Adaptive Learning with Artificial Barriers Yielding Nash Equilibria in General Games

Abstract

Artificial barriers in Learning Automata (LA) is a powerful and yet under-explored concept although it was first proposed in the 1980s [1]. 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 [2]. 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 (LR−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 LR−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 LR−I .Reward-ǫPenalty (LR−ǫP ) scheme proposed by Lakshmivarahan and Narendra [3] almost four decades ago, is the only LA scheme that was shown to converge to the optimal mixed Nash equilibrium when no Saddle Point exists in pure strategy, and the proofs were limited to only zero-sum games. In this paper, we tackle a general game as opposed to the particular case of zero-sum games proposed by Lakshmivarahan and Narendra in [3], and provide a novel scheme and proofs characterizing its behavior. Furthermore, we provide experimental results that are in line with our theoretical findings.

Learning Automata with Artificial Reflecting Barriers in Games with Limited Information

“…The proofs of the theorems reported in this paper as well as some other additional theoretical results are omitted here due to space limitations and will be published in an extended version of the current article. A preprint of the extended version (Hassan, Oommen, and Yazidi 2022) is available on arXiv.…”