Soc Choice Welfare (1984) 1:177-186
Soeial Choiee
ndWelfare
© Springer-Verlag 1984
Expected Subjective Utility:
Is the Neumann-Morgenstern Utility the Same
as the Neoclassical's ?
Yew-Kwang Ng
Department of Economics, Monash University, Clayton, Victoria 3168, Australia
Received August 23, 1983 / Accepted June 26, 1984
Abstract. Using axioms no stronger than those for the Neumann-Morgenstern
expected utility hypothesis, with the recognition of finite sensibility, it is shown
that the utility function derived by the N - M method is a neoclassical subjective
utility function, contrary to the belief otherwise by prominent economists. This
result is relevant for issues of utility measurability, social choice, etc. since it is
subjective utility that is relevant for social choice. The relevance of individual risk
aversion to the form of social welfare functions and the rationality of "pure" risk
aversion are also discussed.
I. Introduction
The expected utility hypothesis of von Neumann and Morgenstern (1947, N - M
hereafter) says that the choice involving risk of an individual (or other decisionmakers) satisfying some reasonable axioms can be represented by the maximization
of the expected value of a cardinal utility function (unique up to a positive afine
transformation) the existence of which can be proved. This much is well accepted.
What is controversial is whether the utility function derived by the N - M method can
be taken to reflect the true subjective utility of the individual in the sense of the
neoclassical economists such as Edgeworth. A firm negative answer has been given:
"what relationship, if any, does the N - M cardinal utility theory have to that of the
neoclassical utility theorists? It is generally (though not universally) agreed that
there is none - the two utility measures have nothing in common insofar as their
cardinality is concerned" (Baumol 1977, p. 431; see also Banmol 1951, 1958).
The quotation is from a prominent economist and the chapter from which the
quotation comes is described by another well-known economist as "the most useful
general readings on the subject-matter" (Green 1976, p. 230). The importance of that
view cannot be doubted. On the other hand, the majority of economists who use the
N - M utility indices in theoretical and applied studies (e.g. agricultural economics)
treat the indices as having subjective significance (i.e., except for the undetermined zero
178
Y.-K. Ng
point and the arbitrary scale, the N - M utility measures the subjective utility of an
individual). How can this apparent inconsistency be reconciled?
Baumol is right, at least formally. The N - M hypothesis does not show that the
utility function derived, though predictive of the objective choice of the individual,
actually reflects his subjective utility in the sense of the neoclassical. On the other hand,
most practising economists are also right, at least effectively, in accepting the N - M
utility indices as subjective. This is so since, as is shown below, using the same set of
axioms as the N - M hypothesis, with only the recognition of finite sensibility (that
human beings are not infinitely sensitive or perfectly discriminative) 1, it can be shown
that the utility function derived from the N - M method is in fact a subjective utility
function in the sense of Edgeworth.
Our result implies that the N - M method can be used to derive cardinal utility
functions for individuals with subjective significance, with relevance for the issues of
utility measurability, social choice, etc. This substantially increases the importance of
the N - M method of utility measurement since the resulting utility indices are not just
objective indicators of choices involving risk but are themselves the subjective utilities
of individuals (apart from the choice of units and the determination of zero points).
This is important since in social choice presumably it is the subjective utility that is
of normative interest. However, while our result enhances the importance of the N - M
method in subjective utility measurement, it does not in itself solve the problem of
interpersonal comparison of utility. Nevertheless, given that each individual subjective
utility function is measurable by the N - M method, solution of the interpersonal
comparison problem can be much easily tackled by methods such as the finite sensibility approach, especially if methods of indirect measurement (Ng 1975, Sect. 9) are used.
(Objections by Arrow 1951/1963, pp. 116-8, Sen 1970, p. 94, etc. to the use of a just
perceivable increment of pleasure as an interpersonal unit are replied to in Ng 1975,
Sect. 5 and in Ng 1984.)
