Global COE Hi-Stat Discussion Paper Series 298

Research Unit for Statisticaland Empirical Analysis in Social Sciences (Hi-Stat)

Extending the Original Position:
Revisiting the Pattanaik Critique ofVickrey/Harsanyi Utilitarianism

Hi-Stat Discussion Paper

Peter J. Hammond

April 2013

Hi-Stat
Institute of Economic Research
Hitotsubashi University
2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan
http://gcoe.ier.hit-u.ac.jp

Extending the Original Position:
Revisiting the Pattanaik Critiqueof Vickrey/Harsanyi Utilitarianism
Peter J. Hammond∗
Abstract
Harsanyi’s original position treats personal identity, upon which eachindividual’s utility depends, as risky. Pattanaik’s critique is related tothe problem of scaling “state-dependent” von Neumann–Morgensternutility when determining subjective probabilities. But a unique social welfare functional, incorporating both level and unit interpersonal
comparisons, emerges from contemplating an “extended” original position allowing the probability of becoming each person to be chosen.
Moreover, the paper suggests the relevance of a “Harsanyi ethical typespace”, with types as both causes and objects of preference.

∗

Department of Economics, University of Warwick, Coventry CV4 7AL, UK. e-mail:p.j.hammond@warwick.ac.uk

“By nature, men are nearly alike; by practice, they get to be wide apart.”
Confucius, Analects ch. 17 from www.analects.org.

1

Introduction

1.1

Vickrey, Rawls, and Harsanyi

Pattanaik (1968, pp. 1157–8) reminded economists and social choice theorists
how the philosopher Hare (1961) defined ethics as being about universal
prescriptive statements. An example is the Golden Rule of reciprocal ethics,
which is usually stated as: “Do unto others as you would have them to dounto you.” An improved version, however, would seem to be: “Do unto others
as you should want them to do unto you.” After all, one should recognizethat a person’s own wishes, even for themselves, may not always be for thebest.
Be that as it may, an intuitively appealing way to introduce universalizability when discussing policy choices is through impersonality. That is, weseek normative statements about policy that remain valid no matter whichis the individual in society whose position we think of occupying. This approach can be linked to contractarianism through what eventually became
Rawls’ (1951, 1958, 1971) device of an “original position”, where each individual is put “behind the veil of ignorance”. As Pattanaik points out, thisadds to universalizability by requiring normative statements about policy toremain valid even when a person’s identity remains unknown.2
Even before Rawls really elaborated his theory, Vickrey (1945, 1960) andHarsanyi (1953, 1955) had already independently formulated a different approach to the original position. In it, each individual faces an equal probability of becoming any person in the society. Then the standard hypothesesof Bayesian rationality imply that, in this original position, each individual
should want to maximize the expected value of a von Neumann–Morgensternutility function defined over lotteries whose outcomes combine social stateswith personal identities.
2

See Mongin (2001) for a careful discussion of the ethics of impartiality.

1

1.2

Pattanaik, Kolm and Broome

This Vickrey/Harsanyi version of the original position is the subject of Pattanaik’s (1968) important critique. Amongst other things, he carefully distinguished two different variations, namely an “objective” approach due to Vickrey, as opposed to a “subjective” approach due to Harsanyi. Both present
their own difficulties, however. In particular, this original position fails to determine the social preference ordering uniquely. One difficulty is in knowing
which von Neumann–Morgenstern utility function, within the relevant cardinal equivalence class, should be used to represent that person’s welfare in
the original position. More fundamentally, Pattanaik (1968) raised the issueof whether that utility function should represent that person’s own attitudes
to risk, as well as their preferences between different social states.
Here the fundamental difficulty is that the expected value of each relevant von Neumann–Morgenstern utility function represents only lotteries
over social states, with personal identity fixed. In fact, for the original position argument to work, different von Neumann–Morgenstern utility functions
for different personal identities need to be scaled appropriately so that their
expected values are indeed defined over lotteries whose outcomes include
variable personal identities as well as social states. This difficulty also lies at
the heart of the debate between Broome (1993, 1994) and Kolm (1994a, b)
over the difference between a cause of preference and an object of preference.

1.3

Extending the Original Position

This paper sets out to resolve many of these issues by considering a suitably
extended original position. The extension consists in requiring that preferences apply not just in the Vickrey/Harsanyi original position where there isan equal chance of becoming any one person, nor just in arbitrarily biased
original positions with unequal probabilities of becoming different individuals in society, but also in decision problems where these biased probabilities
themselves become objects of choice. There is an obvious relationship here
to the extension of the Anscombe and Aumann (1963) theory of subjective probability that allows state-independent utilities to be derived even forstate-dependent consequence domains, as set out in Hammond (1998, 1999).

2

1.4

Ethical Types

As already suggested, another issue to be discussed here concerns the debate
between Broome and Kolm on the validity of the alleged distinction between
causes of preference on the one hand, and objects of preference on the other.
Here it is argued that the construction of a suitable type space could allow
those types to be simultaneously both causes and objects of preference.

