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Archive Oct 2008: LearnedfromTV
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Game Theory Article
by LearnedfromTV on 10/25/08
This is a rough draft of an article I'm working on, the first in a handful of game theory articles. Some of them will deal with naked ace theory in PLO, and the scenario this one covers is similar. Important Disclaimer; the work here and in future articles is my own, but I should reference Chen and Ankenmann's Mathematics of Poker, which is the bible of poker game theory and covers, in some form, anything I'll write about.
This article will cover a simple contrived multi-street nuts-or-nothing betting scenario that can be solved using game theory.
The rules are simple. Pot-limit structure where you must bet pot. The game: two opponents Player A and Player B each hold a single spade. Pot contains 1 unit. Player A has and is known by B to have, the Ten of Spades. Player B's spade is unknown.
One street version
Since B has perfect information about A, the game begins with a checking to B, who then bets pot or checks behind.
So B has a 4/12 chance of beating the T, and an 8/12 chance of being beaten; if there were no betting, B's range has 33% showdown equity, A's has 66%.
In order to find the equilibrium solution for the play of this game, we must find the frequencies A can choose that will make B indifferent between his two options, and vice versa.
That is, B needs to bluff with a frequency that makes A indifferent to calling; and A needs to call with a frequency that makes B indifferent to bluffing.
We know B bets every time he has J,Q,K,A. To bet only those and check all losers would be expoitable; A would never call, and would claim the pot the 66% of the time B checks behind.
If we let B bluff x of the remaining 8 times, we find that if A calls he wins 2 units x/(x+4) of the time, and loses 1 unit 4/(x+4). So his total profit is determined by (2x-4)/(x+4), which equals 0 when x = 2
This makes sense. Since he's laying 2-1, he should have a vbet:bluff ratio of 2:1.
What is A's optimal response?
His goal is to make B indifferent to bluffing. The pot size is 1 unit and B's bluff costs 1 unit, so A makes him be indifferent to bluffing by calling 50% of A's bets.
What is the overall equilibrium strategy:
A bets all his winners and 1/4th his losers (say 9 and 8, it doesn't matter), and A calls half the bets.
12 cases:
1/2 the time B checks; A wins 1 unit, B 0 units
1/12 of the time, B bluffs and A calls; A wins 2 units, B loses 1
1/4 of the time, B bets and A folds; A wins 0 unit, B wins 1 units
1/6 of the time, B value bets and A calls; A loses 1 unit, B wins 2 units
So A wins 1/2 + 1/6 + 0 - 1/6 = 0.5 units
And B wins 0 - 1/12 + 1/4 + 1/3 = 0.5 units
A and B both have 50% equity. B gains 16.66% equity from the street of betting + the information advantage
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Two street version:
For simplicitly, call the first street the "turn" and the second street the "river".
Let's assume (for now)that B will always value bet the first street (ignoring that it is possible that the most profitable strategy would include checking the turn with the nuts some % of the time, in order to be more likely to gain a river bet)
Because B has perfect information, A's options are to check-fold the turn, check-call then check-fold, or check-call twice. B will always value bet his winners on both streets. When he has a loser he can check twice, bluff once and then check, or bluff twice (He can't check and then bluff, because he doesn't have a river valuebetting range after checking)
B's optimal bluff:value bet frequency on the first street is the strategy that will make A indifferent to calling or folding, given that the result of calling is being placed in the river situation where A will often face another bet.
If A check-folds he nets 0 units; if he check-calls he will be placed in the river situation described above. The pot will now be 3 units, but the exact same equilibrium strategy determined above will hold, provided that B has enough bluff hands in his range; that is, provided he bluffs at least 2 hands on the
turn. B will bet all his winners and 2 of his losers (2:1 value bet:bluff ratio), and A will call half the time.
B will bluff the turn with some number x of his 8 bluffing hands. If called, he will then check behind x-2 of those hands. So A's river outcomes are
(x-2)/(4+x) he wins 2 units (the 1 unit in the pot before the turn, and the unit B bet on the turn)
1/(4+x) he wins 5 units (the 1 unit in the pot before the turn, the unit B bet on the turn, and the 3 units B bets on the river)
2/(4+x) times he loses 4 units (the 1 unit he calls on the turn and the three units he calls on the river)
3/(4+x) he loses 1 unit (the 1 unit he called on the turn before folding the river)
So his total outcome is [2(x-2)+5-8-3]/(x+4). For him to be indifferent between calling and folding to the turn bet, we need to set this equal to 0 (the outcome if he folds the turn) and solve for x.
