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  #71  
Old 06-10-2004, 07:57 AM
PairTheBoard PairTheBoard is offline
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Default Bluffing Best or Worst Folding Hands?.

This is news to me Jerrod. From the little I've picked up here I had thought that it was always best to bluff with your worst hands. So that's not true? Which works best in actual Poker?

Also, why is it better for B to bluff with his BEST folding hands while it's best for A to bluff with his WORST folding hands in this game.

PairTheBoard
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  #72  
Old 06-10-2004, 11:20 AM
karlson karlson is offline
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Default Re: Bluffing Best or Worst Folding Hands?.

Since A has the option to check, his non-worst hands have some chance to win if he doesn't bluff - hence the "bluff with your worst hands" type of solution common to these types of problems.

However, B is going to fold all hands below X that he's not bluffing with. Since they never have a chance to win if you decide not to bluff with them, it doesn't matter if you bluff with your worst hands or not. So, you might as well bluff with your best hands (below your limping cutoff) just in case A deviates from optimal strategy and calls with a bad hand.
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  #73  
Old 06-10-2004, 01:23 PM
well well is offline
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Default How I found the solution

Indeed, my solution contained an error.
[ QUOTE ]
Player A
[0,1/12] : raise
[1/12,3/4] : check
[3/4,1] : raise, when called

[0,2/3] : fold
[2/3,1] : call, when raised

Player B
[0,19/36] : raise
[19/36,7/12] : check
[7/12,2/3] : call, fold when raised
[2/3,3/4] : call, call when raised
[3/4,1] : raise

With this, the value of the game for player B is 17/72

[/ QUOTE ]

For the honest raise I actually found [5/6,1] (just as in the other posted solution).
And of course, instead of check it should read fold by player B...

So I like it I got it right, but what I am interested in,
is how others found this solution.
I only read some beginners examples on the net,
and tried to work my own way to solutions to other problems.

To find this solution, I first thought what would be the order of
actions for both players when going from 0 to 1.

For A would this be {bet_fold,check_fold,check_call,raise_call},
and for B {fold,raise,call_fold,call_call,raise}
(The first two of B could be swapped, but probably wouldn't make a difference,
and surely swapped will not be better.)

For the strategies (of A, then B) I used
alpha = {a,b,c} and
beta={w,x,y,z},
where the variables play as the "next action"-points.

Now I need to express player B's EV.
But it wildly depends on the order of the values of a,..,z.
For instance y could be between a and b or between c and 1.

So I wrote a program (in Matlab) that tried every set of positions of the variables of alpha in beta.
With this restriction, the payoff-matrix is easily computed, and solutions can be sought by
solving grad(EV)=0-vector.
In order for such a solution (if found) to be (co-)optimal, I figured that the following
criteria would be suffictient:

- 0<a<b<c<1 and
- 0<w<x<y<z<1.

The programme came with the solution posted.

My question is:
- is this anything like you (Craig and Jerrod) solved it?
- are my criteria indeed correct and sufficient?

Thanks in advance,

Next Time.
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  #74  
Old 06-10-2004, 02:52 PM
Jerrod Ankenman Jerrod Ankenman is offline
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Default Re: Bluffing Best or Worst Folding Hands?.

[ QUOTE ]
Since A has the option to check, his non-worst hands have some chance to win if he doesn't bluff - hence the "bluff with your worst hands" type of solution common to these types of problems.

[/ QUOTE ]

In this game (particularly), it also doesn't matter for A either because once he has raised, he can no longer fold. But what you're saying is the right concept; when you are folding a range of hands and some bluffs need to come out of that range, it's best to bluff with the BEST hands you would have otherwise folded, just in case your opponent loses his mind and calls with something stupid.

