Hi, first I want to state that I realize that the nash equilibrium/unexploitable play is rarely the most optimal in poker. However I think it's important that I develop a solid understanding as a baseline, and adapt to an exploitable strategy from there.

Here is the situation; we are heads up, in a razz game. We get to 7th and the boards are something like KK23 against T987. This is a very common situation, and clearly T987's only value is a bluff catcher, and both players realize this. For the sake of discussion, assume that KK2A can never be value betting a worse hand. In this case, he is completely polarized to either nuts or bluff.

My default strategy as KK23 is to never bluff, because in my experience 95% of the population will fold their bluff catcher exactly 0% of the time. Conversely, my default as T987 is to call 100% because I find the average opponent bluffs here far too often to make hero folding ever correct.

The problem I'm running into now is that the games are getting more and more advanced. Even the average recreational player has figured out that this is a terrible spot to bluff, so the strategy of never folding has become very -EV compared to how it used to be. Likewise, I'm running into a fair amount of players who are capable of hero folding in this spot, so against them my strategy of never bluffing is rendered insufficient, to say the least.

So what I'm looking for is a formula to calculate the nash equilibrium frequency for both bluffing as the KK2A and hero folding as the T987. This way I have something to revert to so I can be certain I'm unexploitable. (the key word here being unexploitable, not maximizing EV). I'm not comfortable with just "throwing in a fold here and there" or "bluff until they catch you and then don't do it again, that way they always call after that." Although it may suffice against some opponents, I'm looking for something more concrete... specifically, a percentage so that I can look at a watch, and make my decision based on that.

What factors do I need to be taking into consideration here? Do the equites matter, or is it strictly a pot size/ pot odds question?

I think that's pretty simple theory of poker stuff. Forget what your equity is exactly with A234 there but let's say 40 percent. 6.5bb in the pot, you bet 1bb, you should be bluffing 1/7.5 of the time to make him indifferent to calling or folding. So you bet the river .40/(1-(1/7.5))=46 percent of the time, 40 percent with the nuts and 6 percent as a bluff. So pick 2 or 3 cards to bluff with.

yah but thennn u gotta factor in the times a234kk makes a 94low and other guy makes a 87 and the 94 value bets.

So Chips you are saying hero should just bluff 6% of the time...shouldn't it be < 1/7.5 of the time?

You want the ratio of bluffs and (winning) value bets to be the same as the pot odds you're offering him. With 6.5bb in the pot you're offering him 7.5:1 pot odds. If you're going to vbet 40% of the time, then you have

7.5:1 = 40:X
and X is the percentage of the rivers you should bluff. So X should be 5.33%

(There's a small discrepancy with Chip's results here because you should be bluffing at a 7.5:1 ratio, which is 1/8.5, not 1/7.5. I posted this not to nitpick his results, but to show why the answer isn't 1/X where X is the pot odds you're giving him.)

Man as soon as I saw this topic I thought "Rusty's got this one."

Lots of tricky wording here, I just want to make sure I'm understanding right. So the times that we brick we need to bluff 11.8%. The 5.33 number just represents the percentage of the time that we birck AND bluff, not the times that we will be bluffing when we brick.. right? What's more important is the actual percentage we need to be bluffing when we miss, not the percentage of our range that is comprised of bluffs with a random river.

Also, Chips you mentioned that our opponent will be indifferent between calling and folding, but from what I understand anytime a player deviates from nash equilibrium they will be exploiting themselves. Is this correct? In other words, say we bluff at a nash equilibrium frequency. If our opponent is calling 100%, he is still making a mistake. It's just less of a mistake than it would be if we bluffed 0%

The 5.33 is the percentage to bluff of all rivers. Imagine there are 1000 possible river cards (bear with me). We will hit 400 times and miss 600 times. So we value bet 400, and bluff 53 of them (5.3%, close enough). In terms of how often to bet when we brick, it's 53/600 which is 8.88%.

He can't do better with a strategy other than blindly calling or blindly folding. That's what equilibrium means.

Yes and no. If we employ our NE then yes, it doesn't matter what he does. This is not actually always the case with an NE, but it sometimes (often?) is. Our NE guarantees we will break even or better.

But also, he does best if he employs his own NE, because it prevents us from being able to profit by deviating from ours and exploiting him.

His NE is similarly found - In order to be unexploitable he needs to fold with the same frequency as the pot odds he's given, in this case, 1/8.5 and call the rest.

Let's look at it from his point of view for a minute. Let's say we bet into him only when we have him beat, never bluffing. His EV would be:
EV = -1*.4 + 6.5*.6 = 3.5

Or say we bet into him every time, his EV would be
.6*7.5 - .4*1 = 4.1

If we are bluffing optimally, then he will win 1/8.5 times and lose 7.5/8.5 times, so
EV = 7.5*1/8.5 - 7.5/8.5*1 = 0

So you see his NE guarantees him 0EV or better. If we deviate from our NE at all, then he profits.

Edit: I've looked over it a few times and I don't see any mistakes, but there may be some. YMMV.

Rusty, it's not *quite* that simple, because to calculate his optimal unexploitable calling frequency and his (and our) EV, we need to take into effect the probability than we outdraw him on the river -- that number does matter to him.

Just to take it to the absurd limits, if we outdraw him 100% of the time, his unexploitable calling frequency is 0%; but if we outdraw him 0% of the time, his unexploitable calling frequency is 100%.

Also, his optimal strategy may not be his Nash Equilibrium, and it depends on how often we outdraw him. If he calls enough to make us indifferent to whether we bluff or not, he's paying us off on our value bets more than his optimal strategy would call for. I think his optimal strategy is to call just a little bit less than that, enough so that we have an incentive to bluff, and then set his calling percentage so that the amount he gains by folding losers is balanced by the amount he loses by not calling at the Nash Equilibrium.

I think.

EDIT: To put it another way... Given the choice between your opponent never putting money in the pot with a loser, or always putting money in the pot with a loser, which would you rather incentivize?

By optimal here I mean NE - it's a dumb convention a bit, but it arose from Mathematics of Poker and it's just kind of what we go for now.

I'm not sure if you're right about our chance of having outdrawn him changing his NE. I'll have to think about it tomorrow.

Oh hell maybe it does. His hand changes our NE (sometimes).

I'm rethinking what I said above, and I can think of one thing that's perhaps a huge concern in a raked cash game which makes things a lot simpler.

If any fraction at all of a potential river bluff is raked, there's a big range of strategies for which the bluff is -EV for both players. With this in mind, the bluff should be discouraged, and the common Razz river strategy of calling with any bluff-catcher is probably not too bad, because it should in theory reduce the bluffing percentage to zero. It's not hard to see that both players have a large interest in not giving that extra rake to the house.

In theory, calling with a bluff-catcher more often than your Nash equilibrium but less than 100% should still in theory decrease the bluffing percentage to 0. That however, assumes perfect information and perfect belief on your opponent's part; if he sees you fold to a bet even once, he's likely to try bluffing more often than he really should, and that costs both of you in the long run.

So you sort of end up with a game of "chicken" where you as the bluff-catcher cannot blink (fold to a bet) because by doing so, you may induce the other guy to a strategy of bluffing which costs both of you money. Convince him by any means possible that bluffing isn't worth it; easiest way is to never fold.

A lot of times the rake is capped by the time you get to the river. Like if the rake is 5% capped at $3 then any pot over $60 won't incur extra rake with more bets. In this hand we're talking about a 6.5bb pot so anything over 5/10 would already have maxed the rake. Maybe 3/6 too since I think the cap is lower? I'd have to get out their charts.