STOCHASTIC

1. STOCHASTIC DOMINANCE
G. Hanoch and H. Levy
Reprinted from The Review of Economic Studies 36, 335-346 (1969).
The Efficiency Analysis of Choices
Involving Risk'
I. INTRODUCTION
The choice of an individual decision-maker among alternative risky ventures, may be
regarded as a two-step procedure. Firstly, he chooses an efficient set among all available
portfolios, independently of his tastes or preferences. Secondly, he applies his individual
preferences to this set, in order to choose the desired portfolio. The subject of this paper
is the analysis of the first step. That is, it deals with optimal selection rules, which minimise
the efficient set, by discarding any portfolio that is inefficient, in the sense that it is inferior
to a member of the efficient set, from point of view of each and every individual, when all
individuals' utility-functions are assumed to be of a given general class of admissible
functions.
It is assumed throughout that any utility function is of the Von-Neumann-Morgenstern
[21] type, i.e., it is determined up to a linear transformation, and is non-decreasing, and
that individuals maximize expected utility. 2 The risks considered are random variables X
with given (cumulative) probability distributions F(x) = Pr {X g x}. F(x) is thus a nondecreasing
function, continuous on the right, with F(— oo) = 0, F(oa) =1. It may be
discrete, continuous or mixed, with finite or infinite range. The analysis is not influenced
by whether these probabilities are " objective " or " subjective ". However, the distributions
are assumed to be fully specified, with no vagueness or uncertainty attached to their
specification.
The present analysis is carried out in terms of a single dimension (e.g., money), both
for the utility functions and for the probability distributions. However, the results may
easily be extended, with minor changes in the theorems and the proofs, to the multivariate
case. 3
Section II gives a necessary and sufficient condition for efficiency, when no further
restrictions are imposed on the utility functions.
Section III states and proves the optimal efficiency criterion in the presence of general
risk aversion, i.e., for concave utility functions.
Section IV analyzes the conditions, under which the well-known and widely used
mean-variance criterion (n, a) is a valid efficiency criterion.
Finally, Section V gives some conclusions, remarks, and suggestions for further
theoretical and empirical analysis related to the present results.
II. UNRESTRICTED UTILITY—THE GENERAL
EFFICIENCY CRITERION
Given two risks (random variables), X and Y, with the respective (known, cumulative)
probability distribution functions F(x), G(y), we say that XD Y, or FDG—X dominates Y,
1 The authors are grateful to A. Beja and S. Kaniel for valuable comments. The referees commented
on an early draft, that similar results appeared in other works, unpublished and unknown to us at that
time: a paper by Hadar and Russel [7), a Thesis by J. Hammond [8], and a book by Pratt, Raiffa and
Schlaifer [17]. There is also some overlap with Quirk and Saposnik [18], and Feldstein [5]. However, the
present paper gives a more general treatment and some significant modifications to most of these results.
2 The axiomatic basis for this assumption is given in: Herstein and Milnor [10], Marschak [15],
Von-Neumann and Morgenstern [21].
3 This, however, raises many special and interesting problems, and will be dealt with more extensively
in a forthcoming paper by the first author.
STOCHASTIC DOMINANCE