Disutility as a Probability: A Reconsideration of the Symmetry Approach to the Committee Decision Problem

Von Neumann and Morgenstern formulated the expected-utiiity approach in terms of a simultaneous utility-probability axiomatization, which makes utility unique up to a linear transformation. This note illustrates, that there are situations in which it is convenient to consider utility or, more precisely, disutility as a probability. The disutility caused by the best decision is zero: that caused by any other decision is nonnegative and usually positive: the only further condition which disutilities must satisfy in order to qualify as probabilities is that they add up to 1, for which a linear transformation of the utility function can be used. Disutility may thus be interpreted as the "probability of injury." This idea is illustrated by means of the symmetry approach to the committee decision problem as formulated by Van den Bogaard and Versiuis [van den Bogaard, P. J. M., J. Versluis. 1962. The design, of optimal committee decisions. Statistica Neerlandica 16 271-289.] and Theil [Theil, H. 1963. On the symmetry approach to the committee decision problem. Management Sci. 9 380-393; Theil, H. 1964. Optimal Decision Rules for Government and Industry. American Elsevier Publishing Co., Inc., New York, and North-Holland Publishing Company. Amsterdam.]. Each committee member has his own preferences, but the committee as a whole is responsible for making a decision. The committee's debate is described in terms of a Markov chain, and the symmetry approach appears to correspond to the steady state of Markov chain theory. An application to a three-member committee is discussed.