So the classic river toy game assumes player A either has nuts or air, and player B has bluffcatchers only. And from there, we can work out the optimal value:bluff ratio for player A, and the calling frequency for player B.

My question is what if the assumption is tweaked so that, while player B's range is still mostly bluffcatchers, he now has one combo of nuts (let's say 20 bluffcatchers to 1 nutted combo)? How would that affect player A's river shoving range ie value:bluff ratio?

If it'a checked to A and A has the option of shoving or checking back, it should be fairly straightforward to solve this.... But if A is betting less than a full stack, B can raise or bet when checked to, etc, then things get much more complicated.

only one can have the nuts or they split

In the “classic” river toy game, hero has nuts or air and villain either calls or folds hero’s bet. The ratio of value bets to bluffs is calculated to make villain indifferent to either action. For a pot size bet, the ratio of value bets to bluffs is 2 to 1, the same as the pot odds given to villain.

OP hypothesizes a value bet less than 100% equity. You can use standard EV analysis to find the corresponding value/bluff ratio. For OP’s example; villain has a win probability against the value bet of 1/21. That results in a bluffing percentage of 30% compared to the 33% for the 100% value equity case. For a bet =2*Pot, the value/bluff ratio is 1.5 to 1 for 100% equity, which increases to 2 to 1 if hero’s value equity is only 90%.

Imo the ratio doesnt change (otherwise he will start exploit you). You get some value from the bluffcatchers with your nuts, but you lose some value to his nutnut (two different events). When the sum of these exshowdown values gets below zero you should start checking.

Thanks! Could you show the calculation? My intuition is to bluff less but I'm not sure about the maths

Here is the equation for villain EV when hero bets Bet into a pot of Pot and bluff equity = 0.

EV_vill.= B*(Pot+Bet) +(1-B)*((1-E)*(Pot+Bet)-E*C)

B= hero bluff frequency
E= hero value bet equity
C=hero call amount (<= Bet)

You can set the EV to 0 and solve for B. I’m lazy so I set this up in Excel and used its Goal Seek function to get the solution. The equation can easily be extended to include some equity for a bluff bet.

Hi statman, I have a few concerns with that system that maybe you can clear up.

First thing I notice is that bluffs should be profitable in position on the river, or else your opponent is calling too much; we shouldn't bluff in position to exploit that.

Out of position, bluffs should be 0ev if our opponent is playing correctly, but it's not the whole story if our opponent can raise.

Last thing is that I think making your opponent's entire strategy 0ev is next to impossible if your opponent is playing anything close to an equilibrium strategy.

Position does not matter here. Profitability of each players range should stay the same even if you flip them.

Obviously we should still be betting all of our nuts. I believe the solution here simply has the bluffcatching player defending his weak bluffcatchers at a lower frequency since he has the nuts to defend with sometimes. The polarized player will still always shove his nuts with a balanced number of bluffs to make the bluffcatching player indifferent with his weaker bluffcatchers, which comprise the majority of his range. The bluffcatching player will always call with the nuts, and call with his weaker bluffcatchers at a frequency which make the polarized player indifferent to bluffing.

This becomes a little more interesting when stacks are large enough that the optimal bet-size is not all-in, or when the bluffcatcher has some hands to beat some of the "polarized" player's value hands.

The optimal bet size will be less than all-in if the defending player is allowed to fold enough of his weaker bluff-catchers that the polarized player is not getting enough value from his nuts because he is chopping too often when called.