Thin call?

[Aisthesis]

I'm going to try to look here at the hand frequency issue brought up by RT from the standpoint of EP.

Assumptions:

1) Before the river hits, there are exactly the same number of possible big wraps as there are middle wraps, but LP just doesn't have a little wrap (like 3467) due to the raise.

2) EP's side cards are blanks, such as with KK33.

Now, from the knowledge standpoint of EP, if a 6 hits, or any card making the middle straight, there are 4 instances of the big straight draw for every 3 of the middle straight draw. Similarly if a J hits.

p1 is going to be the probability that EP chooses for a call, and the task is to choose p1 such that LP is indifferent to bluffing.

Ok, LP will bet the made nut straight at pot 100% of the time, and will bluff the straight he doesn't have with a probability of p2.

Now, one of the scare-cards hits on the river, EP checks and LP bets pot.

What's the probability that LP is bluffing?

3 times, LP really has the straight. 4*p2 times, he's bluffing.

EP is indifferent to a call when he wins 1/3 of the time, hence when the ratio of value-bets is 2:1. This is the case when (4/3)*p2 = 1/2, hence p2 = 3/8. That vindicates RT's suggestion.

But I'm still a bit confused on EP. I think it still has to be p1 = 1/2 to prevent LP from always or never bluffing.

Ok, I think I'm starting to get the way side-cards influence the situation.

Suppose LP, who doesn't know the side cards of EP, is bluffing the missed straight 3/8 of the time, but now EP has KK63. My hypothesis: EP now calls the middle straight 100% of the time and the big straight never.

EP knows the following if the river is a 7 and LP bets: LP will bluff the missed straight 3/8 of the time, and (what LP doesn't know) there are 4*4*4 instances of the big straight over and against 4*3*3 of the little one. Hence, 9 of the little one for every 16 of the big one.

So, 9 times, LP has the nuts, and (3/8)*16 = 6 times LP is bluffing. EP is thus 3:2 on the call, so it's an obvious call, since it's better than 2:1.

But suppose EP now has KK63 and a Q hits. Now there are an exactly equal number of big and middle straights in the deck from the perspective of EP.

LP is still bluffing a missed straight with a probability of 3/8. So, if we run this situation 16 times, LP will have the straight 8 times and will have missed only 3. That's worse than 2:1, so EP should now always fold.

[RoundTower]

I didn't mean anything about side cards, I meant:
you will be value betting the nut straight on 9 cards out of the 40.

So you need to be bluffing 4.5 out of the 40 to make the guy indifferent, you choose to bluff 4.5 out of the 12 "scare cards" 6-8, 4.5/12 is 3/8 which is 0.375.

Let zero be the point where you are thinking about calling the turn. So if you fold the turn you get $0. Your expectation then if EP always calls is

value bet: 9(+1050)
unsuccessful bluff: 4.5(-840)
check behind scare card: 7.5(-210)
blank on river: 19(-210)

added up this gives 9450 - 3780 - 1575 - 3990 equals 105, divided by 40 gives a profit of about $2.50 per hand for the turn call.

If EP always folds
value bet: 9(+420)
bluff: 4.5(+420), etc

again you get 105 (obviously since we chose a strategy for you that made you indifferent to whether EP would call or not).

So yes the call of a pot sized bet with one card to come is (marginally) profitable if you have 9 clean outs and there are 21 total "scare cards", even if your opponent plays perfectly.

Ciaffone and Reuben go into this in detail in their book: they use a different game where you get your last card down. In Omaha it's much more complicated. For example your opponent may make the same straight, or because your last card is face up some cards are more likely to be scary than others. But the solution to what you were trying to figure out is in the bold print above; I'm almost certain there is a list of similar figures in C&R.

[Troll_Inc]

[ QUOTE ]
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.

[/ QUOTE ]

These are all good arguments, but there are good counterarguments.

1. The useful outcome of this analysis really isn't game theory. For example, I believe you can use something similiar to MOP's (pg 48) method of analysis to arrive at the answer that player B (setman) should call if player A (the drawer/bluffer) bluffs more than 33% of the time.

If this number is correct...
In my limited PLO experience (80k hands @ PLO100 or under) I and my opponents either bluff/call way too little or way too often. All kidding aside from those of you that play in higher stakes games and offer advice on this forum it seems there that also the advice imparted is often that there is some extremely correct way to play a line which reverts, i.e. either you do or don't do something >80% of the time or less than <20%. So if you have an answer that is 33% of the time, then apparently it isn't intuitive.

Further evidence that it isn't intuitive is that noone said, oh that answer is easy...it's 33%. So either people that know the answer don't want to say or it's not intuitive.

2. I don't really see this sort of thing analyzed in print. I'd much rather read about it than run through the numbers as I find math tedious. But since it appears to be unknown and novel, I find the analysis interesting eventhough it involves tedious math. There are times that I would much rather work through these things, in my spare time, than play or analyze specific hands that I've been in. (I do that too, and this specific specifically popped up for me on how to play a hand. I felt that when I flopped top set OOP on a drawy board I might be playing it wrong when one of the draws hit.)

