Game theory is a fascinating field and one topic are the so-called two person zero-sum games. A game is called a zero-sum game, if the gains of one player is the loss of the other one and vice versa. If the game is played sufficiently - infinity times - often, loss and gains of both players will become zero. The stimulus now is, that the game isn't played infinity but only finite times. In this case there is a winner and a looser.
The playing field here is a matrix, whose elements are covered with different positive and negative numbers, which are generated with a random number generator. Now the player looks for a row, which is "advantageous" to him, and the HP-41CX calculates a column, respectively. If the so defined element is positive the player wins this round. In case it is negative the HP-41CX is the winner. If the element equals zero, the game ends drawn.
In the simple method used by me the game can have a "value" different from zero. Now, if you would play this game ad infinitum, one of the players will win, and from this fact the "value" of the game is determined. Ideally no one would win - this would be a "zero"-sum game.
An Example For Better Understanding :
Let the following matrix be given :
In this case row 1 would be advantageous to the player, because one time he wins ([r1, c1] = 1) and in the other the game would end in a draw ([r1, c2] = 0). If he chooses row 2 he would win if his opponent chooses column 1 ([r2, c1] = 1), but he looses if the other one chooses column 2 ([r2, c2] = -1).
From the point of view of the other player column 2 is the right one, for the game ends drawn ([r1, c2] = 0) or he wins ([r2, c2] = -1). Column 1 is not acceptable for him, because he looses in both cases.
In the above example for the first player it's advantageous to choose row 1, and for the other one column 2 is favourable. If both players make their choices concerning these considerations the game will end in a draw ([r1, c2] = 0). This matrix is more advantageous to the first player, for he wins in two cases, looses in one and in one other the game ends drawn - correspondingly the other player is in disadvantage.
The above matrix is a very simple one. It's of more interest and more challenging, if the matrix becomes larger in size and the number's values increase. Then it's no longer easy to decide which row or column is the most advantageous or which chooses the other player !
Interestingly, there exists an optimal strategy for solving this problem (and the program masters it, of course ;-) ) ! Noteworthy is, that a choice is possible, which on first sight looks "unfavourable". But, of course, one has to take into account the possible opponent's choices and the arising consequences ! ;-)
I don't want to go into detail here concerning the theory of two person zero-sum games. (In case I should have much time I will write some lines. ;-) ) There is plenty of literature available, for example G. Owen, Game Theory, Springer Verlag.
Course of Game :
Starting the program the first matrix is generated. In the display appears "1. MATRIX" and then "1. ROW:". Now the elements of this row are displayed separated by commas These numbers must be written on a sheet of paper by the player, because the HP-41CX's display cannot contain the whole matrix but only one row per time. ;-) If the matrix is fully constructed "READY" is displayed.
Now the HP-41CX calculates the column which is the most favourable for it and the player makes his choice of a row, respectively. After a new BEEP the row must be entered. Then it is displayed, which column was computed by the HP-41CX and afterwards the element determined by row and column. If the value of this element is positive the player wins this round. In case the value is zero "DRAW" is displayed. The HP-41CX wins, if the value is negative. In the last case the game ends at once and the number of tries and the degree of difficulty are displayed.
In the first or second case the player is asked, whether he wants to continue. In the display appears "CONTIN ? Y/N". Now "Y" must be pressed to continue the game, otherwise "N". If the last round ended in a draw, a new matrix of same level is generated. If the player had won the last round, the difficulty is increased. There are 21 levels (degrees of difficulty) for the matrices.
ERWSP must be the first program in extended memory (in CAT 4 ERWSP must be first displayed). There is no need for jumping by hand to any addresses !
After ERWSP is copied into extended memory the main program GAME9 can be loaded. Before starting the program SIZE 122 must be executed. Additionally, GAME9 creates the data file "DATA" with size 28 registers in extended memory, which is removed at program termination.
