So let's say we have the folowing situation:

We are heads up, in late position on the river, and the player op checks to us. Board is 22234 rainbow and both player's ranges on this river before any bets are AA-77. For simplicity's sake only a pot sized bet is possible, and oop player can call or fold.

Now what is the starting point in order to construct a range for betting in position? Game theory says in order to bet optimally one has to add 1 bluff (zero equity hand) for every 2 value hands (the alpha ratio).

However, aren't value hands relative to oop's calling range? So let's say oop player has a calling range of AA-JJ, then my value range should be AA-KK. Should I add 77 here as a bluff?

What if oop's calling range is AA-88. Then I should bet JJ (50%) and AA-QQ for value against that range. Now should I add the bluffs dictated by the alpha ratio here?

Now, I also undesrstand that "if oop calls too much, you should never bluff, and if he folds too, you should bluff (always, if maximally explotatively, right?). Question here is, too much compared to what?

In the same vein, another situation. Let's say I bet, in position, 100% of the pot. OOP now must call with 33% equity against my range. Now if he folds hands that have equal or better equity than 33% that would be a mistake, and if he calls with hands that have less than 33% equity against my range would be a mistake too. How can oop construct a range based on my range, if my range is based in his range?

I would probably start by asking, “what would happen if I bet my whole range here?”
And the answer to that question would give us villains MDF.

now take look at his MDF range. If u can bet and be good more than 50% of the time when called my hus MDF range, u have a value bet.

That wouldn't give MDF. MDF should be half of the range that beats IP's bluffs. IP's best bluff depends on how often he should go for value, which depends on how much of IP's range calls.

Set up a system of equations and use algebra.

This is exactly the problem. This is a problem that has always bugged me. I can run the above scenario in crev and get equilibrium strategies, but that's not what I am looking for, since most opponents don't play perfectly and don't adjust perfectly.

If most opponents don't play perfectly, how do you ever figure If a bluff is correct or profitable? Because for knowing that I would need to know villains mdf, and for that I need to know heros bluffing frecuency, and so on.

How do we solve this sort of circular reasoning when trying to constructor ranges? Which should be the starting point so to speak?

You first solve the game assuming both players are perfect, which is what solvers do, and then any deviation your opponent makes is either a mistake or an exploit against a mistake you are making.

But how did people design (exploitive) ranges before solvers and gto poker were even a thing?

Well finding an exploitative play is easy, you just figure out how your opponent will react to your play and play whatever is better against that with your hand. So if they underdefend, you bluff like crazy. What solvers do actually is exploit themselves back and forth until they converge to a pair of unexploitable strategies

Simplify it a bit and assume that each player has one combo each of 7s7c-AsAc, so that there are no chops.

Start with OOP’s response to a bet and work backwards. Apply MDF to all hands that can beat a bluff (88+). So 3.5 / 7 combos should call. AA, KK, QQ, and 50% of JJ. Different solvers will produce different calling ranges here because there’s multiple equilibria, but the range above cannot be exploited.

Now we solve for IP’s response. Generally the trick here is to find the weakest hand you can value bet, then just balance with the bottom of your range.

IP equity when checked back:


IP equity against OOP calling range:


Looks like KK is the weakest hand that can value bet. You can calculate the EV of a bet and compare it to a checkback to confirm.

Now we simply balance our KK+ value range with the correct amount of bluffs (as determined by pot odds). Our goal is to make hands in between our value and our bluffs "indifferent". 2 combos of value needs to balance with 1 bluff to give villain 33%, so we simply bluff 77. This ensures OOP cannot exploitatively overfold.

IP betting range: KK+, 77
OOP calling range: QQ+, 50% JJ

How do you know 88 beats bluffs? You'd have to know the opponent doesn't bluff with 88 or better.

Assuming a fact (88 beats bluffs) and searching for a contradiction is totally valid, but I wonder how you would approach a scenario that doesn't work out so cleanly. What if you change both players ranges to [0, 1], where the higher number wins? It's not easy to guess the best hand IP bluffs with now that ranges are non-discrete.

All I know for certain is that 77 does not beat a bluff.

The [0, 1] game is interesting. So each player is dealt a random number between 0-1. OOP must check, and IP can bet pot or check behind.

I would go about it the same way. I would just apply MDF to the entire range. I assume OOP will call with any 0.5 or greater. So IP value bets with hands that have higher EV than checking back, and balances with the bottom of their range.

Therefore IP should value bet 0.75+ and bluff anything lower than 0.125.

Is this correct?

IP gains relative to this strategy by checking back 0 through 0.125, because they are all 0 EV bluffs.

Ahh right, the bluffs have equity. Ok this is a bit trickier to solve.

I think the solution is to make the best bluff and the weakest value hand indifferent.

I ran this through a spreadsheet and used google's built in sheet solver to minimize IP's EV relative to OOP's calling range. I'm only using 100 datapoints so not really continuous, but close enough.

IP should value bet 0.79+ and bluff with 0.12-
OOP should call 0.56+

But ok this is kind of cheating. Linear distributions are super hard to solve in general. Is there a more elegant way to solve this @browni3141?

I just use algebra. Part of my point is that I don't think there's a pure heuristic way of figuring out these strategies that doesn't use math. OP's example is easy because the ranges are discrete and narrow, as well as happen to work out to using exactly 2 value and 1 bluff combo.

Let 'v' be the distance of the worst value hand from the nuts. This hand should have exactly 50% equity against villain's calling range. We want to make villain's calling range indifferent, so to keep a 2:1 value:bluff ratio our best bluff is v/2. MDF is applied to the region which beats the best bluff in order to make the best bluff indifferent between betting/checking, and for a PSB MDF has him defend the top half of that region, so villain calls with:
[(v/2+1)/2, 1] = [(v+2)/4, 1]
Our value range is the top half of that region:
[((v+2)/4+1)/2, 1] = [(v+6)/8, 1]
Our best value hand was defined as 1-v before, so we can set:
(v+6)/8 = 1-v
v+6 = 8-8v
v = 2/9

IP should bet [0, 1/9], [7/9, 1], otherwise check. OOP should call [5/9, 1]

It was bothering me that I did slightly more work than necessary, so I did slightly more work to fix it.

Oh that is clever. I like the geometry. I think you're right that these purely heuristic methods are just oversimplified.

There's slight differences between our answers, but increasing the precision of my spreadsheet does seem to converge with your answers. So I think yours is exact.

Nicely done!