Thin call?

[Aisthesis]

Ok, this is a purely theoretical situation brought up by Troll's analysis on my river bet question. I'm just going to modify the situation a little bit and see if it gets us anywhere.

Let's call if a 5/10 game with full stacks. 3 players limp, and button raises max to $65. Blinds fold, and the 3 limpers all call. So, the flop pot is now $210. Flop comes K52r, and it's checked to LP, who also checks.

Turn is now an offsuit 9, so the board is still rainbow. Now UTG bets pot of $210, and it folds around to LP, who has 9TJQ.

Now, on this board, there's really no draw with more than 9 outs, and LP himself happens to have the best one of those. So, he is quite certain here that UTG has a set that he had intended to check-raise on the flop. I'm going to discount hands like KQJ9 or KQT9 mainly just in light of LP's own hand. I just mention them as the one other alternative, but I'm just going to assume that EP has some set, probably KK.

Here's where Troll's bluffing ideas come in. LP actually doesn't have odds to call this bet, since he only has 9 outs. And it makes little sense to raise, since KK will clearly just push on this board, and that's the most likely set anyway.

But is there a strategy here that would make this call correct?

I don't know the answer, but I suspect that with Troll's bluffing ideas it's probably pretty close but a good call. I'll start to try to answer this in a separate post.

[Aisthesis]

Ok, to start with, some further assumptions. First, since we're talking about calling, EP is going to know (whatever set he has) that LP doesn't have KK. And since LP raised, EP will put him on a wrap including either TJQ OR 678. If LP bets something like a 3 or 4, EP is always going to call, and LP "knows" this.

But EP doesn't know which wrap LP has. It's just 50-50 either way. And all EP can really do is just call purely randomly (we'll also ignore his sidecards, which I'll just assume are blanks--let's say 33 just so that he has no blockers or redraws, which should cause some changes in his call frequency) if one of those straights hits.

The questions are thus: Assuming LP makes a pot-sized bet here 100% of the time if he does hit his straight, how often should he make a pot-sized bluff on the straight he doesn't have? Secondly, how often should EP call when the middle or big straight hits?

If we can figure that and then figure LP's EV on the river, we should then be able to figure out whether his turn call was correct or not.

[Aisthesis]

Now, EP's call frequency has to make LP indifferent to bluffing the river he doesn't have. Hence, EP is forced to call either straight 50% of the time. Let's just say he does this randomly.

So, the second question is: How often should LP bluff? His bluffing frequency should make EP indifferent to calling. If he never bluffs, then EP should never call the bet. And if he always bluffs, EP should always call, because then EP wins 50% of the time, whereas he's getting 2:1 on the call.

So, let's say LP bluffs the river that he doesn't have (6, 7, or 8) with a probability of p. What is the value of p such that EP is indifferent to calling or folding?

We now have a river pot of $630. Knowing that LP bluffs with a probability of p, let's look at EP's best call frequency (if EP thinks p is too high or too low, he doesn't have to stick with the 50%).

Let's say a 7 hits on the river, EP checks, and LP bets pot. EP is indifferent to calling if he wins 1/3 of the time. LP thus needs to bluff the straight he doesn't have exactly half as often as he bets the straight he does have. So, p = 50%. That way, EP will win exactly 1/3 of the time if he calls.

Summary to this point: EP should call either straight half the time, and LP should bluff the straight he doesn't have half the time.

I'll calculate EV for LP in a separate post.

[Aisthesis]

Now the question is whether LP, with this optimal strategy, can get back the $210 he put in on the turn.

Note that if EP always folds to either straight, then LP effectively had not just 9 outs but 21, so the call was definitely warranted. And if EP always calls, he gets huge implied odds on the made straight (which is the only one that LP bets), and hence the call is also clearly going to be justified. But it's unclear whether it is or not with these optimal strategies for both players.

From the standpoint of the turn, we have 12 known cards, leaving 40 in the deck. Of these 19 are effective blanks (or pair the board), and LP just loses $210 in those cases.