1.5

OutlineSection 2 below introduces the notation and basic framework to be usedthroughout the subsequent sections of the paper, sometimes with suitableembellishment. A particular feature worth mentioning here is the inclusionof a special individual consequence representing non-existence. This permits the original position to include a variable number of individuals, thus
allowing for the case when population is also affected by the social decision, as in Hammond (1988) and the works cited there, especially Sidgwick(1887), Meade (1955), Dasgupta (1969) and Parfit (1984). Specifically, one
can convert “different numbers” problems in social choice to “same numbers”
problems by adding as many non-existent individuals as necessary.
Thereafter the paper moves on in Section 3 to discuss Vickrey’s original
position where there is an even chance lottery of facing each individual’s
objective circumstances. This objective version of the original position obviously omits some relevant subjective circumstances, notably how the person
would wish to make choices in the inherently risky original position.
In an attempt to address this failure, Harsanyi introduced “ethical” subjective preferences. These are the main subject of Pattanaik’s critique, andof the latter part of this paper. As Section 4 reminds us, for the original position argument to work, different von Neumann–Morgenstern utility
functions for different personal identities need to be scaled appropriately sothat their expected values are indeed defined over lotteries whose outcomes
include variable personal identities as well as social states.
Next, Section 5 explains in more detail how this scaling issue is really a
special case of the difficulty that has to be confronted when contemplating decisions under uncertainty in case the consequence domain is state-dependent.
After all, the domain of possible consequences that could result from beinga particular person named Chris, say, are generally quite different from thepossible consequences of being Pat, Robin, or Sam. Not least, each of them3

could be either a man or a woman. The following Section 6 briefly discusses
how the scaling issue is straightforward to resolve provided that one considers an extended original position where even the (biased) probabilities of
becoming different individuals can be chosen.
Section 7 goes on to introduce the notion of an ethical type space, andbriefly explores the relevance of some analogies with the kind of type spacethat Harsanyi (1967) used to develop the modern theory of games of incomplete information.
The concluding Section 8 contains a brief summary and suggestions forfurther work.

2

Notation and Basic Assumptions

2.1

Individuals and Their Personal Consequences

As in Hammond (1988) and Blackorby et al. (2005), for instance, we allow
for a variable population that can be influenced by policy, so is effectively
an object of choice. Accordingly, we consider a finite set M large enough toinclude all potential individuals. Let m := #M denote the size of this set.
The actual population is assumed to be a subset N ⊆ M of size n ≤ m.
Let Y denote an arbitrary domain of possible personal consequences.
The members y of the set Y are assumed to represent everything that could
be relevant to a single individual when making an ethical policy decision that
affects them.
One important personal consequence, naturally, is whether that person
ever comes into existence. So we include a particular consequence y0 ∈ Ythat represents non-existence. We assume that any other personal consequence y ∈ Y \ {y0 } entails that particular individual existing as an ethically
relevant person over a time interval whose specification can be included in
the description of y, if those dates are also ethically relevant.

4

2.2

Social States as Personal Consequence Profiles

For each individual i ∈ M , let Yi ⊆ Y denote the subdomain of personalconsequences that could be relevant to person i. We assume that y0 ∈ Yi foreach individual i ∈ M , meaning that social preferences are always definedeven over what may be purely hypothetical social states in which i nevercomes into existence.
Next, we identify social states with personal consequence
Q profiles
y M = hyi ii∈M which belong to the Cartesian product Y M := i∈N Yi ofthe personal consequence subdomains. As in the theory of public goods, thisproduct domain allows us to consider in principle varying just one individual’s
consequence at a time, even though feasibility may impose the constraintthat some “public” components of individuals’ consequences must all change
together.

2.3

Simple Consequence Lotteries

Let ∆(Y ) denote the space of simple consequence lotteries η, each ofwhich attaches a positive probability η(y)P to each member y of a finite
support of elements in Y . Obviouslyy∈Y η(y) = 1, where the sum iswell-defined because only a finite set of terms are non-zero.

2.4

Expected Utility of Single-Person Situations

For each i ∈ M , consider single-person situations where any social decision that is taken affects only individual i, and results in a simple consequencelottery or random outcome η ∈ ∆(Yi ). Under standard axioms for normativedecision-making with risky consequences, the appropriate ethical objective
in such single-person situations is to maximize the expectation
XEη [ui (y)] ≡η(y)ui (y)(1)
y∈Yof each von Neumann–Morgenstern utility function (NMUF) ui : Yi → R ina cardinal equivalence class. This class is defined so that two functions
ui and ũi are cardinally equivalent if and only if they coincide on Yi up toa positive affine transformation of the formũi ≡ αi + ρi ui ,where ρi > 0.
5(2)

In particular, for each quadruple of personal consequences a, b, c, d ∈ Y , theratioui (a) − ui (b)(3)
ui (c) − ui (d)of utility differences must equal the (constant) marginal rate of substitution−δq/δp between small probability shifts δp from consequence b to a, and δqfrom consequence c to d, as one moves along the tangent to an indifference
surface in the three-dimensional subset ∆({a, b, c, d)}) of lotteries in ∆(Y )whose support is the set {a, b, c, d)}. These ratios of utility differences, ofcourse, are preserved by any positive affine transformation of the function
y 7→ ui (y).

