I'm talking specifically about full-street [0,1] games, which are directly applicable to river betting with symetric ranges, but there are many more applications.

We already know that if betting is capped at 1 bet, the first player will simply bet his best hands for value and some of his worst as bluffs. We also know that if we push the betting cap up to 2 bets, the first player will need to begin to check-raise some hands, to prevent the second player from simply value-betting when checked to. We also know that the first player cannot simply raise the strongest of the hands he checked originally, as they are too weak to raise. Instead, he must check-raise the very top of his range, and some bluffs. This way, when he checks and is bet into, he can raise with a range which is threatening to the second player's value-bets.

So, we know that we should bet out with the nuts if betting is capped at 1 bet, but we should check-raise the nuts if it's capped at 2 bets. It's my theory that we must be able to put in the last bet with our nut hands. So, if the betting were capped at 3 bets, we would want to bet-3bet with our strongest hands. If betting is capped at 4 bets we would want to check-raise-cap with our nut hands. If betting were capped at 5 bets we would want to bet-3bet-cap with our nut hands, etc etc etc. (And of course for every line taken, there would be an appropriate bluffing region to go along with it.) This would ensure that we can properly value-raise our opponents next-to-last raise. (For example, if betting cap is 4 bets, when we c/r and get 3bet, we need to be able to cap both for value and rarely as a bluff)

Thoughts so far?

Would these value regions alternate? In other words, for 4 bet cap, would we check-raise-cap our strongest region then bet-3bet-call the next strongest region, then check-raise-call the next strongest region, etc, or just check-raise our strongest region and cap the strongest of those, call the weaker and fold the weakest?

If our nut-region alternates between betting out and check-raising as the betting cap is increased (bet for odd caps, c/r even caps) what happens when there is no betting cap?

Can we apply this to NL in the manner of; if our stack is smaller than the pot maybe we would just open-shove our nut hands, but if we had 3X pot we would want to check-shove our nut hands instead? Or does our opponent's ability to under-bet or over-bet make this totally irrelevant?

While the answer depends on the precise specification of the game, there is a general principle that you bluff with the weakest possible hands because otherwise you are wasting strength.

Suppose you map out the full tree of possible betting paths. It's reasonable to assume that you arrive at every terminal node with some strong hands and some bluff hands. That may not be true, there can be corner solutions such that you either never end up at a node or never end up with one or the other type of hand, but generally you want to use every possible path. Also "strong" may not be very strong in some cases, it just means a hand that will often win if it gets to the node.

Now think about assigning starting hands to choices. If you ever end up with a bluff hand at a node, when your hand has enough strength to be a strong hand at another node, you wasted some strength. Generally you can rearrange things to use that strength elsewhere. Again, this isn't mathematically certain, just a rule from experience of solving a lot of these kinds of problems.

This leads to general advice to do pure bluffs with the absolute weakest hands, and save hands with any strength at all to be non-bluff hands on some possible betting paths.

This principle stops working when you get away from 0,1 games. If there are multiple dimensions of hand strength (like a small pair in hold'em which has some strength but is less likely to improve to a really good hand than mid suited connectors) and additional information revealed between rounds (also like poker) then the pure-bluff principle is less powerful. It's still worth considering, however, you hate to find yourself bluffing when your hand has the strength to win in other circumstances.

i'm not sure if the GTO play with the nuts alternates in the way you describe, but it is a very interesting conjecture and it seems reasonable to me.

one thing which i think is worth pointing out from a practical standpoint, is that, after a few raises go in, hand ranges get pretty tight. in my experience, card-elimination effects can then be very important, and are often the dominating factor in the play of any particular hand combo. in this case, there are really no longer any well-defined contiguous betting regions.

If the continuous [0,1] game is descretized to e.g. [0, 0.001, 0.002, .... 0.999] do we expect the solution to be of similar structure or could it be radically different? (I can't solve continuous games...)

I strongly suspect the solution remain almost the same.

Or you can try to solve [0, 0.01, 0.02, .... 0.99] game and then add a lot of state for each bucket X that have a different strategy from the previous or next bucket. (so solve [0, 0.01, 0.02, .... 0.06, 0.062, 0.064, 0.066, 0.068, 0.07, 0.08 .... 0.99]), and possibily iterate few times. Every iteration add only a fixed number of new bucket for every change in strategy, so this should be computationally feasible.
(and you can merge bucket too, if they share the same strategy)

(as a side note, has anyone done studies on the [0-1] game with infinite stack, infinite round of betting and optimal bet size? or with finite stack size but no restriction on number of possible bet and size of the bet? where can I find such research?)


The conjecture seems reasonable and interesting to me too.

The problem for infinite stack (so no betting cap) don't seem a real problem, there should be an infinite number of region that start with check or with a bet, I think it's not well defined the strategy for "the last hand".
Eg. you have that hand in 0.999-0.9999 start with checking, hand in 0.9999-0.99999 with a bet, hand in 0.99999-0.999999 a check, and so on.

It would be almost identical, only there would be a possability of having to use mixed strategies to get the exact frequencies. (For example you may have to bluff hand .983 30% and check it 70%)