Program Listing ERWSP :
01 LBL "ERWSP" 02 RCL 04 03 2 04 XY? not equal 05 GTO 01 06 ISG 04 07 "" (NOP) 08 GTO "HP" 09 LBL 01 10 X<>Y 11 3 12 XY? not equal 13 GTO 01 14 RCL 07 15 2 16 X=Y? 17 GTO 02 18 ISG 04 19 "" (NOP) 20 GTO "HP" 21 LBL 02 22 ISG 07 23 "" (NOP) 24 5 25 STO 06 26 GTO "HP" 27 LBL 01 28 X<>Y 29 4 30 XY? not equal 31 GTO 01 32 RCL 07 33 3 34 X=Y? 35 GTO 02 36 ISG 04 37 "" (NOP) 38 5 39 STO 06 40 3 41 STO 07 42 GTO "HP" 43 LBL 02 44 ISG 07 45 "" (NOP) 46 7 47 STO 06 48 GTO "HP" 49 LBL 01 50 X<>Y 51 5 52 XY? not equal 53 GTO 01 54 RCL 07 55 3 56 X=Y? 57 GTO 02 58 ISG 04 59 "" (NOP) 60 3 61 STO 07 62 5 63 STO 06 64 GTO "HP" 65 LBL 02 66 ISG 07 67 "" (NOP) 68 7 69 STO 06 70 GTO "HP" 71 LBL 01 72 X<>Y 73 6 74 XY? not equal 75 GTO 11 76 RCL 07 77 3 78 X=Y? 79 GTO 01 80 X<>Y 81 4 82 X=Y? 83 GTO 02 84 ISG 04 85 "" (NOP) 86 3 87 STO 07 88 5 89 STO 06 90 GTO "HP" 91 LBL 02 92 ISG 07 93 "" (NOP) 94 9 95 STO 06 96 GTO "HP" 97 LBL 01 98 ISG 07 99 "" (NOP) 100 7 101 STO 06 102 GTO "HP" 103 LBL 11 104 X<>Y 105 7 106 XY? not equal 107 GTO 11 108 RCL 07 109 3 110 X=Y? 111 GTO 01 112 X<>Y 113 4 114 X=Y? 115 GTO 02 116 ISG 04 117 "" (NOP) 118 4 119 STO 07 120 7 121 STO 06 122 GTO "HP" 123 LBL 02 124 ISG 07 125 "" (NOP) 126 9 127 STO 06 128 GTO "HP" 129 LBL 01 130 ISG 07 131 "" (NOP) 132 7 133 STO 06 134 GTO "HP" 135 LBL 11 136 X<>Y 137 8 138 XY? not equal 139 GTO 01 140 RCL 07 141 4 142 X=Y? 143 GTO 02 144 ISG 04 145 "" (NOP) 146 4 147 STO 07 148 7 149 STO 06 150 GTO "HP" 151 LBL 02 152 ISG 07 153 "" (NOP) 154 9 155 STO 06 156 GTO "HP" 157 LBL 01 158 X<>Y 159 9 160 XY? not equal 161 GTO 11 162 RCL 07 163 4 164 X=Y? 165 GTO 01 166 X<>Y 167 5 168 X=Y? 169 GTO 02 170 ISG 04 171 "" (NOP) 172 4 173 STO 07 174 7 175 STO 06 176 GTO "HP" 177 LBL 02 178 ISG 07 179 "" (NOP) 180 11 181 STO 06 182 GTO "HP" 183 LBL 01 184 ISG 07 185 "" (NOP) 186 9 187 STO 06 188 GTO "HP" 189 LBL 11 190 RCL 07 191 4 192 X=Y? 193 GTO 01 194 X<>Y 195 5 196 X=Y? 197 GTO 02 198 TONE 0 199 TONE 3 200 TONE 5 201 TONE 7 202 TONE 9 203 "GRATULATION" 204 AVIEW 205 PSE 206 PSE 207 GTO "ENDE" 208 LBL 02 209 ISG 07 210 "" (NOP) 211 11 212 STO 06 213 GTO "HP" 214 LBL 01 215 ISG 07 216 "" (NOP) 217 9 218 STO 06 219 GTO "HP" 220 END
Programmlisting SPIEL9 (Hauptprogramm) :
01 LBL "SPIEL9" 02 "MATRIX GAME" 03 AVIEW 04 CLRG 05 CF 05 06 "DATA" 07 28 08 CRFLD 09 E 10 SEEKPT 11 11 12 SAVEX 13 X<>Y 14 STO 05 15 ISG X 16 "" (NOP) 17 STO 04 18 STO 07 19 ISG X 20 "" (NOP) 21 STO 06 22 FIX 0 23 CF 29 24 26 25 SEEKPT 26 TIME 27 E2 28 / 29 SAVEX 30 LBL "HP" 31 2,025 32 STO 10 33 LBL 14 34 RCL 10 35 SEEKPT 36 , 37 SAVEX 38 ISG 10 39 GTO 14 40 27 41 SEEKPT 42 GETX 43 ISG X 44 "" (NOP) 45 X<>Y 46 SEEKPT 47 X<>Y 48 SAVEX 49 SIGN 50 STO 08 51 RCL 05 52 CLA 53 " " (1 space) 54 ARCL X 55 ". MATRIX :" 56 AVIEW 57 RCL 04 58 X^2 59 LASTX 60 + 61 E1 62 + 63 E3 64 / 65 11 66 + 67 STO 09 68 RCL 04 69 E3 70 / 71 ISG X 72 STO 10 73 LBL 03 74 RCL 08 75 " " (1 space) 76 ARCL X 77 "
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Kai Schröder, 6.6.2001
