A dynamic programming analysis of a modified Nano backgammon board game
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Date
2014
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Mathematics,Obafemi Awolowo University
Abstract
This study identified the salient problems involved in the analysis of the full backgammon game especially with respect to the nano backgammon and in this connection constructed an appropriate distance measure of closeness to winning, generated associated recurrence equation from state to state and devised an optimal strategy to maximize the probability of winning over finite number of stages for the players. This was with a view to providing a unified analysis of a unique version of the nano backgammon game.
Having identified the problem as amenable to dynamic programming methodology, two distance measure criteria namely Euclidean and a particular Minkowski with p = 0.5 were adopted by two players (player I and player II) in determining the rational movement of the checkers on the board as driven by throws of a fair die. For illustrative purpose, simulation runs of the modified nano backgammon game played by different starting players, at different starting states (167257 and 027127) using different strategies were presented. The problem of maximization was considered using a generated recurrence equation to obtain the optimal solution via the dynamic programming technique. The total duration of time to win, number of wins and proportion of wins by the two players among other factors, were noted. Two hypotheses were formulated and the proportion of wins by player I was tested at 0.05 level of significance using the t-test statistic. As an alternative, a hypothesis was formulated and the relative dependence of duration of time to win on starting states, starting players, winning players and criteria were tested at 0.01 significant level using the χ2 test statistic.
For the different starting states 167257 and 027127, the total duration of time before conclusion of the game depended on who played first using the Euclidean distance measure criterion. In contrast, while with the starting state 167257, the total duration of time before conclusion of the game appeared not to depend on who started the game if the particular Minkowski distance measure criterion was applied. This was not the case with the starting state 027127. The study also noted that, with respect to the starting states considered, the proportion of wins by player I did not depend on who played first using the particular Minkowski distance measure criterion but with Euclidean distance measure criterion. It depended on who played first. In addition, the duration of time to win depended on starting state, starting player and winning player at 0.01 significant level.
The study concluded that there was significant association of the duration of time to first win and chance of winning with starting state, starting player and winning player for a specific distance measure criterion.
Description
xviii,389p
Keywords
Modified Nano, Backgammon, Board game, Minkowski
Citation
Karokatose,G.B (2014). A dynamic programming analysis of a modified Nano backgammon board game. Obafemi Awolowo University