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#71
10-21-2005, 04:39 AM
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Re: Classic Type Game Theory Problem
[ QUOTE ]
I'm not familiar enough with measure theory to define all the terms for you as rigorously as you would like. But wikipedia should provide the intuition. You'll have to consult a textbook or someone else for the formalism. [/ QUOTE ] Wait, wait, wait so you're all like you don't know what r.v.s are and they may seem non-intuitive and I'm like I think it means blah and you're like [img]/images/graemlins/confused.gif[/img]. 
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#72
10-21-2005, 09:31 AM
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Re: Classic Type Game Theory Problem
[ QUOTE ]
[ QUOTE ] Are you, by chance, an algebraist? [/ QUOTE ] No. Don't insult me. [/ QUOTE ] Them's fighting words... [img]/images/graemlins/wink.gif[/img]
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#73
10-21-2005, 11:53 AM
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Re: typed solution
[ QUOTE ]
I've already done the overall EV calculation and it's: EV(A) = 0.5062 EV(B) = 0.4938 and I'm pretty sure it's right. In case it is, can I be mentioned in a footnote in your published work as the first person to correctly determine the EV? Thanks. [/ QUOTE ] And I thought my number was low (for A). Did you use the solution I mostly described above? To clarify, if b is above .5, B always redraws; if b is in [sqrt(2)-1,.5], B redraws only if A stands; if b is lower than sqrt(2)-1, B randomizes if A stands, while A stands on [0,1 - sqrt(1-2b)]. B randomizes using probability b/sqrt(1-2b). And this gave you the numbers above? alThor PS For those that asked, I will mail the pdf as soon as I can get to it. Running late today.... EDIT: PPS I solved for lowball, as others appear to have, in case it wasn't obvious.
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#74
10-21-2005, 12:02 PM
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Re: Classic Type Game Theory Problem
If you think an understanding of measure theory is required to solve probability problems, then I can't help you. No other posters here had any difficulty understanding David's problem, and the vast majority I'm sure haven't studied real analysis.
EDIT: Most people would be unable to construct the real numbers, starting with the Peano postulates. That doesn't mean that they can't solve problems using real numbers or have an intuitive understanding of what they are.
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#75
10-21-2005, 01:06 PM
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Re: Classic Type Game Theory Problem
[ QUOTE ]
If you think an understanding of measure theory is required to solve probability problems, then I can't help you. No other posters here had any difficulty understanding David's problem, and the vast majority I'm sure haven't studied real analysis. [/ QUOTE ] Irrelevant. The less you know, the more you know. [ QUOTE ] EDIT: Most people would be unable to construct the real numbers, starting with the Peano postulates. That doesn't mean that they can't solve problems using real numbers or have an intuitive understanding of what they are. [/ QUOTE ] Again, irrelevant.
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#76
10-21-2005, 01:16 PM
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Re: typed solution
Yes, well almost. You're playing lowball and the original question was phrased assuming highball, which doesn't affect the calcs, but you're also forgetting that A's optimal strategy gets broken into two parts...my EV calc is based on the first post of the correct, complete solution posted by BBB in the "Poker Theory" forum, which I copy below (while correcting a small typo):
[ QUOTE ] So, to summarize: PLAYER A: If y < 0.5, stand pat if holecard is better than 0.5, draw if worse than 0.5 If 0.5 < y < 0.586, stand pat if holecard is better than 1-y, draw if worse than 1-y. If y > 0.586, stand pat if holecard is better than sqrt(2y-1), draw if worse than sqrt(2y-1) PLAYER B: If A draws, stand pat if y > 0.5, draw if y < 0.5 If A stands pat, draw if y < 0.586, randomly draw z = (1-y)/sqrt(2y-1) of the time if y > 0.586. [/ QUOTE ]
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#78
10-21-2005, 02:56 PM
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Re: typed solution
[ QUOTE ]
but you're also forgetting that A's optimal strategy gets broken into two parts... [/ QUOTE ] Either forgetting, or too lazy to type the obvious. [img]/images/graemlins/smile.gif[/img] [ QUOTE ] my EV calc is based on the first post of the correct, complete solution posted by BBB in the "Poker Theory" forum [/ QUOTE ] Ok, I did post the cutoff values for the strategies 25 hours earlier, but you're right I didn't explicitly post every part of the strategy. Maybe I just got lucky. [img]/images/graemlins/smile.gif[/img] I'm not about to get into a pissing match over it. I'll also assume that my estimate of EV is flawed, until I have a chance to look at it, which I won't have time for today. alThor
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#79
10-21-2005, 05:36 PM
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Re: Classic Type Game Theory Problem
[ QUOTE ]
It's not a nitpick. I'm a mathematician. The statement of the problem doesn't even make sense to me. [/ QUOTE ] Either this is self-parody or this is weak. Even if we go with what you're saying that you can't have uniform distributions on infinite sets (a point debated elsewhere in this thread, I tend to go with those who say you can, but hey, "I'm a physicist"), it's pretty clear that we can approximate this arbitrarily well with a discretized version of the interval [0,1] for the purposes of determining probabilities of winning/EV and the like. You can't make this reasonable assumption, state it, and then crank on the problem?
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#80
10-21-2005, 07:05 PM
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Re: typed solution
[ QUOTE ]
I'm not about to get into a pissing match over it. [/ QUOTE ] I'm not into pissing matches either and I don't want to take away from your fine contributions but in all fairness I felt it necessary to point out BBB's explicit mention of the correct strategy for A when B's upcard is in the interval [0.5,0.586]. It was far from obvious since at least 2 fine mathematicians had trouble with it at first.
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Linear Mode
