[7stud]

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Suppose this is the situation in a three handed $20-40 stud game...antes: $9

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What kind of 20/40 game has only 9+5=$14 in antes??

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$3 ante/player X 3 players = $9. The $5 bringin is shown in the diagrams, and it is included in the 3rd street total along with the other bets.

I probably should have assumed 8 players with a $24 ante, which would add $15 to the total pot, so the calculations would be:

Based on the betting assumptions I made, these are p2's choices:

1) call and play the hand three way with a .31 chance of winning $449 + $15:

........Expected value = .31 * $464 = $144

2) or reraise and play the hand heads up with a .42 chance of winning $374 + $15:

........Expected value = .42 * $389 = $163

So, p2's $20 bet increases his profitability $19.

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I think raising to limit the field (which I think is an overated and often misapplied concept) balences out in a games with a reasonable ante

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Since there is always some chance p1 and his King, does not have a pair of Kings, one could argue raising is either slightly wrong or very right. For instance, if p1 had a King high straight flush draw, which would also be a raising hand:

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7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
Ks Qs 9s 159206 31.84 340782 68.16 12 0.00 0.318
Ts Ad Th 159182 31.84 340804 68.16 14 0.00 0.318
Jd 4d Jh 181596 36.32 318398 63.68 6 0.00 0.363

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and the p2's reraise succeeded in knocking out p3:

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7-card Stud Hi: 500000 sampled outcomes
cards win %win lose %lose tie %tie EV
Ks Qs 9s 209593 41.92 290382 58.08 25 0.01 0.419
Ts Ad Th 290382 58.08 209593 41.92 25 0.01 0.581

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p2's chances of winning the hand would vault from 32% to 58%, and the following increase in Ev:

1) call and play the hand three way with a .32 chance of winning $464:

........Expected value = .32 * $464 = $148.50

2) or reraise and play the hand heads up with a .58 chance of winning $389:

........Expected value = .58 * $389 = $225.50

In that case, p2's $20 bet increases his profitability $77--although I think the assumption about the total pot size would have to be lowered, so that is high.