Identifying Critical Positions Based on Conspiracy Numbers

Abstract

Research in two-player perfect information games has been one of the focuses of computer-game related studies in the domain of artificial intelligence. However, focus on an effective search program is insufficient to give the “taste” of actual entertainment in the gaming industry. Instead of focusing on effective search algorithm, we dedicate our study in realizing the possibility of applying strategy changing technique. However, quantifying and determining this possibility is the main challenge imposed in this study. For this purpose, the Conspiracy Number Search algorithm is considered where the maximum and minimum conspiracy numbers are recorded in the test bed of simple Tic-Tac-Toe and Othello game application. We analysed these numbers as the measures of critical position identifier which determines the right moment for possibility of applying strategy changing technique. For Tic-Tac-Toe game, the conspiracy numbers are analysed through operators formally defined in this article as $\uparrow tactic$ and $\downarrow tactic$ while variance of the conspiracy numbers are analysed in Othello game. Interesting results are obtained with convincing evidences but future works are still needed in order to further strengthen our hypothesis.

Appendices

A Appendix

Experiment Design

This Othello study is conducted by the default opening position on 8$\,\times \,$8 board with 64 squares as shown in Fig. 7. Figure 8 denotes the significant squares that are given special names.

Fig. 8.

Othello opening position.

C-squares are adjacent to the corner D while X-squares are diagonally adjacent to the corner D. A-squares and B-squares are the edges of the board [23]. E-squares are adjacent to A-squares and B-squares. The evaluation function is defined as follows (Fig. 9):

Fig. 9.

Significant squares on board.

Table 8. Fitness value of each significant square.

• D-squares are good and high priority positions.

• C-squares and X-squares are bad positions which may give chances to the opponent player accessing corner D.

• A-squares are good positions if there exist no opponent’s discs at the adjacent squares.

• B-squares are bad positions where the opponent player might has chances to access to A-squares.

Table 8 shows the fitness values that have been assigned to each significant square on board.

B Appendix

Game Analysis using CN Variance and Position Scoring

*Applicable to all the figures

Fig. 10.

Game set 1 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

Fig. 11.

Game set 2 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

Fig. 12.

Game set 3 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

Fig. 13.

Game set 4 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

The trend line drawn in position scoring Figs. 10(c), 11(c), 12(c) and 13(c) is closely related to the CN$_{\tiny \textit{variance}}$ trend line at Figs. 10(a), 11(a), 12(a) and 13(a) whereas the trend line drawn in position scoring Figs. 10(d), 11(d), 12(d) and 13(d) is closely related to the CN$_{\tiny \textit{variance}}$ trend line at Figs. 10(b), 11(b), 12(b) and 13(b). CN$_{\tiny \textit{variance}}$ trend line in Figs. 10(a), (b), 11(a), (b), 12(a), (b) and 13(a), (b) can be used to predict the trend of position scoring in Figs. 10(c), (d), 11(c), (d), 12(c), (d) and 13(c), (d).

Increasing CN$_{\tiny \textit{variance}}$ trend line in Figs. 10(a), (b), 11(a), (b), 12(a), (b) and 13(a), (b) will lead to decreasing trend in position scoring which can be observed by Figs. 10(c), (d), 11(c), (d), 12(c), (d) and 13(c), (d). However, trend line observation is not applicable when there are infinity CN$_{\tiny \textit{variance}}$ values.

Fig. 14.

Game set 1 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

Fig. 15.

Game set 2 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

C Appendix

Incorporating Strategy Changing for Black or White Player Only

*Applicable to all the figures

The trend line drawn in position scoring Figs. 14(c), 15(c), 16(c) and 17(c) is closely related to the CN$_{\tiny \textit{variance}}$ trend line at Figs. 14(a), 15(a), 16(a) and 17(a) whereas the trend line drawn in position scoring Figs. 14(d), 15(d), 16(d) and 17(d) is closely related to the CN$_{\tiny \textit{variance}}$ trend line at Figs. 14(b), 15(b), 16(b) and 17(b). CN$_{\tiny \textit{variance}}$ trend line in Figs. 14(a), (b), 15(a), (b), 16(a), (b) and 17(a), (b) can be used to predict the trend of position scoring in Figs. 14(c), (d), 15(c), (d), 16(c), (d) and 17(c), (d).

Fig. 16.

Game set 3 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

Fig. 17.

Game set 4 (a) Black player’s Conspiracy numbers variance (b) White player’s Conspiracy numbers variance (c) Black player’s position scoring (d) White player’s position scoring. A trend line is drawn in each figure.

Increasing CN$_{\tiny \textit{variance}}$ trend line in Figs. 14(a), (b), 15(a), (b), 16(a), (b) and 17(a), (b) will lead to decreasing trend in position scoring which can be observed by Figs. 14(c), (d), 15(c), (d), 16(c), (d) and 17(c), (d). However, trend line observation is not applicable when there are infinity CN$_{\tiny \textit{variance}}$ values.