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Heads Up NL Discussion of heads up NL Texas Hold'em cash games

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Old 02-03-2018, 12:57 PM   #1
bitzu
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GTO freq decreasing EV

Given the definition of Equilibrium that no player can unilaterally change their strategy to increase their EV, how come that a suboptimal betting frequency will increase the EV of the bettor?

Let's assume the pot is 100 and Villain's range is 2 combos of AA and 2 combos of QQ and we have KK only. He value bets pot size with a GTO range, meaning he bets both combos of AA and 1 QQ combo. We call 50% (GTO frequency). In this case, the EV for both Hero and Villain is 0 (Both will win half of the pot). However, if Villain decides to bet all his combos, our 50% calling frequency will result in a -0.25 EV.

Villain EV when villain bets all combos:
1/2*100 + 1/2*(1/2*200 - 1/2*100) = 125
Villain EV when villain bets GTO frequency:
1/2*100 + 1/2*(2/3*200 - 1/3*100) = 100

So where does Hero get back the EV lost in this spot? Am I missing something?
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Old 02-06-2018, 03:28 AM   #2
DelusionalFish
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Re: GTO freq decreasing EV

You should try the Poker Theory forum.
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Old 02-06-2018, 10:37 AM   #3
fityfmi
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Re: GTO freq decreasing EV

Quote:
Originally Posted by bitzu View Post
Let's assume the pot is 100 and Villain's range is 2 combos of AA and 2 combos of QQ and we have KK only. He value bets pot size with a GTO range, meaning he bets both combos of AA and 1 QQ combo. We call 50% (GTO frequency). In this case, the EV for both Hero and Villain is 0 (Both will win half of the pot).
The guy with 2 combos of AA and 2 combos of QQ wins 3/4 of the pot, and the guy with KK wins 1/4. You correctly identified that the GTO frequenzies will be to bet AA every time, to bet QQ half the time, and to call KK half the time.

EV{AA,QQ}:
(0.75)(0.5)(1) + (0.75)(0.5)(2/3)(2) + (0.75)(0.5)(1/3)(-1) + (0.25)(0) = 0.75

EV{KK}:
(0.75)(0.5)(0) + (0.75)(0.5)(2/3)(-1) + (0.75)(0.5)(1/3)(2) + (0.25)(1) = 0.25

Formula explained:
first term: (bet %)(fold %)(EV)
second term: (bet %)(call %)(value %)(EV)
third term: (bet %)(call %)(bluff %)(EV)
forth term: (check %)(EV)

If the {AA, QQ} range decides to bet every time, the EV will still be 0.75, as long as the {KK} range keeps on only calling 50%. You don't lose EV by having wrong frequenzies with indifferent combos (QQ,KK), as long as the opponent playes GTO. In this spot, the only way to lose EV vs. a GTO opponent would be to fail to bet AA 100%.
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Old 02-15-2018, 03:25 PM   #4
just_grindin
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Re: GTO freq decreasing EV

Quote:
Originally Posted by bitzu View Post
Given the definition of Equilibrium that no player can unilaterally change their strategy to increase their EV, how come that a suboptimal betting frequency will increase the EV of the bettor?

Let's assume the pot is 100 and Villain's range is 2 combos of AA and 2 combos of QQ and we have KK only. He value bets pot size with a GTO range, meaning he bets both combos of AA and 1 QQ combo. We call 50% (GTO frequency). In this case, the EV for both Hero and Villain is 0 (Both will win half of the pot). However, if Villain decides to bet all his combos, our 50% calling frequency will result in a -0.25 EV.

Villain EV when villain bets all combos:
1/2*100 + 1/2*(1/2*200 - 1/2*100) = 125
Villain EV when villain bets GTO frequency:
1/2*100 + 1/2*(2/3*200 - 1/3*100) = 100

So where does Hero get back the EV lost in this spot? Am I missing something?
Yes you're missing the part where players are allowed to change their strategies so if 1 player deviates the other can deviate as well.

When the bettor only bets AA, the caller can fold all the time. The bettor can then improve EV by betting QQ. Once the bettor bets QQ at the equilibrium frequency, the caller can can begin calling with KK with the equilibrium calling frequency. Once the bettor bets QQ past the equilibrium frequency the caller can always call.

It's only if 1 strategy stays fixed at the equilibrium frequency and the other is allowed to change do you get odd questions like "why does the caller only call at 50% when he could call at 100% and still be 0 EV" or "why doesn't the bettor bet 0% QQ".
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