Introduction to GTO
Target of Game Theory
-to maximally exploit our opponent
(maximize our own EV and minimize EV of our opponent)
GTO- Nash Equilibrium
-when two players try to maximize their own EVs, the strategies will eventually settle in balanced scenario:
Nash-equilibrium
-> here neither player can unilaterally deviate from his strategy to increase his EV
GTO- Principle of Indifference
-where we leave no room for our opponent to exploit us
-> 2 scenarios:
1) optimal bluff frequency
2) optimal defense frequencies
Optimal Bluff Frequency
-on the river, when we have a range that consists of value hands and bluffs, and our opponent has a bluff catcher, we want to keep our value-to-bluff at the ratio where our opponent has no option to control his EV-> any option he chooses (fold/call) has the same result for him over the long-run.
* the EV for opponent when he folds is 0
How do we calculate this frequency?
-optimal bluffing frequency can be derived from the betsize and the odds we offer to our opponent
eg. hero bet $80 into $100 -> villain has the odds of 180:80 or 31%
-> if we offer more than 31% equity to our opponent (too many bluffs), he will call 100%
-> if we offer less than 31% equity, he will fold 100% (not enough bluffs)
* therefore our optimal bluff-frequency = value:bluff ratio is 69:31 where we have 31% of bluffs
bet 1/4 pot: optimal bluff frequency = 5:1 or 16% therefore value:bluff ratio = 83:16
bet 1/3 pot: optimal bluff frequency = 4:1 or 20% therefore value:bluff ratio = 80:20
bet: 1/2 pot : optimal bluff frequency = 3:1 or 25% therefore value:bluff ratio = 75:25
bet full pot : optimal bluff frequency = 2:1 or 33% therefore value:bluff ratio = 66:33
bet 2x pot: optimal bluff frequency = 1:1 or 50% therefore value:bluff ratio = 50:50
Optimal Defense Frequency
- on the river, we want to call exactly as often that our opponent has a break-even bet with his bluffs and value, so that his EV is 0
*the EV for our opponent when he gives up a bluff is 0
How do we calculate this frequency?
-optimal calling frequency is derived from the betsize and the odds our opponent is getting from his bet
eg. villain bet $80 into $100, he offers himself 100:80 or 80/180 which is 44%
-> if we fold more than 44%, our opponent will bluff 100%
->if we fold less than 44%, he stops bluffing because his bluffs are -EV
* therefore our optimal defense frequency is 1-(odds villain is getting on his bluff)
= 100% -44% = 56%
*if we know our opponent bluffs a ton of folds a lot, we can use this information to exploit them:
-vs station: we bet bigger with our value hands and bet thinner for value
-vs opponent who folds a lot: we widen our bluffing range.
Scenario
villain's defense frequency = 58%
villain's range consists of: 99(3) 55(3) QQ(1) AQs(3) JTs(4) AJss(1) ATss(1) AQo(9)
-his calling range: 0.58*25=14.5 combos therefore
* villain should call all his sets [99,55,QQ] and of 7.5 AQ combos (preferably not blocking the NFD)
* our best line:
Calculate EV for shoving
1) Calculate equity of our AA vs his call range: 25%
2) Calculate EV:
EV(shove) = P(Vfolds)*(Money in the pot) + P(Vcalls)*((Our EQ*money if we win)-(our bluff shove if villain calls and we lose))
EV(shove)= (0.42*420)+(0.58*(0.25*1040)-320)= +$141.6
Calculate EV for check/call
1) What is villain's betting range when checked to?