II. AnMy~s
Edgeworth recognized that human beings are not infinitely sensitive and took it as
axiomatic ("a first principle incapable of proof') that the "minimum sensible" or the
just-perceivable increment of pleasure, of all pleasures for all persons, are equatable
(Edgeworth 1881, pp. 7ft., 60ff.).2 With the usual continuity (of preference) assumption, one cannot pin-point (even just in principle) a "minimum sensible" or minimal
preference but one can pin-point (in principle) a maximal indifference. (The distinction
is only of technical interest since a minimal preference is continuous with a maximal
indifference.) Ignoring the question of determining the zero point, and ignoring the
distinction between preference and welfare (on which see Ng 1983), I shall thus take
"a utility function subjective in the sense of Edgeworth" to mean one that uses the
same number to measure the utility difference of a maximal indifference. Such a
function must be cardinal in the sense of being unique up to a positive aline transformation. But not any cardinal function in this sense is subjective in the sense of
1 Infinite sensibility contradicts both common sense and psychological studies
2 Ng (1975) derives this is a result from other axioms and discusses other issues related to this
paper
Expected Subjective Utility
179
Edgeworth. In particular, the N - M utility function is cardinal but may or m a y not
be subjective. What we want to prove is precisely that, given our axioms (no stronger
than those for the N - M hypothesis except for our recognition of finite sensibility), the
utility function derived by the N - M method is in fact subjective (in the sense of
Edgeworth).
With finite sensibility, a perfectly rational individual m a y have intransitive indifference. For example, suppose an individual prefers two spoons of sugar (x) to one (y)
in his coffee. If we increase the amount of sugar continuously from one spoon, we will
reach a point y' (say 1.8 spoons) for which the individual cannot tell the difference
between x and y'. There m a y exist another point y" (say 1.6 spoons) for which the
individual is indifferent to y' but he prefers x to y'.
With indifference being intransitive, the preference of an individual is no longer an
order (satisfying completeness and transitivity) but can be a semi-order (satisfying
completeness and weak transitivity 3), with his "underlying preference" (or intrinsic
preference, his preference if he were infinitely sensitive; for an operational definition,
see Ng 1975) an order, such that his (explicit and intrinsic) preferences can be represented by a utility function satisfying the following 4
Vx, y, [(x_Ry~ V(x) - g(y) ____0;
yPx~g(y)
- g ( x ) > k;
x R y ~ U (y) - U (x) <= k where k is some positive constant].
(1)
The notations used are: U = utility, P = "is preferred to", R = "is at least as
preferable as" (preference or indifference), x, y . . . are alternatives, and the underlined
_R, etc. are the underlying preference relations. With a utility function satisfying (1), the
number k is used to measure the utility difference of a maximal indifference. Such a
utility function is thus subjective in the sense of Edgeworth and is cardinal in the sense
of being unique except for the choice of the precise value of k and for the zero point
(i.e. unique up to a positive afine transformation).
It has been argued that a just perceptible improvement m a y differ across different
experiences even for the same individual. ' O n many occasions, a just perceptible
improvement in musical performance means much more to me than a just perceptible
quantity of drink' (Mirrlees 1982, p. 69). This I think m a y be due to the difference in
time period involved and/or to possible future effects. A just perceptible quantity of
drink m a y last only a fraction of a second while the musical performance probably a
couple of hours. Moreover, Mirrlees m a y also value the recollection of a high quality
musical performance. As it is, our analysis is formulated in an a-temporal basis and
hence does not cover the complication introduced by differences in time period. With
time as a dimension, one should then select a just perceptible improvement over a just
perceptible length with neither future nor external effect as the standard unit, and use
indirect measurement (Ng 1975, Sect. 9) for experiences difficult to be so measured.
