Why shove the river when your range is more than 50% value heavy?

Question

From an example in Mastering Small Stakes No Limit Holdem (pg. 383):

Notice that Hero [on the river] has nut hands slightly more often than bluffs. When this is the case, as long as Hero does not have more bluffs than value bets, he should bet as much as possible.

Question: Why is this the case?

Attempt:

1. When hero bets as much as possible on the river, he lays his opponent odds x that approach (but never touch) 50%.

2. Suppose shoving lays your opponent 33% odds, and the top 40% of your range are value hands. Then I can see your opponent having a psychologically hard time calling with bluff beaters that technically beat the 33% percentile of your range (this might require him to call an all in with ace high, for example).

3. But now suppose that shoving lays your opponent 45% odds. Then he is only rational to call with hands that beat the 45th percentile of your range, which means (at the very least) he is only rational to call with premium hands. What is the benefit of this?

In general, why should one shove on the river when your range has more than 50% value hands and less than 50% bluff hands?

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EDIT: Upon reflection, a more reasonable rule of thumb seems to be as follows:

1. When your river range is more than 50% value, you should bet small so as to entice your opponent to call with a marginal hand.

2. When your river range is more than 50% bluffs, you should bet large, so as to minimize the chances that your opponent calls (betting large will force him to make the difficult decision of calling a large bet with nothing but a bluff beater).

Answer 1

The book is quite correct -- if your range contains more value than bluffs, then the correct bet sizing is to move all-in.

To see why, consider the classic polar versus bluffcatcher scenario. You have either nuts or air; your opponent's bluffcatchers only beat your air.

The equilibrium is reached in this game under the following conditions. If the pot is p and your bet size is b, then your opponent needs to call p / (p+b) of the time to make you indifferent to bluffing with your air.

Likewise, if you have V value hands, you need to mix in B = V b / (b+p) bluffs to keep your opponent indifferent between calling and folding.

Notice that when your bet size b becomes large enough, the number of bluffs will approach the number of value hands. This means that as soon as your value hands constitute more than 50% of your range, it becomes possible to bet your entire range for large enough bet sizes. Once this happens, you can continue to increase your bet size such that your opponent is no longer being offered the correct odds to call, and you win the entirety of the pot. If he does call, he's making a mistake, and you gain even more (in expected value).

This highlights the tremendous value of having bluffs in your range. Had the action gone check check on the river, you'd win slightly more than 50% of the time. As you bet larger and larger, the presence of a bluffing possibility allows you to capture progressively more and more of the pot. If you can make a large enough bet, the pot is yours.

Now, in the "real world" the situation is rarely so clean as the polar versus bluffcatcher case, but the same general principles hold. When you have a range advantage over your opponent, particularly when you can hold the nuts but they can not, large bet sizes are in order.

Exploiting your opponents means giving them the opportunity to make the biggest possible mistakes. Large bet sizes do just that.

It's also important to remember that you're trying to maximize your expected value for your entire range, not just a particular hand. Sure, making a quarter pot bet with your quads might get you a little more value from marginal hands than jamming them, but you give up tons of value on bluffing hands that could benefit by being jammed along with your quads/boats/sets etc.

Answer 2

Dont vary your bet sizing based on whether you are value betting or bluffing. It is a leak that is easily exploited.

I kinda disagree with how the book approaches bluffing (maybe im misinterpreting something and missing some context). You should always have more value hands than bluff hands, otherwise anyone who calls you down is profitable.

"as long as Hero does not have more bluffs than value bets, he should bet as much as possible"

I think this means, as the amount of bluff hands approach the amount of value hands, the higher your bet needs to be. By increasing your bet size, you allow yourself to bluff with more hands, thereby gaining equity from more pots whereby you wouldnt have otherwise.

"When your river range is more than 50% value, you should bet small so as to entice your opponent to call with a marginal hand"

I dont think this is always true. By betting small you are allowing your opponent to profitably call with a wide range, while limiting the number of bluffs you can have. E.g. if you are betting 1/3-1/2 pot with 60% value hands and 40% bluffs, villain will be +EV simply by calling you down with everything. (Math breakdown: for 1/2 pot river bet, V has 25% odds to call, meaning he only has to call correctly 1 in 4 times to break even. This means that if youre bluffing more than 1 in 4 times, or if less than 75% of your hands are value hands, he is profitable calling you down with 100% of his hands.)