2x - 4 + 5 - 8 - 3 ----> 2x - 10 = 0 -----> x = 5
So B's strategy is to bluff 5 of his 8 losers on the turn, along with betting the 4 winners, and check behind the other 3.
A's goal is to make B indifferent to bluffing or checking the turn. If we let A's turn calling frequency = y, B's outcome for bluffing the turn is to win 1 unit (1-y)% (when A folds) and to enter the river game y% (when A calls).
In the river game, A calls half the time.
If B bluffs the turn and A folds, B gains 1 unit. If B bluffs the turn and A calls, the outcome is dependent on the river action. When he checks behind he loses 1 unit, when he bluffs and gets called (half the time) he loses 4 units, and when he bluffs and succeeds (half the time) he gains two units.
So if A calls turn y% of the time, the net outcome of B's bluff is
1-y +1 units
3y/5 -1 units
y/5 -4 units
y/5 +2 units
(1-y) - 3y/5 - 4y/5 + 2y/5 = (1-y) - y = 1 - 2y; y=1/2.
So, add it all up and we get the following outcomes:
1/4 of the time B checks turn; A +1 unit, B 0 units
3/8 of the time B bets and A folds the turn, A 0 units, B +1 unit
1/8 of the time A calls the turn, B checks the river, A +2 units, B -1 unit
1/12 of the time A calls twice and loses; A -4 units, B +5 units
1/24 of the time A calls twice and wins; A +5 units, B -4 units
1/8 of the time A calls the turn and folds the river; A -1 units, B +2 units
So A's total win is 1/4 + 0 + 1/4 - 1/3 + 5/24 - 1/8 = 0.25
B's total win is 0 + 3/8 - 1/8 + 5/12 - 1/6 + 1/4 = 0.75
In other words, A's share of the pot goes from 66% with no betting to 50% with one street to 25% with 2 streets.
I'll leave the 3-street version for another time.
This article will cover a simple contrived multi-street nuts-or-nothing betting scenario that can be solved using game theory.
The rules are simple. Pot-limit structure where you must bet pot. The game: two opponents Player A and Player B each hold a single spade. Pot contains 1 unit. Player A has and is known by B to have, the Ten of Spades. Player B's spade is unknown.
One street version
Since B has perfect information about A, the game begins with a checking to B, who then bets pot or checks behind.
So B has a 4/12 chance of beating the T, and an 8/12 chance of being beaten; if there were no betting, B's range has 33% showdown equity, A's has 66%.
In order to find the equilibrium solution for the play of this game, we must find the frequencies A can choose that will make B indifferent between his two options, and vice versa.
That is, B needs to bluff with a frequency that makes A indifferent to calling; and A needs to call with a frequency that makes B indifferent to bluffing.
We know B bets every time he has J,Q,K,A. To bet only those and check all losers would be expoitable; A would never call, and would claim the pot the 66% of the time B checks behind.
If we let B bluff x of the remaining 8 times, we find that if A calls he wins 2 units x/(x+4) of the time, and loses 1 unit 4/(x+4). So his total profit is determined by (2x-4)/(x+4), which equals 0 when x = 2
This makes sense. Since he's laying 2-1, he should have a vbet:bluff ratio of 2:1.
What is A's optimal response?
His goal is to make B indifferent to bluffing. The pot size is 1 unit and B's bluff costs 1 unit, so A makes him be indifferent to bluffing by calling 50% of A's bets.
What is the overall equilibrium strategy:
A bets all his winners and 1/4th his losers (say 9 and 8, it doesn't matter), and A calls half the bets.
12 cases:
1/2 the time B checks; A wins 1 unit, B 0 units
1/12 of the time, B bluffs and A calls; A wins 2 units, B loses 1
1/4 of the time, B bets and A folds; A wins 0 unit, B wins 1 units
1/6 of the time, B value bets and A calls; A loses 1 unit, B wins 2 units
So A wins 1/2 + 1/6 + 0 - 1/6 = 0.5 units
And B wins 0 - 1/12 + 1/4 + 1/3 = 0.5 units
A and B both have 50% equity. B gains 16.66% equity from the street of betting + the information advantage
-------------------------------------------------------------------------------
Two street version:
For simplicitly, call the first street the "turn" and the second street the "river".
Let's assume (for now)that B will always value bet the first street (ignoring that it is possible that the most profitable strategy would include checking the turn with the nuts some % of the time, in order to be more likely to gain a river bet)
Because B has perfect information, A's options are to check-fold the turn, check-call then check-fold, or check-call twice. B will always value bet his winners on both streets. When he has a loser he can check twice, bluff once and then check, or bluff twice (He can't check and then bluff, because he doesn't have a river valuebetting range after checking)
B's optimal bluff:value bet frequency on the first street is the strategy that will make A indifferent to calling or folding, given that the result of calling is being placed in the river situation where A will often face another bet.