2.5

Zero Utility of Non-Existence

A convenient normalization will be to put ui (y0 ) = 0, thus attaching a utility of zero to non-existence.3 Then ui and ũi are equivalent if and only ifthey coincide on Yi up to a positive linear transformation of the formũi ≡ ρi ui , where ρi > 0. In particular, following the above interpretation of(3), for each pair of personal consequences y, y 0 ∈ Y , the ratio ui (y)/ui (y 0 ) of
utilities equals the (constant) marginal rate of substitution −δq/δp between
small probability shifts δp from consequence y0 to y, and δq from consequence y0 to y 0 . These utility ratios are also preserved by any positive linear
transformation of the utility function ui taking the formũi ≡ ρi ui ,where ρi > 0.

3

(4)

By contrast, Blackorby, Bossert and Donaldson (2005) meet Parfit’s (1984) “repugnant
conclusion” by introducing a critical level of utility ci for each individual i ∈ M , defined sothat ui (y0 ) = ci . This allows utility to be normalized in a different way so that ui (y) > 0
if and only if the existing individual i ∈ N regards consequence y as better than having
never come into existence at all. One implication of their critical level approach is that,compared to the case when we ascribe a zero level of utility to non-existence, it becomes
harder to interpret the critical and zero levels of utility “objectively” in the same way
for different individuals. By contrast, in Hammond (1988) I argue that the “repugnant
conclusion” is not so repugnant after all if one reckons the relevant utilities of parents,
including their preferences for children, and perhaps even more relevantly, for reproductive
freedom.

63

Vickrey’s Objective Original Position

3.1

The Original Position Lottery

In Vickrey’s version of the original position, each personal consequence in thespace Y is identified with an observable “objective” circumstance such as income, educational qualifications, date of birth, marital status, etc. We arealso extending Vickrey’s version by including non-existence as a possible objective circumstance, represented by the specific consequence y0 ∈ ∩i∈M Yi ⊆Y . Accordingly, given any social state y M ∈ Y M , all individuals behind theveil of ignorance can be regarded as facing the same equal chance lottery of
becoming any of the individuals j ∈ M , associated with the corresponding
personal consequence yj ∈ Y . That is, all individuals — whether actual or
potential — are put in the position of facing the same lottery ηyM ∈ ∆(Y )over personal consequences defined by
ηyM (E) :=1#{j ∈ M | yj ∈ E} for each finite E ⊆ Y .
m(5)

Suppose now we pay special attention to the set M of all individuals whoever come into existence. Then ηyM (Y \ {y0 }) = n/m and, for each finite
E ⊆ Y \ {y0 }, one has
ηyM (E) :=

3.2

1
1
#{j ∈ N | yj ∈ E} = #{j ∈ N | yj ∈ E} ηyM (Y \ {y0 }). (6)mnIndividuals’ Expected Utilities

Though each social state y M ∈ Y M is supposedly objective, in principle
different individuals may have different preferences over the lotteries ηyM .
Indeed, Vickrey (1960) reinterprets the original position as one confronted
by a potential immigrant to a society, who has an equal chance of acquring
the personal consequence yj of each individual j ∈ M . Thus, each existing
individual i ∈ N is still free to have preferences represented by the expectedvalue of his or her own NMUF ui . The only new requirement is that ui
should be defined over the whole domain Y of possible personal consequences,instead of just over the restricted domain Yi of possible consequences thatare specific to person i.

7

3.3

Vickrey’s Social Welfare Function

Taking the expectation of the different individuals’ utility functions w.r.t.
ηyN leads to the Vickrey social welfare function wim : Y M → R for a societyof m individuals. This can be written as the average utilitarian objectivewim (y M ) :=1 X
ui (yj ).
m j∈M

(7)

Because the population of potential individuals is fixed at size m, the function
wim is cardinally equivalent to the total utilitarian objectiveXX
ui (yj ),(8)ui (yj ) =wi (y M ) =
j∈N

j∈M

where the last equality holds because of the normalization ui (y0 ) = 0, whichholds for all i ∈ M \ N .

3.4

Risky Social States

Finally, there is an straighforward extension to the entire space ∆(Y M ) of
all possible risky social states, each of which is described by an objective
joint probability distribution λM of the personal consequences of all the mdifferent potential individuals in the set M , including the state y0 of nonexistence. Indeed, in this extension, each individual j ∈ M faces his or
her own personal lottery λj ∈ ∆(Yj ) that must be equal to the marginal
distribution on Yj induced by λM on Y M . Working back to the original
position, each individual i ∈ N then faces the identical compound lotteryηλM ∈ ∆(Y ) over personal consequences defined byηλM (E) :=1 X
λj (E) for each finite E ⊆ Y .
m j∈MThat is, for each finite E ⊆ Y , one hasYM
λj (E) = λ
E×i∈M \{j}

(9)


Yi .