Given that the preference of an individual under certainty can be (which does not
mean "must be") represented by a utility function satisfying (1) (defined as Axiom A 0),
3
Requiring two levels of preference to dominate a level of indifference such that
x P y P z ~ x P z , r P x Iy P z or r P x P y Iz ~ r P z
4 To ensure representation, either one of the following (in addition to completeness and weak
transitivity) is sufficient: (i) the preference is defined over a countable set, (ii) the preference is
continuous and defined over a connected set, (iii) there exists a "preference-dense" countable
subset (see Ng 1979/1983, Appendix 1 B)
180
Y.-K. Ng
we wish to show (i) that his preference under risk is the maximization of the expected
value of that utility function, (ii) that the utility function derived by the N - M method
must satisfy (1) and is hence subjective in the sense of Edgeworth, if either of the
following sets of axioms is satisfied. The first set refers to the underlying preference and
the second set works only with the traditional explicit preference. Both sets of axioms
are similar to the set of axioms yielding the N - M result. But since we recognize finite
sensibility, we obtain the additional result that the utility function derived must satisfy
(1), i.e. must be a neoclassical subjective utility function.
In the following, a prospect or lottery L = (x, y; e,/3) denotes the mutually exclusive outcomes (or alternatives) x and y with c~(0 < c~ < 1) the probability of x and
/3( = 1 - c0 the probability of y. Other lotteries are similarly denoted, e.g./3~ = 1 - ex.
Though not explicitly covered, the principle of our results (as is the N - M result) can
be easily extended to lotteries with more than two possible outcomes at the cost of
only notational complication. Since we do not exclude cases where c~ = 0 or c~ = 1,
cases of certain outcomes are just special cases in this formulation.
A x i o m Set I
A 1. c~1 > e2 ~ D _RL2 where D = (b, w; e 1,/~1), L2 = (b, w; e2,/32), with b any best and
w any worst outcome in the relevant set of alternatives. 5
A2. V x, (b, w; c~,/3) I_x for some e: O <_ c~ <_ 1,
A3. Vx, y: (xI_ L x and y_/U) implies ((x,y; e,/3) I_(LX, LY; ~,/3))
A 4 . Vx, y, ((x,y; ~,/3) !(L~,Lr; c~,/3)where L~ = (b,w; c~,fix), L r = (b,w; c~Y,/3Y))implies
((x,y; ~,/3) I_(b,w; ~ + tie r, e/3x + ~3fir)).
1 ~)).
1
A 5. V r, x, y, z, (r I x & z P y) implies ((r,z,. ~,~)
1 1 P_ (x,y; ~,
Axioms A 1 - A 4 are used in the N - M hypothesis except that they refer to the
explicit preference assuming infinite sensibility while the ones here refer to the underlying or intrinsic preference (the preference if the individual were infinitely sensitive).
Since with infinite sensibility, the underlying preference coincides with the explicit
preference, the two sets are thus comparable. A 5 can be deduced as an implication of
the set of axioms for the N - M hypothesis which assumes infinite sensibility. 6 Hence,
our set of Axioms A O - A 5 is no stronger than the set for the N - M hypothesis, except
for our recognition of finite sensibility.
A 1 says that, for any two lotteries with b and w as the only two alternative
outcomes, the one lottery that has a higher probability of winning the best outcome
is preferable and vice versa. A 2 assumes some form of continuity. Originally stated in
the N - M hypothesis as true for any x , y , z : x R y R z instead of just for b,w and any
x (in place of y), it has attracted some queries including one by Alchian (1953). If you
s The assumption that there exists a best and a worst alternative is not really needed. Whenever we wish to consider the range of preferences outside the limits set by b and w, we can always
choose a new pair b' and w'. Alternatively, we may adopt the convention discussed by Samuelson
(1966, p. 135)
6 Apart from the counterpart to A 1-A 4, the set of axioms for the N - M hypothesis includes
a (stronger) counterpart to A 0, an axiom specifying that the individual has a complete ordering
on the set of prospects. An easy way to see that A5 is implies by the N - M set of axioms is to
note that the latter implies (2) below which implies A 5 with infinite sensibility
Expected Subjective Utility
181
prefer two candy bars (2c) to one candy bar (1 c), and one candy bar to being shot in
the head (S), Alchian doubts that there is any positive probability a of being shot in
the head such that (S, 2c; c~,fi) 1(1 c). Green (1971/1976, p. 218) defended the axiom by
saying, "I should regard the second candy bar as compensation for the (positive)
probability that someone in the middle of Sahara desert firing a revolver in the
direction of my head in Toronto could hit his target." I went further by declaring my
willingness to take "the extra candy bar in exchange for letting someone in North
Oxford fire [an ordinary revolver, not a guided missile] at me in the centre of Oxford
... or any other probability of death no larger than (0.1)99" (Ng 1975, p. 559).
