The Annals of Applied Probability

Connectivity and equilibrium in random games

Constantinos Daskalakis, Alexandros G. Dimakis, and Elchanan Mossel

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Abstract

We study how the structure of the interaction graph of a game affects the existence of pure Nash equilibria. In particular, for a fixed interaction graph, we are interested in whether there are pure Nash equilibria arising when random utility tables are assigned to the players. We provide conditions for the structure of the graph under which equilibria are likely to exist and complementary conditions which make the existence of equilibria highly unlikely. Our results have immediate implications for many deterministic graphs and generalize known results for random games on the complete graph. In particular, our results imply that the probability that bounded degree graphs have pure Nash equilibria is exponentially small in the size of the graph and yield a simple algorithm that finds small nonexistence certificates for a large family of graphs. Then we show that in any strongly connected graph of n vertices with expansion (1+Ω(1))log2(n) the distribution of the number of equilibria approaches the Poisson distribution with parameter 1, asymptotically as n→+∞.

In order to obtain a refined characterization of the degree of connectivity associated with the existence of equilibria, we also study the model in the random graph setting. In particular, we look at the case where the interaction graph is drawn from the Erdős–Rényi, G(n, p), model where each edge is present independently with probability p. For this model we establish a double phase transition for the existence of pure Nash equilibria as a function of the average degree pn, consistent with the nonmonotone behavior of the model. We show that when the average degree satisfies np>(2+Ω(1))loge(n), the number of pure Nash equilibria follows a Poisson distribution with parameter 1, asymptotically as n→∞. When 1/nnp<(0.5−Ω(1))loge(n), pure Nash equilibria fail to exist with high probability. Finally, when np=O(1/n) a pure Nash equilibrium exists with constant probability.

Article information

Source
Ann. Appl. Probab. Volume 21, Number 3 (2011), 987-1016.

Dates
First available in Project Euclid: 2 June 2011

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1307020389

Digital Object Identifier
doi:10.1214/10-AAP715

Zentralblatt MATH identifier
05925659

Mathematical Reviews number (MathSciNet)
MR2830610

Subjects
Primary: 91A
Secondary: 60 68Q

Keywords
Game theory graphical games connectivity phase transitions random constraint satisfaction problems

Citation

Daskalakis, Constantinos; Dimakis, Alexandros G.; Mossel, Elchanan. Connectivity and equilibrium in random games. Ann. Appl. Probab. 21 (2011), no. 3, 987--1016. doi:10.1214/10-AAP715. http://projecteuclid.org/euclid.aoap/1307020389.