(10)

So, extending to the entire space ∆(Y M ) of all possible risky social states
the social welfare function given by (7) or, equivalently, by (8), we obtain a8

(complete and transitive) preference ordering % on ∆(Y M ) represented by the
expectation in the original position compound lottery of (7) or, equivalently,of (8). Specifically, i’s version of the social welfare function becomesWi (λM ) :=

1 X
Eλ ui
m j∈M j4

Harsanyi’s Subjective Original Position

4.1

Harsanyi’s Ethical Types(11)

Vickrey (1945, 1960) regarded the consequence domain Y as if it consisted
only of objective circumstances. Yet, as Harsanyi (1955) certainly recognized,
subjective features of individuals such as their tastes or preferences are also
relevant in ranking these objective circumstances. After all, consider thefollowing radical adaptation of an example discussed by Sen (1970). There
are two individuals of whom one (labelled C) is a devoted cricketer enthusiast,
whereas the other (labelled B) is a fanatic follower of baseball. Decisions in
the original position, such as whether these two people should accompany
each other to a cricket match or to a baseball game, then require going
behind the veil of ignorance where one would be forced to choose between:(i) becoming C and being dragged reluctantly to watch a baseball game; (ii)or becoming B and being forced to face up to the unfathomable mysteries ofcricket. Furthermore, as Pattanaik (1968) points out, if ethical preferences
are to determine decisions in the original position, they must also allow for
individuals’ own subjective attitudes to the risk they face in the equal chancelotteries of the original position.
Despite these evident difficulties, Harsanyi (1955) postulated “ethical”
preferences. One way of modelling these is through a space H of Harsanyi
ethical types h which parametrize the family y 7→ u(y; h) of NMUFs whose
expected values represent ethical preferences over ∆(Y ). In the original position, for each j ∈ M the utility uj (yj ) of j’s objective personal consequence
becomes replaced by the utility u(yj ; hj ) of j’s subjective personal consequence (yj , hj ) ∈ Y × H.

9

4.2

Spurious Observer Independence

One key difference from Vickrey’s original position is Harsanyi’s claim thatthe new objective function can be expressed as
XX
w(y M ; hM ) =u(yj ; hj ) =u(yj ; hj ),
(12)
j∈M

j∈N

independent of i. That is, the objective is the same no matter which individual i is taken as the observer. However, this is not really the case, and in fact
we have so far failed to determine uniquely the original position preferences
for even one observer.
This is because, as Pattanaik (1968) makes clear, the parametric familyof utility functions y 7→ u(y; h) is not determined uniquely. Indeed, after
taking into account the normalized utility u(y0 ; h) = 0 of non-existence, eachethical type h determines only an equivalence class of such utility functions,defined up to separate positive linear transformations of the formũ(y; h) ≡ ρ(h)u(y; h) for all y ∈ Y ,(13)

where ρ(h) is a positive constant for each h ∈ H. So far, therefore, our theory
is unable to distinguish between the function w(y M ; hM ) given by (12) andany alternative function w̃(y M ; hM ) given byX
Xw̃(y M ; hM ) =ρ(hj )u(yj ; hj ) =ρ(hj )u(yj ; hj ),
(14)
j∈M

j∈N

where ρ(hj ) (j ∈ M ) is an arbitrary family of positive constants. In particular, given any finite subset X ⊆ Y M and any fixed profile h̄M ∈ H M of
Harsanyi ethical types, one make any individual d ∈ M a dictator over theset X of all degenerate lotteries in ∆(X) by setting all the ratios ρ(hj )/ρ(hd )(j ∈ M \ {d}) sufficiently close to zero.

10

5

Consequence Domains

5.1

Personal Consequences

The key difficulty with Vickrey’s version of the original position describedin Section 3 is that the objective consequences y ∈ Y do not describe completely what is relevant to ethical decisions. As explained in Section 4, thesame objective consequence y ∈ Y has a different ethical significance depending on the Harsanyi ethical type h ∈ H of the individual who is going
to experience y.
What we would really like is for each potential individual j ∈ M to face
the same extended common domain Z := Y × H whose members z = (y, h)are pairs of personal consequences that combine objective circumstances withpersonal ethical types. In effect, therefore, we simply replace Y in Section 3
by the new space Z, and the assessment by each existing individual i ∈ Nof the individual welfare function ui : Y → R to use in the original positionby the new function u∗i : Z → R. After retaining the obvious normalizationu∗i (y0 , h) = 0 for all h ∈ H, we arrive at the modified formXXwi∗ (z M ) =u∗i (zj ) =u∗i (zj ),
(15)
j∈M

j∈N

of equation (8).
Indeed, the lotteries we have considered so far are limited to the particularsubdomain of ∆(Y × H) whose members ζzM , in an obvious adaptation ofequation (5) from Y to Z = Y × H, take the formζzM (F ) :=1#{j ∈ M | zj ∈ F } for each finite F ⊆ Z.
m

(16)

In particular, the probability of any finite F ⊆ Z must be some integer
multiple of 1/m.
Nevertheless, there is an obvious extension to the space ∆(Z M ) of riskysocial states, each described by an objective joint distribution λM of thepersonal consequences of the m different potential individuals in the set M .
Then each j ∈ M faces a personal lottery λj ∈ ∆(Zj ), equal to the marginal
distribution on Zj induced by λM on Z M . That is, following (10), for eachfinite F ⊆ Z one hasYM
λj (F ) = λ
F×
Zi .
(17)
i∈M \{j}11Working back to the original position, all individuals i ∈ N face theidentical compound lottery ζλM ∈ ∆(Z) over personal consequences definedby
1 X
λj (F ) for each finite F ⊆ Z.
(18)
ηλM (F ) :=m j∈MEach such lottery, however, is restricted to the domain
(
)
1 X
1 X
∆(Zj ) := ζ ∈ ∆(Z) | ∃λj ∈ ∆(Z) (j ∈ M ) : ζ =λj . (19)
m j∈M
m j∈M

