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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