This idea is only a few weeks old in my mind, so here's some salt before I ramble: (***)

Most of us by now know about the indifference equations that were once believed to rule poker strategy. However, I think that these equations are only applicable to the river. When considering indifference before the river, these equations do not work because of the possibility that individual hands may improve or regress on the next card, as well as the possibility of winning or losing money on future streets.

So? How do we make our opponent indifferent before the river? I believe that the answer is to make ourselves indifferent, thus rendering suboptimal any and all deviations from optimal play by our opponents.

This is self imposed indifference and it is key in maintaining a minimum ev vs any and all deviations by our opponents.

By having a default strategy that utilizes this concept of self imposed indifference, we have a baseline from which to deviate and exploit if given the opportunity.

How did I get here?

https://forumserver.twoplustwo.com/1...d+indifference

https://forumserver.twoplustwo.com/1...olved-1707286/

Now? I'm still down the rabbit hole; won't you come join me?

No, I don't think that's right.

I agree the basic indifference equations only apply to the river. However Tipton in his "Expert HU NLH Vol. 2" shows how they can be modified to apply to the turn.

Sorry but I can't get past your first statement. My understanding of GTO play is EV of a bluffing range = 0 = EV of a bluff catching range. Anything different means neither opponent is playing GTO.

semibluffs are +ev. strong bluffcatchers are +ev. thus the average ev of both a bluffing range and that of a bluffcatcher range are positive values.

the worst semibluff is 0ev. the worst bluffcatcher is 0ev.

This is only true heads up out of position on the river for the bluffs. This is only true of bluffcatchers heads up on the river when those bluffcatchers cannot beat any value hands.

Just checking in. Love these kinds of threads, makes you think.

Worst bluffcatcher is 0ev, some can be +ev due to blocking effect and thus played as pure calls.


OTR most bluffs are usually 0EV in most scenarios, some bluffs can be +EV.

OTT and OTF all "bluffs" are +EV, if you think bluffing is close to 0EV, then check, which almost surely is +EV.

This is true if we define bluffcatcher as a hand that can only beat a bluff, but in real poker ranges overlap and there are bluffcatchers that can beat a fraction of the bettors value range and are thus profitable, which brings up the average ev of the bluffcatching range.
This is true heads up out of position on the river. Heads up in position on the river all bluffs should be as profitable as checking back the bluff with the most showdown value. That is unless one or more players made terrible mistakes earlier in the hand thus rendering the bottom of the in position player's range entirely unshowdownable.

Perfect. Thanks.

Here's an example of what maintaining indifference means to me. Can you please let me know if this is in any way relates to what you mean by self imposed indifference as I read and re-read your post but am not sure I get what you are on about. Giving an example would help, thanks.

Say hero is on OTT and has enough behind for two pot sized bets to be all in OTR. We know from the indifference equations that if hero goes all in OTR his bluff:value ratio is 1/(1+1) = 1/2. So he has 33% bluff and 67% value in his betting range where his bluffs have 0% equity and value have 100% equity.

Now let's go to the turn and imagine his value range has 100% equity and his bluffs 0%. Then he would need to have 67%*67% = 45% value and 55% bluffs in his turn betting range. OTR he would drop 1/3 of his betting range (his turn bluffs) in order to have the correct bluff:value ratio (1:2).

Now the problem is that hero doesn't have a perfectly polarised range OTT. His value has equity <100% and his bluffs have equity >0%.

One example of how to get around this is to extrapolate the approach described by Janda in his book. Hero examines his range OTT and selects the combos he wants to bet for value. Let's say he selects 32 value combos with total equity of 80%. He then needs to select X bluff combos with total equity of Y to ensure the following indifference relationship holds:

(32*.80 + X*Y)/(32+X) = 0.45
25.6 + X*Y = 14.4 + 0.45X
11.2 = X(0.45-Y)
Y = 0.45 - 11.2/X

So if hero decides to bluff with 30 combos for example he needs to make sure the 30 combos have a total equity of 0.45 - 11.2/30 = 8%. He can pick whatever combos to bluff with but needs to make sure the total equity is 8%. This gives him freedom to combine good draws with air hands to build his bluffing range.

This tells me indifference holds OTT, if adjusted to account for equities.

The thing is that those value hands will not necessarily be value hands on the river. No matter the card that falls, except in very narrow range situations, there will always be improvement and regression within ones range.

Thus planning on having y value hands and x bluffs in your river range, before the river falls, cannot be correct.

Agreed, some of the value combos will regress and some of the bluff combos will improve. In my example 32 value combos and 30 bluff combos are bet OTT for a total of 62 combos. Now 1/3 of those combos need to be dropped to create a balanced river betting range of ~ 40 combos. That yields ~ 26 value and 13 bluff combos to bet on the river. So yes there are less value hands OTR however indifference is maintained. Note those 26 river value combos consist of value combos from the turn that held onto their value + bluff combos that improved. They do not include the value combos that regressed.

Where's the proof that you have 26 value combos on every possible runout? Sometimes you might have less or more, but that's ok because we give up more or give up less on certain river cards depending on how the ranges stack up against each other. Bluffing frequency is maintained but the number of value and bluff combos may vary. This maintains indifference, not a predetermined number of value combos. Since every rivercard is different, and ranges are asymmetric, I think the full solution will be much more complicated than above.