5.2

State-Dependent Consequence Domains

It makes little sense, however, to assume that the extended consequence
domain Z is really the same for each individual. After all, in our earlier example, can we really imagine a common consequence domain which includes
not only the cricket enthusiast’s experience when watching baseball, but also
the baseball fan’s experience at a cricket match? Thus, we really need toallow different consequence domains Zh for each ethical type h ∈ H. Indeed,
one might argue that Zh ⊆ Y × {h} for each h ∈ H, in which case the twodomains Zh , Zh0 are not only different when h 6= h0 , but pairwise disjoint.
Now, in the theories of subjective probability developed by Savage (1954)
and by Anscombe and Aumann (1963), a key assumption is that, given any
fixed consequence z̄ ∈ Z, there exists a “constant act” that could produce z̄
as the consequence in all possible uncertain states of the world. If we liken
ethical types in the original position to uncertain states of nature, this would
imply that the consequence domains satisfy Zh = Z for all h ∈ H, andso are state-independent. But in the setting of this paper, state-dependent
consequence domains Zh must clearly be allowed.
Because the consequence domains Yh may be entirely disjoint for differenth ∈ H, the problem we face here is formally identical to the subjective expected utility model with state-dependent consequence domains that was set
out in Hammond (1999), as well as Section 6 of Hammond (1998). Actually,following the key pioneering ideas of Drèze in the 1960s, most of the previous literature on decision theory with state-dependent consequence domains
has considered state-dependent utility — see, for example, the work thatis collected, discussed, and surveyed in Drèze (1987), Karni (1985, 2008),
Drèze and Rustichini (2004), etc. These works have typically sought to extend Savage’s theory of subjective probability; in the original position we are12considering, however, with an equal probability of becoming any potential orexisting individual, there are clearly objective probabilities only.
Still, one could try applying the kind of extended preference ordering
considered in Karni, Schmeidler and Vind (1983) to the present context.
This makes it seem natural to consider “biased” original positions with unequal probabilities of acquiring different Harsanyi ethical types. Or, following
Hammond (1999), even an extended original position where these biased
probabilities can be chosen. This is our next topic.

6

An Extended Original Position

6.1

Biased Original Positions

So far, in each original position we have considered, there has been an equal
probability 1/m of becoming any potential individual j ∈ M . Now we
consider and even compare biased original positions with generally different
probabilities µj of becoming each j ∈ M , where µ ∈ ∆(M ). Or more exactly,
we consider the domain whose members are profiles (y M , hM ) ∈ Y M × H M
of consequence–type pairs (y, h) ∈ Y × H, together with lotteries µ ∈ ∆(M ).
Then each triple (µ, y M , hM ) ∈ ∆(M ) × Y M × H M together determines the(simple) probability measure on consequence–type pairs (y, h) ∈ Y × H defined byπ(µ,yM ,hM ) := µ ({j ∈ M | (yj , hj ) ∈ E})

6.2

for each finite E ⊆ Y × H. (20)Comparing Biased Original Positions

In the extended original position we consider, we postulate that each individual i ∈ N has preferences over biased original positions represented bytriples (µ, y M , hM ) ∈ ∆(M ) × Y M × H M . Specifically, we assume that these
preferences correspond to preferences over the induced lotteries π(µ,yM ,hM ) ∈∆(Y × H). Moreover, we postulate that these preferences are represented
by the expected value E[vi (y, h)] of an extended NMUF vi : Y × H → R
defined over consequence–type pairs.
To be explicit, the expected utility to person i of the biased original
position lottery determined by the triple (µ, y M , hM ) isX
Wi (µ, y M , hM ) := Eπ(µ,yM ,hM ) vi (y, h) =µ({j})vi (yj , hj ).
(21)
j∈M

13

This implies that vi is the von Neumann–Morgenstern utility function definedon Y ×H whose expected value determines individual i’s preferences, not onlyin the Vickrey/Harsanyi unbiased original position, but also in every biased
original position, and also in comparing different biased original position.

6.3

Personal Interpersonal Utility Functions

Using information about comparisons between such biased original positions
allows the simultaneous derivation of an interpersonal utility function (unique
up to a cardinal ratio scale) defined on person-dependent consequence domains. Like the “fundamental” interpersonal notion of utility consideredby Tinbergen (1957) and Kolm (1972, 1994), and later endorsed by Rawls(1982), this function incorporates both level interpersonal comparisons, representing who one would prefer to be with certainty, as well as unit interpersonal comparisons, representing marginal rates of substitution between theprobabilities of becoming different kinds of person. Also, unlike the general“state-dependent” utilities that have appeared in the literature cited above,
the ideas in Hammond (1999) imply that this fundamental utility must be
equal for indistinguishable individuals.
Nevertheless, one must emphasise a key difference from Tinbergen and
Kolm, as well as Rawls. Nowhere do we claim that our notion of “fundamental” interpersonal utility must be equal for distinguishable individuals whenfacing an extended original position. This point may be the key to resolving
the difference between Broome and Kolm regarding whether there is a valid
distinction between the causes and objects of preference.

