I seem to be drawn again recently to this [0,1] game model and thought I'd attempt an analysis of river bets from that perspective. One thing that should make it more accurate is that there are no cards left to draw, so there are only winning and losing hands.

What I'd ultimately like to get to is the following scenario: Pot of 200 and stacks of 800 remaining with 2 players in. Now EP can bet 100 or 200 or check. Then LP can bet 100 or 200 to a check and push to either bet. EP can also CRAI if LP bets. I'm not at all sure that I can get this solved, but I'm trying some simplifications first, which I think just might get the full problem solved.

Here's what I tried first: EP just checks his entire range by assumption, and then I stipulated which bet was allowed for LP.

Before getting to the solution I got, one note: Since in practice there's always some kind of betting sequence beforehand, the percentages given apply only to villain's hand. If you're supposed to bet half-pot on top 31.5% (solution I get), then it doesn't mean you should bet top 31.5% of YOUR range but rather top 31.5% of VILLAIN'S range, then the other way around when you get to the other player, etc.

Ok, now here's the comparison: If LP is allowed only the half pot bet, here's what it looks like by my calculations:

LP VB: [0,31.5%]
LP bluff: [89.5%,1]
EP value CR (VCR): [0,5.7%]
EP bluff CR (BCR): [63.0%,66.7%] total of 3.6%
LP call CR: [0,11.5%]

And here's what happens when the bet is full pot:

LP VB: [0,23.3%]
LP bluff: [88.4%,1]
EP VCR: [0,5.8%]
EP BCR: [46.5%,50%] total of 3.5%
LP call: [0,11.6%]

What I'm still not sure about is how to put these together so that now LP has a choice of whether to bet full pot or half pot.

What I thought was interesting here is that LP is actually SUPPOSED to bluff a bit more often on full pot, although not much. And it makes sense since more LAG-ish types are more prone to full pot the river. And the VB range for LP is substantially tighter, although I'm not entirely sure whether this really matters in practice, since you can usually boil it down to a few possible hands by the river given the previous betting.

Also interesting to me is that on optimal play EP VCRs and BCRs almost exactly the same number of times regardless of whether the bet is half-pot or full pot. And LP calls the CR the same amount. And with the percentage for the VCR at 5.8%, that's so low that it should in practice almost always be the nuts. Exceptions are allowed only when you're pretty sure of getting a call on a decent range of worse hands. In any case, any VB here always means that you think you'll get a call from at least as many losers as winners, and if villain bets the river, he already probably has a pretty decent hand.

I have a different theory question.

Is it possible for me to be one of the best players in the world at my stakes and not understand a single word you just said?

Ok, here's what the [0,1] game is: 2 players are dealt random numbers between 0 and 1, and the low number wins at showdown (if it goes to showdown).

Sticking with this game for the moment, what it allows you to do is to give exact "optimal" percentages for value bets, bluffs, etc. And... (to pick up on dropthe72's remarks) once you get that far, it also allows you to calculate the best way to exploit mistakes. Since there's no drawing in this game, it obviously doesn't capture certain hugely important aspects of poker. But of course, on the river, there's also no drawing. On other streets, I think the game also has plenty of relevance, but you do have to bear in mind the fact that underdog hands can catch up in poker.

What I'm trying to do ultimately is figure out when to half-pot and when to full-pot the river--optimally to begin with, and I think that's a good start for thinking about how to exploit mistakes. In situations where you can narrow your opponent's range down very closely and also have a read, those facts are probably going to make things pretty much crystal clear without all of this theoretical superstructure. But where you can't (like flushy boards with a straight--where you have a large number of hands that can still be playing, calling, CRing), I think it does help. Possibly even on paired boards with multiple possible fulls as well as possible quads. But anyhow, while those are the aspects I'd ultimately like to explore (somewhat like we did in the SB/BB simulations), I'd first like to get to the optimal solutions in the simple model.

Now for the numbers in brackets as well as the basic set-up. The originally proposed game with full-pot, half-pot and check-raises for both players with lots of stack-depth, is so complex that I can't solve it directly. So, I started with some much simpler games.

In the one, EP actually isn't allowed to bet (checks his entire range), and LP can only half-pot. For calculations I actually assumed a pot of 2 and stack-depths of 8 because I do allow EP to check-raise all-in (you can multiply the 2 by whatever factor you need to get a simulation for the game you're playing--for example, pot of 200 if that's about what river pots look like in your game, or pot of 20 if that's what river pots look like, etc.).

When you solve this game, the answer turns out to be (or at least the one I get is) that LP value bets half-pot on the interval [0,31.5%]. In other words, he half-pots here his best 31.5% of "hands." And LP bluffs on [89.5%,1]. Translated: He bluffs his worst 10.5% of hands. Of course, if he never bluffs, then EP calls (or CRs) according to pot odds somewhere in the interval [0,31.5%] (like about [0,22%] for calls). But because of the bluff option, EP has to call with a large number of worse hands. If he doesn't, then LP should bluff more often.

Then I looked at what happens when you change the rules of the game so that LP can only bet full pot. Here, he value bets fewer hands, bluffs more, and EP calls fewer hands. EP's optimal CR range is about the same, etc.

Does that help?

Yes that is basically correct.

