The Consistent Assessment and Fairing of Preference Functions

When a decision maker is assessing a preference (utility) function for assets (wealth), it is natural for him to start by making some quantitative assessments of the certainty equivalents of a few simple gambles and some qualitative statements specifying any regions in which he feels risk-averse or risk-seeking and any regions in which he feels decreasingly or increasingly risk-averse or risk-seeking. Several questions then arise. Does any preference function exist which satisfies all the quantitative and qualitative restrictions simultaneously, that is, are the restrictions consistent? If so, how far do they determine the preference function? How might one fair a "smooth" function satisfying the restrictions? This paper is addressed to these questions. First the problem is introduced in some detail, and the concepts involved reviewed. Then the case is considered where the qualitative restrictions only specify regions of risk-aversion or risk-seeking. It turns out in this case that all the restrictions are linear in certain quantities, so that the existence problem is essentially one of satisfying linear constraints. Furthermore, finding the maximum or minimum solution at a specified point is exactly a linear programming problem. Also discussed briefly are the possibility that some smoothing problems might simply introduce a nonlinear objective function (though the general smoothing problem is more complicated) and the problem of making the derivative of the preference function continuous (which is not always possible). If regions of increasing or decreasing risk-aversion are also given, the problem becomes much more difficult.