| >Essentially everyone who has understood the question answers "no" to
>question 4. It seems that the only thing affecting your answer to
>question 3 should be which you prefer between the sports car and the
>luxury cruise.
Essentially everyone is wrong.
To see why, tighten up the phrasing a little
3. Which would you rather have:
(a) An 89 percent chance of winning a Mystery Prize plus
an 11 percent chance of winning a non-transferable
lease for 1994 on a sports car.
(b) An 89 percent chance of winning a Mystery Prize plus
an 11 percent chance of winning a non-transferable
luxury cruise for two leaving on 10-Jan-1994.
Now suppose you are told that the Mystery Prize is a non-transferable
lease for 1994 on a sports car? Or that it is a non-transferable luxury
cruise for two leaving on 10-Jan-1994. I think most rational choosers would
prefer b in the first case and a in the second.
If you leave the phrasing alone, the answer is not so obvious, but I still
think many rational choosers would prefer one sports car and one vacation
to two of either. The original phrasing involves us in a lot of confusing
detail about values of the prizes in trade or as gifts.
So in general, the answer to question 4 has to be "Maybe."
The usual approach to problems like this is to assign subjective (or
occasionally objective) utilities to the outcomes.
One I start thinking of it that way, it comes as no surprise to me that
my subjective utility for a sports car is not half my utility for two sports
cars. And in general, my utility for a sports car depends on what I have
already, including that 89 percent chance at the Mystery Prize.
Applying utilites to the first two questions is similarly illuminating.
I can just define three symbols
U0 my personal utility for winning $0
U1 my personal utility for winning $1,000,000
U5 my personal utility for winning $5,000,000
Since only utility difference will enter into our final calculations, we can
set the scale of our utilities by
U0 = 0
If I give the usual answers reported in .0, then the answer a to question 1
allows me to deduce that
1.00 * U1 > 0.89 * U1 + 0.10 * U2
and the answer b to question 2 allows me to deduce that
0.10 * U5 > 0.11 * U1
The paradox occurs because the former equation can be rearranged to give
0.11 * U1 > 0.10 * U5
No values of U1 and U5 can satisfy both inequalities.
The basic conclusion is a psychological one: people are not very good at
assigning utilities and computing expectations off the top of their heads.
I have seen similar studies which reached similar conclusions.
There is a more philosophic conclusion available, that no assignment of
utilites will avoid all similar paradoxes. I have seen some evidence
for this, but I don't think this examplequalifies.
I can avoid the philosophic conclusion by saying "Oops, I misspoke myself.
After looking this over carefully, I realize I have no preference for 1a
over 1b, or 2b over 2a. These inequalities above should really be equalities."
This leads to the conclusion that my personal utilities will satisfy
U5 = 1.1 * U1
This is intuitively in the right ballpark. Certainly I prefer having $5,000,000
to having $1,000,000. But the utility of $5,000,000 is certainly not 5 times
the utility of $1,000,000. I could have a pretty good time on $1,000,000. I
find it hard to imagine how much more fun $5,000,000 would buy me. Note to
future readers: you may wish to scale these amounts to reflect the inflation
between 1993 and your time.
Alternatively, by working it out on paper, I may realize that I really prefer
1b to 1a. This would make my personal U5 somewhat greater than 1.1*U1. My
current subjective guess is that my U5 is about 2*U1. The ratio varies
depending on what expect to earn and save, what I have in the bank and what
I expect to do with the money.
|
| Re .1:
> If you leave the phrasing alone, the answer is not so obvious, but I
> still think many rational choosers would prefer one sports car and one
> vacation to two of either.
I think you have misinterpreted the statement. The 89% chance and the
11% chance are mutually exclusive.
-- edp
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|
| > 1. Which would you rather have:
> (a) $1,000,000
> (b) an 89 percent chance of winning $1,000,000, a 10 percent
> chance of winning $5,000,000, and a 1 percent chance of
> winning $0. (These are mutually exclusive outcomes.)
I think many people would find (b) confusing and therefore would choose
(a) since they would probably not be strongly motivated to analyse (b),
(I assume the prizes were not real).
> 2. Which would you rather have:
> (a) An 11 percent chance of winning $1,000,000
> (b) A 10 percent chance of winning $5,000,000
Since these two options are easy to compare then most people will
choose the slightly smaller chance of winning lots more money.
Peter
|
| on intuitive level, i'll always will take one dollar in cash rather than
any chance of winning 1 million bucks, this i based on the good
premise that a bird in the hand is better than 10 on the tree.
so based on this, i'll pick option (a) any time.
offcourse there is no math here, but this is how i feel about it.
\nasser
|