7

Causes versus Objects of Preference

7.1

Broome versus Kolm on Fundamental Preferences

Broome’s (1993) critique of Kolm (1972) denied that a cause of preference
could be an object of preference. Actually, Kolm’s claim regarding fundamental preferences can usefully be split into two parts. The first claim is thata cause of preference can be an object of preference. We will investigate how
constructing a suitable type space could help justify this claim. But a second
important claim that Kolm makes, acknowledging inspiration from Tinbergen (1957), is that, for a suitably constructed type space, all individuals will
14

share identical “fundamental preferences” over (random) consequence–typepairs. This second much stronger claim remains seems to lack a formal justification, rather like the common prior assumption in Harsanyi’s theory ofgames of incomplete information.

7.2

Ethical Type Spaces

In the approach set out in Sections 4–6, we would like each individual j’s
ethical type hj ∈ H to be both:
1. a cause of preference, represented by the parameter hj of the NMUF
ui (y; hj ) that individual i in any biased original position applies tolotteries faced by individual j;
2. an object of preference, such that the expected value Eui (y; hj ) of the
NMUF ui (y; hj ) represents i’s preferences over lotteries in ∆(Y × H)
faced by individual j.
Recall that in mathematical analysis a Polish space Z is metric, complete,and separable. Given any Polish space Z, let U(Z) denote the (Polish space)of continuous bounded normalized NMUFs u : Z → [−1, 1] equipped withthe sup metric defined for all u, v ∈ U(Z) byd(u, v) := sup |u(z) − v(z)|.

(22)

z∈Z

Note that the normalization has been chosen to allow the existence of personal consequences z ∈ Z for which u(z) < 0, indicating that they are so badas to be worse than non-existence.
Ideally, then, our aim should be to construct an ethical type space H of“causes of ethical preference” h ∈ H. Moreover, this space should have thedefining property that H and U(Y × H) are homeomorphic as metric spaces.
That is, there should exist a continuous bijective mapping ψ : H → U(Y ×H)
whose inverse ψ −1 : U(Y × H) → H is also continuous. This mapping
identifies each ethical type h ∈ H with a unique utility function ψ(h) withthe property that y 7→ ψ(h)(y) = u(y; h) can also be regarded as a function(y, h) 7→ u(y; h) in the space U(Y × H) of utility functions defined on thedomain Y × H. Conversely, each utility function u ∈ U(Y × H) is identified
with a uniaue ethical type h = ψ −1 (u) ∈ H.

15

It remains to be seen if this is possible, perhaps following the ideas set
out first by Armbruster and Bog̈e (1979), and Bog̈e and Eisele (1979), followed by Mertens and Zamir (1985) and then by Brandenburger and Dekel(1993), amongst many others, for constructing a complete Polish type spacein a game of incomplete information. In this game-theoretic setting, types
are belief hierarchies about belief hierarchies, where beliefs are σ-additive
probability measures.
For the time being, however, it may be enough to consider an incomplete
ethical type space H which is homeomorphic only to some proper subset ofU(Y × H). That is, the mapping ψ : H → U(Y × H) is an injection rather
than a bijection. For example, H could even be some (large) finite dimensional parameter space, implying that the range ψ(H) of the mapping ψ isa finite parameter family of utility functions y 7→ u(y; h). Then, viewed asa parameter vector, the ethical type h can still be regarded as a cause ofpreference; on the other hand, when h is regarded as an argument of any
function (y, h) 7→ u(y, h) in the range ψ(H) ⊂ U(Y × H), then it becomesas an object of preference. We are assuming that there is a homeomorphism
between causes of preference and a subspace of the space of utility functions that treat those causes as objects of preference; the distinction between
causes and objects of preference therefore loses its significance.

7.3

Broome versus Kolm Revisited

This approach, however, leaves each individual i ∈ N with not only theirown ethical type hi ∈ H, but also their own corresponding utility functionui ∈ U(Y ×H). We have definitely not shown the existence of a truly “fundamental” utility function which is independent of i, which is what Tinbergen
and Kolm presume. It remains to be seen whether it is actually possible toconstruct a single ethical type space such that all differences in individual’sutility functions can be ascribed to differences in their ethical types. Unless,that is, one treats each individual i ∈ N as having his or her own idiosyncratic ethical type space Hi , disjoint from Hj for each j ∈ N \ {i}, which ismoreover homeomorphic to a subset of the space of personal utility functions
ui ∈ U(Y × H ∗ ), where H ∗ := ∪i∈N Hi .

16

8Concluding Remarks

8.1

Summary

Pattanaik (1968) criticized Harsanyi (1953, 1955) for failing to give a properaccount of how to determine the interpersonal comparisons which determine
preferences in his version of the original position, as well as Vickrey’s (1945,1960) related version. The main contribution of this paper is to show how one
can circumvent Pattanaik’s critique by considering preferences that represent
rational choice in an extended original position that allows the individual tochoose even the probabilities of becoming different individuals in various
biased original positions. Of course, Hare’s case for making universalizable
prescriptive statements by considering only an unbiased original position
retains its normative appeal. Nevertheless, considering the extended original
position serves to uncover an interpersonal utility function that, given thenormalization requiring the utility of non-existence to be zero, is unique up
to a cardinal ratio scale.