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References

  • [1] Achlioptas, D. and Peres, Y. (2003). The threshold for random k-SAT is 2kln2−O(k). In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing 223–231 (electronic). ACM, New York.
  • [2] Alon, N. and Spencer, J. H. (2000). The Probabilistic Method, 2nd ed. Wiley, New York.
  • [3] Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T. and Roughgarden, T. (2004). The price of stability for network design with fair cost allocation. In FOCS’04: Proceedings of the 45th Symposium on Foundations of Computer Science. IEEE Computer Society, Washington, DC.
  • [4] Arratia, R., Goldstein, L. and Gordon, L. (1989). Two moments suffice for Poisson approximations: The Chen–Stein method. Ann. Probab. 17 9–25.
  • [5] Bárány, I., Vempala, S. and Vetta, A. (2005). Nash equilibria in random games. In FOCS’05: Proceedings of the of the 46th Symposium on Foundations of Computer Science 134–145. IEEE Computer Society, Washington, DC.
  • [6] Beeri, C., Fagin, R., Maier, D. and Yannakakis, M. (1983). On the desirability of acyclic database schemes. J. Assoc. Comput. Mach. 30 479–513.
  • [7] Correa, J. R., Schulz, A. S. and Stier-Moses, N. E. (2004). Selfish routing in capacitated networks. Math. Oper. Res. 29 961–976.
  • [8] Czumaj, A., Krysta, P. and Vöcking, B. (2002). Selfish traffic allocation for server farms. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing 287–296 (electronic). ACM, New York.
  • [9] Czumaj, A. and Vöcking, B. (2002). Tight bounds for worst-case equilibria. In SODA’02: Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms 413–420. SIAM, Philadelphia, PA.
  • [10] Daskalakis, C. and Papadimitriou, C. H. (2006). Computing pure nash equilibria in graphical games via Markov random fields. In EC’06: Proceedings of the 7th ACM Conference on Electronic Commerce 91–99. ACM, New York.
  • [11] Dilkina, B., Gomes, C. P. and Sabharwal, A. (2007). The impact of network topology on pure nash equilibria in graphical games. In AAAI-07: Proceedings of the 22nd Conference on Artificial Intelligence 42–49. AAAI Press, Menlo Park, CA.
  • [12] Dresher, M. (1970). Probability of a pure equilibrium point in n-person games. J. Combin. Theory 8 134–145.
  • [13] Elkind, E., Goldberg, L. A. and Goldberg, P. (2006). Nash equilibria in graphical games on trees revisited. In EC’06: Proceedings of the 7th ACM Conference on Electronic Commerce 100–109. ACM, New York.
  • [14] Friedgut, E. (1999). Sharp thresholds of graph properties, and the k-sat problem. J. Amer. Math. Soc. 12 1017–1054. With an appendix by Jean Bourgain.
  • [15] Goldberg, K., Goldman, A. J. and Newman, M. (1968). The probability of an equilibrium point. J. Res. Nat. Bur. Standards Sect. B 72B 93–101.
  • [16] Gottlob, G., Greco, G. and Scarcello, F. (2003). Pure nash equilibria: Hard and easy games. In TARK’03: Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge 215–230. ACM, New York.
  • [17] Gottlob, G., Leone, N. and Scarcello, F. (2002). Hypertree decompositions and tractable queries. J. Comput. System Sci. 64 579–627.
  • [18] Hart, S., Rinott, Y. and Weiss, B. (2008). Evolutionarily stable strategies of random games, and the vertices of random polygons. Ann. Appl. Probab. 18 259–287.
  • [19] Kearns, M. J., Littman, M. L. and Singh, S. (2001). Graphical models for game theory. In UAI’01: Proceedings of the 17th Conference in Uncertainty in Artificial Intelligence 253–260. Morgan Kaufmann, San Francisco, CA.
  • [20] Kleinberg, J. (2006). The emerging intersection of social and technological networks: Open questions and algorithmic challenges. In FOCS’06: Proceedings of the of the 47th Symposium on Foundations of Computer Science. IEEE Computer Society, Washington, DC.
  • [21] Koutsoupias, E. and Papadimitriou, C. (1999). Worst-case equilibria. In STACS 99: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science 1563 404–413. Springer, Berlin.
  • [22] Littman, M. L., Kearns, M. J. and Singh, S. (2001). An efficient, exact algorithm for solving tree-structured graphical games. In NIPS 817–823. Neural Information Processing Systems Foundation, La Jolla, CA.
  • [23] Ortiz, L. E. and Kearns, M. J. (2002). Nash propagation for loopy graphical games. In NIPS 793–800. Neural Information Processing Systems Foundation, La Jolla, CA.
  • [24] Papadimitriou, C. (2001). Algorithms, games, and the internet. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing 749–753 (electronic). ACM, New York.
  • [25] Papavassilopoulos, G. P. (1995). On the probability of existence of pure equilibria in matrix games. J. Optim. Theory Appl. 87 419–439.
  • [26] Powers, I. Y. (1990). Limiting distributions of the number of pure strategy Nash equilibria in N-person games. Internat. J. Game Theory 19 277–286.
  • [27] Raghavan, P. (2006). The changing face of web search: Algorithms, auctions and advertising. In STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing 129. ACM, New York.
  • [28] Rinott, Y. and Scarsini, M. (2000). On the number of pure strategy Nash equilibria in random games. Games Econom. Behav. 33 274–293.
  • [29] Roughgarden, T. (2002). The price of anarchy is independent of the network topology. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing 428–437 (electronic). ACM, New York.
  • [30] Roughgarden, T. (2001). Stackelberg scheduling strategies. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing 104–113 (electronic). ACM, New York.
  • [31] Roughgarden, T. and Tardos, É. (2002). How bad is selfish routing? J. ACM 49 236–259 (electronic).
  • [32] Roughgarden, T. and Tardos, É. (2004). Bounding the inefficiency of equilibria in nonatomic congestion games. Games Econom. Behav. 47 389–403.
  • [33] Stanford, W. (1995). A note on the probability of k pure Nash equilibria in matrix games. Games Econom. Behav. 9 238–246.
  • [34] Vetta, A. (2002). Nash equilibria in competitive societies with applications to facility location. In FOCS’02: Proceedings of the 43rd Symposium on Foundations of Computer Science 416–425. IEEE Computer Society, Washington, DC.