Rational Behavior under Complete Ignorance

Rational behavior under complete ignorance is described by means of requirements such as invariance of choice with respect to modifications of states of nature. Possible criteria necessarily involve intransitivities of indifference, and are incompatible with the ascribing of personal probabilities to events. Characterization of criteria shows that in first order approximation they take into account only the extremal possible outcomes of each choice; effects linked to events also come into play, although only in the second order, whereas an axiom system like that of Arrow and Hurwicz, which requires transitivity of indifference, excludes their being taken into account at all. IN PERSONAL PROBABILITY THEORIES, there is no such thing as "non-probabilizable uncertainty." According to these theories, the choices of a rational decision maker in an uncertain environment are explainable by a mathematical expectation of utility criterion. The decision maker thus assigns, consciously or unconsciously, to any given event, a certain probability determined by the a priori information he possesses or, in the absence of such information, by considerations of symmetry known as the principle of insufficient reason. This principle, in actual fact, does not permit equal treatment of all events; we shall investigate the decision criteria which do permit such treatment. Arrow and Hurwicz [1] have shown that, among such decision criteria, those whose weak preferences are transitive vary only according to the maximum and minimum values of the outcomes of each decision. They thus have the advantage of being easily stated but the drawback of not automatically preferring, over any given decision, any other which weakly dominates it. We shall prove that decision criteria can be made to take weak dominance into account while yet giving up only transitivity of indifference, a property which is, for that matter, seldom genuinely present, even in the case of choice under certainty, since it turns out to be generally noncomparability rather than true indifference. The criteria characterized by the axioms of Arrow and Hurwicz will then prove to be an approximation of those fulfilling ours.