99,55,QQ,JdTd,JhTh,JcTc (10 combos out of 19=53%)
2) Calculate Hero's equity against betting range: 7:3 so 30%
3) Calculate Hero's EV
EV(x/c) = P(Vchecks*Money won+ P(Vbets)*((our equity*money won if we win)-money lost if we call and loses)
= (0.47*$420)+(0.53*((0.3*1040)-310) = +$198.46
*therefore we should check/call always
-check/fold is out of the option as the EV of that is 0
-to maximally exploit our opponent
(maximize our own EV and minimize EV of our opponent)
GTO- Nash Equilibrium
-when two players try to maximize their own EVs, the strategies will eventually settle in balanced scenario:
Nash-equilibrium
-> here neither player can unilaterally deviate from his strategy to increase his EV
GTO- Principle of Indifference
-where we leave no room for our opponent to exploit us
-> 2 scenarios:
1) optimal bluff frequency
2) optimal defense frequencies
Optimal Bluff Frequency
-on the river, when we have a range that consists of value hands and bluffs, and our opponent has a bluff catcher, we want to keep our value-to-bluff at the ratio where our opponent has no option to control his EV-> any option he chooses (fold/call) has the same result for him over the long-run.
* the EV for opponent when he folds is 0
How do we calculate this frequency?
-optimal bluffing frequency can be derived from the betsize and the odds we offer to our opponent
eg. hero bet $80 into $100 -> villain has the odds of 180:80 or 31%
-> if we offer more than 31% equity to our opponent (too many bluffs), he will call 100%
-> if we offer less than 31% equity, he will fold 100% (not enough bluffs)
* therefore our optimal bluff-frequency = value:bluff ratio is 69:31 where we have 31% of bluffs
bet 1/4 pot: optimal bluff frequency = 5:1 or 16% therefore value:bluff ratio = 83:16
bet 1/3 pot: optimal bluff frequency = 4:1 or 20% therefore value:bluff ratio = 80:20
bet: 1/2 pot : optimal bluff frequency = 3:1 or 25% therefore value:bluff ratio = 75:25
bet full pot : optimal bluff frequency = 2:1 or 33% therefore value:bluff ratio = 66:33
bet 2x pot: optimal bluff frequency = 1:1 or 50% therefore value:bluff ratio = 50:50
Optimal Defense Frequency
- on the river, we want to call exactly as often that our opponent has a break-even bet with his bluffs and value, so that his EV is 0
*the EV for our opponent when he gives up a bluff is 0
How do we calculate this frequency?
-optimal calling frequency is derived from the betsize and the odds our opponent is getting from his bet
eg. villain bet $80 into $100, he offers himself 100:80 or 80/180 which is 44%
-> if we fold more than 44%, our opponent will bluff 100%
->if we fold less than 44%, he stops bluffing because his bluffs are -EV
* therefore our optimal defense frequency is 1-(odds villain is getting on his bluff)
= 100% -44% = 56%
*if we know our opponent bluffs a ton of folds a lot, we can use this information to exploit them:
-vs station: we bet bigger with our value hands and bet thinner for value
-vs opponent who folds a lot: we widen our bluffing range.
Scenario
villain's defense frequency = 58%
villain's range consists of: 99(3) 55(3) QQ(1) AQs(3) JTs(4) AJss(1) ATss(1) AQo(9)
-his calling range: 0.58*25=14.5 combos therefore
* villain should call all his sets [99,55,QQ] and of 7.5 AQ combos (preferably not blocking the NFD)
* our best line:
Calculate EV for shoving
1) Calculate equity of our AA vs his call range: 25%
2) Calculate EV:
EV(shove) = P(Vfolds)*(Money in the pot) + P(Vcalls)*((Our EQ*money if we win)-(our bluff shove if villain calls and we lose))
EV(shove)= (0.42*420)+(0.58*(0.25*1040)-320)= +$141.6
Calculate EV for check/call
1) What is villain's betting range when checked to?
99,55,QQ,JdTd,JhTh,JcTc (10 combos out of 19=53%)
2) Calculate Hero's equity against betting range: 7:3 so 30%
3) Calculate Hero's EV
EV(x/c) = P(Vchecks*Money won+ P(Vbets)*((our equity*money won if we win)-money lost if we call and loses)
= (0.47*$420)+(0.53*((0.3*1040)-310) = +$198.46
*therefore we should check/call always
-check/fold is out of the option as the EV of that is 0
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