High Level Thinking: Calling a Pot-Sized Bet
Monday, 26 May 2014
Warning – not for the faint-hearted. PokerStars’ Isaac Haxton shares his insights on being unexploitable.
If you’ve spent any time reading about poker strategy or watching poker training videos, you’re probably familiar with the maxim ‘you must call a pot-sized bet half the time to avoid being exploited’. In this article, I’m going to examine where this piece of wisdom comes from, how to apply it, and when it can lead you astray.
Like many poker rules of thumb, ‘call a pot-sized bet half the time’ is really the solution to a toy game, a simplified poker-like game that can easily be analyzed to provide insight into real poker scenarios. The toy game in question is as follows: Player One has either the nuts or air with equal probability. Player Two always has a bluff-catcher, a hand that beats air but loses to the nuts. There is $1 in the pot and Player One can either bet $1, or check and go to showdown.
The most important thing we can know about a game like this is its Nash Equilibrium, the pair (or pairs) of strategies such that either player can write down his strategy and commit to playing it unchanged and there is nothing the other can do to improve his own strategy and ‘exploit’ it. In order to find that equilibrium point, we can begin by writing down the values of the different actions available to each player:
Player One
EV(check with air) = $0
EV(check with nuts) = $1
EV(bet with air) = $1*(frequency P2 folds) - $1*(frequency P2 calls)
EV(bet with nuts) = $1*(frequency P2 folds) + $2(frequency P2 calls)
Since (frequency P2 folds) = 1-(frequency P2 calls), we can simplify the above to:
EV(bet with air) = $1 - $2*(frequency P2 calls)
EV(bet with nuts) = $1+$1*(frequency P2 calls)
Player Two
EV(fold) = $0
EV(call) = ($2*(frequency P1 bets air) - $1*(frequency P1 bets nuts))/(frequency P1 bets air + frequency P1 bets nuts)
Notice that the values of checking and of folding are fixed, but that the values of betting and of calling depend on what the opposing player does. Also notice that EV(bet with nuts) >= EV(check with nuts), for any non-negative call frequency, so we know (frequency P1 bets nuts) = 1.
Betting with air is better than checking if P2 calls less than half the time. So if P1 knows P2 will call less than half the time, he should always bluff. But now, if P2 knows P1 is always bluffing, we see that calling is worth $0.50, so P2 should always call. But if P2 always calls, now P1’s bluffs are back to losing, P1 should stop bluffing, then P2 should stop calling, then P1 should start bluffing again, and around and around and around.
The solution is that P1 has to bluff only some of the time and that P2 has to call only some of the time. More specifically, each side must pick a frequency such that the other is indifferent between his two options. In other words, to find P1’s bluff frequency, we set P2’s EV(call) = EV(fold):
($2*(frequency P1 bets air) - $1*(frequency P1 bets nuts))/(frequency P1 bets air + frequency P1 bets nuts) = 0
Substitute in (frequency P1 bets nuts) = 1
($2*(frequency P1 bets air) - $1))/(frequency P1 bets air + 1) = 0
and solve for
(frequency P1 bets air) = 1/2
And similarly, to find P2’s call frequency, we set P1’s EV(bet with air) = EV(check with air):
$1 - $2(frequency P2 calls) = $0
and solve for
(frequency P2 calls) = 1/2
Hence, the common wisdom is that ‘you should call a pot-sized bet half the time to make your opponent indifferent to bluffing’. Really, though, what we have just demonstrated is much narrower than that. It only applies if the payoffs for each action are exactly as above. To illustrate how to use this insight, I’ll give one example of a scenario where it applies and another where it will lead you astray.
Consider a pretty common NLHE scenario: one player raises from the button, another three-bets from the BB, and the first player calls. The flop comes A-2-2 rainbow, and the three-bettor c-bets. The button calls. The turn is an off-suit 6, and it goes bet, call again. The river is a 7, and there’s a pot-sized bet left to play. The bettor is probably betting for value with something like A-Q or better and bluffing with hands like 9-8 suited, and the caller is considering calling down with hands like A-T. Does this look enough like the toy game to apply what we learned there? The bettor’s candidate bluffs are actual air, hands that have no value whatsoever as checks. The bettor’s value-bets are not quite the nuts. The caller might occasionally have hands like A-7 suited that beat some value-bets, but this will not change the solution (rework the toy game with the defender holding a hand that beats the aggressor’s value-bets 5% of the time, if you’re curious or untrusting). Applying the principles from the nuts or air, toy game in this situation seems appropriate. In order to play unexploitably, the button should call a pot-sized river shove here half the time.
Now let’s consider another NLHE scenario: one player opens UTG in a six-handed game, and another calls from the button. The flop comes down Kh-9h-2c. The opener c-bets and the button calls. The turn is the 3s, and it goes bet, call again. The river is the 7h, and now the opener checks and the button is once again considering shoving for $1 into a pot of $1. How does this scenario compare to the nuts/air game? For the sake of simplicity, let’s say the button isn’t trying to value-bet a hand as weak as A-K, and would have raised before the river with K-9 or a set, so the button’s entire value range is flushes. Furthermore, let’s assume UTG wouldn’t check a flush, so the button’s value-bets are the pure nuts. In practice, there may be a little more subtlety to what is going on with the button’s value range, but the important way in which this scenario differs from the nuts or air game has to do with the button’s bluffs.
What is the worst hand the button can get to the river with? Depending on the player it might be something as weak as 9-8 or something as strong as K-J, but it’s almost certainly a pair. UTG, on the other hand, could pretty easily have been bluffing flop and turn with one of the Q-J/Q-T/J-T gutshots, or even something like Ac-5c. The worst hands in button’s range, the ones he might use as bluffs, will win fairly often if he just checks back. To pull a number out of thin air, let’s say these hands are good 30% of the time. To put it in the terms of the toy game:
EV(check with “air”) = $0.30
This is an important difference! Now if we set EV(bet with “air”) = EV(check with “air”) and solve for the defender's fold frequency we see:
$1 - $2(frequency P2 calls) = $0.30
(frequency P2 calls) = 0.35
UTG only needs to call this bet 35% of the time to keep the button indifferent between bluffing and checking down with the bottom of his range. If UTG were to call this pot-sized bet half the time, he would be playing very exploitably! In response, the button should never bluff, and he gets to extract much more money with his value-bets than the parameters of the situation should allow.
So, before the next time you shrug and pay off the top half of your range to a pot-sized river bet, make sure you think through the situation and confirm it’s an appropriate spot to apply the concept!