Complexity and Optimality of the Best Response Algorithm in Random Potential Games

  • Stéphane Durand
  • Bruno GaujalEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)


In this paper we compute the worst-case and average execution time of the Best Response Algorithm (BRA) to compute a pure Nash equilibrium in finite potential games. Our approach is based on a Markov chain model of BRA and a coupling technique that transform the average execution time of this discrete algorithm into the solution of an ordinary differential equation. In a potential game with N players and A strategies per player, we show that the worst case complexity of BRA (number of moves) is exactly \(N A^{N-1}\), while its average complexity over random potential games is equal to \(e^\gamma N + O(N)\), where \(\gamma \) is the Euler constant. We also show that the effective number of states visited by BRA is equal to \(\log N + c + O(1/N)\) (with \( c \leqslant e^\gamma \)), on average. Finally, we show that BRA computes a pure Nash Equilibrium faster (in the strong stochastic order sense) than any local search algorithm over random potential games.


Nash Equilibrium Markov Chain Model Local Search Algorithm Payoff Vector Congestion Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was partially supported by LabEx Persyval-Lab.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Univ. Grenoble AlpesGrenobleFrance
  2. 2.InriaGrenobleFrance

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