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#31
06-09-2004, 12:11 PM
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Re: If he did it every time he had at least 3 heads...
[ QUOTE ]
It looks pretty clear, that even if I'm not sure which game we're playing, there is still a greater chance that the "forth" result is tails. The question is, how much greater. [/ QUOTE ] No, your friend could decide to make the statement only when all 4 tosses were heads. The problem is underspecified. The economist Knight distinguished between risk and uncertainty. Risks have knowable probabilities. Uncertainties may not. You don't always have enough information to assign probabilities to outcomes in a meaningful/consistent fashion. Imagine we play rock-paper-scissors, but I get to make my choice after yours. You can't determine a probability distribution for my choice, then make an optimal play based on it. 
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#32
06-09-2004, 02:40 PM
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Re: If he did it every time he had at least 3 heads...
[ QUOTE ]
No, your friend could decide to make the statement only when all 4 tosses were heads. The problem is underspecified. The economist Knight distinguished between risk and uncertainty. Risks have knowable probabilities. Uncertainties may not. You don't always have enough information to assign probabilities to outcomes in a meaningful/consistent fashion. Imagine we play rock-paper-scissors, but I get to make my choice after yours. You can't determine a probability distribution for my choice, then make an optimal play based on it. [/ QUOTE ] Thanks, the link for Knight's biography and writings is really helpful. I should read some of it. Anyway, I think your reasoning here leads to a somewhat strange conclusion. If the problem is indeed underspecified as you say, then if my friend offers me money on my bet here, I have no information to decide what is a better choice: heads or tails. Therefore, when guessing at it, I am practically looking at it as a 50-50 bet (as in your rock-paper-scissors example, where it's 1/3-1/3-1/3). So, at the bottom-line, according to this logic, I'm treating this as a the simplest coin-tossing problem. To make it more clear: if I'm asked to make a bet on a problem like this many times (by different people, say, so I don't know anything about any of them), without having any information about their "strategy", my best EV move is to toss a coin myself, every time, to make a choice. This IS a solution to the original problem, underspecified or not. Do you agree?
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#33
06-09-2004, 03:20 PM
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Re: If he did it every time he had at least 3 heads...
After a little more thought, I understand why 50-50 is not better than any other way of solving it.
It is basically the same as betting on the come of coin-tosses. Your EV will not be higher if you always bet heads, or always tails or 87-13 or 50-50 or whatever. This is what you mean by underspecified, if I understand it correctly.
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