The Exact Computational Complexity of Evolutionarily Stable Strategies

While the computational complexity of many game-theoretic solution concepts, notably Nash equilibrium, has now been settled, the question of determining the exact complexity of computing an evolutionarily stable strategy has resisted solution since attention was drawn to it in 2004. In this paper, I settle this question by proving that deciding the existence of an evolutionarily stable strategy is $\Sigma_2^P$ -complete.

[1]  Troels Bjerre Sørensen Computing a proper equilibrium of a bimatrix game , 2012, EC '12.

[2]  Bernhard von Stengel,et al.  Leadership games with convex strategy sets , 2010, Games Econ. Behav..

[3]  J. M. Smith,et al.  The Logic of Animal Conflict , 1973, Nature.

[4]  Christos H. Papadimitriou,et al.  On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence , 1994, J. Comput. Syst. Sci..

[5]  K. Ko,et al.  On the Complexity of Min-Max Optimization Problems and their Approximation , 1995 .

[6]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.

[7]  J. Nash Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[8]  Christos H. Papadimitriou,et al.  Computing correlated equilibria in multi-player games , 2005, STOC '05.

[9]  Kousha Etessami,et al.  On the Complexity of Nash Equilibria and Other Fixed Points , 2010, SIAM J. Comput..

[10]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[11]  Paul G. Spirakis,et al.  Equilibrium Points in Fear of Correlated Threats , 2008, WINE.

[12]  Vincent Conitzer,et al.  Computing the optimal strategy to commit to , 2006, EC '06.

[13]  Peter Stone,et al.  A polynomial-time nash equilibrium algorithm for repeated games , 2003, EC '03.

[14]  Vincent Conitzer,et al.  Fast Equilibrium Computation for Infinitely Repeated Games , 2013, AAAI.

[15]  Kousha Etessami,et al.  The computational complexity of evolutionarily stable strategies , 2008, Int. J. Game Theory.

[16]  M. Schaefer,et al.  Completeness in the Polynomial-Time Hierarchy A Compendium ∗ , 2008 .

[17]  Noam Nisan,et al.  A Note on the computational hardness of evolutionary stable strategies , 2006, Electron. Colloquium Comput. Complex..

[18]  The myth of the Folk Theorem , 2010 .

[19]  Kristoffer Arnsfelt Hansen,et al.  The Computational Complexity of Trembling Hand Perfection and Other Equilibrium Refinements , 2010, SAGT.

[20]  C. E. Lemke,et al.  Equilibrium Points of Bimatrix Games , 1964 .

[21]  Kevin Leyton-Brown,et al.  Polynomial-time computation of exact correlated equilibrium in compact games , 2010, EC '11.

[22]  Vincent Conitzer,et al.  Commitment to Correlated Strategies , 2011, AAAI.

[23]  Siddharth Suri Algorithmic Game Theory: Computational Evolutionary Game Theory , 2007 .

[24]  Vincent Conitzer,et al.  New complexity results about Nash equilibria , 2008, Games Econ. Behav..

[25]  Christos H. Papadimitriou,et al.  Algorithms, games, and the internet , 2001, STOC '01.

[26]  Rahul Savani,et al.  Hard‐to‐Solve Bimatrix Games , 2006 .