On classification of the genetic and evolution rock-paper-scissor algebras - Ganikhodjaev, Nasir; Ftameh, Kalid (American Institute of Physics Inc., 2018)
In recent decades in game theory, evolutionary and dynamically aspects have increased in popularity. An evolutionary zero-sum game generated by a Volterra quadratic stochastic operator V is called a Rock-Paper-Scissors (RPS) game if the operator V is a non-ergodic transformation. An algebra with genetic realization (genetic algebra) is an algebra A over the real numbers R which has a finite basis and a multiplication table is generated by the Volterra quadratic stochastic operator. An evolution algebra over the real numbers R is an algebra with a basis on which multiplication is defined by the product of distinct basis terms being zero and the square of each basis element being a linear form in basis elements. We assume that the structure constants in the linear form are defined also by Volterra operator. We consider genetic and evolution algebras generated by non-ergodic Volterra operator. It is known that a zero-sum game generated by Volterra operator be a RPS game if and only if the operator is a non-ergodic transformation. We will call the genetic (evolution) algebra generated by non-ergodic Volterra operator RPS genetic (respectively RPS evolution) algebra. In this paper we investigate the problem of isomorphism of two RPS genetic (evolutionary) algebras and establish necessary and sufficient conditions when two such algebras will be isomorphic.