3. While PLO is a complex game, I think many basic situations, if analyzed, can lead to lines that people don't really use. There appear to even be ones that the "PLO pros" deride that under certain circumstances are profitable.

For example in this thread , the villain's JJTT hand has implied odds if the hero is willing to push his whole stack in on the turn x% of the time. Do you know what x% of the time is? If it's only 50% and your poker skills as villain get a read on the hero that you think he'll do it "almost all the time" and you think that number is about 90% of the time, then you better take that opportunity or you aren't maximizing your profits.

Again this isn't game theory, it's discovering exploitive strategies and utilizing them. You are correct in that game theory for PLO is a waste of time until all the exploitive strategies are discovered and both you and an opponent are utilizing them. And that will be a long time from now because PLO is so complex.

*For this same reason, once bots ruin online NLHE play, PLO will still be safe, but that is another discussion.

[grizy]

I took a different route to the solution, but it's the same as RoundTowers. I am going to post it anyway because it illustrates a point Pete brought up before about mixed strategies and constant adjustments... Furthermore, it is interesting because here, there is no mutually optimal solution except maybe a razor's edge where villain would be indifferent.

If we do discount the

Scare cards on that board: any 678QJT, for a total 21 (assuming hero
holding one of them), 9 of which gives the goods.

Hero:
9/40=probability hero makes the hand
let bluff=probability hero bets pot on river with a scare card that didn't
help him.

villain:
let call = probability villain calls the pot sized river bet

Hero's EV on river=
9/40 * (630+630*call) <<== hero's river bet EV when he makes the hand PLUS
12/40 * bluff * (630-1260*call) = <<=== hero's EV when he misses but there
is scare card PLUS
0 <<=== hero's EV when a blank drops or he opts not to bluff

Let bp=probability the pot bet on river is a bluff
bp=(bluff*12)/((bluff*12)+9)
villain's EV on calling a river bet when one of the 21 scare cards show
therefore is:
1890*bluff*12/((bluff*12)+9)-630.
Solve for the EV > 0 and we have EV > 0 when
bluff > 3/8.

Since game is zero sum, no Nash can exist unless bluff=3/8.

Plug 3/8 into hero's EV, and the river EV is 212.625, for a net of 2.625 dollar profit on the 210 dollar turn call.

If you derive it, you can see for almost all values of call, (except where call=1/2, but villain is indifferent, therefore that cannot be a stable equilibrium), dEV(hero)/dbluff = 189-378*call != (does not equal) 0. This implies there is no Nash equilibrium, therefore no mutually optimal strategy, except maybe a razor's edge where call can change randomly, therefore forcing bluff to change. The implication of this is well... there is no set solution and constant adjustments to the flow and mood of the game are needed.

Also, as Troll noted, people either call too little or too much, because as your expectations of "bluff" changes, you should be either calling all the time or never. This is consistent with game theory.

[Aisthesis]

Yes, I get the same.

But you guys haven't said anything about my suggested Nash equilibrium for EP's decision (call or fold).

Do you agree with my 50% there--with the appropriate adjustments depending on EP's blockers?

[Aisthesis]

I like this. It's a simpler way of calculating it while getting the same result.

However, I'd like to see what happens with blockers in EP's hand. If EP has at least one blocker to the big straight, I'm going to suggest he call the big straight always and the middle straight never. Similarly for a blocker on the middle straight.

Now:
value bet: 8 (+1050)
unsuccessful bluff: 4.5(-840)
check behind scare card: 7.5(-210)
blank on river: 20(-210).

Now we can subtract $1,260 from your $105 for a total loss for LP of 1,155/40 = about 29.

The same will be the case for 1 blocker on the little straight.

The loss will also be HUGE if EP has 2 blockers for one of the straights.

Then we can subtract yet again 1,260 from LP's EV and get 2,415/40, which is roughly a loss of 60.

Finally, I'm pretty sure EP needs to call always if he has a blocker to both straights, but let's check.

value bet: 8(+1050)
unsuccessful bluff: (3/8)*11(-840)
check behind scare card: (5/8)*11(-210)
blank on river: 21(-210)

The items to calculate are:
(3/8)*11*840 = 33*105 = 3465, and
(3/8)*11*210 = 866.25

So, we now get -341.25/40 = -$8.50.

[Aisthesis]

Finally, the question arises whether LP can still make this strategy work profitably with 9 outs if EP adopts this superior strategy. I don't think he can, but let's check:

From the standpoint of LP, there are actually 42 completely unknown cards in the deck. He knows that EP has a set but doesn't know his sidecards. EP is going to try to use that fact to his advantage.

Since LP has the big straight draw (or whichever draw), there are 9 + 12 = 21 cards of the 42 that are relevant here. The rest are blank sidecards.

So, EP has two blank sidecards 1/4 of the time, 2 relevant sidecards 1/4 of the time, and 1 relevant and 1 blank sidecard 1/2 the time.

Hence, 1/4 of the time, LP has an EV of $2.50 according to RT's calculation for a total EV of about $.60.

