# A Note on the computational hardness of evolutionary stable strategies

@article{Nisan2006ANO, title={A Note on the computational hardness of evolutionary stable strategies}, author={Noam Nisan}, journal={Electron. Colloquium Comput. Complex.}, year={2006}, volume={TR06} }

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. Formally this shows that existence of evolutionary stable strategies is hard for a complexity class called co − D, slightly strengthening (and greatly simplifying) the known NP-hardness and co-NP-hardness. En route we show that even recognizing an evolutionary stable strategy is co-NP complete.

## 19 Citations

### The Exact Computational Complexity of Evolutionarily Stable Strategies

- Computer ScienceWINE
- 2013

This paper proves that deciding the existence of an evolutionarily stable strategy is $\Sigma_2^P$ -complete, which means that the solution to the Nash equilibrium problem is known.

### Computational Complexity of Multi-player Evolutionarily Stable Strategies

- Mathematics, Computer ScienceCSR
- 2021

It is shown that deciding existence of an ESS of a multi-player game is closely connected to the second level of the real polynomial time hierarchy, and that deciding whether a given strategy is an E SS is complete for ∀ R .

### On the computational complexity of evolution

- EconomicsArXiv
- 2015

This paper shows that recognizing an ESS even in doubly symmetric bimatrix games is also coNP-complete, which implies that recognizing asymptotically stable equilibria of the replicator dynamic in this class of games isAlso a co NP-complete problem.

### Existence of Evolutionarily Stable Strategies Remains Hard to Decide for a Wide Range of Payoff Values

- EconomicsCIAC
- 2017

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.

### The computational complexity of evolutionarily stable strategies

- MathematicsInt. J. Game Theory
- 2004

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.

### Algorithms and complexity of problems arising from strategic settings

- Computer Science, Mathematics
- 2019

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.

### NP = coNP

- Economics
- 2015

It is well-known that the problem of recognizing an ESS in a symmetric bimatrix game is coNP-complete. In this paper, we show that recognizing an ESS even in doubly symmetric bimatrix games is also…

### A Game-Theoretic Approach to Hypergraph Clustering

- Computer ScienceIEEE Transactions on Pattern Analysis and Machine Intelligence
- 2013

It is proved that the problem of finding the equilibria of the clustering game is equivalent to locally optimizing a polynomial function over the standard simplex, and a discrete-time high-order replicator dynamics to perform this optimization, based on the Baum-Eagon inequality is provided.

### Evolutionary stability implies asymptotic stability under multiplicative weights

- Economics, MathematicsArXiv
- 2016

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).

### A Case Study of Agent-Based Models for Evolutionary Game Theory

- MathematicsArXiv
- 2021

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.

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### Appendix: Rationality of x and

- Appendix: Rationality of x and