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# A. Pre-Flop Income Rates

These are the computed income rates (* 1000) used for all 169 distinct hand types (13 paired, suited and unsuited). Each table is labeled IRx where x is the number of players (hands dealt) in the simulation (so there are x-1 opponents). Each entry is indexed IRx[row][col] and the cards are suited when row>col. This means that for 2 players the income rate for a 3 and 2 of the same suit is IR2[3][2] = -279, and for a 3 and 2 of different suits is IR2[2][3] = -351.

In all simulations a pair of aces had the highest income rate (a gain of 2.043 with 7 players, meaning an investment of \$1 would return a profit of \$2.043, on average). In the 7-player simulation, a 2 and 7 of different suits had the lowest income rate (a loss of \$0.495 for every \$1 invested) and 88 of the 169 different hand types returned non-negative income.

There is a strong correlation between these rankings and the pre-flop hand rankings given in Sklansky and Malmuth [14]. They break the pre-flop hands into 9 groups, ranked by their strength (call this ranking of hands SM). If we take IR7 (most reflective of the full game they are assuming) and break it into the same number of groups with the same number of hands per group (5 for group 1, 5 for group 2, 6 for group 3, and so on) we note that all but 16 of the 169 hand types are either in the same group or are only one away. While most of the hands in the middle groups are shifted by one class, the top three groups are virtually identical. Details of the comparison by groups can be found in Table A.4.

The most interesting similarity is that the top two groups contain the same hands but there is only one different hand in the third group. In IR7, KTs (King and Ten of the same suit) replaces JTs (Jack and Ten of the same suit). In fact, there appears to be a trend favoring big cards in IR7. However, any minor discrepancy could be due to the simple-minded approach of the simulations.

Table A.1: IR2: income rates for 1 opponent
 2 3 4 5 6 7 8 9 T J Q K A 2 7 -351 -334 -314 -318 -308 -264 -217 -166 -113 -53 10 98 3 -279 74 -296 -274 -277 -267 -251 -201 -148 -93 -35 27 116 4 -263 -225 142 -236 -240 -231 -209 -185 -130 -75 -17 46 134 5 -244 -206 -169 207 -201 -189 -169 -148 -114 -55 2 68 153 6 -247 -208 -171 -138 264 -153 -134 -108 -78 -43 19 85 154 7 -236 -200 -162 -125 -91 324 -99 -72 -43 -6 37 104 176 8 -192 -182 -143 -108 -75 -43 384 -39 -4 29 72 120 197 9 -152 -134 -122 -84 -50 -17 16 440 28 65 106 155 215 T -104 -86 -69 -56 -19 12 47 81 499 102 146 195 254 J -52 -35 -19 0 11 46 79 113 149 549 161 212 271 Q 2 21 34 55 72 86 121 153 188 204 598 228 289 K 63 79 98 116 132 151 168 200 235 249 268 647 305 A 146 164 180 198 198 220 240 257 291 305 323 339 704

Table A.2: IR4: income rates for 3 opponents
 2 3 4 5 6 7 8 9 T J Q K A 2 -121 -440 -409 -382 -411 -432 -394 -357 -301 -259 -194 -116 16 3 -271 -42 -345 -312 -340 -358 -371 -328 -277 -231 -165 -87 54 4 -245 -183 52 -246 -269 -287 -300 -308 -252 -204 -135 -55 84 5 -219 -151 -91 152 -200 -211 -227 -236 -227 -169 -104 -24 118 6 -247 -177 -113 -52 256 -145 -152 -158 -152 -145 -74 9 99 7 -261 -201 -129 -65 3 376 -76 -79 -68 -66 -44 48 148 8 -226 -204 -140 -73 -2 66 503 0 15 24 45 84 194 9 -191 -166 -147 -79 -5 68 138 647 104 113 136 177 241 T -141 -116 -91 -69 -4 75 150 235 806 226 255 295 354 J -89 -67 -41 -12 7 82 163 248 349 965 301 348 410 Q -29 -3 22 51 80 108 185 274 379 423 1141 403 473 K 47 76 101 128 161 199 230 318 425 473 529 1325 541 A 175 211 237 266 249 295 338 381 491 539 594 655 1554

Table A.3: IR7: income rates for 6 opponents
 2 3 4 5 6 7 8 9 T J Q K A 2 -6 -462 -422 -397 -459 -495 -469 -433 -383 -336 -274 -188 -39 3 -180 21 -347 -304 -365 -418 -447 -414 -356 -308 -248 -163 -1 4 -148 -69 67 -227 -273 -323 -362 -391 -334 -287 -223 -133 32 5 -121 -38 31 122 -198 -230 -270 -303 -309 -259 -200 -103 64 6 -174 -95 -10 64 206 -151 -175 -204 -217 -235 -164 -72 23 7 -208 -135 -47 35 108 298 -87 -106 -112 -128 -124 -26 72 8 -184 -164 -83 2 93 168 420 -5 6 -10 -10 22 126 9 -146 -128 -111 -26 64 153 245 565 134 118 118 151 189 T -88 -68 -46 -29 59 155 268 383 765 299 305 336 373 J -38 -15 1 30 51 147 256 377 536 996 380 420 462 Q 35 49 72 99 127 162 268 384 553 628 1279 529 574 K 117 141 167 190 223 261 304 423 591 669 764 1621 712 A 269 304 333 363 313 365 416 475 644 720 815 934 2043

Table A.4: Comparison between SM and IR7
 Group Size Matches in Group Size Matches in IR7 grouping IR7 grouping 1 5 5 6 10 2 2 5 5 7 17 6 3 6 5 8 16 1 4 8 4 9 85 69 5 17 11

Next: B. Expert-Defined Values Up: Dealing with Imperfect Information Previous: Bibliography   Contents
Denis Papp
1998-11-30