Jerrod
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  #75  
Old 06-10-2004, 03:39 PM
Jerrod Ankenman Jerrod Ankenman is offline
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Default Re: How I found the solution

[ QUOTE ]
Indeed, my solution contained an error.
[ QUOTE ]
Player A
[0,1/12] : raise
[1/12,3/4] : check
[3/4,1] : raise, when called

[0,2/3] : fold
[2/3,1] : call, when raised

Player B
[0,19/36] : raise
[19/36,7/12] : check
[7/12,2/3] : call, fold when raised
[2/3,3/4] : call, call when raised
[3/4,1] : raise

With this, the value of the game for player B is 17/72

[/ QUOTE ]

For the honest raise I actually found [5/6,1] (just as in the other posted solution).
And of course, instead of check it should read fold by player B...

[/ QUOTE ]

B is bluff-raising way too many hands in this solution - more than half!

[ QUOTE ]
So I like it I got it right, but what I am interested in,
is how others found this solution.


[/ QUOTE ]

Here's how I solved it.

B has the option of raising. So A's calling strategy will just be to make B indifferent to bluff-raising. Since B is putting in 2 to win 1, A needs to call (1)/(1+2) or 1/3 of the time.

Then B can value raise 1/2 of the time that A calls and make profit. So B's value raise is his 1/6 of best hands.

B's ratio of bluffs to bets is 1/3, so he bluffs with 1/18 of the hands he would fold with.

Ok, now in addition to raising and folding, B's going to call sometimes. When he calls, A has the option of raising. If this occurs, A will be betting 1 unit to win 2. So of the times that B calls, B has to call a raise at least 2/(2+1) or 2/3 of the time.

A's raises in the top 1/6 of hands are always profitable because B never has a hand in that range; A can then value raise hands above the midpoint of B's calling range.

So now you can get two variables going and write some kind of equation for A's limping threshold and B's raising threshold and solve it if you wish.

However, I resorted to a little trickery.
We know that when we're done, B will be indifferent to raising at the point 5/6. When B is at 5/6, two bets are going in all the time when A has a hand between 5/6 and 1. A will call B's raise an additional 1/6 of the time and lose (because he calls 1/3 of the time). So we know that if B limps at 5/6, A will raise with 1/6 of hands that B beats, so that B will be indifferent to raising at that point.

Ok, so now A raises 2/6 of the time total (1/6 that win and 1/6 that lose) and his bluffs make up 1/4 of the total hands he raises.

So he bluffs 1/12 of the time, and value raises with hands worse than 5/6 the remaining 2/6 - 1/12 - 1/6 = 1/12.

So his value raising threshold is 5/6-1/12 or 3/4.

This point is the midpoint of A's limp-call area, so he limp calls with 8/12 - 10/12, and since his limp-fold area is half as wide as his limp-call area, he limp-folds with 7/12 and above, and fill in his bluff-raise range to the right of that, etc.

Jerrod
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  #76  
Old 06-10-2004, 06:46 PM
PairTheBoard PairTheBoard is offline
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Default Re: Bluffing Best or Worst Folding Hands?.

ok I see what you guys are saying. But why can't you look at B's bluff raise similarly to A's by way of B's limping hands. B has a range of hands he will limp with. Once in that range he is not folding, similiar to A having the check option. Why shouldn't B bluff raise with his worst limping hands?

And how should I apply this to Poker? I'm on the river and bet into. I have a range of hands to fold,call,and raise. Should I bluff Raise with my worst folding hands, my best folding hands, or my worst calling hands?

It seems to me that it depends on my assesment of my opponents likely holdings. Am I more likely to push out a better hand by bluff raising with my worst, or to get a worst hand to call by bluff raising with better hands. Also, I'm usually going to gain a lot more with a successful bluff than inducing a bad call. Maybe that's the point Aisthesis was getting at in #4 when he talked about A getting more Bang from his Buck when he bluffs.

PairTheBoard
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  #77  
Old 06-10-2004, 07:40 PM
Bozeman Bozeman is offline
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Default Re: Thoughts on #4

Sorry, it looks like I was not careful enough in saying that there was agreement. I looked at most of the points in well's result, found them to agree, and then just scanned the rest to make sure no strange #'s appeared. I found JA's results exactly, and apparently well made a typographical error in that one of his 3/4's should have been 5/6.

