Convergence Issues in Competitive Games

  • Vahab S. Mirrokni
  • Adrian Vetta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3122)


We study the speed of convergence to approximate solutions in iterative competitive games. We also investigate the value of Nash equilibria as a measure of the cost of the lack of coordination in such games. Our basic model uses the underlying best response graph induced by the selfish behavior of the players. In this model, we study the value of the social function after multiple rounds of best response behavior by the players. This work therefore deviates from other attempts to study the outcome of selfish behavior of players in non-cooperative games in that we dispense with the insistence upon only evaluating Nash equilibria. A detailed theoretical and practical justification for this approach is presented. We consider non-cooperative games with a submodular social utility function; in particular, we focus upon the class of valid-utility games introduced in [13]. Special cases include basic-utility games and market sharing games which we examine in depth. On the positive side we show that for basic-utility games we obtain extremely quick convergence. After just one round of iterative selfish behavior we are guaranteed to obtain a solution with social value at least \(\frac13\) that of optimal. For n-player valid-utility games, in general, after one round we obtain a \(\frac{1}{2n}\)-approximate solution. For market sharing games we prove that one round of selfish response behavior of players gives \(\Omega({1\over \ln n})\)-approximate solutions and this bound is almost tight. On the negative side we present an example to show that even in games in which every Nash equilibrium has high social value (at least half of optimal), iterative selfish behavior may “converge” to a set of extremely poor solutions (each being at least a factor n from optimal). In such games Nash equilibria may severely underestimate the cost of the lack of coordination in a game, and we discuss the implications of this.


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  1. 1.
    Brucker, P., Hurink, J., Werner, F.: Improving local search heuristics for some scheduling problems. Discrete Applied Mathematics 72, 47–69 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Even-dar, E., Kesselman, A., Mansour, Y.: Convergence time to Nash equilibria. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Fabrikant, A., Papadimitriou, C., Talwar, K.: On the complexity of pure equilibria. In: STOC (2004)Google Scholar
  4. 4.
    Goemans, M., Li, L., Mirrokni, V.S., Thottan, M.: Market sharing games applied to content distribution in ad-hoc networks. In: MOBIHOC (2004)Google Scholar
  5. 5.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior 13(1), 111–124 (1996)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Monderer, D., Shapley, L.: Potential games. Games and Economic Behavior 14(1), 124–143 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nisan, N., Ronen, A.: Algorithmic mechanism design. Games and Economic Behavior 35(1-2), 166–196 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Papadimitriou, C.: Algorithms, games, and the internet. In: STOC 2001 (2001)Google Scholar
  9. 9.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2, 65–67 (1973)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Roughgarden, T., Tardos, E.: Bounding the inefficiency of equilibria in nonatomic congestion games. Games and Economic Behavior 47(2), 389–403 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Roughgarden, T., Tardos, E.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Schuurman, P., Vredeveld, T.: Performance guarantees of local search for multiprocessor scheduling. In: IPCO (2001)Google Scholar
  13. 13.
    Vetta, A.: Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In: FOCS (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Vahab S. Mirrokni
    • 1
  • Adrian Vetta
    • 2
  1. 1.Massachusetts Institute of Technology 
  2. 2.McGill University 

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