I'll leave out division by 40 until the very end, so that makes a subtotal on EV of -19*210.

Scenario 1: LP makes his straight.

This happens 9 times, and of those he wins $420 half the time (EP folds), and wins $1,050 the other half (EP calls pot). When EP folds, that makes a subtotal of 9*210, and when EP calls, we get a subtotal of 45*210/2.

Switching to increments of 105 for ease of calculation, that gives us (18 + 45)*105 = 63*105 as subtotal on scenario 1.

On the blanks and paired boards, LP has lost only 38*105. So, with only the bluffs left, LP has made 15*105 with the call up to this point.

Scenario 2: LP misses, but a scare-card hits (6, 7, or 8).

This happens 12 times. Half the time, LP just loses his turn bet of $210 here. So, -12*105.

The other 6 times: He wins 420 half the time (= 12*105) and loses 630 + 210 (= 5*105) the other half.

So, the total EV here is -12*105 + 12*105 - 15*105 = -15*105.

Wow! This is amazing. If all this is correct, then the call is EXACTLY break-even with perfect strategies for both players!!

That means that the call is definitely good on either of 2 scenarios: First, if EP calls less than half the time and allows LP to bluff profitably on the river he doesn't have. Or, secondly, if EP calls more than half the time and gives LP superior implied odds when he does make his hand.

[RoundTower]

[ QUOTE ]
So, let's say LP bluffs the river that he doesn't have (6, 7, or 8) with a probability of p. What is the value of p such that EP is indifferent to calling or folding?

We now have a river pot of $630. Knowing that LP bluffs with a probability of p, let's look at EP's best call frequency (if EP thinks p is too high or too low, he doesn't have to stick with the 50%).

Let's say a 7 hits on the river, EP checks, and LP bets pot. EP is indifferent to calling if he wins 1/3 of the time. LP thus needs to bluff the straight he doesn't have exactly half as often as he bets the straight he does have. So, p = 50%.

[/ QUOTE ]
I haven't followed the whole thread perfectly but this looks like a mistake. There are 12 live cards that are 6, 7 or 8, and there 9 live cards that are T, J, or Q. So LP should bluff 3/8 of the time he hits a 6 through 8, not 1/2 the time.

[RoundTower]

also there are some basic mistakes/omissions, e.g. you have the pot size wrong on the flop, and not given any reason why he can't have for example 3467 which gives you 17 outs almost all to the nuts.

also by this
[ QUOTE ]
Summary to this point: EP should call either straight half the time, and LP should bluff the straight he doesn't have half the time.

[/ QUOTE ]
you should mean EP should be equally likely to call either straight. The percentage of the time he should call when a scare card comes doesn't matter if LP makes him indifferent to calling. But in fact in the third post you seem to have used it to mean that EP calls 50% of the time when a 6-8 or T-Q hits, which is wrong.

I like the idea of the analysis and I think you are nearly there but have to redo all of the third post because of this flaw.

[wazz]

Excuse me for my ignorance. I've read MOP and quite enjoyed it; I have a strong background in maths so didn't really find any of it difficult going. However, I fail to see the value of a lot of the in-depth game theory discussion. This game is simply too complex and convoluted to be able to accurately quantify bluffing/v-betting %s on the spot, especially when you deal with the hugely incomplete information of our opponent's though process. Sure, you can come up with some good guidelines for how often to bluff when you've missed your draw, but intuitionally the very good players will be very close to this by nature of being good players. How can such in-depth analysis of an isolated type of drawing hand be of any real value on the tables? I mean, a lot of people will be saying to themselves 'based on the game theory discussions, I will bluff my missed draw 20% of the time.... hold on, let me look at my watch. Ok, the second hand is in the first 12 seconds, I bet.' While you can clearly aid your analytical skills by looking at these situations, I question the value of spending more time doing this than playing and analyzing your game, especially when you allow an artificial external factor to justify your play.

