Quote:
BUT these books imply in ALL cases the caller must play much tighter than the shover even when they both have equal stacks.
Are the books utilizing the Gap Concept and Fold equity to require the caller to play much tighter than the shover when the two all-in participants have equal stacks? (but this has nothing to do with ICM)
The exact same question for short-stack 'GTO Equilibrium Play'.
I quess I'm missing something very fundamental. Can anyone clarify this? Thanks.
Long time since I've thought about poker so hopefully this makes sense
Consider this simple game:
Two players get dealt a real value between 0 and 1.
Player 1 shoves all-in with Q% of his range.
Player 2 can now either fold (and
win/lose nothing) or call, in which case he
wins $1 if his value is greater than player 1's, else he
loses $1 if his value is less than player 1's.
What range, P%, should player 2 call with?
With the help of wolfram alpha, here's the solution:
Player 1's range: "
UniformDistribution[{1-p, 1}]".
Player 2's range: "
UniformDistribution[{1-q, 1}]".
PDF[UniformDistribution[{1-p, 1}], x] = 1/p
CDF[UniformDistribution[{1-q, 1}], x] = (q+x-1)/q
P(player 1 wins) = Integrate[PDF[UniformDistribution[{1-p, 1}], x] * CDF[UniformDistribution[{1-q, 1}], x], {x, 1-p, 1}] =
Integrate[1/p * (q+x-1)/q, {x, 1-p, 1}] = 1-p/(2*q)
EV = p * ( P(player 1 wins)*
(+1) + (1-P(player 1 wins))*
(-1) ) + (1-p) *
(0) = p*(P(player 1 wins)*1 - (1-P(player 1 wins)) = p*((1-p/(2 q))-(1-(1-p/(2 q)))) =
p*(1-p/q)
So now to find the p that maximizes EV:
d/dp p*(1-p/q) = 1-(2*p)/q
1-(2*p)/q=0 ---> p=q/2
So player 2 maximizes his EV when he calls with exactly half the range player 1 shoves with - this is Sklansky's "Gap Concept".
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If you want then you can quite easily change the EV formula above to account for blinds or ICM equities.
If you don't get how this works then try redoing the steps with player 1 shoving 100% of his range... You should see then that the EV formula is a balancing act between how much you fold and how good a hand you have when you call; with the two extremes being folding every hand and calling every hand (both of which are obviously 0EV)... It turns out that your EV is maximized when 50% of the time you fold and 50% of the time you call (and win 75% of the times you call). You can see it's 75% of the time without any need for the PDF/CDF integral too: 50% of the time player 1 will have a worse hand than player 2 (and lose 100% of the time), and the other 50% of the time it will be a coinflip (and player 2 will lose 50% of the time).
Juk
Last edited by jukofyork; 07-03-2017 at 08:44 PM.