The reason I haven't attempted to apply this yet is that I don't know the solution. In fact, I barely have a clue for when EP half-pots, full-pots, and CRAI. But to illustrate how it would apply, I'll just assume a solution (that's definitely wrong since I haven't figured it out but should be moderately close):

Let's suppose also as a start is that one of you and I are in a FR 5/10 game with full stacks, unraised pot with 4 players seeing the flop, which comes Jh3h2c with heart draw (I'll leave out straight flush here, although that gets quite interesting). I've limped in MP1, and you have called OTB. I bet the flop at 30, and you call. Turn pot 90. Turn is 6h. I bet 60, and you call. River pot is now 210 with 800 in stack depth, so we have exactly the situation in terms of stacks. River is 9c.

Now there's a very wide range that we both can have. I can have TPGK or TPTK, 54s no flush, set, or any flush. You can have the same except for AJ, which you would have raised, and sets are also reduced in frequency because you often would have raised flop or turn with a set. You also would often have raised at least non-nut flushes on the turn, and you might also have raised nut flush draw on the flop. Also, as last to act, you have absolutely nothing (flopped gutshot, naked Ah, complete air) some percentage of the time because you thought this was a scary board where you could likely take it away from me.

Now let's suppose the optimal solution looks like this: Note: please don't get bogged down on all these numbers (they're invented anyway at this point). Their only purpose is to give a rough idea of how we're playing, so think of it more as general orientation for what we're doing on our best, worst and mediocre hands.

EP CRAI for value: [0,5%]
EP value bet full pot: [5%,15%]
EP vaue bet half pot: [15%,25%]
EP bluff half pot: [97%,1]
EP bluff full pot: [92%,97%]
EP bluff CRAI to full pot: [48%,50%]
EP bluff CRAI to half pot: [65%,67%]
LP VB full pot to check: [3%,20%]
LP VB half pot to check: [0,3%] and [20%,40%]
LP call CRAI: [0,10%]
LP value shove to full pot: [0,10%]
LP value shove to half pot: [0,20%]
LP bluff shove to half pot: [60%,67%]
LP bluff shove to full pot: [45%,50%]
LP bluff half pot to check: [92%,1]
LP bluff full pot to check: [86%,92%]
LP call full pot: [10%,45%]
LP call half pot: [20%,60%]
EP check-call half pot: [5%,48%]
EP check-call full pot: [5%,65%]

Now let's translate to the various hand ranges.

Let's first consider what EP wants to do. Your range is now any KJ (you would have laid down QJ or JT on the turn), sets, straight, any flush and basic air. The air often has Ah and occasionally a pair. And the sets, straights and non-nut flushes are all discounted somewhat because you didn't raise the turn, nut flush somewhat because you didn't raise flop.

If I have the nuts, according to the model, I'm checking. And I'm only betting at all on top 25% or your possible hands. That should mean that I'm checking roughly anything except a flush or straight in EP. So, my bets are all flushes or straights, and I'm half-potting the lower half, full-potting the top half of those (except the nut flush). I also have to bluff something if I want you ever to call without a flush. So, I full pot bluff roughly QJ and half pot bluff JT. I'm always beat if you call and I have either of these.

Now let's look at LP's ("your") action. To a check, LP value bets full pot on all flushes except the nuts. LP value bets half pot on the nut flush, any straight, and 33. Note that for the LP value bets, we're looking not at LP's range but at MY range. LP checks behind on KJ and 22 basically as well as some of the air, depending on frequency of air. But on a lot of the air, LP will also bet half-pot or full-pot (bluffing half-pot pretty rarely but full pot somewhat more often). The full-pot bluff will often be with naked Ah since that means I can't have the nuts and hence didn't go with the CRAI for value.

LP's action to a FP bet: Since I bet FP, I have a flush but it isn't the nuts--or I'm bluffing. So, value shove on nut or K-high flush. Call on any other flush, straight, and probably 33 (only to prevent me from bluffing more often--I always lose to your value bets and win only against your bluffs). Lay down KJ and 22. Bluff shove Ah. I call your shove with about 86s or better. Again, I'm beat every time you made a value shove but win against your naked Ah bluff.

LP's action to half-pot bet: I have a low flush or a straight on my value bets. Again, you bluff shove Ah and value shove now any flush bigger than 7-high. Because of your bluffs, I still have to call with the 5- or 6-high flush.

I think that's covered all the bases on my hypothetical optima--at least enough to give an idea of where this would be going in practical contexts. It also shows, I think, that the whole thing is mainly going to have applications on fairly rich boards where both players can have a fairly wide range. One spot where I think (?) it might also have some relevance is in raised pots on paired boards where there's maybe only a feeler bet prior to the river--so that you have in the mix something like overpairs, trips, various boats, also quads and sometimes flushes as well.

I also think that the above analysis shows how the percentages are relevant: One question I have without calculating it is whether EP actually does typically want to check the nuts on the river or whether there's more value in a bet, whether it be half- or full-pot (I do note that to half-pot, LP should be shoving quite a bit).

This is definitely good stuff. I hope you'll keep posting.

I dont know how the solutions are calculated as it gets more complex- is it possible to do a computer analysis? Or is it always necessary to solve it manually?