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The following hand data is taken from *Mastering Small Stakes No Limit Holdem* by Jonathan Little. Both hero and villain have known starting hand ranges that narrow as they get to the river. Below, I post only the ranges as they are at the river (since it seems irrelevant what the starting ranges were, unless I'm missing something).

**Hero's River Range.** Suppose hero is on the river and decides to bet all in $141 into a pot of $121 with the following range:

The author then comments:

And then here is the range that villain should call down with according to the author:

Notice here that villain is calling with (roughly) the top 35% of his range (technically, it's the top 39% of his range; the author later mentions that to achieve a perfect 35%, villain can remove some combinations of AJo).

**Question:** Why should villain only call with the top 35% of his range in this example? Shouldn't villian call with 100% of his premium hands (for value), and with 100% of his bluff beaters (expecting to win 35% of the time with them, per hero's strategy)?

Notice that in this particular case, 100% of villain's range are in fact bluff beaters or better (far greater than 35%), in that they beat hero's QJo bluff on the given board (Ah, Kc, 5d, 3h, 7s)!

So again: why should villain call with the top 35% of his range when 100% of his hands are bluff beaters or better? More generally, why should a rational poker player call with the top X% of his range when being laid X% pot odds (by an opponent playing perfectly GTO)?

**EDIT.** I thought about this some more. To make the math easier, let's now assume that hero bluffs with 100% of QJo (and not just the right amount of them to achieve a 35% bluffing frequency). Then it follows that hero's range will consist of 44% bluffs and 56% premium hands. Assume now that hero bets in such a way that he lays villain with 44% pot odds (instead of the 35% talked about above).

Then the following principle is true:

(P) Villain should call with any hand that gives him 44% or more equity against hero's range (KK+, AK, and QJo).

Here is the equity of each of villain's hands that he could call with:

As you can see, it is rational for villain to call with 100% of his river range! This is far greater than the top 44%.

This assumes that (i) villain wins every time he calls with the top 35% of his range and (ii) villain loses every time he calls with the bottom 65% of his range. Yet as my (in fact: the author's own) example demonstrates, this is false. If villain called with 100% of his range, he would win at least every time hero had QJo (which is approximately 35% of the time). In fact, villain will win far greater than 35% of the time with a 100% calling range since sometimes hero will have a pair of kings, while villain will have a pair of aces (and so forth). How does one account for this? – George – 2018-11-26T00:26:22.667

I clarified this point above with an equity table (see my original answer). It demonstrates that villain is rational to call with 100% of his range. – George – 2018-11-26T01:46:49.993

@George, very interesting I see your point. I still feel like there is something missing here, can I ask what page this is on so I can read about it for myself? I am reading this book too. – Clarko – 2018-11-26T07:52:54.807

It starts on pg. 373. – George – 2018-11-26T16:42:47.463