As for your supposed improvement, I can only guess that you made an arithmetic error (easy to do here), or miscalculated the EV or size of one of the areas.

As for how I reached the solution, sop: define several decision points, and then note that at each decision point a player is indifferent to 2 possibilities. For example, if b is Player B's minimal value raise, then EVraise(b)=EVcall(b). These 7 eqs. then let you solve for the 7 decision points. To get game value, I look at the square [0,1] x [0,1]. The diagonal denotes a tie if showdown is reached, and each region divided by the diagonal and the lines corresponding to decision pts. has a particular expectation value and a probability of occuring equal to its area, so I just add them up. Many terms, so I occasionally make arithmetic mistakes here since I am not using Mathematica or Maple.

Craig
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  #78  
Old 06-10-2004, 10:04 PM
Dan Mezick Dan Mezick is offline
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Default Re: The [0,1] game and poker

Hi David,

How about some closure on hypothetical #2? That question didn't get much traction but I'm still very interested in your analysis on this!
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  #79  
Old 06-10-2004, 10:38 PM
Jerrod Ankenman Jerrod Ankenman is offline
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Default Re: Bluffing Best or Worst Folding Hands?.

[ QUOTE ]
ok I see what you guys are saying. But why can't you look at B's bluff raise similarly to A's by way of B's limping hands. B has a range of hands he will limp with. Once in that range he is not folding, similiar to A having the check option. Why shouldn't B bluff raise with his worst limping hands?

[/ QUOTE ]

Because he does better by limping with those hands than by bluff-raising with them.

[ QUOTE ]
And how should I apply this to Poker? I'm on the river and bet into. I have a range of hands to fold,call,and raise. Should I bluff Raise with my worst folding hands, my best folding hands, or my worst calling hands?

[/ QUOTE ]

Well, in real poker, you should bluff-raise hands that you would otherwise fold that block your opponent from having very strong hands, like hands that contain the ace of the three-flush suit.

[ QUOTE ]
It seems to me that it depends on my assesment of my opponents likely holdings. Am I more likely to push out a better hand by bluff raising with my worst, or to get a worst hand to call by bluff raising with better hands. Also, I'm usually going to gain a lot more with a successful bluff than inducing a bad call. Maybe that's the point Aisthesis was getting at in #4 when he talked about A getting more Bang from his Buck when he bluffs.

PairTheBoard

[/ QUOTE ]

What's important to understand are a couple of things:

--When you bluff-raise, you expect to lose the pot 100% of the time when you are called. You bluff-raise your best hands in preference to your worst ones in case your opponent does something stupid, but against an optimal-strategy-playing opponent, your EV is not changed at all by which hands you bluff-raise with. You bluff raise because you want your opponent to call when you value raise. If you never bluff-raise, he can simply fold hands worse than 5/6 to your raises and do better.
--You bluff-raise with hands that you would otherwise fold. You're not trying to induce worse hands to call you. That would be a value raise.
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  #80  
Old 06-10-2004, 10:45 PM
Jerrod Ankenman Jerrod Ankenman is offline
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Default Re: Improved raising strategy for A in #4

[ QUOTE ]
Interesting in your answer, however, is that it would seem that there are lots of co-optimal bluff-raising strategies for A. Hence, neither 1/12 nor 2/9 is any kind of "magic number" for when A would want to bluff-raise. Does that mean that 1/12 is just the minimum amount of bluffing to force B down to 2/3 for the limp-call?

If so, I guess it also means that if A starts bluffing more, B could make even more profit by reducing his calling criteria after the limp.

[/ QUOTE ]

No; 1/12 is the only co-optimal bluff-raising amount for A. Any other bluff-raising amount can be exploited by B. Now, it turns out in this game, that if B plays his optimal strategy, then A can raise-bluff with all kinds of hands and make the same EV, because he's indifferent to checking and raise-bluffing with most hands. But this is not optimal, because B can in turn exploit it.

Think about Roshambo (rock-paper-scissors). We know that if A plays [1/3,1/3,1/3], then all strategies for B have the same EV. But they are not optimal.

Jerrod
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