I've simplified the game in order to get some valid solutions--but even the simplified version didn't prove quite so simple.

Anyhow, here's the game I tried setting up: Pot is 200 and stacks are 200. EP always checks ("according to the rules" for the moment). Then LP can check behind, half-pot or full pot. If LP half-pots, then EP can CRAI.

I kept getting contradictory solutions here but finally figured out why: LP never bets half-pot here. If the CR isn't allowed, LP bets medium strength hands at HP but it's in order to control the pot WHEN HE'S BEAT. With the CR allowed, he can't do that anyway, so he just never bets half pot, and the solution I get is:

LP VB (full pot): [0,25%]
LP bluff (full pot, too): [87.5%,1]
EP call: [0,50%]

So, basically, playing optimally in this situation (assuming EP check), you never half-pot but only full pot top 1/4 of villain's range, bluff half that % on hands guaranteed to lose if called, and EP still has to call with the top half of his range because of the bluffs.

If the terminology is still incomprehensible (after reading the explanations in the other game theory thread), please let me know and I'll be happy to explain.

And of course, I'd also be very happy to hear from anyone who has read MOP and can check my results.

Note: This solution is still of minimal practical value since EP obviously can and will bet some hands. But my suspicion at this point is that both players have no need for half-pot bets here. If that's the case, then I would draw the practical conclusion of always shoving or checking here. There might be some opponents' mistakes that would be exploited by a half-pot bet, but I don't see why they can't also be exploited by increasing or eliminating the bluffing range here--and then it just gives you fewer things to think about, too.

I haven't read the book and can't do the math but I sort of have an intuitive sense that these numbers make sense.

Are you saying this result translates directly to poker? That is, you should never do anything other than shove or check behind if you have a PSB left in your stack going to the river and you're in position?

It seems to me that there is one mistake that an opponent can make -- namely, the willingness to call a half pot bet but not a full pot bet with a much wider range, plus an unwillingness to check/shove bluff -- that would make a half-pot bet superior. (In fact this combination of factors is pretty common at lower limits.) Or am I missing something?

On the river, the difference due to the drawing aspect is gone. The problem with "direct translation" is only that you have to put your opponent on a hand range--it's not a random distribution of hands. So, you always have to view these percentages in terms of the relative ranking of hands within your opponent's range.

All I'm saying at this point is that "you should never do anything other than shove or check behind" if your opponent always checks to you and otherwise plays optimally--and that's actually pretty weird in practice.

My next step will be to get a solution where EP is allowed to bet full or half pot or check, and LP can call or fold to FP but also raise HP. The only difference here is that if EP checks, LP can still act, whereas that wasn't the case in the first game.

I'm pretty sure that it isn't EP's unwillingness to bluff CRAI that defeats the HP bet--that happens rarely since LP almost always calls for 1 to win 5. It's the fact that you allow him to CRAI on his strongest hands that way, and you have to call most of the time given pot odds if there's any chance at all that he's bluffing--so, ok, you can lay down your own bluffs, but that's about it.

I also think one thing that's making this a bit difficult is viewing it in terms of what happens this particular hand (which may be fine in situations where you're only going to play <100 hands with villain anyway) and what happens in the long run. Short-term, you can indeed get lots of flat calls with mediocre hands here on HP, and, yeah, you're probably ok doing this against a straightforward station-ish kind of player until you have evidence that he's also capable of a CR.

But also note that we're only talking about the range [25%,33.3%] for LP, so 8% of all hands. EP actually SHOULD be calling HP on top 2/3 of his hands and FP on top 50% (if not, you bluff more). Well, I guess I shouldn't say that quite so strongly, since if EP isn't allowed to CR, then LP full pots [0,16.7%] and HPs [16.7%,33.3%], and that's a wide enough range to be significant. Those are probably optimal value bets against a player incapable of CRing. The reason for the 16.7% is that full pot still breaks even for LP at 25%, but in the range [16.7%,25%], he gets more value from half pot as long as there's no threat of a CR.

Also, the station type is going to bet out something here always when he has a big hand, so he always has a weak hand if he checks. So that probably means that LP should actually always HP top 1/3 of EP's range and never FP.

That would at this point mean: Vs. a station, bet half-pot on all top 1/3 of his range (and bluff often, too, since he's very weak--he's a station, so he was drawing almost always and missed since he didn't bet).

Vs. an opponent at least sometimes capable of trickiness, stick with FP (subject to further investigation of somewhat more complex games).

Those aren't bluffs, which serve to force EP to call with a larger range than your VBs. On the VBs, you should win at least 50% of the time on a call (otherwise they have no value)--and it's the bluffs that force the calls with losers.

Now suppose LP just full pots [0,37.5%] and never bluffs [87.5%,1]. Costs EP 200 for a chance at 400. So, he needs to win 1/3 of the time. So, he makes the strategy adjustment of reducing his call range to [0,25%] rather than [0,50%].

As EP, your call range is always top 2/3 of the hands LP will bet.

k3nt, I just realized that I may have misinterpreted your question. Anyhow, it made me start wondering whether my solution really was a Nash equilibrium because it would seem that with no change in EP's strategy, LP could make more by betting [0,37.5%] rather than [0,25%] and [87.5%,1].

