to ensure pure nash equilibria in cost sharing games. Math. Oper. Res. 39(4), 1252–1296 (2014)
Gourvès, L., Monnot, J.: On strong equilibria in the max cut game. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 608–615. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10841-9_62
Gourvès, L., Monnot, J.: The max k-cut game and its strong equilibria. In: Kratochvíl, J., Li, A., Fiala, J., Kolman, P. (eds.) TAMC 2010. LNCS, vol. 6108, pp. 234–246. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13562-0_22
Hajek, B.E.: Performance of global load balancing of local adjustment. IEEE Trans. Inf. Theory 36(6), 1398–1414 (1990)
Hoefer, M.: Cost sharing and clustering under distributed competition. Ph.D. thesis (2007)
Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-49116-3_38
Kun, J., Powers, B., Reyzin, L.: Anti-coordination games and stable graph colorings. In: Vöcking, B. (ed.) SAGT 2013. LNCS, vol. 8146, pp. 122–133. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-41392-6_11
Rahn, M., Schäfer, G.: Efficient equilibria in polymatrix coordination games. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9235, pp. 529–541. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48054-0_44
Roughgarden, T., Tardos, E.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)
Valiant, G., Roughgarden, T.: Braess’s paradox in large random graphs. Random Struct. Algor. 37(4), 495–515 (2010)
Acknowledgements
The first author thanks Remco van der Hofstad for a helpful discussion on random graph theory and, in particular, the results in [2].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kleer, P., Schäfer, G. (2019). Topological Price of Anarchy Bounds for Clustering Games on Networks. In: Caragiannis, I., Mirrokni, V., Nikolova, E. (eds) Web and Internet Economics. WINE 2019. Lecture Notes in Computer Science(), vol 11920. Springer, Cham. https://doi.org/10.1007/978-3-030-35389-6_18
Download citation
DOI: https://doi.org/10.1007/978-3-030-35389-6_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35388-9
Online ISBN: 978-3-030-35389-6
eBook Packages: Computer ScienceComputer Science (R0)