[EM] Utility and probability (does Condorcet's approach to voting have ramifications for individual choice?)
David Catchpole
s349436 at student.uq.edu.au
Mon Aug 2 19:35:36 PDT 1999
I was thinking about the usual method which is used to find a measure of
utility for a probabilistic event (mean over probability of the
utility of specific outcomes) and thought- hey! Why do we use the mean? I
can think of a set of principles which would imply that the median
(weighted to probability) value of utility makes more sense.
Consider that you have a choice between opening two boxes, with two
probabilistic sets of outcomes. *As a utility-maximising actor, you wish
to choose the box which has the best probability of giving you a better
deal than the other box* (THE principle I use).
The difference in probabilities of "best deals" between the boxes (1 and
2) is
/
< (1) (2) (1) (2) /| (1) (2)|
> p p (U - U ) / |U - U |
< i j i j / | i j |
\
ij
(the sum over the probabilities of the difference in utilities divided by
the modulus of the difference in utitilities).
(2)
Comparing a probabilistic box to a certain box with utility U, the
difference is
/
< (1) (1) (2) /| (1) (2)|
> p (U - U ) / |U - U |
< i i / | i |
\
i
When the preference for the boxes is equivalent, this is equal to zero,
so U is the median utility.
I guess I should extend this to multiple and probabilistic choices, but
one can infer the similarity to Condorcet's approach to public choice,
especially if we think of the thing as a comparison of the votes of time
travelling elves who represent the probability of an event. (?)
Any comments?
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