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Exploiting Short-Stacks who Steal a Lot

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loonychune

02-19-2009, 08:43 PM Post subject: Exploiting Short-Stacks who Steal a Lot

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Join Date: Jan 2009

Posts: 145

I'm going to discuss our expectation vs a short-stack's pre-flop raising and folding to 3-bet ranges in late position. It so transpires that the best way to exploit an aggro short-stack is to shove with a lot of air. What follows doesn't take into account the fact that a short-stack who plays well is likely to adjust his opening range and calling 3bet range when you play back. Also, i'm only speculating as to a short-stacks opening and folding to 3bet ranges: bare in mind this is largely a mathematical exercise.

Let's say we have plenty of information on our short-stack. We deduce that his late position opening range is 17.2% of hands, something that might look like:

22+,A7s+,K9s+,QTs+,JTs,ATo+,KTo+,QJo

We're also pretty sure that he folds all of these hands to a 3-bet shove except these 4.2% of hands:

TT+,AQs+,AKo and AQo 50% of the time.

This means he calls with 24.4% of the hands he opens, i.e. probability he calls, p(call) = 0.244; probability he folds, p(fold) = 1 - p(call) = 0.756

The game is $1/$2. Pre-flop: short-stack is $40 deep. He open raises for $6 in the CO and the Button and SB fold. You are in the BB and must decide how to act. The pot is $9.

The equity of a random hand versus his calling range is:

Code:

 117,464,054,400  games   142.802 secs   822,565,891  games/sec

	equity 	win 	tie 	      pots won 	pots tied	
Hand 0: 	73.276%  	72.71% 	00.57% 	   85408646638 	664832731.00   { TT+, AQs+, AKo, AQo half the time }
Hand 1: 	26.724%  	26.16% 	00.57% 	   30725742300 	664832731.00   { random }

i.e. 26.16% - the probability that our random hand wins is 0.2616.

Our expectation for shoving then is:

<FTR> = {expectation when he folds} + {expectation when he calls}
<FTR> = p(he folds)*(pot) + p(call)*[p(we win)*(size of pot when he calls shove) - cost of shoving]
<FTR> = 0.756*($9) + 0.244*[0.2616*($81) - $38]
<FTR> = $6.80 + (-$4.10)
<FTR> = +$2.70

What about if EVERY TIME WE DO THIS we are (by some miracle) holding 72o. Against his range we now have just 21.07% equity, i.e. p(we win) = 0.2107.

<FTR> = 0.756*($9) + 0.244*[0.2107*($81) - $38]
<FTR> = $6.80 + (-$5.11)
<FTR> = +$1.69

Let's say that he instead calls our shove with 9.2% of hands, i.e. with 53% of the hands he raises with (p(call) = 0.53, p(fold) = 0.47). Against the range,

22+,ATs+,KQs,AQo+

a random hand has 32.6% equity; p(win) = 0.326.

<FTR> = 0.47*($9) + 0.53*[0.326*($81) - $38]
<FTR> = $4.23 + (-$6.14)
<FTR> = -$1.91

In fact, to make this play against this range, we need to find a range of hands that whose probability is greater than or equal to p(win) where,

<FTR> = 0.47*($9) + 0.53*[p(win)*($81) - $38] >= 0

i.e. p(win) >= 0.37 or 37%

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