Adaptive Learning with Artificial Barriers Yielding Nash Equilibria in General Games

Artificial barriers in Learning Automata (LA) is a powerful and yet under-explored concept although it was first proposed in the 1980s. Introducing artificial non-absorbing barriers makes the LA schemes resilient to being trapped in absorbing barriers, a phenomenon which is often referred to as lock in probability leading to an exclusive choice of one action after convergence. Within the field of LA and reinforcement learning in general, there is a sacristy of theoretical works and applications of schemes with artificial barriers. In this paper, we devise a LA with artificial barriers for solving a general form of stochastic bimatrix game. Classical LA systems possess properties of absorbing barriers and they are a powerful tool in game theory and were shown to converge to game's of Nash equilibrium under limited information. However, the stream of works in LA for solving game theoretical problems can merely solve the case where the Saddle Point of the game exists in a pure strategy and fail to reach mixed Nash equilibrium when no Saddle Point exists for a pure strategy. In this paper, by resorting to the powerful concept of artificial barriers, we suggest a LA that converges to an optimal mixed Nash equilibrium even though there may be no Saddle Point when a pure strategy is invoked. Our deployed scheme is of Linear Reward-Inaction ($L_{R-I}$) flavor which is originally an absorbing LA scheme, however, we render it non-absorbing by introducing artificial barriers in an elegant and natural manner, in the sense that that the well-known legacy $L_{R-I}$ scheme can be seen as an instance of our proposed algorithm for a particular choice of the barrier. Furthermore, we present an $S$ Learning version of our LA with absorbing barriers that is able to handle $S$-Learning environment in which the feedback is continuous and not binary as in the case of the $L_{R-I}$.