If A check-folds he nets 0 units; if he check-calls he will be placed in the river situation described above. The pot will now be 3 units, but the exact same equilibrium strategy determined above will hold, provided that B has enough bluff hands in his range; that is, provided he bluffs at least 2 hands on the
turn. B will bet all his winners and 2 of his losers (2:1 value bet:bluff ratio), and A will call half the time.
B will bluff the turn with some number x of his 8 bluffing hands. If called, he will then check behind x-2 of those hands. So A's river outcomes are
(x-2)/(4+x) he wins 2 units (the 1 unit in the pot before the turn, and the unit B bet on the turn)
1/(4+x) he wins 5 units (the 1 unit in the pot before the turn, the unit B bet on the turn, and the 3 units B bets on the river)
2/(4+x) times he loses 4 units (the 1 unit he calls on the turn and the three units he calls on the river)
3/(4+x) he loses 1 unit (the 1 unit he called on the turn before folding the river)
So his total outcome is [2(x-2)+5-8-3]/(x+4). For him to be indifferent between calling and folding to the turn bet, we need to set this equal to 0 (the outcome if he folds the turn) and solve for x.
2x - 4 + 5 - 8 - 3 ----> 2x - 10 = 0 -----> x = 5
So B's strategy is to bluff 5 of his 8 losers on the turn, along with betting the 4 winners, and check behind the other 3.
A's goal is to make B indifferent to bluffing or checking the turn. If we let A's turn calling frequency = y, B's outcome for bluffing the turn is to win 1 unit (1-y)% (when A folds) and to enter the river game y% (when A calls).
In the river game, A calls half the time.
If B bluffs the turn and A folds, B gains 1 unit. If B bluffs the turn and A calls, the outcome is dependent on the river action. When he checks behind he loses 1 unit, when he bluffs and gets called (half the time) he loses 4 units, and when he bluffs and succeeds (half the time) he gains two units.
So if A calls turn y% of the time, the net outcome of B's bluff is
1-y +1 units
3y/5 -1 units
y/5 -4 units
y/5 +2 units
(1-y) - 3y/5 - 4y/5 + 2y/5 = (1-y) - y = 1 - 2y; y=1/2.
So, add it all up and we get the following outcomes:
1/4 of the time B checks turn; A +1 unit, B 0 units
3/8 of the time B bets and A folds the turn, A 0 units, B +1 unit
1/8 of the time A calls the turn, B checks the river, A +2 units, B -1 unit
1/12 of the time A calls twice and loses; A -4 units, B +5 units
1/24 of the time A calls twice and wins; A +5 units, B -4 units
1/8 of the time A calls the turn and folds the river; A -1 units, B +2 units
So A's total win is 1/4 + 0 + 1/4 - 1/3 + 5/24 - 1/8 = 0.25
B's total win is 0 + 3/8 - 1/8 + 5/12 - 1/6 + 1/4 = 0.75
In other words, A's share of the pot goes from 66% with no betting to 50% with one street to 25% with 2 streets.
I'll leave the 3-street version for another time.
Not enough hours in the day
by LearnedfromTV on 10/03/08
I'm in the midst of a organzing a bunch of projects, and finding myself with only 24 hours in a day, so I decided to write a post in the hope that it will help me prioritize.
Job 1 right now, and until the end of the year is Supernova Elite. I have 607k vpp as of this moment, having done ~115k each of the past two months, which is solid but not enough to get me there if I repeat it the next few months. 4k/day is relatively easy, consistely doing 4500+ is a lot harder; and I need somewhere around 4500/day, depending on how many days off for holidays, to get it. 9 tabling plo gets 550/hour, 12 tabling around 700, and I've recently taken up 18ish tabling the 114 sit and goes, which gets ~1k/hour. When the PCA Steps kick in full speed that will give a boost which will be much needed in December. Right now I give myself about a 70% chance of getting elite; if I get close enough I'll play sick marathons if necessary to close it, but too many bad days (losing sessions early in day often leads to 3k vpp or worse days; family stuff left me getting 1k a couple days ago). Too much of that happens and ~mid November I'll decide it's hopeless. There's only 80 something days left, and I'll basically have to play ~8 hours a day to succeed.
