The expected utility hypothesis as immunity against the 'legal' utilization of the money pump and underachiever methods.

• Author: Persky, Allan

1. Introduction

   The expected utility hypothesis is somewhat controversial.(1) Not only has it met difficulty in experimental studies,(2) but its theoretical status and traditional defense have been questioned [6; 3]. In particular, the argument that non-linear preferences allow an individual to make book against himself has been refined in Green [3]. Green shows that an individual whose preferences arc not quasi-convex can be manipulated into making book against himself, but an individual with quasi-convex preferences that are not linear will not make book against himself, even though his choices are subject to certain infelicities.(3)

   The present paper applies Green's model and the expected utility hypothesis to the administrator of a government institution. Instead of an individual who is being manipulated as in Green [3], we assume that the administrator attempts to exploit the institution for his own benefit. Expected utility theory is customarily aimed at describing individual choice under uncertainty, while the scope of this paper is public choice rather than individual choice. We adopt individual choice theory for our analysis with the following situations in mind: (i) The administrator controls the institution's actions in a particular matter. (ii) The administrator technically lacks authority as in (i), but the limitations on his control are just a formality. For example, in practice the administrator makes a recommendation, and someone else rubber-stamps it. (iii) The administrator does not control the institution's actions, but he influences them, and his influence is important enough to study. For example, a United States senator cannot simply enact laws at his pleasure, but his influence is still a matter of importance. We will concentrate on the cases where, as a practical matter, the administrator has control of a particular action of the institution.(4) To simplify exposition in this context, we will speak of an administrator running an institution.

   Suppose that within the institution, an administrator has, in practice, control over a certain institutional action. Then he still faces potential difficulties from both outside and inside the institution. For example, if his decision is overtly dishonest, then the legal system may intervene and indict him for crimes, or his fellow administrators may make trouble for him in the future by not rubber-stamping his decisions any more or by pressing for his removal. We assume that by exerting his authority according to preferences that satisfy order, continuity, and attraction for (first order) stochastic dominance, the administrator is likely to avoid such difficulties for a sufficiently long period to be of interest. To see the point of this assumption, it is helpful to interpret these mathematical properties. The order assumption represents ability to make decisions. The leader makes decisions, and he makes them according to a definite system. He does not allow the institution to drift while he pursues his personal interests. The institution is led, not neglected. Furthermore, attraction for stochastic dominance means that the leader must oppose any program that is certain to reduce the institution's profit. He cannot simply sell or steal the institution's property or order his subordinates to do so. Finally, the continuity assumption means that there is always some program that is both preferable to the status quo and not certain to succeed; and there is also some program that is both inferior to the status quo and not certain to fail.(5) Continuity can be interpreted as level-headedness. The leader is bold enough to risk failure and cautious enough to be dissuaded by it. Under failure of order, continuity, or attraction for stochastic dominance, the administrator can neglect or grossly mismanage the institution for his own benefit. It can be deduced from Green [3], however, that preferences satisfying order, continuity, and attraction for stochastic dominance do not protect the institution. The administrator can still exploit it.

   How can a leader who faces the constraints of order, attraction for stochastic dominance, and continuity exploit his institution? The answer is by manipulating the probabilistic nature of the institution's situation. Suppose the institution's initial situation is represented by probability distribution H, and random events move the institution from H to [F.sub.2]. Then [F.sub.2] can be better than, equivalent to, or worse than H; and moreover, the leader did not choose [F.sub.2], so he can deny the blame for it, if it is worse than H. It is interesting that this obvious point allows subtle possibilities for manipulating the institution. We will consider two such possibilities: the money pump method and the underachiever method.

   In the money pump method, the administrator runs the institution according to public preferences that are not quasi-convex. This allows the administrator to weaken the institution's position for his own benefit. In the underachiever method, the administrator runs the institution according to public preferences that are quasi-convex, but not linear. This prevents him from making the institution's situation absolutely worse,(6) but it allows him to make the institution's situation inferior to one it could have had in the absence of his manipulation. In contrast, if an administrator runs an institution according to linear public preferences, then he cannot manipulate it by either method or by methods available when order, continuity, or attraction for stochastic dominance are not satisfied.

2. The Green Model

   Call the model in this section the Green model. It is essentially the model from Green [3].

   To establish economic interpretation, consider a government institution that takes in money and pays out money. There are many examples: tax collecting agencies, postal services, public transportation services, universities, etc. Consider the numerical difference between the money an institution takes in and the money it pays out. It is natural to call this difference the profit of the institution even though this profit is not legally available for distribution.

   Let X be a random variable that represents the institution's profit. Assume that X is certain to be in some interval I = [min, max] where min [is less than] max. Let D[min, max] be the set of all(7) probability distributions for X that assign a probability of zero to every set outside [min, max]. By a probability distribution, we mean a cumulative distribution function F. Thus for any number x,

   P[X [is less than or equal to] x] = F(x). (1)

   Note that D[min, max] is a convex set. For any F and G in D[min, max] and any [infinity] in [0, 1], [infinity] F + (1 - [infinity])G is in D[min, max]. Also any F in D[min, max] can be constructed as a convex combination

   F = [summation of] [[infinity].sub.i] [F.sup.i] where i = 1 to n (2)

   where n is a positive integer, each [[infinity].sub.i] is in [0, 1] or (0, 1), the [[infinity].sub.i]'s sum to 1, and each [F.sup.i] is in D [min...