Approaches to Criteria Combination - Probabilistic Composition of Preferences, Theory and Applications - page 20

distribution assigning the value 1 to {c} and another assigning the value ½ to {c}

and freely assigning the value ½ to other element of C. If the evaluator prefers the

first of these distributions, assigns to {c} a value closer to 1 than to ½ , that means a

value larger than ¾ . If the other distribution is preferred, then that value is smaller

than ¾ . Indifference means that ¾ is the value assigned and this will be the capacity

of {c}.
If the Von Neumann and Morgenstern assumptions are satis ed, indifference

will appear after a while. After the capacities of the unitary sets are obtained,
capacities for the sets of size two will be determined.
Suppose, for instance, the capacities of {c 1 } and {c 2 } are found to be 0.3 and
0.2. Then the capacity of {c 1 ,c 2 } is between 0.3 and 1. To determine this capacity,
the decision maker will be asked about preference between a distribution assigning
the value 1 to that set and another assigning the value 0.7 to a freely chosen subset
of its complement. From preference for the value 0.7 for the complement follows
that the capacity of {c 1 ,c 2 } for the decision maker is closer to 0.3 than to 1, what
means that it is between 0.3 and 0.65. From preference for the other distribution
follows a capacity from 0.65 to 1. From indifference, follow the final value of 0.65.

So, if, for instance, the interval from 0.3 to 0.65 follows from the answer
obtained, the next choice will be between a distribution assigning to the set a value
of 0.3 and another assigning the value 0.65. If the preference is for that assigning
0.65, we get a restriction to the interval between 0.475 and 0.65. If the preference is
for the other, the capacity is between 0.3 and 0.475. If the answer is indifference,
the value is 0.475. And so on.
After a logical sequence of such questions, we would eventually

find a capacity

representing the preference of the decision maker.

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