I don't think this is the case, because, while the pure bluffs on my suggested optimal strategy only break-even, they do pick up some value on the folds (value they wouldn't have with a check). But let's test it:

On the raises [0,25%] the results are obviously the same. So, we only need to compare what happens on the bets [25%,37.5%] vs. [87.5%,1]. I'm going to normalize the pot to 100 for easy calculation. For easy reference, I'll cal your strategy the value bluff strategy and mine the naked bluff strategy.

1a) LP has [25%,37.5%] on value bluff strategy
EP has [0,25%]: -100, hence -25 weighted
EP has [25%,37.5%]. +50 splitting the original pot, hence +6.25 weighted
EP has [37.5%,50%]. +200, hence 25 weighted.
EP has [50%,1]. 100, hence 50 weighted
Total EV over this range on value bluff strategy: 56.25

1b) LP has [25%,37.5%] on naked bluff strategy (and checks)
EP has [0,25%]: 0
EP has 25%,37.5%]. Again +50 and weighted 6.25
EP has [37.5%,1]. +100, hence weighted 62.5
Total over this range is 68.75.

So, on the value bluff strategy, LP loses 12.50 by betting rather than checking on [25%,37.5%].

How about on the range [87.5%,1]?

2a) value bluff strategy
EP has [0,87.5%]. 0
EP has [87.5%,1]. Split the pot of 100 for EV of 12.50
Total: 12.50

2b) naked bluff strategy
EP has [0,50%]. -100 for EV of -50
EP has [50%,1]. 100 for EV of 50.
Total: 0

So, the total EV for LP is the same on both.

Another simplified game: EP is now allowed to bet, but only full pot. LP can fold or call to a bet, and LP can bet or check behind to a check. Stacks are 100 with pot of 100

Here's the solution I get:

EP VB: [0,13.3%]
EP bluff: [93.3%,1]
EP c/c: [13.3%,53.3%]
LP call bet: [0,50%]
LP VB: [0,33.3%]
LP bluff: [83.3%,1]

Note that EP value bets far less than in the game where we didn't have someone left to act behind. In practice the full pot bet here should be something like only nuts or second nuts on most boards for EP--but EP does bet out his very strongest hands and bluff his very weakest.

LP can bet much looser, and that's really the reason why EP VBs so rarely, I think: to have plenty of good hands with which he can win to an LP VB. And of course the fact that EP bets out his very best means that LP's VBs are top 1/3 rather than top 1/4 in the game where EP always checked.

Interestingly, if we make the pot 200 and remaining stacks 100, so that only a half-pot bet is allowed, here's what I get:

EP VB: [0,30.0%]
EP bluff: [90.0%,1]
EP c/c: [30.0%,70.0%]
LP call bet: [0,66.7%]
LP VB: [0,50.0%]
LP bluff: [83.3%,1]

Here, EP bets out almost as much as he would if LP couldn't bet to a check. In the game where EP checked always, LP only bet here [0,33.3%], so only 3% more often than EP bets out half-pot.

Ais, back to my question, with perhaps a more naive question thrown in:

How does EP know whether LP is betting [0,25%] + [87.5%,100%] or [0,37.5%]. All that EP knows is that LP is betting 3/8ths of his hands.

Also, I'm not quite good enough to make a good argument here, but it feels like the value bluff has to be better than the pure bluff. For the simple reason that you always lose the extra money when called in the pure bluff strategy, but you actually suck out a win part of the time in the value bluff strategy. Those times that you suck out a win on a larger pot size have to improve the overall EV. Don't they???

I'm probably missing something really basic here, as I have very little experience in this area. Oh well if you're not willing to make a fool of yourself, you rarely learn anything really new, right?

Thanks for your help here.

First, how does EP know? The hands get shown down sometimes, too. At the beginning, EP doesn't even necessarily know that LP bets 3/8, and LP doesn't necessarily know that EP calls 1/2.

Basically, the optimal solutions (on which we ASSUME that each player knows the other's strategy but not the other player's hand) can be viewed as what play gravitates toward if they play some very large number of hands.

If both players can do some figuring between hands with notes on what is shown down, it probably generally won't take more than maybe 20 or so hands to get there. But if both players start blind and just take stabs, it is likely to take a lot more--or forever, if one player misses some options.

On your second question, about value bluffs vs. pure bluffs: My EV calculation above should prove conclusively that they're equal.

I think where you're getting confused is that you're comparing only the cases where LP bluffs and EP calls--i.e., LP has [25%,37.5%] and EP has [0,50%] on the value bluffs with the case where LP has [87.5%,1] and EP has [0,50%] on the pure bluff.

You also have to consider the folds, though. On the folds, LP gets the same result on value bluffs as he would have by checking (namely winning 100), but not on the naked bluffs, since he folds out a lot of winners.

The amount that LP gains through value bluffs on the calls is, as it turns out, exactly the same amount that he loses by just checking behind on his worst hands that almost never win at showdown.

If you figure EV through all ranges, those 2 strategies turn out to be exactly equal if EP just sticks to his strategy of calling on [0,50%].

But the value bluff strategy is unstable because EP can change strategies and increase his EV. With the pure bluff strategy, there's no way for EP to improve his EV over that of just calling [0,50%].