Complexity Results about Nash Equilibria
<|reference_start|>Complexity Results about Nash Equilibria: Noncooperative game theory provides a normative framework for analyzing strategic interactions. However, for the toolbox to be operational, the solutions it defines will have to be computed. In this paper, we provide a single reduction that 1) demonstrates NP-hardness of determining whether Nash equilibria with certain natural properties exist, and 2) demonstrates the #P-hardness of counting Nash equilibria (or connected sets of Nash equilibria). We also show that 3) determining whether a pure-strategy Bayes-Nash equilibrium exists is NP-hard, and that 4) determining whether a pure-strategy Nash equilibrium exists in a stochastic (Markov) game is PSPACE-hard even if the game is invisible (this remains NP-hard if the game is finite). All of our hardness results hold even if there are only two players and the game is symmetric.   Keywords: Nash equilibrium; game theory; computational complexity; noncooperative game theory; normal form game; stochastic game; Markov game; Bayes-Nash equilibrium; multiagent systems.<|reference_end|>
@article{conitzer2002complexity,
    title={Complexity Results about Nash Equilibria},
    author={Vincent Conitzer, Tuomas Sandholm},
    journal={In Proceedings of the 18th International Joint Conference on
  Artificial Intelligence (IJCAI-03), Acapulco, Mexico, 2003},
    year={2002},
    number={CMU-CS-02-135},
    archivePrefix={arXiv},
    eprint={cs/0205074},
    primaryClass={cs.GT cs.CC cs.MA}
}