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The axioms of utility 115
the highest probability of reward A. We used this axiom in our explanation
of utility earlier, where we reduced the conference organizer’s
decision to a comparison of the two lotteries shown in Figure 5.13.
Clearly, if the conference organizer’s preferences conform to this axiom
then she will prefer lottery 1.
Axiom 6: Compound lottery axiom
If this axiom applies then a decision maker will be indifferent between
a compound lottery and a simple lottery which offers the same rewards
with the same probabilities. For example, suppose that the conference
organizer is offered the compound lottery shown in Figure 5.14(a). Note
that this lottery offers a 0.28 (i.e. 0.4 × 0.7) probability of $60 000 and a
0.72 (i.e. 0.4 × 0.3 + 0.6) probability of −$10 000. According to this axiom
she will also be indifferent between the compound lottery and the simple
lottery shown in Figure 5.14(b).
It can be shown (see, for example, French 3 ) that if the decision maker
accepts these six axioms then a utility function exists which represents
his preferences. Moreover, if the decision maker behaves in a manner
which is consistent with the axioms (i.e. rationally), then he will choose
Figure 5.13
Figure 5.14
Lottery 1:
0.75
0.25
−$10000
0.4
0.6
0.28
$60000
0.72
0.7
0.3
−$10000
(a)
−$10000
(b)
Lottery 2:
$60000
$60000
−$10000
0.5
0.5
$60000
−$10000