Hey there, this isn't strictly probability, but I know there's a ton of really smart people in this forum. Hoping someone can help me understand a concept from Janda's Applications of No-Limit Holdem.

In the chapter "The Value to Bluff Raise Ratio," Janda discusses how many combos you must bluff on the flop to make an unexploitable river bet. In the example, one player has a range of nuts and air, and their opponent has bluff-catcher. The player with the polarized range makes pot-sized bets on the flop, turn and river. Janda says the bettor's range on the flop is 20% nuts and 80% air, so on the river they bet 30% of their flop range.

Janda begins at the river, and works backward through the hand to show how many bluff combos should be bet on the flop. When the polarized player bets on the river, they offer their opponent 2-1 odds. Therefore, their betting range should be 2/3 nuts and 1/3 air to make the other player indifferent to calling. Because this player is indifferent to calling, we can "look at our river bet from the perspective our opponent always folds when we bet...since our opponent folds every time we bet, these are all winning bets." Because the river bet is always a winning bet, there must be more bluff combos on the turn to maintain value-to-bluff ratio. Janda says to maintain the ratio, 15% bluffs should be added on the turn, so the player is betting 30% of their flop range for "value" (a winning bet on the river), and 15% bluffs which will check on the river and forfeit the pot. In summary, the player bets turn with 2/3 of their range for value (the winning river bet which = 30% of the flop range), and 1/3 as a bluff (the 15% of flop hands that will check and forfeit the pot on the river). He goes on to apply similar reasoning to add additional bluff combos to the flop betting range.

Quite frankly, none of this makes any sense to me. First, I don't understand the importance of perceiving the river bet as always winning.

Second, in my mind there is a simple example that goes against everything he wrote. Using the same game structure as his example, let's say Player A has 20 nut combos and 10 air combos on the flop. If they bet the flop, their opponent is indifferent to calling or folding, because they are getting 2-1 pot odds, and facing a range that is 2/3 nuts and 1/3 air. Player A can continue betting on the turn and river, with the same 30 combos, and retain their unexploitability. I don't see the need to add additional bluff combos, and I don't understand how Janda's method of starting at the river and working backwards justifies this.

I have been reading Applications very slowly, and I'm honestly disappointed. It's riddled with so many errors, and so poorly and haphazardly explained I'm losing faith in the content. Hoping I'm just being dumb and someone can straighten me out


edit: pages 105-7 if you have the book

Let's do it.

Say we're playing against each other on the flop. I make a PSB with a range of 20 nut combos and 10 air combos. Are you calling or all you folding? What's your play and why?

First, I'm embarrassed I made rude comments about the book. I was frustrated after grappling with this chapter all day, and didn't realize you would see the post. I appreciate you writing the book and taking the time help me and countless other posters. I was just upset with myself for not understanding the content and decided to vent which I regret.

Both calling and folding appear to be viable because of the ranges and odds I'm being offered. But now that I'm playing it out, I'm noticing something. After the flop bet the pot is 3 PSBs and after the turn bet it's 9 PSBs. On the river I'll be facing a neutral EV decision, but in order to get to that decision I have to pay both flop and turn bets. The river decision is neutral EV, but I had to pay for the privilege to arrive there. Because I'm losing money with this line, the co-strategy between players can't be at equilibrium.

However, if the bettor sometimes gives up and forfeits the pot by checking back bluffs, we can reach an equilibrium where my decisions to call or fold on the flop and turn are 0 EV, instead of me paying to reach a neutral river decision as I described above.

Am I on the right track? I know I didn't directly answer your question, but as I started working through the problem I began seeing things differently. I'm hoping this is the correct way to conceptualize it.

^Sorry for that inane response, I think I have it figured out. Your question got me thinking enough to realize I didn't know how to think about the situation. Here's what I'm thinking now.

1. When we bet the river our opponent's EV is 0. When they fold, we win the pot. Thus, our bet essentially wins the pot.

2. On the turn, our opponent is getting 2-1. If we don't add in bluffs, they will make a correct fold because they can't win on the river when we bet. Therefore, we value bet and bluff in a 2-1 ratio, and they call knowing 1/3 of our range checks the river and forfeits the pot.

3. On the flop, the same logic follows. Our opponent is getting 2-1, so 1/3 of our flop betting range should be bluffs that check the next street and forfeit the pot.

Is this the correct way to think about the hand? I think I'm getting caught up in bad thinking and old terminology, which confuses things when I'm trying to learn these new concepts.

If you're used to older terms this is basically the concept of RIO.

Seems like you get it but I'm not sure if you're still missing the idea that the threat of future bets allows the aggressor to have more bluffs on earlier streets in a pure polar vs bluffcatcher scenario.

Curious what your answer would be to the question Janda posed in post #2. Is calling the flop in his scenario -EV, 0 (indifference), or +EV? What is your calling frequency? Going a step further, what is the EV of calling the flop bet? If you can answer this I think you understand the concept.

Thanks for the response.

In Janda's scenario, the bluffcatcher is indifferent to calling the flop bet. They are entitled to 1/3 of the pot on the flop, when the bettor checks 1/3 of his range on the turn. The same logic follows on the turn and river. Because the bluffcatcher is entitled to 1/3 of the pot on each street, they get back 13 PSBs. The total cost for them to call flop, turn, and river is 13 PSBs. There is no reason for either player to change their strategy.

The correct calling frequency on each street is 1/3 of the bluffcatcher's range. If they have 27 combos on the flop, they should call with 9 combos versus a PSB. They should call with 3 combos on the turn. On the river, they can call with their remaining combo. But at any point they can fold, because their total EV for the hand is 0.

On the flop, the EV of calling is 1 PSB. The polarized player bets 67.5% of their flop range, and 1/3 of that range will check the turn and forfeit the pot. Thus, the bluffcatcher is entitled to 1/3 of 3 PSBs.

The reason my example doesn't make sense is that the river is neutral EV, but the bluffcatcher had to pay on the flop and turn. I think that's what you mean by reverse implied odds; they lost 4 PSBs before getting to the river and facing a neutral EV decision.

I'm feeling more confident in this explanation.

Just to clarify - the bluffcatcher is entitled to 1/3 of the 3 PSB pot after calling. They paid 1 PSB to retain that equity, so the total EV = 0.

Ya I think you got it.

My point was going to be if you call me when I bet 20 value hands and 10 air hands on the flop for 1 PSB, then what are you going to do on the turn if I bet 20 value hands and 9 air hands and just check-fold 1 air hand? Well, now you'll fold lose the pot the 29 times I bet it, and win it the 1 time I bet it... clearly an awful deal, which means you should have folded on the flop. This is why you can only look at the immediate odds you're being laid to determine if a call is +EV or -EV on the river... everywhere else position, total equity, the distribution of equity (e.g which range has more nut type hands) stack depth, etc all play a role in determining whether or not you can make a profitable call with a mediocre hand. That's also why the terms "value bet" or "bluff" are for the most part dated terms that don't really make sense unless you are on the river.