Naked before God: HELP ME SKLANSKY!

[PairTheBoard]

Thanks. That's what I was looking for.

PairTheBoard

[JoshuaD]

Wow, I thought that when I first looked at the thread, and it seemed way too simple to be right.

[Kristian]

Sorry about the late answer, I have been on a short vacation. In the mean time, I have given the matter some more thought, and also discussed it with an equally geeky friend of mine.
My statement that the strategies posted used undefined probabilities was wrong, and I am sorry about posting stuff that I had not thought through.

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I am also quite sure that the professional mathematicians you mention can confirm that they make some additional assumptions before presenting their solution.

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I suggest you be less sure of things that are not true. The only assumption necessary is that the distribution is continuous (the CDF is continuous), which I mentioned above.

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...but in this matter you are wrong. The problem certainly needs a further assumption in order to devise an above 50% strategy, but the assumption needed is much weaker than I first thought.
What you need to know is the interval of possible outcomes, since every strategy posted requires that you are able to choose a seperation point of the distribution with a positive probability of outcomes on either side of this point. If you don't know this interval, then it is impossible to construct a >50% strategy unless you start considering some sort of 'distribution of possible distributions', and this would very quickly make the problem very boring.
Note however, that it is enough to know that the distribution has positive probabilities on the whole real axis. This would, for instance, be the case with the normal distribution with any parameters. Of course, if there are no bounds on the parameters God chooses for his normal distribution, he can put the probability of success as close to 50% as he likes.

[Kristian]

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It is not impossible to consider P(X>0,Y<0). You can consider it all day. It's a number. It exists. It is between 0 and 1.


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You are right, sorry [img]/images/graemlins/smile.gif[/img]

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But God can never choose a distribution which makes your probability of winning less than or equal to .5. This is why your probability of winning is always bigger than 50%.


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You are wrong. He can choose 'any' distribution, and you will have no chance at all of finding that crucial seperation point in this distribution if you know nothing further about it.

[pzhon]

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What you need to know is the interval of possible outcomes, since every strategy posted requires that you are able to choose a seperation point of the distribution with a positive probability of outcomes on either side of this point. If you don't know this interval, then it is impossible to construct a >50% strategy unless you start considering some sort of 'distribution of possible distributions', and this would very quickly make the problem very boring.

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That is still wrong. Let me quote a paragraph from elsewhere in this thread:

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Let R be a random variable from a distribution that has positive probability on any interval... Then R has a positive probability of being in the middle of God's distribution, so S(R) works strictly greater than 50% of the time.[/list]Once again, there are strategies that work strictly more than 50% of the time against any continuous distribution. If you don't believe that, try to find a continuous distribution against which guessing X is greater with probability 1/2 + arctan(x)/pi wins at most 50%. Please post such a distribution or retract your objection to the proof.

[Kristian]

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That is still wrong. Let me quote a paragraph from elsewhere in this thread:

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Let R be a random variable from a distribution that has positive probability on any interval... Then R has a positive probability of being in the middle of God's distribution, so S(R) works strictly greater than 50% of the time.[/list]

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But here the further assumption has been made that any interval has a possible probability. I agree that that is enough to make a &gt;50% strategy.

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If you don't believe that, try to find a continuous distribution against which guessing X is greater with probability 1/2 + arctan(x)/pi wins at most 50%.


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I don't understand this, please clarify.

[jason1990]

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But God can never choose a distribution which makes your probability of winning less than or equal to .5. This is why your probability of winning is always bigger than 50%.


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You are wrong. He can choose 'any' distribution, and you will have no chance at all of finding that crucial seperation point in this distribution if you know nothing further about it.

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As pzhon has suggested, you either need to submit to our mathematical prowess [img]/images/graemlins/shocked.gif[/img] and admit you are wrong, or "play God" and defeat one of these strategies.

So you are God. I will tell you beforehand what I'm going to do. (As God, you know it anyway.) I am going to take the number you give me, X, and compute 1/2 + arctan(X)/pi. Whatever this number is, call it p, I'm then going to flip a biased coin that has probability p of being heads. After I flip it, if it's heads, I'm going to guess that X is the larger number. If it's tails, I'm going to guess that Y is the larger number.

So now, as God, what distribution will you choose? You can choose any continuous distribution you like. So please choose it. And then please prove that my probability of winning is not more than 50%.

This is the challenge that pzhon has laid down for you. If you cannot complete it, then you have no ground to dispute the claims in this thread other than uninformed speculation. (And, yes, I am claiming the challenge cannot be completed. Hence, I'm claiming that you are uninformed and speculating.)

[Kristian]

Please disregard that last post. I misunderstood your first point, and had the second one explained in the last post by Jason1990. [img]/images/graemlins/confused.gif[/img] [img]/images/graemlins/blush.gif[/img]

[Izverg04]

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University of Maryland, Professor Michael Brin.


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How did this thread go so long without anyone mentioning Google, as in Michael Brin, grandfather of Google.

[Moonsugar]

some infinities are bigger than others