This article is composed of two parts. In the first part we deal with the probability that a player gets each place in a sequence of fair games when the movement of his fortune follows a Markov Process. We first find the equations of the relationship among the probabilities according to the initial fortune and then solve them with an iterative method. In the second part, we examine whether conservative strategies can give us more advantage to win a high payout in repeated multi-player games than the myopic strategy. We conduct simulations and find that in certain payout structures a player has higher expected payout with conservative strategies than with the myopic strategy.