Generalized Utilitarianism and Harsanyi's Impartial Observer Theorem

Abstract

Harsanyi's impartial observer must consider two types of lotteries: imaginary identity lotteries ("accidents of birth") that she faces as herself and the real outcome lotteries ("life chances") to be faced by the individuals she imagines becoming. If we maintain a distinction between identity and outcome lotteries, then Harsanyi-like axioms yield generalized utilitarianism, and allow us to accommodate concerns about different individuals' risk attitudes and concerns about fairness. Requiring an impartial observer to be indifferent as to which individual should face similar risks restricts her social welfare function, but still allows her to accommodate fairness. Requiring an impartial observer to be indifferent between identity and outcome lotteries, however, forces her to ignore both fairness and different risk attitudes, and yields a new axiomatization of Harsanyi's utilitarianism. 2010 The Econometric Society.

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Cowles Foundation
for Research in Economics
at Yale University
Cowles Foundation Discussion Paper No. 1578
September 2006
GENERALIZED UTILITARIANISM AND
HARSANYI’S IMPARTIAL OBSERVER THEOREM
Simon Grant, Atsushi Kajii, Ben Polak and Zvi Safra
This paper can be downloaded without charge from the
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http://ssrn.com/abstract=931408
An index to the working papers in the
Cowles Foundation Discussion Paper Series is located at:
http://cowles.econ.yale.edu/P/au/DINDEX.htm

September 11, 2006
Generalized Utilitarianism
and
Harsanyi’s Impartial Observer Theorem

Abstract
We provide an axio matization of gene ralized utilitarian social welfare functions in the context of Harsanyi’s
impartia l observer th eorem. To do this, we reformulate Harsanyi’s problem such that lotteries over
identity (accidents of birth) and lotteries over outcomes (life chances) are independe nt. We s how how to
accommodate (…rst) Di amond’s critique concerning fairness and (second) Pattanaik’s critique concerning
di¤erin g attitud es towar d r isk. In ea ch ca se, we show what separates them from Harsanyi by showing what
extra axioms return us to Harsanyi. Thus we provide two new axiomat izations of Harsanyi’s u tilitariani sm..
Keywords: generalized uti litarianism, impartial observer, social welfare function, fairness, ex ante
egalitarianism.
JEL Classi…cation: D63, D71
Simon Grant
Department of Economics
Ri ce University
Atsushi Kajii
Insitute of Economic Research
Kyoto University
Ben Polak
Department of Economics & School of Management
Yale University
Zvi Safra
Tel Aviv University and College of Management

We thank Jurgen Eichberger, Edi Karni, Bart Lipman, Stephen Morris, Heve Moulin, David Pearce, John Quig-
gin, John Roemer and John Weymark for many helpful comments. Zvi Safra thanks the Israe l Science Foundation
(grant no. 1299/05) and the Henry Crown Institute of Business Research for their support.

