Friday, March 2, 2012

The Quiz Question That Won't Die (Q #48)

Okay, once more into the breach. Here's the math that shows shoving all-in with AKo preflop against the two TAgs in Quiz Question #48 is negative EV.

First, let's look at TAg #1, who was the original opener in MP:
  • TAg #1 Original Raising Range= 15.20% (22+,ATs+,KTs+,QTs+,JTs,ATo+,KJo+,Qjo) 
  • TAg #1's 4bet calling range= 0.90% (KK+) 
  • Probability that TAg #1's will call our shove= 5.9%  (this is just 4bet calling range divided by original opening range)
  • Therefore, probability that TAg #1 will fold to all-in shove= 94.1% (this is just 100% minus the probability that we get called)
Similarly, we can then make some assumptions and calculations about TAg #2:
  • TAg #2 Original ReRaising Range= 5.00% (99+,AQs+,Ako)
  • TAg #2's 4bet calling range= 0.90% (KK+) 
  • Therefore, probability that TAg #2's will call our shove= 18.0%
  • Therefore, probability that TAg #2 will fold to all-in shove= 82.0%
All of this means:
  • Probability that TAg #1 and TAg #2 will both fold = 77.1% (To calculate this, just multiply the two probabilities that the TAgs will fold together; in this case: 82% x 94.1%)
  • Therefore, probability that we get called by at least one TAg =  22.9% (Again, just 100% minus the number in the previous line)
And we know that:
  • Amount we're risking when shoving = $1,500
  • Amount we win if everyone folds = $195 (dead money in the pot right now)
  • Average amount we could win if our shove gets called by one player = $1,590 (this is the calling amount plus the average dead money left in pot)
Next, per pokerstove:
  • Equity of our AK against calling range of one opponent = 18.5%
Finally, we apply the basic EV calculation:
  • 77.1% of the time everyone folds and we win $195
  • 22.9% of the time we get called by at least one player:
    • 18.5% of the time when we get called by one player our AK wins and we win $1590
    • 81.5% of the time when we get called by one player, our AK is beat and we lose $1500
Therefore:
  • Our jamming expected value = negative $61.74
  • If both opponents call, our EV is even less than this amount
  • If I loosen up both of the opponents' calling ranges further to, say, AK+ and QQ+, our EV becomes even more negative (-$103.69) 
  • Jamming all-in is therefore a mathematically bad play
  • QED
All-in for now...
-Bug

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