The Exact Computational Complexity of Evolutionarily Stable Strategies

@inproceedings{Conitzer2013TheEC,
  title={The Exact Computational Complexity of Evolutionarily Stable Strategies},
  author={Vincent Conitzer},
  booktitle={WINE},
  year={2013}
}
  • V. Conitzer
  • Published in WINE 11 December 2013
  • Computer Science
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. 
Existence of Evolutionarily Stable Strategies Remains Hard to Decide for a Wide Range of Payoff Values
TLDR
A reduction robustness notion is introduced and it is shown that deciding the existence of an ESS remains coNP-hard for a wide range of games even if the authors arbitrarily perturb within some intervals the payoff values of the game under consideration.
Settling Some Open Problems on 2-Player Symmetric Nash Equilibria
TLDR
It is shown that this problem of finding a non-symmetric Nash equilibrium (NE) in a symmetric game is NP-complete and the problem of counting the number of non-Symmetric NE in a symmetry game is #P-complete.
Computational Complexity of Multi-player Evolutionarily Stable Strategies
TLDR
It is shown that deciding existence of an ESS of a multiplayer game is closely connected to the second level of the real polynomial time hierarchy, and as a special case that deciding whether a given strategy is an LSS is complete for ∀R.
Algorithms and complexity of problems arising from strategic settings
TLDR
This thesis deals with an evolutionary setting where it is shown that for a wide range of symmetric bimatrix games, deciding ESS existence is intractable, and presents a general framework for constructing approximation schemes for problems that can be written as an Existential Theory of the Reals formula with variables constrained in a bounded convex set.
A Case Study of Agent-Based Models for Evolutionary Game Theory
TLDR
This short paper presents a game with complex interactions and examines how an agent-based model may be used as a heuristic technique to find evolutionarily stable states.
Algorithm for Evolutionarily Stable Strategies Against Pure Mutations
TLDR
This work presents an algorithm for the case where mutations are restricted to pure strategies, and presents experiments on several game classes including random and a recently-proposed cancer model based on the first general optimization formulation for this problem.
Computing Nash Equilibria for District-based Nominations
We study political parties that strategically place their candidates in districts so to maximise the number of their nominees that get elected. In each district, voters rank the nominated candidates
alpha-Rank: Multi-Agent Evaluation by Evolution
TLDR
Proofs are introduced that not only provide a unifying perspective of existing continuous- and discrete-time evolutionary evaluation models, but also reveal the formal underpinnings of the $\alpha$-Rank methodology.
Evolutionary stability implies asymptotic stability under multiplicative weights
TLDR
It is shown that evolutionarily stable states in general (nonlinear) population games are asymptotically stable under a multiplicative weights dynamic (under appropriate choices of a parameter called the learning rate or step size, which is demonstrated to be crucial to achieve convergence, as otherwise even chaotic behavior is possible to manifest).
α-Rank: Multi-Agent Evaluation by Evolution
We introduce α-Rank, a principled evolutionary dynamics methodology, for the evaluation and ranking of agents in large-scale multi-agent interactions, grounded in a novel dynamical game-theoretic
...
...

References

SHOWING 1-10 OF 39 REFERENCES
The complexity of computing a Nash equilibrium
TLDR
This proof uses ideas from the recently-established equivalence between polynomial time solvability of normal form games and graphical games, establishing that these kinds of games can simulate a PPAD-complete class of Brouwer functions.
The computational complexity of evolutionarily stable strategies
TLDR
It is shown that determining the existence of an ESS is both hard and hard and coNP-hard, and that the problem is contained in $$\Sigma_{2}^{\rm p}$$ , the second level of the polynomial time hierarchy.
A Note on the computational hardness of evolutionary stable strategies
  • N. Nisan
  • Mathematics
    Electron. Colloquium Comput. Complex.
  • 2006
We present a very simple reduction that when given a graph G and an integer k produces a game that has an evolutionary stable strategy if and only if the maximum clique size of G is not exactly k.
A polynomial-time nash equilibrium algorithm for repeated games
TLDR
This approach draws on the "folk theorem" from game theory and shows how finite-state equilibrium strategies can be found efficiently and expressed succinctly in a polynomial-time algorithm.
Settling the complexity of computing two-player Nash equilibria
We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by
Fast Equilibrium Computation for Infinitely Repeated Games
TLDR
This paper proposes an algorithmic framework for computing equilibria of repeated games that does not require linear programming and that doesn't necessarily need to inspect all payoffs of the game, and demonstrates that most of the time it succeeds quickly on uniformly random games.
The Computational Complexity of Trembling Hand Perfection and Other Equilibrium Refinements
TLDR
It is shown that it is NP-hard and Sqrt-Sum-hard to decide if a given pure strategy Nash equilibrium of a given three-player game in strategic form with integer payoffs is trembling hand perfect.
Equilibrium Points in N-Person Games.
  • J. Nash
  • Economics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1950
One may define a concept of an n -person game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each n -tuple of pure
Computing correlated equilibria in multi-player games
We develop a polynomial-time algorithm for finding correlated equilibria (a well-studied notion of rationality due to Aumann that generalizes the Nash equilibrium) in a broad class of succinctly
On the Complexity of Nash Equilibria and Other Fixed Points
TLDR
It is shown that the (exact or approximate) computation of Nash equilibria for 3 or more players is complete for FIXP, which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits (straight line programs) over basis with rational constants.
...
...