Heads Up Game Theory exercise

[jay_shark]

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 .

[jay_shark]

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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 .

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I gotta disagree with this result.

BB should always fold 1 thru 62, raise 1/8 of the time with 63, and always raise 64 thru 100

By doing this, the SB is indifferent to raising or folding with 1 thru 62, and should raise with 63 thru 100.

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Mykey , you are wrong . The sb should be raising with hands 44 and up . The BB should be calling with hands 63 and up .

[Klyka]

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Mykey , you are wrong . The sb should be raising with hands 44 and up . The BB should be calling with hands 63 and up .

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But this is still not accounting for bluffs, so that is also wrong, as I pointed out in my last post.

[jay_shark]

Klyka , we were only discussing value raises .

So from a closed interval [0,1], hero should bet with numbers 0.4375+ and villain should call with 0.625+ .

[mykey1961]

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Mykey , you are wrong . The sb should be raising with hands 44 and up . The BB should be calling with hands 63 and up .

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But this is still not accounting for bluffs, so that is also wrong, as I pointed out in my last post.

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This is completely accounting for bluffs. The whole point is to choose a level that negates the gain by a bluff.

if the BB raises with 64+ and 63 1/8 of the time, then SB's EV for folding and raising with a 1 thru 62 is the same.

Fold = 0 EV
Raise = (61.875*1.5)-(37.125*2.5) = 0 EV


If the SB knows the BB will only call with 63 or higher, why would the SB Raise with 44 thru 62 when that's worse than folding with 44 thru 62?

The beauty of the BB calling with 63 1/8 of the time, and 64 or higher is that raising with less than 63 gains the SB nothing.

Correction to the 10x answer BB should fold with 1 thru 86 and 50% of the time with 87, and otherwise call.

I had an error in my previous calculations due to the conversion of a fold EV = 0 vs a fold EV of -0.5

I prefer the EV = -0.5, but I'll try to communicate using the fold EV = 0 "standard"

[Klyka]

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Klyka , we were only discussing value raises .

So from a closed interval [0,1], hero should bet with numbers 0.4375+ and villain should call with 0.625+ .

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Yes, that is true if you only account for value bets, and that's the scenario I touched upon in my post as an answer to your discussion. But the question is, why? Why just discuss value betting, when only value betting is clearly wrong? We need bluffs in order to force BB to call with lesser hands than our value betting range. That should be clear, so I don't really understand why not to discuss it...

Mykey, I must be missunderstanding you. Your solution is not accounting for bluffs, since you don't have any bluffing range. The bluffing range has to consist of your very worst hands, and your value betting range of your best hands. If there isn't a gap in between the two ranges, then BB can just call profitably with 2/3 of your betting hands.

EDIT: Also, Mykey, BB can not bet or raise.

[TNixon]

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I think you have a problem with these lines.
c1 = c2 = rand() assigns the same rand to both c1 and c2


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That is intentional, it basically just saves a line, because the while loop will continue to regenerate c2 until it is *not* equal to c1.

and do..until is not part of C or C++, which is what that code is.

Give me just a *little* bit of credit here, huh? Programming is how I make my living. And the code I posted was *extremely* basic. I do understand how while loops work.

[img]/images/graemlins/smile.gif[/img]

[Klyka]

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.

[mykey1961]

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I think you have a problem with these lines.
c1 = c2 = rand() assigns the same rand to both c1 and c2


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That is intentional, it basically just saves a line, because the while loop will continue to regenerate c2 until it is *not* equal to c1.

and do..until is not part of C or C++, which is what that code is.

Give me just a *little* bit of credit here, huh? Programming is how I make my living. And the code I posted was *extremely* basic. I do understand how while loops work.

[img]/images/graemlins/smile.gif[/img]

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my bad,
my eyes didn't see the while (..) in the second line.
my eyes saw an if (..)

[mykey1961]

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Mykey, I must be missunderstanding you. Your solution is not accounting for bluffs, since you don't have any bluffing range. The bluffing range has to consist of your very worst hands, and your value betting range of your best hands. If there isn't a gap in between the two ranges, then BB can just call profitably with 2/3 of your betting hands.

EDIT: Also, Mykey, BB can not bet or raise.

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I was looking at this from the viewpoint of the BB.

Once the BB has determined the optimal strategy, any possible bluffing by the SB doesn't effect the EV.

From SB's viewpoint, the only reason to bluff is to potentially draw the BB away from his optimal strategy in an attempt to exploit the SB bluffing.

Unless we know how suboptimal the BB is playing, it's rather difficult to calculate when bluffing is better than folding.