Here's a copy of the figures Babe gave for her WinPoker play:
···
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Game EV # of Hands Played* # of RF Actual Payback %
8/5BP 99.16 66,164 5 100.34
(198,492)*
9/6BPD 99.64 34,016 0 98.46
(102,048)*
9/6DDB 98.98 230,004 21 99.16
(690,012)*
9/6DJPT 100.34 55,557 6 100.20
(166,671)*
10/7DB 100.17 143,778 10 100.98
(431,133)*
FPDW 100.76 111,904 5 100.33
(335,712)*
FPKBJW 100.64 290,592 30** 100.42
(871,776)*
Downtown
DW 100.92 51,920 3 101.30
(155,160)*
8/5JoB
$3200Prog 102.30 101,715 6** 101.81
(305,145)*
DWBonusP 99.45 28,467 2 99.95
(85,401)*
Total "Draws": 1,114,117
Total "Hands": 3,342,351*
Total Royal Flushes: 88
Average # of Hands played to snag a RF: 37,981
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I've just started analyzing them busy weekend but so far it looks
like an interesting case of "cross-normalization." Overall, the
average return excluding royals is only 0.04% off the expected value.
But broken down by game, it varies from +2.3% to minus 1.2%. And it's
not due to small sample size the +2.3% comes over 300K hands.
Further, the game is a JB progressive, where the next most
significant hand after the royal, the 4OAK, has a cycle of about 425
hands. So that's a +2.3% variation after 700 cycles.
Another extreme example is her Joker experience 870K hands but
still a 1% shortfall to expected return for hands below the RF.
As a reminder to what these numbers represent - they are the
accumulated results of her practice sessions on WinPoker. She played
3-play on all the games, and she always accepted proposed strategy
corrections, so they represent the results of playing perfect
strategy over 3 million hands and 1 million deals.
Iguana, can you apply that formula for expected returns you and Harry
were discussing to some of these examples? I wonder the odds are
under that method that JOB hands less than RF would be 2.3% off after
300K hands, for example?
And, is there a theoretical basis for "cross-normalization"? That is,
if you play a variety of games, can you expect your money results to
even out over the whole group? I mentioned the other day that I had
analyzed the results of a heavy player who had a major shortfall to
expectation after around one million hands. In that case, the cross-
normalization was actually negative (the sum of all games was off
more than the average individual game because of adverse patterns in
paytables and denominations), and that player has tentatively decided
to play fewer types of games to increase the chances of normalization.
Here's a table summarizing Babe's non-royal results:
Game Expected Actual diff #hands
8/5BP 97.178% 98.327% 1.150% 198,492
9/6BPD 97.740% 98.460% 0.720% 102,048
9/6DDB 97.020% 96.728% -0.292% 690,012
9/6DJPT 98.482% 97.324% -1.158% 166,671
10/7DB 98.508% 99.128% 0.620% 431,334
FPDW 98.995% 99.140% 0.145% 335,712
FPKBJW 98.705% 97.670% -1.035% 871,776
DW 99.201% 99.761% 0.560% 155,760
$3200Pr 8/5 JB 94.513% 96.778% 2.265% 305,145
DWBonusP 97.549% 98.079% 0.530% 85,401
Total/Average 97.840% 97.884% 0.043% 3,342,351