1 Introduction
This paper revisits Harsanyi’s (1953, 1955) utilitarian impartial observer theorem. Consider a
society of individuals I: The society has to choose among di¤erent social policies, each of wh ich
induces a probability distribution or ‘lottery’` over a set of social outcomes X. Each individual i
has preferences %
i
over these lotteries. These preferences are known, and they di¤er.
To help choose among social policies, Harsanyi proposed that each individual should imagine
herself as an ‘impartial observer’ who does not know which person she will be. That is, the
impartial observer faces not only the real lottery ` over the social outcomes in X, but also a
hypothetical lottery z over which identity in I she will assume. In forming preferences % over all
such ‘extended lotteries’, the impartial observer is forced to make interpersonal comparisons; for
example, she is forced to compare being person i in social state x with being person j in social
state x
0
.
Harsanyi assumed that when the impartial observer imagines herself being person i she adopts
person i’s preferences over the outcome lotteries. He also assumed that all individuals are expected
utility maximizers, and that they continue to be so in the role of the impartial observer. Harsanyi
argued that these “B ayesian rationality” axioms force the impartial observer to be a (weighted)
utilitarian. More formally, over all extended lotteries (z; `) in which the identity lottery and
the outcome lotteries are independently distributed, the impartial observer’s preferences admit a
representation of the form
V (z; `) =
X
i
z
i
U
i
(`)
where z
i
is the probability of assuming person i’s identity and U
i
(`) :=
Z
X
u
i
(x) ` (dx) is person
i’s von Neuman-Morgenstern expected utility for the outcome lottery `.
Harsanyi’s utilitarianism has attracted many criticisms.
1
We confront just two: one associated
1
Sen(1970, 1977), Weymark (1991) and others have observed that Harsanyi’s utilitarianism is not ‘welfare’
utilitarianism in the nineteenth-century sense. Harsanyi’s representation is additive in individuals’von Neumann-
Morgenstern utilities but not necessarily in individuals’‘welfares’. We will return to this issue in section 7.
There have been other criticisms be yond the scope of this paper. For example, i t is unclear tha t placing di¤erent
individuals in the role of t he imp artia l o bserver will lead them to agree on the appropriate inter personal comparisons.
Eve n within Harsanyi’s utilitarian for m, the impartial observe r has to decide among a¢ ne transforma tions which
von Neumann-Morgens tern utility fu nction to use for each individual, and which is t he appropriate weighted identity
lottery (see, for example, Mongin (20 01)). It is also unclear that actual individuals will (or even should) feel bound
1

with Diamond (1967) concerning fairness; and one associated with Pattanaik (1968) concerning
di¤erent attitudes toward risk. To illustrate both criticisms, consider two individuals, i and j and
two social outcomes x
i
and x
j
. Person i strictly prefers outcome x
i
to outcome x
j
, but person
j strictly prefers x
j
to x
i
. Perhaps, there is some (possibly indivisible) good, and x
i
is the state
in which person i has the good while x
j
is the state in which person j has it. Sup pose that the
impartial observer would be indi¤erent being person i in state x
i
and being person j in state x
j
;
hence u
i
(x
i
) = u
j
(x
j
) =: u
H
. She is also indi¤erent between being i in x
j
and being j in x
i
;
hence u
i
(x
j
) = u
j
(x
i
) =: u
L
. And she strictly prefers the …rst pair (having the good) to the
second (not having the good); hence u
H
> u
L
.
To illustrate Diamond’s criticism, consider the two extended lotteries illustrated in tables (a)
and (b) in which rows are the people and columns are the outcomes.
x
i
x
j
x
i
x
j
i 1=2 0 i 1=4 1=4
j 1=2 0 j 1=4 1=4
(a) (b)
In each, the impartial observer has a half chance of being person i or person j. But in table (a),
the good is simply given outright to person i: outcome x
i
has probability 1. In table (b), the good
is allocated by tossing a coin: the outcomes x
i
and x
j
each have probability 1=2. Diamond argued
that a fair-minded person might prefer the second allocation policy since it gives each person a
“fair shake”.
2
But Harsanyi’s utilitarian impartial observer is indi¤erent to such considerations
of fairness. Each policy (or its associated extended lottery) involves a h alf chance of getting the
good and hence yields the impartial observer
1
2
u
H
+
1
2
u
L
. The impartial observer cares only about
her total chance of getting the good, not how this chance is distributed between person i and j.
To illustrate Pattanaik’s criticism, consider the two extended lotteries illustrated in tables (c)
by their evaluations in the role of the impartial observer once they ‘resume’ their ro le as real people (see, for
example, Broome (1991)).
2
Societies often use both simple lotteries and weighted lotteries to allocate goods (and bads), presumably for
fairness considerations. Examples include the draft, kidney machines, ove rsubscribed ev ents, schools, and public
housing, and ev en whom should be thrown out of a lifeboat! For a long list and an enlig htening discussion, see
Elster (1989).
2

