Kind of an odd question... PL5CD and game theory

[kurto]

I'm curious if people have formal ways of determining when to call with "bluff catchers."

If I have good notes that certain people never bluff and I know the hand range that they'll bet post draw with.. its fairly easy to determine whether to call or fold.

But there are a number of players who will play every draw... and will bluff quite often if its checked to them. They either have a straight/a flush or complete air. You raise with Trips and do not improve.

If you check and the pot is decent sized, they will bet at it a decent amount of time. Do you have some random ritual that says whether you call or not?

Similar thought- post draw you have nothing but AA. The person in position will bet 100% of the time its checked to him. His starting requirements to play are 8 card draw or any pair. Do you call 100% of the time here?

Thanks.

[flafishy]

Against an unknown player, if I've raised the pot with AA, I get one caller, I don't improve and I'm first to act, I'll always lead out with a bet. Most of the time, villain will fold or sometimes call with a smaller pair. Every now and again, villain will call with a better hand. You'll get raised with a stronger hand or even on a bluff fairly rarely. I've found that it's a profitable play long-term to bet out, and no matter what happens, you get solid information for your notes.

In 5CD in particular, I think it's always better to bet out than to check-call. If you're planning on throwing out one bet postdraw anyway, use it to try to get a lookup from a weak hand.

[kurto]

So you don't find that, for instance when you're betting into someone who drew one card.... they're only going to call if you're beat? I find (at the $50 tables) that MOST people will not reraise with just 2 pair (the few that do almost always have AAxx). So if I open with AA and someone calls and draws 1... if they can't beat me they're folding. If they call with weak 2 pair... I may have gotten away cheaper?

(btw- I'm not trying to be contrary. I'm just trying to rationalize what your saying with what I've observed.

I'll have to meditate on this...)

[MarkGritter]

I started writing up an explanation, but Chen and Ankenman's is better than mine, so read their book. [img]/images/graemlins/wink.gif[/img] I'll just present the game-theoretical result and a couple examples.

Suppose the drawing hand makes only pot-sized bets.

The optimal strategy for the bluffer is to bluff 1/(pot size+1) = 1/2 of the time when he has nothing. You should call with 1/2 of your hands to make the draw indifferent to bluffing or not bluffing.

Check (does this actually work):

Suppose the draw makes his draw 10% of the time and bluffs the other 90%. You call 50% of the time and fold the other 50%.

EV to you = 0.10 * 0.50 * -1 +
0.10 * 0.50 * 0 +
0.90 * 0.50 * 2 +
0.90 * 0.50 * 0 =
-0.05 + 0.9 =
0.85.

Now suppose your opponent never bluffs.
EV to you = 0.10 * 0.50 * -1 +
0.10 * 0.50 * 0 +
0.90 * 1 =
0.85

So your strategy is indifferent to whether your opponent always bets or always bluffs. If he bluffs with the appropriate frequency:

Your EV =
0.10 * 0.50 * -1 + (call value-bet)
0.10 * 0.50 * 0 + (fold to value-bet)
0.90 * 0.50 * 0.50 * 2 + (he bluffs, you call)
0.90 * 0.50 * 0.50 * 0 + (he bluffs, you fold)
0.90 * 0.50 * 1 (he checks behind, you win the pot)
= -0.05 + 0.45 + 0.45 =
0.85

(Now, you should occasionally check-raise when you make a boat or quads which complicates stuff, but in terms of a basic strategy calling 1/2 of the time to catch bluffs is unexploitable.)

[flafishy]

Kurto, my bad. I keep answering PL5CD questions with answers based on the way I play limit.

MarkGritter, I don't have a clue what your conclusion is.

[kurto]

[ QUOTE ]
Kurto, my bad. I keep answering PL5CD questions with answers based on the way I play limit.

MarkGritter, I don't have a clue what your conclusion is.

[/ QUOTE ]

Whew. I was thinking I was missing something. I thought it was going to really expensive quick (especially if they reraise me a pot sized bet and I have to call when they're bluffing!)

[Murakawa]

Opponents don't always make Pot sized bets. So though it works theoritically, your notes are where you go. Firstly, if you've never seen an opponent bluff, unless he seems to bet betting frequently, I ususally don't call.

Also, most draw players don't know how to bluff. Sometimes they are good, and sometimes they are bad. Many players make small bets because they don't want to lose much if they are caught, and make big bets with their real hands to extract money. This is a good strategy that they employ for the wrong reasons. You bet small as a bluff because it looks like a bet that you want called, it looks like a value bet. You bet big with your real hands because they'll put you on a desperate bluff. That's not everyone, of course. And I won't call a big bet without a hand unless I'm in a tournament anyway.

Of course, if you're against good players this doesn't work. People do not call big bets as the pots are too small to fight for, and are willing to call small bets to win a big pot, as this can be +EV against a person who only bluffs occasionally.

Here's the best part. Most players always bet the same amount in the same situation. If they have a hand they'll bet small or they'll bet big. If they don't, they'll bet small or they'll bet big. Keep track! Of course good players are more difficult to read but my trust notes don't let me down that often.

[kurto]

Are you suggesting that if someone takes a 1 card draw (let's say for the sake of argument that they don't have 2 pair here)... that when they bet post draw: if I call 50% of the time I'm playing correctly regardless of their tendency to bluff?

[MarkGritter]

[ QUOTE ]
Are you suggesting that if someone takes a 1 card draw (let's say for the sake of argument that they don't have 2 pair here)... that when they bet post draw: if I call 50% of the time I'm playing correctly regardless of their tendency to bluff?

[/ QUOTE ]

Yes. If you call their pot-sized bet or bluff with a straight or flush draw 50% of the time, they can't exploit your play.

You actually asked a somewhat more practical question. Against an opponent who you are sure is bluffing too much, you can just switch to always calling. Or if the opponent is bluffing too little you can just switch to always folding. And that will earn you more in the long run. But note that an opponent can be bluffing about 45% of the time here and not be bluffing "too much"!

But it doesn't really make sense to call, for example, 80% of the time or 20% of the time, unless you need to disguise the fact that you are trying to exploit your opponent's tendencies.

[MarkGritter]

You can extend the analysis for various bet sizes. From
TMoP:

[ QUOTE ]

Let a = 1/(pot size + 1)
Y should bet all "nut" hands and bluffs "a" of his dead hands...
X should call with 1-a of his hands total


[/ QUOTE ]

The key point is that if Y tries to vary his bet sizes, then X can still use his optimal strategy!

Suppose Y bets the pot with the nuts and bluffs half the pot with losing hands. Even if X doesn't realize what's going on, he can play the following strategy:

Call 1/2 of the time when Y bets the pot.
Call 2/3 of the time when Y bets half the pot. (a=1/3)

X's EV =
0.10 * 0.50 * -1 + (Y bets the pot, X calls)
0.10 * 0.50 * 0 (Y bets the pot, X folds)
0.90 * 0.667 * 1.5 (Y bets half-pot, X calls)
0.90 * 0.223 * 0 (Y bets half-pot, X folds)
= -0.05 + 0.90 = 0.85

Let's say Y switches gears and starts bluffing the whole pot:

X's EV =
0.10 * 0.667 * -0.5 + (Y bets half-pot, X calls)
0.10 * 0.223 * 0 (Y bets half-pot, X folds)
0.90 * 0.5 * 2 (Y bets the pot, X calls)
0.90 * 0.5 * 0 (Y bets the pot, X folds)
= -0.0333 + 0.90 = 0.867

Oops, Y made things worse for himself by playing a dominated strategy. [img]/images/graemlins/wink.gif[/img]