Game Theory Problem Of The Week

[mykey1961]

[ QUOTE ]
Here is the proof that you should bet with [0,1/9] , [7/9,1]where x is your optimal betting range and that your value bets to bluffs should be in the ratio 2:1 .

EV= 1*(1-x)*(2x+1)/3 +3*(1-x)(2-2x)/3*1/3 -3*(1-x)*(2-2x)/3*2/3 -x*(1-x)/3


EV= (-3x^2+4x-1)/3 after simplifying .
EV' = -2x+4/3
Set EV'=0 and we get 4/3=2x and x =2/3 .

So Mykey , you haven't proved a thing . All you are doing is spewing random numbers because you're incompetent to figure it out yourself . You're relying on simulators as a means to find your answer which is unethical .

[/ QUOTE ]

Proof? you want stinkin proof...

P2 Bets with [0,x] and P1 folds with [0,1-2x] P2 wins 1 ante.
+1*x*(1-2x)
P2 Bets with [0,x] and P1 calls with [1-2x,1] P2 loses 3 antes.
-3*x*2x
P2 Bets with [1-2x,1] and P1 folds with [0,1-2x] P2 wins 1 ante.
+1*2x*(1-2x)
P2 Checks with [x,1-2x] and P1 has with [0,x] P2 wins 1 ante.
+1*(1-3x)*x
P2 Checks with [x,1-2x] and P2 has [1-2x,1] P2 loses 1 ante.
-1*(1-3x)*2x

Ev = +1*x*(1-2x)-3*x*2x+1*2x*(1-2x)+1*(1-3x)*x-1*(1-3x)*2x

EV = -9x^2+2x

Ev' = -18x+2 = 0

x = 1/9

plugging x=1/9 into -9x^2+2x gives an EV of +1/9 just as we expected, with no rounding errors.

[mykey1961]

[ QUOTE ]
[ QUOTE ]
Tnixon , please point to the thread where I made an EV error .

[/ QUOTE ]

If I answer this in the way that I really want to, this thread will turn into a bigger flamefest than it already is.

Stupidity hits us all at some point or other. Some of us can admit when it's happened, and some of us can't. So I'll choose to remain silent.

Leader, see how much restraint I'm showing? I *like* to flame. [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

When battling with math geeks, never take a protractor to a compass fight.

[jay_shark]

Mykey , you really do not understand a thing . Yes you came up with the answer relatively quickly but gave no proof why it was optimal . I was the only one attempting at finding a mathematical solution which you did not have .

It doesn't take anything special to plug random numbers to see what gives the optimal result which is clearly what you did .

Oh and by the way , my EV equation is right . Maybe you don't understand what I did ? I challenge you to select numbers from your simulator to show that you will get the same results .

[jay_shark]

[ QUOTE ]
[ QUOTE ]
Tnixon , please point to the thread where I made an EV error .

[/ QUOTE ]

If I answer this in the way that I really want to, this thread will turn into a bigger flamefest than it already is.

Stupidity hits us all at some point or other. Some of us can admit when it's happened, and some of us can't. So I'll choose to remain silent.

Leader, see how much restraint I'm showing? I *like* to flame. [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

lol Tnixon , I guess we all screwed up at some point .
I'll admit it too .

[mykey1961]

This is Jay's message
[ QUOTE ]

1*33/100*78/99 + 3*33/100*22/99*1/3 -3*33/100*22/99*2/3
+1*66/100*11/99 - 1*66/100*22/99 = 0.11333333 (This is the exact EV without replacement.


[/ QUOTE ]

This was my reply

[ QUOTE ]

Found some problems...
With the 78:
When P2 has [1,11] and bets, there are only 77 numbers P1 will fold with.
When P2 has [79,100] and bets, there are 78 numbers P1 will fold with.

With the 66:

There are 67 cards that P2 will check with

It should be:
+1*11/100*77/99 +1*22/100*78/99 + 3*33/100*22/99*1/3 -3*33/100*22/99*2/3
+1*67/100*11/99 - 1*67/100*22/99 = 0.11111111


[/ QUOTE ]

And Jay accuses me of taking credit for something he has done.

[ QUOTE ]

Mykey , you are such a nit . This is a round off error !!

Stop trying to take credit for something I've done .

[/ QUOTE ]

I wasn't taking credit for having create this.. I was pointing out the errors your formula had.

When the errors in the formula are corrected, the answer becomes 0.1111111

That was proof that difference between my answer (0.111111) and your's (0.1133333) wasn't due to rounding errors.

[mykey1961]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
Tnixon , please point to the thread where I made an EV error .

[/ QUOTE ]

If I answer this in the way that I really want to, this thread will turn into a bigger flamefest than it already is.

Stupidity hits us all at some point or other. Some of us can admit when it's happened, and some of us can't. So I'll choose to remain silent.

Leader, see how much restraint I'm showing? I *like* to flame. [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

lol Tnixon , I guess we all screwed up at some point .
I'll admit it too .

[/ QUOTE ]

If i ever happen to make a mistake, I will admit it, if I can't do that ninja edit thing in time [img]/images/graemlins/smile.gif[/img]

[jay_shark]

Fair enough Mykey .

I thought this was an interesting discussion and I believe we all learned something from this .

Till the next post ....

[TNixon]

[ QUOTE ]
If i ever happen to make a mistake, I will admit it, if I can't do that ninja edit thing in time

[/ QUOTE ]

Hey, I took credit for that mistake too. I didn't try to erase the fact that I had been a moron, just the full extent of how big a moron I really was.

Some mistakes are just too big for public consumption. Chaos could erupt all across the globe over something like that.

[mykey1961]

[ QUOTE ]
Fair enough Mykey .

I thought this was an interesting discussion and I believe we all learned something from this .

Till the next post ....

[/ QUOTE ]

I don't see how to prove that P1's [56,100] call is optimal other than by complete examination of all strategies for P2.

[jay_shark]

If player 1 checks with [12-78] , then there is nothing player 2 can do to increase his EV . So I don't see a point in player 1 check-calling with [57-100] since he should be indifferent to betting or checking with [12-78] .