Bet Sizing Holding The Absolute Nuts
The absolute nuts means you are 100% sure of winning the hand. Therefore, having such a hand at any point guarantees you win at least the current pot. We will assume you have the lead bet on the river. Your options are to check or bet. We will explore the math of sizing a bet so that you maximize expected value.
A general EV equation for a bet by you is the following:
EV = fe*Pot + (1-fe)( eq(Pot + Bet) - (1-eq)Bet),
where fe is fold equity (i.e., fold probability) for a bet of Bet into a pot of Pot with card equity of eq.
We conditioned eq to equal 100% and will assume Pot = 1, so Bet is in pot-size units. Then,
EV = fe + (1-fe)(1+Bet) = 1+ (1-fe)Bet
This simple equation has a complication, namely the fold equity will generally be a function of the bet size. It is reasonable to assume a “lazy” S type function ( ) where fe, the fold equity, is small for small bets and approaches 1 as the bet size reaches multiples of the pot size. (If villain suspects a bluff with a very large bet, there may be a dip at the end).
We’ll use the following fold equity function:
fe = 1-1/(1+exp(4*Bet-4)).
Then
EV =1+Bet/(1+exp(4*Bet-4)).
The fold equity function and EV values for various bet sizes are shown below:
Clearly EV is positive for any bet size. EV is maximized when a 0.8Pot size bet is made and it is equal to 1.55Pot. The EV for various bet sizes is shown in the following graph.
What about the check option? A check might be a good option if villain is likely to bet hoping to bluff you off the hand or if he believes he is ahead. Of course, the danger is that villain may also check thus resulting in a minimum win amount. Here the math has to be supplemented with good poker thinking. Is villain passive or aggressive? What has he done in the past in similar situations? Did the river card possibly help him so he might think he has a winner? Has he bluffed in the past? Is your winning hand pretty well concealed? If he does bet, you’re basically in the same situation as before in deciding now how much to raise rather than bet. Of course, you now have a bigger pot to win.
Let’s now further explore the fold equity function which plays a big part in determining your EV. The one we used says that villain will almost certainly fold as the bet size approaches 2Pot. It came about to ensure that at smaller bets the fold equity was about right, in this case a half-pot bet, for example, had fold equity of about 20%.
Clearly you cannot expect villain to always fold at 2Pot many times with you holding any two cards for he certainly will catch on and change his folding range. Let’s now consider a more aggressive villain who folds as shown below:
Again, fe is about 20% at a half pot bet but rises much more slowly than the previous function, never gets to 100% and actually dips down at about 4.5Pot indicating villain suspects a bluff at such a large bet. A graph of the resulting EV equation follows:
We see here that EV is maximized at 1.61Pot for a bet between 1.25 and 1.5Pot, a bet up to almost twice as large as the previous example with an increase in EV of only about 4% (1.61/1.55). The EV starts to increase again at about 4.0Pot because villain starts to call more frequently for very large bets but we remind the reader that we are assuming hero has the absolute nuts so villain calling very large bets still favors hero.
Summary
We showed how one can incorporate fold equity into an EV analysis assuming hero has a winning hand. A key to the analysis was to recognize that fold equity will usually be dependent on bet size. We explored two fold equity functions – one fairly tight (villain unlikely to call large bets) and one fairly loose (villain calls large bets more frequently). With a 100% equity hand, the more villain calls, the higher the EV until fold equity becomes dominant. The suggested model approach can define the bet size that maximizes EV. If you were restricted to only these two fe functions, a pot size bet will work out pretty well for each. As with any math model, the decision resulting from the analysis should be modified to account for factors not directly considered.