and (d).
x
i
x
j
x
i
x
j
i 1=2 1=2 i 0 0
j 0 0 j 1=2 1=2
(c) (d)
In each, the impartial observer has a half chance of being in state x
i
or state x
j
, and hence a half
chance of getting the good. But in (c), the impartial observer faces this risk as person i, while in
(d), she faces the risk as person j. Pattanaik argued that if person i is more comfortable facing
such a risk than is person j then the impartial observer might prefer to face the risk as person i.
But Harsanyi’s utilitarian impartial observer is indi¤erent to such considerations of risk attitude.
Each of the extended lotteries (c) and (d) again yield
1
2
u
H
+
1
2
u
L
. Thus, the impartial does not
care who faces this risk.
Most attempts to adapt Harsanyi’s axioms to deal with these concerns have focussed on the
independence axiom of expected utility theory. For example, Karni & Safra (2002) relax inde-
pendence for the individual preferences, while Epstein & Segal (1992) relax independence for the
impartial observer.
3
It is not clear, however that the independence axiom pe se is at the crux of
the disagreements between Harsanyi and his critics.
4
In his own response to Diamond, Harsanyi (1975) argued that, even if randomizations were of
value for promoting ‘fairness’(which he doubted), any explicit randomization is super‡u ous since
‘the great lottery of (pre-)life’may be viewed as having already given each child an equal chance
of being each individual. That is, for Harsanyi, it does not matter whether a good is allocated
entirely by ‘accidents of birth’ (as in the extended lottery (a) above), or whether the good is
allocated entirely by individuals’‘life chances’(as in the extended lottery (c)): for Harsanyi, they
are equivalent. The dispute between Diamond and Harsanyi thus seems to rest on whether or not
3
Strictly speaking, Epstein & Segal’s paper is in the context of Harsanyi’s (1955) aggregatio n theorem, not his
impartial observer’s theorem.
4
Broome (19 91) also argues that independence per se is not the key issue for Harsanyi’s utilitarianism. He
expands the outcome set to allow us t o distinguish not just who gets the good but also the means of allocation.
Debates about independence then become debates about “rational indi¤erence”: that i s, which such outcomes
should be viewed as equivalent. Somewha t analogously, we emphasize which lott eries should be viewed as equivalent.
Broome’s own critiques, however, both of the impartial observer theorem and of Diamond’s notion of fair ness (see,
for e xample, Broome 19 84) are on other grounds.
3

we think (imagined) lotteries over identities are indeed equivalent to (real) lotteries over outcomes.
We will argue th at such an equivalence is also at the heart of Pattanaik and Harsanyi’s dispute.
Which of Harsanyi’s axioms yield the equivalence between identity lotteries and outcome lot-
teries? In most formulations of Harsanyi’s theorem, the impartial observer is assumed to form
preferences not just over extended lotteries in which the identity lottery and outcome lottery are
independently distributed but over the entire set of joint distributions 4(I  X) over identities
and outcomes.
5
In such a set up, it is hard even to distinguish the outcome lottery from the
identity lottery since the resolution of identity can partially or fully resolve the outcome. For
example, the impartial observer could face a joint distribution in which, if she becomes person
i then society holds the outcome lottery `, but if she becomes person j then social outcome x
obtains for sure. Harsanyi’s u tilitarianism comes from later restricting the representation to the
set of independent or ‘product’lotteries.
Suppose ins tead that we restrict attention from the outset to product lotteries, 4(I) 4(X),
That is, the impartial observer only forms preferences over extended lotteries in which the outcome
lottery she faces is the same regardless of which identity she assumes. This setting seems closer
to most informal accou nts of Harsanyi’s thought experiment. Suppose we then impose each of
Harsanyi axioms in this simpler setting: in particular, each individual satis…es the independence
axiom for outcome lotteries, that the impartial obse rver respects these individuals’ preferences
(and so inherits this independence), and that the impartial observer herself satis…es independence
for identity lotteries. In this case, we are no longer forced to Harsanyi’s utilitarianism. Instead
(see theorem 2 below), we obtain a generalized (weighted) utilitarian representation:
V (z; `) =
X
i
z
i

i
(U
i
(`))
where z
i
is again the probability of assuming person i’s identity and U
i
(`) is again person i’s
expected utility from the outcome lottery `, but each 
i
(:) is a (possibly non-linear) transfor-
mation of person i’s expected utility. Generalized utilitarianism is well known to applied welfare
economists but has, till now, lacked axiomatic foundations.
5
See, for exampl e, Weymark (1991). An exception is Safra & Weissengrin (2002).
4

