Deriving the Optimal Strategy for the Two Dice Pig Game via Reinforcement Learning

Abstract

Games of chance have historically played a critical role in the development and teaching of probability theory and game theory, and, in the modern age, computer programming and reinforcement learning. In this paper, we derive the optimal strategy for playing the two-dice game Pig, both the standard version and its variant with doubles, coined “Double-Trouble”, using certain fundamental concepts of reinforcement learning, especially the Markov decision process and dynamic programming. We further compare the newly derived optimal strategy to other popular play strategies in terms of the winning chances and the order of play. In particular, we compare to the popular “hold at n” strategy, which is considered to be close to the optimal strategy, especially for the best n, for each type of Pig Game. For the standard two-player, two-dice, sequential Pig Game examined here, we found that “hold at 23” is the best choice, with the average winning chance against the optimal strategy being 0.4747. For the “Double-Trouble” version, we found that the “hold at 18” is the best choice, with the average winning chance against the optimal strategy being 0.4733. Furthermore, time in terms of turns to play each type of game is also examined for practical purposes. For optimal vs. optimal or optimal vs. the best “hold at n” strategy, we found that the average number of turns is 19, 23, and 24 for one-die Pig, standard two-dice Pig, and the “Double-Trouble” two-dice Pig games, respectively. We hope our work will inspire students of all ages to invest in the field of reinforcement learning, which is crucial for the development of artificial intelligence and robotics and, subsequently, for the future of humanity.