Quadratic stochastic operators and zero-sum game dynamics
Nasir GanikhodjaevDepartment of Computational and Theoretical Sciences, Faculty of Science, IIUM, 25200 Kuantan, Malaysia emailRasul GanikhodjaevDepartment of Algebra and Functional Analysis, Faculty of Mathematics, National University of Uzbekistan, 100095 Tashkent, Uzbekistan emailUygun JamilovInstitute of Mathematics at the National University of Uzbekistan, 29, Do’rmon Yo’li str., 100125 Tashkent, Uzbekistan email
ABI
Аннотация
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.
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