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:

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The author then comments:

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

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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:

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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%.

I think the wording here is a bit confusing, so I will restate what the author is saying how I understand it:

  • The villain needs to win more than 35% of the time to be +ev over time
  • The villain expects hero to have 65% value hands and 35% bluffs (because of the pot odds he has been given)

If villain calls with 35% of his/her range, that is essentially the same thing as villain calling 35% of the time.

For example, lets theoretically pretend this exact situation happened 100 times in a row and villain had AJo 16.6667 times (how many times/hands in range, 100/6), KQo 16.6667 times, 55 16.6667 times etc. villain needs to call 35 times out of 100 to beat the perceived 35% bluffing frequency that hero has, so villain will pick the best hands to call with. (in my example, 55 16.6667 times, AJo, 16.6667 times, and KQo 1.6666 times)

My example doesn't match up exactly with the book because of combinations, but hopefully I got my idea across.

Calling all the time as villain would guarantee you lose much more often than you win, because of how nutted hero's range is. Calling with the top 35% of villains range ensures that villain will win if hero is bluffing, but also ensures that he does not lose too often when hero does have the goods.

  • 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 Nov 26 at 0:26
  • 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 Nov 26 at 1:46
  • @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 Nov 26 at 7:52
  • It starts on pg. 373. – George Nov 26 at 16:42

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