The dual theory of choice under risk.

*(English)*Zbl 0616.90005Let a set of random variables (representing payments) on the unit interval be given. A probability mixture operation of these random variables is defined. It is well known that, if a preference on these random variables satisfies some standard axioms and the independence axiom, then there exists a von Neumann-Morgenstern utility function on certain prospects, such that the utility of the random variables equals the expected utility of the payments.

The author introduces an alternative theory of choice under uncertainty, which he calls a dual theory, because the roles of payments and probabilities are interchanged. He first defines a new mixture operation of random variables. Not the probabilities are mixed for each possible payment, but the payments are mixed for each possible probability, which is, in this case, the probability that the payment will be larger than the given value (decumulative distribution function).

If the preference on random variables satisfies the standard axioms and an independence axiom on the newly defined mixtures, then utility of the random variables can be expressed in terms of a function having the probabilities in the above sense as arguments.

Properties in the new theory are similar to the ones in the expected utility theory. The theory is quite natural within a model where payments depend on uncertain states of the world. Some well known paradoxes (due to Tversky, and Allais a.o.) in expected utility theory give no problem in the dual theory, but similar paradoxes appear in the dual theory, which are normal in the original theory. Risk aversion in expected utility theory is determined by the concavity of the utility function on certain payments and in the dual theory risk aversion can be characterized by convexity of the utility generating function. The theory is applied to simple portfolio analysis and it is compared with other “non expected utility theories”.

The author introduces an alternative theory of choice under uncertainty, which he calls a dual theory, because the roles of payments and probabilities are interchanged. He first defines a new mixture operation of random variables. Not the probabilities are mixed for each possible payment, but the payments are mixed for each possible probability, which is, in this case, the probability that the payment will be larger than the given value (decumulative distribution function).

If the preference on random variables satisfies the standard axioms and an independence axiom on the newly defined mixtures, then utility of the random variables can be expressed in terms of a function having the probabilities in the above sense as arguments.

Properties in the new theory are similar to the ones in the expected utility theory. The theory is quite natural within a model where payments depend on uncertain states of the world. Some well known paradoxes (due to Tversky, and Allais a.o.) in expected utility theory give no problem in the dual theory, but similar paradoxes appear in the dual theory, which are normal in the original theory. Risk aversion in expected utility theory is determined by the concavity of the utility function on certain payments and in the dual theory risk aversion can be characterized by convexity of the utility generating function. The theory is applied to simple portfolio analysis and it is compared with other “non expected utility theories”.

Reviewer: C.Weddepohl