Heads Up Game Theory exercise

[jay_shark]

There are two players in a game . Hero posts the sb and villain posts the bb . There are only two choices for this game . You may either raise 3X the bb ,or fold .

The object of the game is to select a card from a deck with numbers 1,2,3,...100 and raise if you think your number is the highest . Your opponent can either call you or fold . Since there is only one deck , your opponent's card must be different in rank than yours .

What numbers should hero raise with ?
What numbers should your opponent call you down with ??
What is your bluffing frequency ?

[jay_shark]

Part B )

Suppose instead of raising 3X the BB , you are allowed to raise up to 10X the BB or fold .

Now answer the same questions as before .

[Nichomacheo]

This is really hard no?

[jay_shark]

Actually the first problem is very easy .

If you're familiar with SAGE then you probably know the answer to the first problem .

[omgwtfnoway]

edit: i see that the bb cannot raise, i'll be back when i've thought about it some.

[dippy111]

you are risking 3bb to win 1.5 if im not mistaken in the definition. Therefore you should raise top 1/3 of numbers = 67+. Knowing that you're raising this range, villain should call with top 1/2 of your range = 83+? Then again, that doesnt account for bluffs. Hmmm...

[Schpacko]

I guess you should raise 51 and above, or is this too easy?

So villain is getting 1:2 for a call. Has to be ahead 33%, so he should call (or raise?) with the the top 33% of the top 50% -> top 16.6%, right?

But i really suck @ things like that, this is probably abolutely ridiculous [img]/images/graemlins/smile.gif[/img]

[jay_shark]

Ok here is the answer to the first problem . I'll let others think about the second .

1) The sb can either raise an additional 2.5 bb's which means the BB is getting 2:1 on his call . If the sb folds at any point for any specific hand then his EV =0 . I'll show that pushing with any number is positive EV .

If you have card #1 , then it's clearly the lowest number form the deck . However , the bb is not aware of this . He must call you if he believes his hand can beat at least a third of yours . Since he's getting 2:1 on his call , he should call with numbers 34,35,36,...100 . Notice that 34 beats precisely 33 numbers and loses to 66 numbers . So , if the sb pushes with any card , then he actually increases his EV . Since this is the case , there is no bluffing frequency for the sb .

The probability that the BB wins given that he calls you will converge to 2/3 as the numbers approach infinity . In this case , the numbers stop at 100 but it still converges to 2/3 fairly quickly .

Just show that 1/3 + 2/3*1/2 = 2/3 . Simply reason that the BB will beat one third of the hands when he calls and the sb shows 1-33 . However two thirds of the time , he will win half of the hands (2/3*1/2) .

Ev(sb) = 1/3*1.5 + 2/3*(3.5*1/3 - 2.5*2/3)
Ev(sb) = 0.166666666

This shows that raising with any number is better than folding , even if your first card is a 1 .

The second problem is a bit harder and algebra intensive but it is pretty neat . The solution hinges primarily on ideas expressed in the first problem but it's still interesting to work out .

[abcjnich]

The solution to this is to read "Mathematics of Poker"

[omgwtfnoway]

jay, if the sb folds his equity is -.5bb, no?