A purely abstract decision problem

jason1990 · #1 06-07-2007, 04:14 PM

There is a bent coin. (Big surprise.) Suppose you really know absolutely nothing about it. You do not even have any information about how coins are generally bent by random people. You really have absolutely no information at all. (This is the abstract part.)

You now get one piece of information. You learn that the coin was just flipped 3 times. The results, in order, were tails, tails, heads.

Here is your decision problem. Your friend offers to lay you 7 to 4 odds if you will bet on heads for the fourth toss. Your friend has the same information (and lack of information) that you do. Do you accept? Why or why not? Is there an objective answer to this question?

TomCowley · #2 06-07-2007, 06:11 PM

I take the bet and expect a small +2/55 unit EV because I think P(heads) is greater than 4/11. The objectivity of the answer concerns the objectivity of choosing a prior distribution with no information. I used the uniform prior.

TomCowley · #3 06-07-2007, 07:06 PM

I don't claim to be qualified to judge the relative merits of different priors- obviously one for this case must be symmetric around 0.5, but that's the only hard constraint I'm willing to put on. It looks like the uniform prior estimates P=2/5=0.4, the Jaynes prior P=1/3=0.333, and the Jeffreys prior 3/8=0.375 (I think). Given the offered odds of .3636, you'll come up with a different answer depending on the different prior, which was probably Jason's point.

BluffTHIS! · #4 06-07-2007, 07:29 PM

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Is there an objective answer to this question?

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It is usually assumed that a coin is fair or not, in stating probability problems. But obviously that is just a convention. The question is that with a random coin, which you don't know is fair or not, how large of a sample size of flips would you need in order to assume it was fair or not? 3 isn't close and I personally wouldn't bet either way at those odds.

PairTheBoard · #5 06-07-2007, 07:41 PM

Well, I figure there are conservatively at least a googolplex of different scenarios by which biases might be produced by way of People Bending Coins. ie. There are at least a googolplex of functions
f1(p), f2(p),...,f_googolplex(p) describing prior distributions for the Bent Coin's true long run P(Heads).

Since I know nothing about the Bending System I must conclude all googolplex of the f_i's are equally likely. Since they virtually cover all possible distributions I will just track down each and every one of them, use it to compute the P(Heads | tails, tails, heads, f_i), average those over the googolplex of them and will have my answer.

I'll have to get back to you.

PairTheBoard

jason1990 · #6 06-07-2007, 08:20 PM

[ QUOTE ]
It is usually assumed that a coin is fair or not, in stating probability problems. But obviously that is just a convention. The question is that with a random coin, which you don't know is fair or not, how large of a sample size of flips would you need in order to assume it was fair or not? 3 isn't close and I personally wouldn't bet either way at those odds.

[/ QUOTE ]
Thank you for the reply. Also thank you for bringing the bold part to my attention. It has reminded me that someone might want to randomly choose between 7 to 4 on heads and 4 to 7 on tails.

Let me emphasize that this is not an option in this puzzle. You just need to decide whether or not to accept the bet as given. You can assume your friend has no inside information at all. He just sees the two tails, wants to bet on tails, and is laying you odds. The question is, do you like those odds? Also, do you think this is just a matter of personal taste, or do you think the information in the problem is sufficient to come to a logical decision?

luckyme · #7 06-07-2007, 08:24 PM

[ QUOTE ]
do you think the information in the problem is sufficient to come to a logical decision.

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if it isn't then a million flips won't allow us to come to a logical conclusion either.

luckyme

BluffTHIS! · #8 06-07-2007, 08:27 PM

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Also, do you think this is just a matter of personal taste, or do you think the information in the problem is sufficient to come to a logical decision?

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This problem would seem to me the equivalent of you turning on the TV to some random game in a sport which I didn't follow and had virtually zero knowledge, excpet that you give me their win/loss stats for their past 3 games only. Which just like your problem I would judge a no bet situation *at those odds*.

The real question here is how good of odds do you need to make a wager in these kinds of situations where you have some very minimal information that either side has won/come up, at least once (making you think, perhaps incorrectly, that the whole game isn't rigged), and for how much of your roll or net worth.

jason1990 · #9 06-07-2007, 08:27 PM

[ QUOTE ]
[ QUOTE ]
do you think the information in the problem is sufficient to come to a logical decision.

[/ QUOTE ]

if it isn't then a million flips won't allow us to come to a logical conclusion either.

[/ QUOTE ]
Abstractly, you may be right. So does this mean you think it is or it isn't?

jason1990 · #10 06-07-2007, 08:37 PM

[ QUOTE ]
This problem would seem to me the equivalent of you turning on the TV to some random game in a sport which I didn't follow and had virtually zero knowledge, excpet that you give me their win/loss stats for their past 3 games only. Which just like your problem I would judge a no bet situation *at those odds*.

The real question here is how good of odds do you need to make a wager in these kinds of situations where you have some very minimal information that either side has won/come up, at least once (making you think, perhaps incorrectly, that the whole game isn't rigged), and for how much of your roll or net worth.

[/ QUOTE ]
You can assume that the amount your friend wants to wager is small compared to your bankroll -- small enough so that bankroll and utility considerations are not an issue.