Two priorities dueling for number 2 are the coaching videos I've been making for pokersavvy and the plo theory coaching program I'm putting together. There's a decent amount of work involved with most of the videos I want to do; selecting hands that fit a theme, incorporating powerpoint, and so on. The shell/outline of the coaching program is in place (viewable in the pokersavvy coaching forum; maybe I'll post it here too), but some of the supplemental work is still in progress. With new students on board in the next week or two, I need to get some more work done on a few projects, particularly in the short term the starting hands sheet and the flop textures sheet, which are mostly ready but appear early in the program.
There also are several articles I want to write, either to be posted here or at Pokersavvy. I've done a bunch of game theory work for plo, mainly focusing on the naked ace play but some other stuff as well, and I will turn that into a series of articles at some point. I also want to write articles on a handful of other topics, such as AAxx play, 3betting, and board texture. And way down the list are artcilaes on key horse concepts.
In the back of my mind is the possibility that all of this work will end up in a theory book I want to write next year. Students who do the full theory program are basically going to get a book's worth of supplementary content, and if the situation is right I'd clean that up and actually write the book. There's a weird market force at work here because the full twenty lesson program is definitely worth a few thousand to dedicated students who can use it to step into midstakes games and win ($3500 at that moment, likely $5k jan 1 or therabouts), but of course packaging that material in a book makes it worth $25 or whatever to the mass market. So I'm not sure yet what I want to do. There's another book possibility out there as well, but nothing concrete or that I can really consider until the year's over.
I'm also doing a handful of things regularly outside poker; working out, playing in tennis and basketball leagues, plus that whole social life thing. So it all adds up to probably not doing everything I'd like between now and the end of the year. Regardless, decent chance I'll be posting more over here in the near future. I'm not sure what might be done at pokersavvy with my articles, and I'll probably be debuting them here, at least the next couple, which I hope to write in the first half of October.
Job 1 right now, and until the end of the year is Supernova Elite. I have 607k vpp as of this moment, having done ~115k each of the past two months, which is solid but not enough to get me there if I repeat it the next few months. 4k/day is relatively easy, consistely doing 4500+ is a lot harder; and I need somewhere around 4500/day, depending on how many days off for holidays, to get it. 9 tabling plo gets 550/hour, 12 tabling around 700, and I've recently taken up 18ish tabling the 114 sit and goes, which gets ~1k/hour. When the PCA Steps kick in full speed that will give a boost which will be much needed in December. Right now I give myself about a 70% chance of getting elite; if I get close enough I'll play sick marathons if necessary to close it, but too many bad days (losing sessions early in day often leads to 3k vpp or worse days; family stuff left me getting 1k a couple days ago). Too much of that happens and ~mid November I'll decide it's hopeless. There's only 80 something days left, and I'll basically have to play ~8 hours a day to succeed.
Two priorities dueling for number 2 are the coaching videos I've been making for pokersavvy and the plo theory coaching program I'm putting together. There's a decent amount of work involved with most of the videos I want to do; selecting hands that fit a theme, incorporating powerpoint, and so on. The shell/outline of the coaching program is in place (viewable in the pokersavvy coaching forum; maybe I'll post it here too), but some of the supplemental work is still in progress. With new students on board in the next week or two, I need to get some more work done on a few projects, particularly in the short term the starting hands sheet and the flop textures sheet, which are mostly ready but appear early in the program.
There also are several articles I want to write, either to be posted here or at Pokersavvy. I've done a bunch of game theory work for plo, mainly focusing on the naked ace play but some other stuff as well, and I will turn that into a series of articles at some point. I also want to write articles on a handful of other topics, such as AAxx play, 3betting, and board texture. And way down the list are artcilaes on key horse concepts.
In the back of my mind is the possibility that all of this work will end up in a theory book I want to write next year. Students who do the full theory program are basically going to get a book's worth of supplementary content, and if the situation is right I'd clean that up and actually write the book. There's a weird market force at work here because the full twenty lesson program is definitely worth a few thousand to dedicated students who can use it to step into midstakes games and win ($3500 at that moment, likely $5k jan 1 or therabouts), but of course packaging that material in a book makes it worth $25 or whatever to the mass market. So I'm not sure yet what I want to do. There's another book possibility out there as well, but nothing concrete or that I can really consider until the year's over.
I'm also doing a handful of things regularly outside poker; working out, playing in tennis and basketball leagues, plus that whole social life thing. So it all adds up to probably not doing everything I'd like between now and the end of the year. Regardless, decent chance I'll be posting more over here in the near future. I'm not sure what might be done at pokersavvy with my articles, and I'll probably be debuting them here, at least the next couple, which I hope to write in the first half of October.
