GTO River Bet Sizing

Game theory. People don’t get it. I don’t really get it, but I probably get it more than 90%+ of poker pros. Part of becoming good at something is being humble enough to realize you suck and don’t know anything. The inspiration for this post can be found here:

http://forumserver.twoplustwo.com/56/medium-stakes-pl-nl/size-matters-1219472/,

where almost nobody gives a reason for their preferred bet size beyond “oh just bet what he calls duehh”. Let’s try to figure out the most profitable optimal betting strategy is for a similar situation, that is, a betting strategy that doesn’t let our opponent improve by deviating from his optimal strategy and that makes us the most money. If the players are bluffing and calling with optimal frequencies, the 1-period (as opposed to multiple periods) “game” of this river situation is in equilibrium and neither player can improve by changing their play.

There’s not really a need for a hand history, so I’ll just describe the hypothetical situation:

1.       There is $100 in the pot on the river.

2.       The players both have $200 remaining in their stacks.

3.       Villain is first to act and checks. Villain’s hand range is entirely made up of “bluff catchers”, or hands that beat all of hero’s bluffs and lose to all of hero’s value bets. Villain only ever calls or folds on the river.

4.       Hero gets to the river with 40 made hands that beat all of villain’s hands, and 60 missed draws that lose to all of villain’s hands. He needs to choose a bluff % and a bet amount.

Any equilibrium strategy for hero will result in villain being indifferent between calling and folding, meaning villain will call 50% of the time and fold 50% of the time. Let’s look at what that means from villain’s perspective.

The most villain ever has to win to call a bet is 50% of the time. If someone bets $100 into a $0 pot, you must have the best hand 50% of the time to call and break even. If someone bets pot, you must have the best hand 33% of the time to call and break even (.33*200 + .66*-100 = 0). If someone bets half pot, you need the best hand a quarter of the time, etc.

Now we’ll examine the EV of different bet sizes. Let’s start with a half pot bet. If we decide that half pot ($50) will be our bet size, we need to have a betting range containing 75% value hands and 25% bluffs to make villain indifferent between calling and folding. Since only 40% of our getting-to-the-river range is made up of value hands, now we figure out what 40 is 75% of and fill the rest in with bluffs. 40/.75 = 53.333. Round that to 53…53-40 = 13 bluffs. If we bet half pot with 40 value hands and 13 bluffs, player B makes the same amount of money calling as he does folding: $0. With the other 47 bluffs, we have to just check back and lose, or else villain can call every time.

Half pot bet

Villain EV for calling = .25*150 + .75*-50 = $0 = villain EV for folding

.75*-50 is the probability that we lose the pot times the amount we lose. .25*150 is the probability that we win the pot times the amount we win.

It’s slightly more complicated to calculate hero’s EV for playing this strategy. Four things can happen:

1.       Bet, get called, win

2.       Bet, get called, lose

3.       Bet, villain folds

4.       Check and lose

We’re checking and losing with 47 hands (also 47% of our range) so that part is easy: .47*0 = 0. When we bet, which we do 53% of the time, we know that villain folds half of the time: .5*.53*100 = +26.5. The other half of the time we bet, he calls. 75% of the time he calls, we win, and 25% of the time he calls, we lose. .5*.53*(.75*150 + .25*-50) = +26.5.

Hero EV for this strategy = .47*0 (check back) + .5*.53*100 (bet, fold) + .5*.53*(.75*150 + .25*-50) (bet, call) = +$53

Just as a demonstration, let’s see what happens if we decide to bluff more and villain changes his strategy to calling every time. Let’s say we decide to value bet 40 hands, bluff 40 hands, and check back 20 hands.

Villain EV for calling every time vs. new strategy = .5*150 + .5*-50 = +$50.

Hero EV for new half pot strategy that lets villain call every time = .2*0 (check back) + .8*(.5*150 + .5*-50) (bet, call) = +$40.

So when we deviate from optimal bluffing frequency and start bluffing too much against a perfect villain, villain’s EV goes from $0 to +$50, and ours goes from +$53 to +$40. Our reaction to him calling every time would be to bluff less and less until we end up back in the first situation.

Let’s move on to different bet sizes. For a full pot bet, villain is indifferent between calling and folding when he has the best hand 33% of the time, so player A will be bluffing 33% of the time. 40/.66 = 60, so our betting range will be made up of all 40 value hands as well as 20 bluffs. We check back 40 missed draws.

Full pot bet

Villain EV = .33*200 + .66*-100 = $0

Hero EV = .4*0 (check back) + .5*.6*100 (bet, fold) + .5*.6*(.66*200 + .33*-100) (bet, call) = +$60

Looks like against a perfect player, in this specific bluff catching scenario, a full pot betting strategy wins more than a half pot strategy. Let’s look at all in for 2x pot now. Villain needs to put $200 in to win the $300 sitting in the middle, so he needs to win 40% of the time to break even on a call. 40/.6 = 67, so hero will be value betting 40 hands, bluffing 27 hands, and checking back 33 hands.

2x pot bet

Villain EV = .4*300 + .6*-200 = $0

Hero EV = .33*0 (check back) + .5*.67*100 (bet, fold) + .5*.67*(.6*300 + .4*-200) (bet, call) = +$67

Betting as large as possible yields the biggest profit against a perfect opponent in this situation. When I figure out GTO (game theory optimal) solutions like this, I like to use them as starting points in my thought process in real situations, and adjust according to mistakes I believe my opponent will make. If I’m playing against someone who will call a large bet too often in this spot because lots of draws missed, I will still bet as big as possible but will check back all of my bluffs. If I’m playing against a very good player who may call and may fold, I simply revert back to GTO strategy and become unexploitable. If I’m bluffing an optimal frequency for my chosen bet size, the very good player can’t do anything to win money against me. I don’t care how much he calls or folds, he’s still not beating me for more money by adjusting his play.

Sometimes in a real game, you can’t make such rock solid assumptions. If I’m betting $1,000 into a $100 pot it could go from optimal to a massive losing play if there’s any chance my opponent slowplayed a monster hand (one that beats some or all of my value betting hands). Most situations aren't as clear cut as in this example. Sometimes the villain check/raises, sometimes villain has monsters, air, and mediocre hands all in his range, etc. 

That’s it for now. Any feedback is appreciated.

 John




Edit: I've been reminded that, while it is correct in this case, it isn't always most profitable to make the opponent indifferent between calling and folding. I might not have gotten that across. The key is figuring out what maximizes value against a perfect opponent. In this case they are the same thing.


Edit2: First comment corrected me. Villain calls 33% for 2x pot, 50% for 1x pot, etc. If you plug in the numbers the end results are the same but that is an important point.