Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 19

X

x ðÞ c c :

Sometimes we can compute also the utility of the expected outcome
u X x ðÞ c c :

A concave utility means risk aversion and a convex utility means risk proclivity,
in the sense that the utility of the distribution on D(C) that gives probability 1 to the
expected outcome is, respectively, greater and smaller than its expected utility.
Thus concavity represents a utility-decreasing evaluation of pure risk-bearing and
convexity the contrary.
To understand the concept, suppose there are two lotteries, one that pays the
expected value with certainty and another that pays the different values with their
different probabilities. The utility of the

first lottery is larger than the utility of the
second for a risk-averse evaluation. On the other hand, giving risk a positive value
would lead to a convex utility. Finally, neutrality with respect to risk would make
indifferent the choice between the certain outcome and the same outcome in the
average, so that

u X

X

x ðÞ x x ¼ x ðÞ x ux ðÞ:

2.3.5 Example of Capacity Determination

Von Newmann and Morgenstern representation theorem provides the basis for the
design of complex tools to derive the capacity of each set. Accepting the above
listed conditions, instead of directly assigning a value to the set, its capacity may be
derived from preferences between distributions. The evaluator will

find easier to
compare simple distributions involving the set than choosing a numeric value for
the capacity of that set. The key idea consists of asking the decision maker
appropriate questions about extreme distributions involving the set, to determine if
its capacity is closer to one of two extreme values than to the other.
The procedure starts by determining the capacities of unitary sets {c}. For each
such set, the evaluator answers a question about preference between two distribu-
tions: one of them assigns the value 1 to {c}; the other, a free choice between the
distributions assigning the value 1 to any other unitary set in C (and consequently 0
to {c}).
If the evaluator prefers the first distribution, we conclude that the decision maker

assigns to {c} a value closer to 1 than to 0, that means a value larger than

½

; if the
other is preferred, we conclude that the decision maker assigns to {c} a value
smaller than

½ ; in the case of indifference, the quest ends, with the value ½ assigned

to {c}.
Suppose the answer to this first question is a preference for the distribution that

assigns the value 1 to {c}. Then we proceed by asking the preference between the