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Quadratic stochastic operators and zero-sum game dynamics
- Source :
- Ergodic Theory and Dynamical Systems. 35:1443-1473
- Publication Year :
- 2014
- Publisher :
- Cambridge University Press (CUP), 2014.
-
Abstract
- In this paper we consider the set of all extremal Volterra quadratic stochastic operators defined on a unit simplex $S^{4}$ and show that such operators can be reinterpreted in terms of zero-sum games. We show that an extremal Volterra operator is non-ergodic and an appropriate zero-sum game is a rock-paper-scissors game if either the Volterra operator is a uniform operator or for a non-uniform Volterra operator $V$ there exists a subset $I\subset \{1,2,3,4,5\}$ with $|I|\leq 2$ such that $\sum _{i\in I}(V^{n}\mathbf{x})_{i}\rightarrow 0,$ and the restriction of $V$ on an invariant face ${\rm\Gamma}_{I}=\{\mathbf{x}\in S^{m-1}:x_{i}=0,i\in I\}$ is a uniform Volterra operator.
Details
- ISSN :
- 14694417 and 01433857
- Volume :
- 35
- Database :
- OpenAIRE
- Journal :
- Ergodic Theory and Dynamical Systems
- Accession number :
- edsair.doi...........43d5b00cc663ce591a4010466bb5d124