Heads Up Game Theory exercise

[mykey1961]

[ QUOTE ]
Mykey, when reading your post again i get the impression that you think that since bluffing will gain SB nothing, he should not bluff. Well, the problem is your'e looking at the bluff in isolation. The point of the optimal bluffing is not for the bluff to show a profit, but to induce BB to call SB's value bets. SB will break even on his bluffs, but the bluffs makes BB call with worse hands, thereby making for an indirect profit.

This, and your assumption that BB can bet and raise, makes your solution wrong.

[/ QUOTE ]

If I (in the BB) choose to fold 1 thru 62, call with 63 1/8 of the time, and always call with 64 thru 100, explain how any potential bluffing can make you some indirect profit?

Since my optimal strategy for the BB is netting me about 0.07BB per hand, I don't see a need to try and exploit your bluffs which could open me up to exploitation.

Also I may have said the BB can bet or raise, but I know it can't, and my solution (for the BB) doesn't require it to bet or raise.

[mykey1961]

[ QUOTE ]
Mykey , you are way behind on this thread .

Read Game Theory Resolution for the answer to this question . It's been corrected by Tnixon and I .

[/ QUOTE ]

The other thread addresses a slightly different problem [0,1] with the assumption that there will never be a tie.

[jay_shark]

c'mon now , I made it easy for the both of us .

Why stress yourself out ??

[Klyka]

Mykey, your'e misstaken about what the optimal strategy is all about. No, SB's bluffing is not meant to "force" BB to make suboptimal exploitive strategy changes.

First of all, without calculating things, I don't know about your calling ratio, but I think you might be right. But the reason BB uses whatever ratio he is using is to make SB indifferent to bluffing, which seemed to be what your ratio is doing. So it might be right. The thing is, SB too has to make BB indifferent to calling. This is done by bluffing optimally.

More on this later, when I have more time.

[mykey1961]

[ QUOTE ]
Mykey, your'e misstaken about what the optimal strategy is all about. No, SB's bluffing is not meant to "force" BB to make suboptimal exploitive strategy changes.

First of all, without calculating things, I don't know about your calling ratio, but I think you might be right. But the reason BB uses whatever ratio he is using is to make SB indifferent to bluffing, which seemed to be what your ratio is doing. So it might be right. The thing is, SB too has to make BB indifferent to calling. This is done by bluffing optimally.

More on this later, when I have more time.

[/ QUOTE ]

SB bluffs with 1..19, folds with 20..62, and raises 63..100 for value

I think that makes the BB indifferent to calling for 20..61

[marv]

I nearly agree with this answer.

This is a jam/fold problem with smallblind=1 bigblind=2.
With stacksize=6 (SB can raise to 3x BB) I get a solution of

SB:
1..16 jam
17 mixed
18..59 fold
60..100 jam

BB:
1..62 fold
63 mixed
64..100 call

while with stacksize=8 (sb can raise by 3x BB), I get

SB:
1..15 jam
16 mixed
17..68 fold
69..100 jam

BB:
1..70 fold
71 mixed
72..100 call

Marv

[jay_shark]

Marv there are many solutions to this . Yours is one of them and so is Mykey's as well as mine .

You can't do any better than folding 1-43, shoving with 44+ .BB would call with 63+ . Other solutions exist such as the following which agrees with Mykey's :

Jam 1-19 , fold 20-62 ,jam 63+ .

[mykey1961]

[ QUOTE ]
I nearly agree with this answer.

This is a jam/fold problem with smallblind=1 bigblind=2.
With stacksize=6 (SB can raise to 3x BB) I get a solution of

SB:
1..16 jam
17 mixed
18..59 fold
60..100 jam

BB:
1..62 fold
63 mixed
64..100 call

Marv

[/ QUOTE ]


Marv,

Assuming the mixed is 50%/50% jam/fold I don't think is correct.

SB:
1..16 jam
17 mixed
18..59 fold
60..100 jam


BB can improve his EV by playing

1..62 fold
63..100 call


BB EV = +0.078914bb's per hand

This seems to be a correct strategy.

SB:
1..15 jam
16 50%/50% jam/fold
17..59 fold
60..100 jam



The error I had in my strategy for SB was not making BB's call / fold decision indifferent for 63.

To do that, I made SB raise 50% with 19 instead of 100%.


SB:
1..18 jam
19 50%/50% jam/fold
20..62 fold
63..100 jam

BB:
1..62 fold
63 12.5%/87.5% call/fold
64..100 call


BB EV = +0.07856BBs per hand

Jay_Shark,

Your strategy for SB

1..43 fold
44..100 jam

allows BB to play

1..62 fold
63..100 call

BB EV = +0.078737BBs per hand

This strategy is a correct one

SB:
1 50%/50% jam/fold
2..44 fold
45..100 jam


Another interesting strategy for the SB


1..19 jam 37/38 and fold 1/38
20..62 fold
63..100 jam


Klyka,

It appears that making the BB indifferent to call vs fold is only important for the times BB has 63.

[marv]

[ QUOTE ]
Marv there are many solutions to this . Yours is one of them and so is Mykey's as well as mine .

You can't do any better than folding 1-43, shoving with 44+ .BB would call with 63+ . Other solutions exist such as the following which agrees with Mykey's :

Jam 1-19 , fold 20-62 ,jam 63+ .

[/ QUOTE ]

Sure, there are lots of equilibria (I didn't mean to imply mkey's answer wasn't), but I thought your orginal answer had SB jamming with everything and BB calling with top 2/3. Since the SB's fold action with a 1 isn't a best response to BB's strategy in that case, and you seemed to be saying that SB had an empty bluffing region there was something amiss.

In the case with stacksize=6, the support of the set of nash equilibria for SB looks like
1..62 jam or fold
63..100 jam

and for the BB
1..62 fold
63 fold and call
64..100 call

which explains all the different answers.

Marv

[marv]

[ QUOTE ]
[ QUOTE ]
I nearly agree with this answer.

This is a jam/fold problem with smallblind=1 bigblind=2.
With stacksize=6 (SB can raise to 3x BB) I get a solution of

SB:
1..16 jam
17 mixed
18..59 fold
60..100 jam

BB:
1..62 fold
63 mixed
64..100 call

Marv

[/ QUOTE ]


Marv,

Assuming the mixed is 50%/50% jam/fold I don't think is correct.

[/ QUOTE ]

No, by mixed I did not mean exactly 50%/50% .

Marv