A 3 says that, in any lottery, any component outcome can be replaced by a lottery
indifferent to it, and the resulting lottery is indifferent to the original one. The reasonableness of a stronger version of A 3 (the Strong Independence Postulate) is explained
in Samuelson (1966, pp. 133-4). Essentially, since outcomes are mutually exclusive,
the usual interdependency between goods does not apply. A 4just means that we may
apply the usual rules of combining probabilities. The only possible outcomes for the
lottery (Lx, LY; c~,fl) are b and w. What are the probabilities of b and w ? They are given
by the combination of probabilities stated in A 4.
A 5 says that the same (50 - 50) probability mix of obtaining (explicitly) preferred
and an indifferent outcome must be an intrinsically preferred lottery. This axiom is in
the spirit of a semiorder that a preference should outweigh an indifference to give at
least an intrinsic preference. The question of the variability of a just perceptible
difference across different experiences has been addressed above. There remains the
question of probability assessment. If the probability figures in A 5 are not definite
numerical figures but refer to some actual uncertain situations, then the figure of ,,1,,
may just mean that people on average judge the probability to be ½but different people
may differ in their judgments. In this case, the difference in probability judgment may
well outweigh the strength of a preference over an indifference to yield a choice
contrary to A 5. However, it does not affect our proof by confining the probability
figures to be exact numerical figures. Then it seems that A 5 is very reasonable. In fact,
a stronger version (A 10 below) requiring explicit preference is also reasonable. This
is so since a rational individual can be taken to have perfect discrimination over
explicit numerical figures. Moreover, A 5 refers to situations where the probabilities
involved are all the same - exactly ~. Thus A 5 seems to be rather compelling. (See
however the discussion of "pure" risk aversion/preference in Sect. III.)
Proposition 1: The preference of an individual satisfying Axioms A O-A 5 is represented
by the maximization of the expected value of a subjective cardinal utility function such
that
L_RE *-~E (L) >_>_E (E),
(2)
where E is the expected value of a utility function satisfying (1). Given our axioms, a
utility function derived by the N - M method or any function satisfying (2), must satisfy
(1), i.e., must be cardinal and subjective in the sense of Edgeworth.
Proof: From the relevant set of alternatives (outcomes), take a best alternative b and
a worst alternative w. Allot the utility number "1" (one) to b and "0" (zero) to w, and
,,~x,, to any x_/(b, w; a x, fix) and define the expected utility of any lottery L = (x, y; a, fi)
182
Y.-K. Ng
as E(L) =- eU(x) + flU(y), and U(x) = ax for all xI_(b,w; a~,fl~) as is possible
from A2. Now consider any L = (x,y; c~,fl) and E = (u,v; a',fl'). F r o m A2,
x ! (b, w; c~~, fix) ~ L~, y_/(b, w; ~r, fly) = Lr, for some c~x and ~r, 0 <- c~~, ~r < 1. F r o m A 3,
L!(L~,LY; a, fl). From A4, Ll(b,w; ~:' + fla y, afl* + fiftY). Similarly, we can derive
El(b,w; a'a" + fl'o~v, a'fl" + fl,flv). F r o m A I ,
LR_E ~ a~~ + fla r >=~'~ + fl'a ~.
Since by definition E(L) = aa ~ + fla y and E(E) = ~'ct~ + fl'a v, we have (2) above.
Establishing (2) itself does not provide anything more than the N - M hypothesis
(except that we use a set of axioms weaker in some aspects). What is more important
for our purpose here is to show that any utility function satisfying (2) must be a
subjective cardinal utility function satisfying (1).