8.2

Unsolved Problems

One of Harsanyi’s claims is that all individuals would want to make thesame decision in a properly defined original position. This requires all individuals to agree on each others’ utility functions, given their “ethical types”
that determine these functions. But these ethical types have to be not only
determinants of preferences, but also objects of the preferences involved in interpersonal comparisons. This is the subject of interchange between Broome
and Kolm.
Resolving the conflicting views expressed in this interchange seems possible if one can construct suitable ethical type spaces. Doing this properly,
however, remains an unsolved problem. This paper approaches the issue
rather like Harsanyi himself did when considering players’ types in a gameof incomplete information. It does not essay the kind of construction madefamous in the work of Mertens and Zamir.
A somewhat related problem occurs when different individuals can haveinconsistent probabilistic beliefs. This leads to issues of the kind raised in
the literature on the contrast between the ex ante and ex post approaches
to welfare economics that Harsanyi himself had hinted at, and which was
initiated by a footnote in Diamond’s (1967) renowned paper on the stock
17

market, followed by Starr (1973), Harris (1978), Harris and Olewiler (1979),
Hammond (1981, 1982), and Nielsen (2009) — amongst others. To analysethese issues along the lines of this paper, the ethical types considered here
will need supplementing by belief types. Again, however, this is an issue for
future work.
Acknowledgements
This is a very welcome opportunity to express in print my deep gratitude to Prasanta Pattanaik for many years of friendship and intellectual support. In particular,
his lectures at Nuffield College, Oxford, during 1969–70 did much to awaken myinterest and broaden my education in social choice theory, as well as its links toethics and political philosophy. So did the opportunity to read the typescript ofwhat became his book Pattanaik (1971), in which, whenever a paragraph raised
a question in my mind, in the next paragraph I found it answered. Furthermore,while I was visiting the University of California at Riverside in May 2007, Prasantahimself raised a question that encouraged me to revisit this topic.
Generous support from a Marie Curie Chair funded by the European Commission under contract number MEXC-CT-2006-041121 is also gratefully acknowledged. This supported the preparation of earlier versions that were presented tothe conference honouring Prasanta at the University of California at Riverside inAutumn 2008, as well as to a workshop on Heterogeneous Beliefs which Mordecai
Kurz organized at Stanford University in August 2009.
This 2013 revision owes much to the warm hospitality provided by the Hitotsubashi Institute for Economic Research, through its Global Centre of Excellence
for the Ethical Foundations of Social Choice Theory, especially to Reiko Gotoh andNaoki Yoshihara. Many thanks also to Takashi Kunimoto for a timely reminderof the relevance of an ethical type space similar to those used in Harsanyi’s theoryof games of incomplete information.

18

References
Anscombe, F.J. and R.J. Aumann (1963) “A Definition of Subjective Probability,” Annals of Mathematical Statistics, 34: 199–205.
Armbruster, W. and W. Böge (1979) “Bayesian Game Theory,” in O. Moeschlin
and D. Pallaschke (eds.) Game Theory and Related Topics (Amsterdam:
North-Holland), pp. 17–28.
Blackorby, C., Bossert, W. and D.J. Donaldson (2005) Population Issues in Social
Choice Theory, Welfare Economics, and Ethics (Cambridge: Cambridge
University Press).
Böge, W. and T. Eisele (1979) “On Solutions of Bayesian Games,” International
Journal of Game Theory, 8: 193–215.
Brandenburger, A., and E. Dekel (1993) “Hierarchies of Beliefs and Common
Knowledge,” Journal of Economic Theory, 59: 189–198.
Broome, J. (1993) “A Cause of Preference Is Not an Object of Preference” Social
Choice and Welfare 10: 57–68.
Broome, J. (1994) “Reply to Kolm” Social Choice and Welfare 11: 199–201.
Dasgupta, P. (1969) “On the Concept of Optimum Population” Review of Economic Studies 36: 295–318.
Diamond, P.A. (1967) “The Role of a Stock Market in a General Equilibrium
Model with Technological Uncertainty” American Economic Review 57: 759–
776.
Drèze, J.H. (1987) Essays on Economic Decisions under Uncertainty Cambridge
University Press: Cambridge.
Drèze, J.H. and A. Rustichini (2004) “State-Dependent Utility and Decision Theory” in S. Barberà, P.J. Hammond, and C. Seidl (eds.) Handbook of Utility
Theory, vol. 2: Extensions (Dordrecht: Kluwer Academic) ch. 16, pp. 839–
892.
Grant, S., Kajii, A., Polak, B. and Z. Safra (2010) “Generalized Utilitarianismand Harsanyi’s Impartial Observer Theorem” Econometrica 78: 1939–1971.
Hammond, P.J. (1981) “Ex-Ante and Ex-Post Welfare Optimality under Uncertainty” Economica 48: 235–250.

