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 [quote="tigg"]For part 1: As I understand utility theory, they draw a distinction between the value of money and the utility of money. The value of a \$10 bill is always \$10, but depending on your circumstances, that \$10 may mean a lot to you or not. In other words, if you give \$10 to Bill Gates you are wasting his time. If you give \$10 to a pauper, he can buy dinner. The same \$10 increase in wealth will have a high utility for the pauper and a low utility for Bill Gates. A utility function maps the value of money to the utility of money for a given person. The standard assumption of utility theory is that utility is a non-linear function of value. (Actuaries will tell you that this assumption is the basis for insurance companies.) In a very simple case, your utility function may be: U(v) = sqrt(v), where v is the money's value and U is utility. For example, the value of \$4 has utility 2. The value of \$16 is 4. etc. Anyway, in the question at hand we have a bet to consider. It is one thing to compute the expected value of the outcome and another to compute the expected utility. Presumably, we want to optimize the expected utility. I'll use U(v) = sqrt(v). If you don't take the bet, your expected utility is: U1 = (sqrt(X)+sqrt(Y))/2. If you do take the bet, U2 = (sqrt(X+Y)+sqrt(0))/2 Since U1 is always bigger than U2. I think you don't want the bet. [/quote]
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Chuck
Posted: Thu Mar 07, 2002 11:03 pm    Post subject: 1

I think this puzzle assume that utility equals money.

After seeing what I have, can I still assume that I have a 50% chance of having the larger amount?
Lepton
Posted: Thu Mar 07, 2002 10:59 pm    Post subject: 0

The communists are out to get me. I never take risks.
tigg
Posted: Thu Mar 07, 2002 9:14 pm    Post subject: -1

For part 1:

As I understand utility theory, they draw a distinction between the value of money and the utility of money.
The value of a \$10 bill is always \$10, but depending on your circumstances, that \$10 may mean a lot to you or not. In other words, if you give \$10 to Bill Gates you are wasting his time. If you give \$10 to a pauper, he can buy dinner. The same \$10 increase in wealth will have a high utility for the pauper and a low utility for Bill Gates.

A utility function maps the value of money to the utility of money for a given person.
The standard assumption of utility theory is that utility is a non-linear function of value.
(Actuaries will tell you that this assumption is the basis for insurance companies.)

In a very simple case, your utility function may be:
U(v) = sqrt(v), where v is the money's value and U is utility.
For example, the value of \$4 has utility 2. The value of \$16 is 4. etc.

Anyway, in the question at hand we have a bet to consider.
It is one thing to compute the expected value of the outcome and another to compute the expected utility.
Presumably, we want to optimize the expected utility. I'll use U(v) = sqrt(v).

If you don't take the bet, your expected utility is:
U1 = (sqrt(X)+sqrt(Y))/2.

If you do take the bet,
U2 = (sqrt(X+Y)+sqrt(0))/2

Since U1 is always bigger than U2. I think you don't want the bet.
Rollercoaster
Posted: Thu Mar 07, 2002 7:34 pm    Post subject: -2

Well, 'answering' a paradox usually raises more questions than it answers, but I'll chime in and risk the verbal beatings. In part II, I have a problem with the statement that there is a 50% chance of winning once you learn the amount in your envelope. It reminds me of the statement, "I have a 50% chance of winning the lottery this week - either I WILL or I WON'T."
test78
Posted: Thu Mar 07, 2002 6:39 pm    Post subject: -3

I don't know if this qualifies as chestnut or not because the questions at least to me semm somewhat differnet than the ones I've seen here ...
Anyway I felt like posting it...feel free to flame me if you must
And yes these are not my puzzles...

Aaron Brown

I. Money, money, money
Two different amounts of money are placed into envelopes. One envelope is selected at random and given to you. The other envelope is given to Paul. Neither you nor Paul know the amounts. Paul offers you a bet, which you may take or leave. The bet is that after opening the envelopes whoever has the larger amount of money gives it to the other, leaving him with nothing.
Call the amounts of money in the envelopes X and Y with X>Y. There is a 50% chance your envelope contains X and a 50% chance it contains Y. If you turn down the bet, your expected winnings are (X + Y)/2 and your standard deviation is (X - Y)/2. If you accept the bet you have the same expectation, (X + Y)/2, but the standard deviation is now (X + Y)/2. [You should check these for yourself!] You turn down the bet because it leaves your expected value unchanged and increases your standard deviation.

Q: Is this the correct decision under normal economic utility theory assumptions?

II. The envelope please. . .
You open your envelope and find \$100. Paul asks you again if you want to bet. Now you reason that you have a 50% chance of winning or losing. If you lose, you lose \$100. If you win, you win more than \$100. So your expected value is positive and you take the bet.
Q: Is it correct that your expected value is positive?
Q: Is Paul's expected value also positive for the same bet?
Q: If so, where does the extra value come from?
Q: And why would you take a bet after you open the envelope, but not before?
Q: Does it matter if Paul sees how much you had in your envelope?