A 0 ensures that the preference of the individual can be represented by a utility
function satisfying (1); it does not ensure that the one derived above and that any one
satisfying (2) must also satisfy (1). However, for the case of certainty (L = (x .... ; 1, 0),
E = (y .... ; 1, 0)), we have from (2),
x _Ry ,~-~U (x) - U (y) >=0
which is the first part of (1). F r o m A 5 and (2), we have for any four alternatives f i x ,
zPy,
½u(r) + ½v( ) > ½V(x) +
Hence, U ( z ) - U ( y ) > U ( x ) - U(r). This is true as long as z P y and r I x for any
r, x, y, z. Calling k the supremum of the absolute value of the utility difference
U(x) - U(r) such that xIr, we have the second part of (1).
Q.E.D.
Remark 1: The intuitive meaning of Proposition 1 is this: Our axioms ensure the
existence of a utility function satisfying (2) (i.e. the N - M hypothesis), and our A5 in
conjunction with (2), ensures that any preference is associated with a utility difference
more than any indifference, making the utility function satisfy (1).
Since most readers are likely to be unfamiliar with the concept of intrinsic preference, it is desirable to have a similar proposition based on axioms involving only the
explicit preference.
Axiom Set II
L1pL2_..~l > az, where L 1 = (b,w; ai, fli), L2 (b,w; ~2 fl2).
gx, (b,w; ~,fl)Ix for some a: 0 -< a _< 1.
Yx, y, ( x l L ~ & y l L y) implies ((x,y; ~,fl) I(LX, LY; ~,fl)).
Vx, y, ((x,y; a, fl) I(IS, LY; a, fl) where L~ = (b,w; ~x, flx), Ly = (b,w; 7Y, flY))
implies ((x,y; a, fl) I(b,w; ¢t~* + fla y, aflx + flflY)).
AIO. Vr, x,y,z, ( r l x & z P y ) implies ((r,z; 7,~)i
i P(x,y; -i,2))'li
A6.
A7.
A8.
A9.
=
It may be noted that, apart from changing all axioms in reference only to the
explicit preference, A 6 specifies only one way relation and only for strict preferences.
This relaxation allows for imperfect discrimination in the perception of probabilities.
In fact, if we are confined to numerical probability figures (over which a rational
individual can be taken to have perfect discrimination) instead of probability assess-
Expected Subjective Utility
183
ment of actual risky situations, this relaxation is not necessary. Correspondingly,
Proposition 2 below can be strengthened to be a more direct counterpart of Proposition 1. But even with the relaxation, we can obtain a sufficiently strong result.
Proposition 2: The preference of an individual satisfying Axioms AO, A 6 - A I O is
represented by the maximization of the expected value of a subjective cardinal utility
function such that
LPE ~ E (r) > E (E),
(2')
where E is the expected value of a utility function satisfying:
Vx, y, (yPx~-~u(y) - U(x) > k;
xRy,--~U(y) - U(x) < k,
(1')
where k is some positive constant. Given our axioms, a utility function derived by the
N - M method, or any function satisfying (2'), must satisfy (1'), i.e. must be cardinal and
subjective in the sense of Edgeworth.
Proof: Similar to the proof of Proposition 1, except that the weaker A 6 allows (2') to
be stated only as a one-way relation.
Q.E.D.
(A more detailed proof is available from the author.)
III. Concluding Remarks
A. Social Welfare Functions and Individual Risk Aversion
Our result implies that the N - M method can be used to derive cardinal utility
functions for individuals with subjective significance, with relevance for the issues of
utility measurability, social choice, etc. For example, Samuelson (1947, p. 228 n) and
Hahn (1982, p. 195) both declare their failure to see why social choice (e.g. with respect
to income distribution) should depend on individual risk aversion (with respect to
income). This dependence is straightforward once our result is recognized. The degree
of risk aversion reveals the degree at which subjective marginal utility of income
diminishes. Since social welfare is a function of individual utilities, how rapidly marginal utilities diminish has obviously important effects on social choices that affect
individual income levels. The reason Samuelson and Hahn fail to see this simple
dependence is probably related to their view similar to that of Baumol quoted in the
introductory paragraph above. Since the utility function derived by the N - M method
is taken as having nothing to do with subjective utility, the diminishing marginal
utility revealed by risk aversion does not imply diminishing marginal subjective utility
and hence does not affect social choice. But our result shows that the N - M utility
function does measure subjective utility, making social choice dependent on individual
risk aversion.