Well, blast, I just realized that I'll need to calculate EV for a few more scenarios to give a complete picture, but I think it's clear that the call is unprofitable if EP adopts this superior strategy, since in all other cases LP loses a lot more than $2.50.

[grizy]

[ QUOTE ]
Yes, I get the same.

But you guys haven't said anything about my suggested Nash equilibrium for EP's decision (call or fold).

Do you agree with my 50% there--with the appropriate adjustments depending on EP's blockers?

[/ QUOTE ]

I addressed this in my post. For any value of bluff, except 3/8, Call is either 1 or 0, there is no other Nash equilibrium possible. If bluff=3/8, then EP is indifferent with his calling %. However, if call is indifferent, there is no reason for him to stay at 1/2 (the only nash equilibrium possible), so the (3/8,1/2) solution is not an evolutionarily stable solution. As call changes (simply because EP feels like it), bluff will have to adjust, and EP will again either always call, or always fold, and bluff changes again.

There is no Nash equilibrium in the problem as given. If you put in blockers, you are changing the fractions, say instance, instead of 9/20, it becomes 8/19. Then 3/8 bluff will no longer be a potential nash equilibrium.

In practical turns, whoever is best at adjusting and observing each other's play will win in the long run and, as constructed, there is no set Nash equilibrium. There ARE games where we will converge on some nash equlibrium, but there is simply none here.

[Aisthesis]

Well, wait a minute. You can say the same thing about LP. If EP calls less than 50% of the time, then LP's bluffing frequency isn't optimal at 3/8 but should be 100%. And if EP calls more than 50% of the time, it should also be 0%.

Anyhow, I can't exactly remember the definition of Nash equilibrium, but I'm inclined to agree that the numbers 3/8 for LP and 1/2 for EP don't quite cut it. Basically, as you say, neither is an evolutionarily stable solution.

I still think the numbers 1/2 and 3/8 are relevant, if not Nash equilibria. They essentially define the boundary (for LP looking at EP) between weak-tight and calling station (the 1/2). And for EP looking at LP, the number 3/8 defines the boundary between bluffer and nut-peddler.

I actually went through and looked at all the situations with blockers, which are in EP's hand and hence unknown to LP. If EP factors these in and adjusts his calling and folding accordingly, then I'm convinced LP can't make a profit regardless of how he plays it.

[grizy]

[ QUOTE ]
Well, wait a minute. You can say the same thing about LP. If EP calls less than 50% of the time, then LP's bluffing frequency isn't optimal at 3/8 but should be 100%.

And if EP calls more than 50% of the time, it should also be 0%.


[/ QUOTE ]

Correct, and therefore there is no Nash equilibrium other than (3/8,1/2)


[ QUOTE ]

Anyhow, I can't exactly remember the definition of Nash equilibrium, but I'm inclined to agree that the numbers 3/8 for LP and 1/2 for EP don't quite cut it. Basically, as you say, neither is an evolutionarily stable solution.

[/ QUOTE ]

(3/8,1/2) IS a Nash equilibrium, it's just not an evolutionarily stable one.

[ QUOTE ]
I still think the numbers 1/2 and 3/8 are relevant, if not Nash equilibria. They essentially define the boundary (for LP looking at EP) between weak-tight and calling station (the 1/2). And for EP looking at LP, the number 3/8 defines the boundary between bluffer and nut-peddler.

[/ QUOTE ]

The numbers are relevant. If LP believes call>0.5, he won't bluff. If EP believes bluff > 3/8, he will always call. The trick here is to lead your opponent to the wrong assumption regarding your call or bluff values.

[ QUOTE ]
I actually went through and looked at all the situations with blockers, which are in EP's hand and hence unknown to LP. If EP factors these in and adjusts his calling and folding accordingly, then I'm convinced LP can't make a profit regardless of how he plays it.

[/ QUOTE ]

You're pointing at the phenomenon of EP having blockers sufficiently changes LP's payoff to make it possible for EP to guarantee LP will lose money on a call. You can derive this by taking the derivative of LP's EV function with respect to bluff. You'll find that EV(LP) is negatively directly correlated with call. This implies for any call greater than some value, it's impossible for LP to make money on the turn call.

However, all this says is it's possible for EP to prevent LP from getting over 210 on river; it does not say anything about EP's optimal behavior on the river. In other words, even though by having blockers, you COULD guarantee a positive EV, you are not necessarily maximizing your EV. in fact, there is a nash equilibrium on bluff and call on the river... it just doesn't give LP enough payoff to justify a 210 dollar call on the turn. It is still entirely possible for LP to mislead EP enough with regard to LP's probability of bluff (say.... 1, for simplicity, you know, make people believe you're a total and utter rock) that EP will fold to every pot bet on every scare card... this can justify a turn call.

Without a function to define how the adjustments to bluff/call probabilities are made, any set, given the absence of an ESS, is a possibility depending on the flow of the game (whether someone is tilting, or has clamped down because he's on the last of his bankroll, stuff like that), it is entirely possible to arrive at seemingly improbable solutions, such as 100% bluff, and 0% call, which would be enough to justify a call.