Modified Engel Algorithm and Applications in Absorbing/Non-Absorbing Markov Chains and Monopoly Game

The Engel algorithm was created to solve chip-firing games and can be used to find the stationary distribution for absorbing Markov chains. Kaushal et. al. developed a matlab-based version of the generalized Engel algorithm based on Engel’s probabilistic abacus theory. This paper introduces a modified version of the generalized Engel algorithm, which we call the modified Engel algorithm or the mEngel algorithm, for short. This modified version is designed to address issues related to non-absorbing Markov chains. It achieves this by breaking down the transition matrix into two distinct matrices, where each entry in the transition matrix is calculated from the ratio of the numerator and denominator matrices. In a nested iteration setting, these matrices play a crucial role in converting non-absorbing Markov chains into absorbing ones and then back again, thereby providing an approximation to the solutions of non-absorbing Markov chains until the distribution of a Markov chain converges to a stationary distribution. Our results show that the numerical outcomes of the mEngel algorithm align with those obtained from the power method and the canonical decomposition of absorbing Markov chains. We provide an example, such as Torrence’s problem, to illustrate the application of absorbing probabilities. Furthermore, our proposed algorithm analyzes the Monopoly transition matrix as a form of non-absorbing probabilities based on the rules of the Monopoly game, a complete information dynamic game, particularly the probability of landing on the Jail square, which is determined by the order of the product of the movement, jail, chance, and community chest matrices. There are more than two players in one game, and the last player who is not bankrupt wins. Strategies in the game include whether to spend as long as possible in prison to avoid paying tolls to other players, known as long jail, or to leave prison immediately to buy as much land as possible, known as short jail. The long jail strategy, the short jail strategy, and the strategy of getting out of Jail by rolling consecutive doubles three times have been formulated and tested. In addition, choosing which color group to buy is also an important strategy. By comparing the probability distribution of each strategy and their profit return on each property and the colour group property, we find which one should be used when playing Monopoly. In conclusion, the mEngel algorithm, implemented using R codes, offers an alternative approach to solving the Monopoly game and demonstrates practical value.