By Tony Guerrera

In another article I wrote for Poker Helper, entitled Don’t Bet the River in Limit Hold’em Just Because You Have a Good Hand, I made the point that you shouldn’t bet on the river after your opponent checks to you in limit hold’em just because you have a good hand. If you have over 50% of your opponent’s hands beaten, you shouldn’t necessarily bet. Whether you bet in position on the river in limit hold’em is really determined by your opponent’s calling distribution; your opponent’s calling distribution is a subset of his hand distribution. A good value bet is one in which you have over 50% of the hands in your opponent’s calling distribution beaten. Well, actually, if you are against opponents capable of check-raising on the river, then things become more complicated. But if you assume that your opponents will just call or fold, which is a good assumption for most games, then a good value bet is one in which you have over 50% of your opponent’s calling distribution beaten.

This concept extends to no limit hold’em. Just as in limit hold’em, a value bet in no limit hold’em is positive EV (expectation value) if you have over 50% of the hands in your opponent’s calling distribution beaten. And again, if we account for opponents who are capable of check-raising on the river, then things become more complicated. Since most players tend to bet their good hands on the river, and since most players aren’t capable of check-raise bluffing, this article assumes we are dealing with opponents who will not check-raise you on the river. In other words, after your opponents check to you on the river and you bet, they are constrained to either call or fold.

The object of poker is to make plays that are +EV, right? And I regurgitated the definition of a +EV value bet, so we are done, right? Wrong and wrong. The object of poker isn’t simply to make plays that are +EV. The object of poker is to make plays that optimize your EV, and our understanding of what a profitable value bet is doesn’t help us to find the optimal value bet in no limit hold’em, a game in which you can control the size of your bet. Therefore, value betting in no limit hold’em requires one more piece of analysis than value betting in limit hold’em.

Whenever you make a value bet in no limit hold’em, your EV is determined by the following formula:

In this formula, B is the size of the bet, P(called) is the probability that your opponent calls, P(win|called) is the probability that you win given that you are called, and P(lose|called) is the probability that you lose given that you are called-in probability, the “|” symbol means “given that.” Actually, this formula holds for value betting in any game when in position on the river against an opponent who will only call or fold to a bet (i.e. he won’t check-raise).

So, here’s what’s so special about no limit hold’em. In limit hold’em, B is a constant. However, in no limit hold’em, B can be any number you choose. To complicate matters, in no limit hold’em, P(called), P(win|called), and P(lose|called) are all functions of B. In other words, the size of your bet will determine your opponent’s calling distribution. In limit hold’em (and limit poker in general), all you need to do is apply this formula is see whether it’s positive or negative. If it’s positive, you bet; if it’s negative, you check. Meanwhile, in no limit hold’em, you not only want your EV to be positive, but you also want it to be as high as possible. As a simple example, suppose your opponent will call a $100 bet 50% of the time, and when your $100 bet is called, you win 80% of the time. Meanwhile, suppose your opponent will call a $200 bet 30% of the time, and when your $200 bet is called, you win 60% of the time. The EV for the $100 bet is given by the following expression:

The EV for the $200 bet is given by the following expression:

Both of these bets are profitable, but the $100 bet is clearly better here.

With this concept in place, let’s go through the actual thought process you’d go through in the heat of battle. Normally, when talking about a hand of hold’em, you need to go from the beginning to the end, and put everything in context of what you know about your opponents; however, in the example that follows, I just give you the board, your opponent’s hand distribution, and your opponent’s calling distribution. Actually determining those distributions is an exercise in studying your opponents, and that’s a topic that deserves an article of its own (or many articles). Just realize that to properly evaluate the EV of a value bet, you need a precise handle on your opponent’s hand and calling distributions.

Suppose the board is J8733, and you have KJ. Your opponent is on the following distribution: [AJ,QJ]||[JT,J8]||[87]. This hand distribution, expressed in terms of the number of combinations available for each holding, is given in the table below:

Holding	Combinations
AJ	8
KJ	6
QJ	8
JT	8
J9	8
J8	6
87	9

In total, there are 53 total combinations of cards that your opponent may hold.

The pot is $200, and both you and your opponent have $500. Your opponent’s calling distribution as a function of a few different bets is given in the table below, along with information about the number of combinations in each distribution that you beat and the number of combinations in each distribution that beat you:

Bet	Calling Distribution	Total Combinations	Combinations You Beat	Combinations that Beat You
$50	[AJ,QJ]||[JT,J8]	44	24	14
$125	[AJ,QJ]||[JT]||[J8]	36	16	14
$200	[AJ,KJ]||[J8]	20	0	14

Applying the EV formula to the $50 bet, we get:

Applying the EV formula to the $125 bet, we get:

And applying the EV formula to the $200 bet, we get:

The $50 bet has the highest EV, so between the $50 bet, the $125 bet, and the $200 bet, the $50 bet is best. Of course, there may be some other bet out there that yields a higher EV than $9.43-finding the optimal bet size when there are so many options is one of the toughest tasks when figuring out your value bet in no limit hold’em.

Finding the optimal bet size may be tough, but there are some general concepts that may help guide you. Assuming that you know what your opponent’s hand distribution is, the whole art of value betting in no limit hold’em has to do with deducing your opponent’s calling distribution as a function of your bet size. In the example I gave above, your opponent seemed to have a narrower calling distribution as the size of the bet increased. In practice, you’ll often see this phenomenon; however, there are two related exceptions. Sometimes, a small bet may be interpreted by some opponents to be a value bet with a very good hand, and when a small bet is interpreted as such, your opponent’s calling distribution will actually be smaller against small bets. Vice versa, a large bet may be interpreted by some opponents to be a bluff, and when a large bet is interpreted as such, your opponent’s calling distribution will actually be wider against large bets. Of course, there are limits to this. For example, suppose an opponent may be willing to call a $200-$250 bet with a wider range of hands than he would call a $100-$200 bet. Such an opponent may shrink his calling distribution once you start betting over $250.

Now that you know the theory of value betting in the situation where your out-of-position opponent only calls or folds after you bet, you can extend the theory to account for the more complicated cases where your opponent check-raise bluffs, possibly forcing you to lay down winners, and where he check-raises with monster hands, possibly forcing you to call when you are beaten. Knowing the theory, you are well-equipped to make good decisions on the river in no limit hold’em, and since the pots and the resultant bets are the largest on the river, this is a great place to add considerable profit to your game. With an understanding of the theory, all that remains is reading your opponents as well as possible to deduce their hand distributions and to then deduce their calling distributions as a function of how much you bet. I hope that your EV is optimized in all of your value betting decisions!

Tony Guerrera is the author of Killer Poker by the Numbers

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