[Aisthesis]

lol, yes, while I'll have to think about it to know if 3/8 is the correct number, your point is one I was thinking about (and ignored for simplicity's sake): Whichever straight hits, it's somewhat more likely that LP has the other one (4 cases of the missed one to 3 of the one that hit).

This definitely has consequences for LP's bluffing frequency, and I'm pretty sure you're right with the 3/8 figure.

There are two reasons why I ignored this in the initial calculation: First, I just thought I'd be close enough anyway.

Second, it seemed like it could get VERY complicated, because if you open that door, then you start to get into issues with EP's sidecards and just which set he actually has. For example, if EP has KK33, that actually makes it less frequent, I think, that LP has the QJT wrap because now KQJT is much less frequent. But if EP has a hand like 9988, more often than not, LP will have the big wrap. Similarly if EP holds KK78, then the big wrap is more frequent, etc. Anyhow, EP's side cards as well as the particular set he has should also cause adjustments to his call frequency: Just as LP should bluff a little less because the straight that he doesn't have hits more often than the straight he has (at least as far as he knows), EP should call a bit more often if his hand suggests that one of the wraps is more probable than the other.

Actually, just thinking about it while writing this post, I'm pretty sure your 3/8 number is right. It's a reduction of exactly 1/4 from the 50% number, and that should be right on the money.

[Aisthesis]

I'd also question the value of spending more time doing this than analysing OTHER PLAYERS' games.

In my experience, there are very few players who will have the "golden call frequency" in EP. Either they're sufficiently loose that there's no reason for them to have a set here, or they're very tight and will almost never call if either straight hits. In the one case, it's very dangerous for LP to bluff at all (but may have a hand with some showdown value if he has the bigger wrap with a pair on either end), and in the other, LP should call and bluff the straight always.

Just as side note: I don't know whether any of you guys followed the whole discussion here a while back when Sklansky and Ankenman got involved in some [0,1] game stuff. Anyhow, I did get rather heavily involved and tried to apply it in HU NLHE. What I discovered was that this, in practice, seemed to give me a very marginal win-rate (just about even with the rake, maybe very very slightly profitable). But I do think finding the "balance points" can be helpful in figuring out how loose, tight or aggro one needs to play against various types of opponents. Once I took that step, the whole study gave me some additional ideas that got my win-rate up much higher than it had been prior to the [0,1] game analysis.

[Aisthesis]

Or 3456. The reason there was because he raised, and I just asssumed for simplicity's sake that he's a fairly tight raiser. One could also assume that he'll raise more QJT hands than 678 hands, like any of the big wraps but 6789 only if it's ss. There are obviously a number of different LP raising strategies. I tried to pick one that seemed both plausible and convenient for the calculation.

I'm not following you on the second part of this. If LP really makes him indifferent, then it actually doesn't matter. If LP bluffs more, he should call more than 50%, and if LP bluffs less, he should call less.

Just as LP's equilibrium strategy is essentially DEFENSIVE against strategy changes of EP, so is EP's calling strategy DEFENSIVE against strategy changes of LP.

The point is this: If EP calls pot 50% of the time, then LP is indifferent to bluffing pot when he misses.

I'm not sure whether this number changes due to the hand frequency issue you brought up in your other post (I don't think it does). I think it changes up or down if EP's side cards are blockers to one of the straights. And maybe it should be a little higher than 50% in general due to the frequency issues you brought up in your other post (if a Q hits, then it's less likely that LP has the big wrap, but if a 6 hits, then it's less likely that LP has the small one).

My intuition is that that issue is irrelevant for EP as long as his side-cards are blanks. But that it IS relevant if he has blockers to one straight or the other.

Anyhow, if EP's side cards are blanks, then he MUST be equally likely to call either straight. If they aren't, then I don't think that's true.

I'm going to have to think about this some more with blank side-cards first. 50% may not be right, for the reasons you mentioned in your other post.