I think we need to go back to this part with an example that leads up to this turn decision and then to the river.

100 big blind no limit holdem 3 handed.

button raises 3x, small blind calls, big blind calls.

flop(9bb) A65r

checks to button, button bets 6bb, small blind folds, big blind calls.

turn(21bb) 7r

checks to button, button bets 21bb, big blind calls.

my favorite rivers here are in order of how much I want to see these cards:

7, 6, 5, A, K, 2, Q, J, T, 3, 9, 4, 8.

That's just a guess at the true order of river card preference. I picked those cards in that order for a few reasons:

(765) allows the opponent to hold many strong hands to pay off the button's huge hands like quads and full houses. (A) just seemed like the obvious next choice despite the blocking qualities. (K2QJT) I think these are in the wrong order. (3948) the straight cards that really hurt the button's value betting range, I think these are in the right order.

I think we should see fewer river bets from the button as we go through the list of my favorite river cards, but bluffing frequency will be maintained.

----

It's about how the marginal hands have a metering effect on the profitability of the opposing marginal hands.

If you can keep the margins where they are,* then you can maintain a minimum ev of zero.

If you can't keep the margins where they are, then you're either gaining ev vs some opponents or you're losing ev vs other opponents. The more aware you are of where the margin is in a particular situation, the more likely you are to be the one gaining ev.

From the example hand: say there's a low frequency turn bluff with T8s at a frequency of 25% bet/75%check, but I mess up and over a large sample I bet 50% of the time.

If 25% is the betting frequency that maintains the margin of the big blinds hands that should fold the turn and those that should call, then my mistake of betting too often has caused the margin to shift; there is ev to be gained by my opponent. I have not maintained the margin and thus I have not maintained self imposed indifference; if my opponent is not satisfied with his minimum ev of zero, then he is free to exploit me for a profit greater than zero. If he declines that profit, while maintaining his minimum ev of zero, then this will be self imposed indifference.

It's a bit like asking the question: should you cut cards or flip coins for even money bets?

There won't be 26 value combos on every possible runout. However the average number of value combos will be 26 across all runouts. Because of this hero can bet pot on the river with 67% frequency and villain will respond as if hero has the 26 value combos and 13 bluffs. Taken across all runouts both players are playing GTO.

Given there is some overlap in ranges cards that hurt your value betting range can be the same cards that turn bluffs into value hands. You can arrive at the river with ~ the same combination of value hands, they just won't be the identical hands that you value bet the turn with.

This.

Remember the reason why you need to be able to bet river with average freq of 67%.

Got it. Both player are playing GTO to begin but in your example hero increases the bet frequency of his marginal bluffing hand and thus deviates from optimal play. Villain now needs to call a bit wider (shift his margin) to keep hero from gaining Ev with his bluffs.

Actually, since given the assumption is that it's a mixed strategy hand, there is no ev gain for hero vs strict nash equilibrium, but there is potential ev gain for the big blind.

I’m assuming that hero’s overall bet frequency (value + bluffs) has gone up by a small amount and that villain’s call frequency remains as if nothing has changed. Villain is thus over-folding and hero is gaining ev with his bluffs. So villain needs to bring things back to equilibrium by calling a bit wider.....what am I missing ?

Assuming the bold is true, that would mean that the big blind wasn't playing his part of the equilibrium from the start since he can increase his ev by changing his strategy. My mistake of betting the T8s 50% instead of the assumed correct frequency of 25% doesn't immediately lose or gain ev vs nash equilibrium because the other action of checking has the exact same ev vs nash equilibrium. However, my mistake of betting the T8s is a liability vs an exploitive agent.

Ok back to this. So it seems that the flop and turn differ in this aspect:

flop: two variable cards to come; one dynamic game street to play; one static game street to play.

turn: one variable card to fall; one static game street to play.

So I see how you go from (river betting range) --> (turn betting range considering all rivers) and that's how you come up with turn strategy even if we're talking about mixed strategies with partial combos in the turn betting range. However, does this property transfer to the flop as well?

I agree an optimal villain will immediately adjust to hero’s mistake and exploit.

“If he declines that profit, while maintaining his minimum ev of zero, then this will be self imposed indifference. “

In this sense, we could define self imposed indifference as declining exploitive profits.

Let me play out a scenario:

1. Both hero and villain start out at perfect equilibrium.
2. With the T8s hand hero changes his bet/check ratio from 25/75 to 50/50 and thus deviates from optimal play. He now has too many bluffs in his range for the bet size he is using.
3. Villain at first does not notice this and continues to fold at the frequency defined by the equilibrium. He is thus over-folding and hero is gains ev with his bluffs. The ev that hero gains is the ev that villain loses.
4. Villain eventually notices hero's mistake and makes an exploitative adjustment by calling 100% of the time with his bluff catchers.
5. Hero notices villains mistake and starts betting with only his value hands.
6. This is an unstable situation where hero and villain exploit and counter exploit.
7. Equilibrium is restored when hero goes back to 25/75 with his T8s OR increases his bet size to account for his increased bluffing.

Where does your idea of self imposed indifference fit with this scenario ?