This thesis intends to explore the optimal strategies in Markov chain games and their relevance to applications. For this, the present study not only analyses the classic optimal bold strategies that gamblers bet everything they can afford, but also takes into account the strategies for capital portfolios and certain criteria that outperform traditional bold strategies. In addition, this thesis builds a P-adic number version by modifying the original model and proposes its associated algorithms.
Speci?cally, it presents a case of the 2-adic version after listing all the algorithms for the 64 strategies, and proves that the bold strategies are optimal in this case(with three strategies exceptionally optimal and still bold). While applying Probability Generating Function (PGF) approach enables us to successfully solve the gambler's ruin process, the solution of the process involving investment is not immediately obvious.
Furthermore, this study introduces various Markov mouse models and ?nds that the survival probability is generally larger under the horizontal splitting context. To develop a general version of an in?nite two- and three-dimensional Markov mouse model, we have ?rst processed the versions of the mouse in both vertical and horizontal models, during the former of which the optimality depends on the probability of being caught, whereas the latter case always outperforms when the number of grids increases. We have also applied the lumped Markov process and the perturbation theory to explore the transition matrix of the in?nite three-dimensional game, and ?nally,have developed a continuous version of the Markov mouse model that can produce the same results with the discrete version in terms of the absorption probability.</p