19

Hammond, P.J. (1982) “Utilitarianism, Uncertainty and Information” in A.K. Sen
and B. Williams (eds.) Utilitarianism and Beyond (Cambridge University
Press, 1982), ch. 4, pp. 85–102.
Hammond, P.J. (1988) “Consequentialist Demographic Norms and Parenting
Rights,” Social Choice and Welfare 5: 127–145.
Hammond, P.J. (1998) “Subjectively Expected Utility” in S. Barberà, P.J. Hammond, and C. Seidl (eds.) Handbook of Utility Theory, vol. 1: Principles(Dordrecht: Kluwer Academic) ch. 6, pp. 213-271.
Hammond, P.J. (1999) “Subjectively Expected State-Independent Utility on
State-Dependent Consequence Domains” in M.J. Machina and B. Munier(eds.) Beliefs, Interactions, and Preferences in Decision Making (Dordrecht:
Kluwer Academic), pp. 7–21.
Hare, R.M. (1961) The Language of Morals (London: Oxford University Press).
Harris, R. G. (1978) “Ex-post Efficiency and Resource Allocation under Uncertainty” Review of Economic Studies 45: 427–436.
Harris, R. G. and N. Olewiler (1979) “The Welfare Economics of Ex-Post Optimality”. Economica 46: 137-147.
Harsanyi, J.C. (1953) “Cardinal Utility in Welfare Economics and in the Theoryof Risk-Taking,” Journal of Political Economy 61, 434–435.
Harsanyi, J.C. (1955) “Cardinal Welfare, Individualistic Ethics, and Interpersonal
Comparisons of Utility,” Journal of Political Economy 63, 309–321.
Harsanyi, J.C. (1967–8) “Games with Incomplete Information Played by ‘Bayesian’ Players, I–III,” Management Science, 14: 159–182, 320–334, and 486–
502.
Karni, E. (1985) Decision Making under Uncertainty: The Case of State-Dependent Preferences. Harvard University Press: Cambridge MA.
Karni, E. (2008) “State-Dependent Preferences” in L. Blume and S. Durlauf(eds.) The New Palgrave Dictionary of Economics, 2nd edn. (Palgrave
Macmillan).
Karni, E. and P. Mongin (2000) “On the Determination of Subjective Probability
by Choices” Management Science 46: 233–248.

20

Karni, E., Schmeidler, D. and K. Vind (1983) “On State Dependent Preferences
and Subjective Probabilities,” Econometrica 51, 1021–1031.
Kolm, S.-C. (1972; 1998) Justice et équité (Paris: Editions du CNRS); translated
as Justice and Equity (Cambridge, MA: MIT Press).
Kolm, S.-C. (1994a) “The Meaning of Fundamental Preferences” Social Choiceand Welfare, 11, 193–198.
Kolm, S.-C. (1994b) “Rejoinder to John Broome” Social Choice and Welfare, 11,
203–204.
Meade, J.E. (1955) Trade and Welfare (Oxford: Oxford University Press).
Mertens, J.-F. and S. Zamir (1985) “Formulation of Bayesian Analysis for Games
with Incomplete Information,” International Journal of Game Theory, 14:
1–29.
Mongin, P. (2001) “The Impartial Observer Theorem of Social Ethics” Economics
and Philosophy 17: 147–179.
Nielsen, C.K. (2009) “Rational Overconfidence and Social Security” available at
http://econ.korea.ac.kr/~ri/WorkingPapers/w0916.pdf
Parfit, D. (1984) Reasons and Persons (Oxford: Oxford University Press).
Pattanaik, P.K. (1968) “Risk, Impersonality, and the Social Welfare Function,”
Journal of Political Economy 76, 1152–1169.
Pattanaik, P.K. (1971) Voting and Collective Choice (Cambridge: Cambridge
University Press).
Perlman, M. (ed.) (1974) The Economics of Health and Medical Care: Proceedings of a Conference Held by the International Economic Association at
Tokyo.
Rawls, J. (1951) “Outline of a Decision Procedure for Ethics” Philosophical Review 60: 177–197.
Rawls, J. (1958) “Justice as Fairness” Philosophical Review 67: 164–194.
Rawls, J. (1971) A Theory of Justice (Cambridge MA: Harvard University Press).
Rawls, J. (1982) “Social Unity and Primary Goods” in A.K. Sen and B. Williams(eds.) Utilitarianism and Beyond (Cambridge: Cambridge University Press).

21

Roberts, K.W.S. (1995) “Valued Opinions or Opinionated Values: The Double
Aggregation Problem” in K. Basu, P. Pattanaik and K. Suzumura (eds.),
Choice, Welfare and Development: A Festschrift for Amartya Sen (Oxford:
Oxford University Press).
Savage, L.J. (1954, 1972) The Foundations of Statistics (New York: John Wiley;
and New York: Dover Publications).
Sen, A.K. (1970) Collective Choice and Social Welfare (San Francisco: HoldenDay).
Sidgwick, H. (1887) The Elements of Politics (London: Macmillan).
Starr, R. (1973) “Optimal Production and Allocation Under Uncertainty” Quarterly Journal of Economics, 87: 81–95.
Tinbergen, J. (1957) “Welfare Economics and Income Distribution” American
Economic Review (Papers and Proceedings) 47: 490–503
Vickrey, W.S. (1945) “Measuring Marginal Utility by the Reactions to Risk,”
Econometrica 13: 319–333.
Vickrey, W.S. (1960) “Utility, Strategy and Social Decision Rules” Quarterly
Journal of Economics 74: 507–535.

22