While we recognize the dependence of social choice on individual risk aversion
with respect to income (through its indication of the degree of diminishing marginal
utility of income), this does not mean that social welfare should not be a linear function
but should rather be a quasi-concave function of individual utilities if individuals are
risk-averse. Such a view is quite generally held (e.g. Pattanaik 1968). Thus the maximin
184
Y.-K. Ng
or leximin (Rawlsian) social welfare function is usually criticized as implying absolute
risk aversion. Commentators usually add that some degree of risk aversion may be
reasonable but not absolute risk aversion, implying that a SWF strictly quasi-concave
in individual utilities can be justified on the ground of general risk aversion by
individuals, and that a leximin SWF would be right if individuals were absolutely
risk-averse. This fallacy (made by myself during a seminar remark) is based on the
confusion of risk aversion with respect to income and risk aversion with respect to
utility.
Due to diminishing marginal (subjective) utility of income, a rational individual
may be risk-averse with respect to income. Then, even for a SWF linear in individual
utilities, it should be strictly quasi-concave in individual income levels if diminishing
marginal utility is the rule. But this does not justify a SWF strictly quasi-concave in
individual utilities (though it does not reject it either). On the other hand, since
alternative outcomes in a risk prospect are mutually exclusive and since utility is by
definition whatever the individual is ultimately interested in, individual risk aversion
with respect to utility does not seem to me to be rational. One can rationally be
risk-averse with respect to income or other objective things since one does not value
income for income sake. Rather, one values income because it contributes to one's
ultimate objective (defined as utility). If the rate at which income contributes to utility
diminishes with income, risk-averseness with respect to income has a rational basis as
the maximization of expected utility implies risk-averseness with respect to income.
But since utility is itself the ultimate objective, risk-averseness with respect to utility
cannot be so rationalized.
B. "Pure" Risk Aversion~Preference
The possible existence of "pure" risk aversion/preference cannot however be ruled out
completely. In Fig. 1 the curve SU relates the level of subjective utility to the level of
income Y, ceteris paribus. If one is risk averse with respect to income due only to the
diminishing marginal (subjective) utility of income, one would be, say, intrinsically
indifferent between x for sure and (x - 30, x + 50; ~,
1 2) if the difference in subjective
utility is the same, i.e. if SU(x) - SU(x - 30) = SU(x + 50) - SU(x).
SU
f
0
Fig. 1
×-30
x+~x+70
Y
Expected Subjective Utility
185
However, if on top of diminishing marginal utility, one has "pure" risk aversion,
one may prefer x to (x - 30, x + 50; 5,
1 ~). One may then be only intrinsically indifferent between x and say (x - 30, x + 70; 5,
1 ~)1 The "objective" utility function representing such a preference will then be something like the curve OU with more rapidly
diminishing marginal "objective" utility than SU. However, such a preference violates
A5. Thus, if SU(x) - SU(x - 30) = SU(x + 50) - SU(x) = k (a maximal indifference), we have (x - 30) I x and (x + 70) P x , and A 5 dictates (x - 30, x + 70; ½, ½)
P_ ( x , x ; ~,~)
1 1 - x. In fact, A5 requires that (x - 30, x + 50 + ~; ½, ½) should be intrinsically preferred to x, where e is any positive number. Thus, for individuals with "pure"
risk aversion/preference, the utility function derived by the N - M method (such as OU
in Fig. 1) may thus not satisfy (1) and may diverge from the neoclassical subjective
utility function (SU in Fig. 1).
"Pure" risk aversion/preference may be due to the anxiety/excitement associated
with uncertain choice. At an abstract level, we may interpret this as really involving
a different outcome. The alternative (x - 30, x + 50; ½, ½) does not just involve a
50 - 50 chance of x - 30 and x + 50 as such but also involves the acompanied
anxiety/excitement. With this interpretation, such "pure" risk aversion/preference
need not be inconsistent with our axioms but the theory is made operationally difficult
to use. Nevertheless, while "pure" risk aversion/preference due to anxiety/excitement
may be very important in such choices as gambling, they are unlikely to be the major
factor in mainly wealth-oriented choices. Thus, while the objective utility function
derived by the N - M method may diverge from the neoclassical subjective utility
function, the divergence is unlikely to be large. While the N - M utility function need
not necessarily exactly mirror the neoclassicals', the relation is certainly there.
