[EM] Utility and probability (does Condorcet's approach to voting have ramifications for individual choice?)

[David Catchpole]

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?