On a Unique Solution of the Stochastic Functional Equation Arising in Gambling Theory and Human Learning Process

Abstract

The term “learning” is often used to refer to a generally stable behavioral change resulting from practice. However, it is a fundamental biological capacity far more developed in humans than in other living beings. In an animal or human being, the learning phase may often be viewed as a series of choices between multiple possible reactions. Here, we analyze a specific type of human learning process related to gambling in which a subject inserts a poker chip to operate a two-armed bandit device and then presses one of the two keys. Through the use of an electromagnet, one or more poker chips are given to the individual in a container located in the apparatus’s center. If a chip is provided, it is declared a winner; otherwise, it is considered a loser. The goal of this paper is to look at the subject’s actions in such situations and provide a mathematical model that is appropriate for it. The existence of a unique solution to the suggested human learning model is examined using relevant fixed point results.