For example, one may be indifferent between $ 20,000 and ($10,000, $ 40,000; ½, ½)
despite the fact that SU($20,000)-SU($10,000)= 1,000k but SU($40,000)SU($20,000) = 900k only, due either to "pure" risk preference, misjudgment, etc.
But given $20,000 I($10,000, $40,000; ½, ½), it is extremely unlikely for
SU($20,000)-SU($10,000) to equal, say, ten times or ~ of SU($40,000)SU ($ 20,000). If the N - M utility function has no relation whatsoever with the subjective utility function, such wide divergences are completely plausible.
"Pure" risk aversion/preference may also be due to a form of double counting (or
double discounting). One may be habitually risk averse due to diminishing marginal
utility of income. This habit may then be carelessly carried over to the discounting of
utility in the presence of risk. Such double discounting is clearly a mistake to be
avoided. For choices involving risk, one is apt to make decisions which appear
reasonable but can be shown to be irrational in some sense (see Allais and Hagen
1979). As one learns more about such pitfalls, one may then be able to make choices
more consistent with the maximization of one's expected subjective utility which is
one's true objective (cf. Marschak 1979).
References
Alchian AA (1953) The meaning of utility measurement. Am Econ Rev 43:26-50
Allais M, Hagen O (1979) Expected utility hypothesis and the Allais paradox. D Reidel,
Dordrecht,The Netherlands
Arrow KJ (1951/1963) Social choice and individual values. J Wiley, New York
186
Y.-K. Ng
Baumol WJ (1951) The N - M utility i n d e x - an ordinalist view. J Polit Econ 59:61-66
Baumol WJ (1958) The cardinal utility which is ordinal. Econ J 68:665-672
Baumol WJ (1977) Economic theory and operations analysis, 4th ed. Prentice-Hall, London
Edgeworth FY (1881) Mathematical psychics. Kegan Paul, London
Green HAJ (1971/1976) Consumer theory. Macmillan, London
Hahn F (1982) On some difficulties of the utilitarian economist. In: Sen A, Williams B (eds)
Utilitarianism and beyond. Cambridge University Press, Cambridge, pp 187-198
Marschak J (1979) Utilities, psychological values, and the training of decision makers. In: Allais
M, Hagen O (eds) Expected utility hypothesis and the Allais paradox. D Reidel, Dordrecht,
The Netherlands, pp 163-174
Mirrlees JA (1982) The economic uses of utilitarianism. In: Sen A, Williams B (eds) Utilitarianism and beyond. Cambridge University Press, Cambridge, pp 63-84
Neumann J yon, Morgenstern O (1947) Theory of games and economic behavior. Princeton
University Press, Princeton
Ng Y-K (1975) Bentham or Bergson? Finite sensibility, utility functions, and social welfare
functions. Rev Eeon Stud 42:545-570
Ng Y-K (1979/1983) Welfare economics: Introduction and development of basic concepts.
Macmillan, London
Ng Y-K (1983) Some broader issues of social choice. In: Pattanaik PK, Salles M (eds) Social
choice and welfare. North Holland, Amsterdam, pp 151-173
Ng Y-K (1984) Some fundamental issues in social welfare. In: Feiwel G (ed) Issues in contemporary microeconomics and welfare. Macmillan, London
Pattanaik PK (1968) Risk, impersonality, and the social welfare function. J Polit Econ
76:1152-1169
Samuelson PA (1947) Foundations of economic analysis. Harvard University Press, London
Samuelson PA (1966) Collected scientific papers, Vol 1. Stiglitz J (ed) MIT Press, London
Sen AK (1970) Collective choice and social welfare. North Holland, Amsterdam