Generalized utilitarianism can accommodate both Diamond and Pattanaik. Diamond’s concern
for fairness can be accommodated if the 
i
-functions are concave.
6
Pattanaik’s concern for di¤erent
risk attitudes can be accommodated by allowing the 
i
-functions to di¤er in their degree of
concavity or convexity.
7
Harsanyi’s utilitarianism can be thought of as the special case where
each 
i
is a¢ ne.
Since generalized utilitarianism nests Diamond’s, Pattanaik’s and Harsanyi’s models, we pro-
ceed to axiomatize each in turn. We show formally that what separates Harsanyi from Diamond
is that Harsanyi assumes that identity lotteries (accidents of birth) and outcome lotteries (life
chances) are ‘equivalent’in the sense of being indi¤erent. By contrast, Diamond assumes a pref-
erence for life chances; that is, in the example above, he prefers that the good be allocated by a
real outcome lottery (as it is in (b),(c) and (d)) than by the imaginary chance of assuming the
right identity (as it is in (a)).
8
What separates Harsanyi from Pattanaik is again that Harsanyi
assumes that identity lotteries and outcome lotteries are ‘equivalent’but this time in the sense
that all axioms are symmetric across the two types of lottery. By contrast, Pattanaik imposes in-
dependence over identity lotteries directly on the impartial observer but allows her only to inherit
independence over outcome lotteries indirectly from the individuals who will actually face those
lotteries.
These two di¤erent notions of equivalence yield two di¤erent new axiomatizations of Harsanyi’s
utilitarianism. Each is built by adding an axiom to those that delivered generalized utilitarianism.
More abstractly, we can also think of these as two new bi-linearity theorems for products of lottery
spaces.
Although restricting attention to product lotteries seems natural and yields the results we
want, it comes at a technical cost in that we can no longer rely on well-known results from decision
6
In our story, we have 
i
(u
i
(x
i
)) = 
j
(u
j
(x
j
)) > 
i
(u
i
(x
j
)) = 
j
(u
j
(x
i
)). Thus, if the -functions
are (strictly) concave, the impartial observer evaluatation of allocation policy (c ) 
i

1
2
u
i
(x
i
) +
1
2
u
i
(x
j
)

>
1
2

i
(u
i
(x
i
)) +
1
2

i
(u
i
(x
j
)) =
1
2

i
(u
i
(x
i
)) +
1
2

j
(u
j
(x
i
)), her eval uation of policy (a). The ar gument comparing
(b) and (a) is si milar.
7
For example, if 
i
is strictly concave but 
j
is lin ear, then the impartial observer’s evaluation of policy (c)

i

1
2
u
i
(x
i
) +
1
2
u
i
(x
j
)

>
1
2

i
(u
i
(x
i
)) +
1
2

i
(u
i
(x
j
)) =
1
2

j
(u
j
(x
j
)) +
1
2

j
(u
j
(x
i
)) = 
j

1
2
u
j
(x
j
) +
1
2
u
j
(x
i
)

,
her evaluat ion of poli cy (d).
8
Indeed, Diamond’s orig inal example was between allocation policies like (a) an d (c). He strictly preferred (c).
5

theory. First, the set of product lotteries 4(I)  4(X) is not a convex subset of 4(I  X). In
particular, we have to be careful that our independence axioms only involve mixtures that remain
in the set of product lotteries. Fortunately, we can adapt some axioms developed by Fishburn
(1982) to study mixed strategies in games.
9
Second, the set of product lotteries does not have a nice recursive structure. With the full set
of joint distributions, it is as if each individual i faces his own ‘personal’outcome lottery. Each
vector of personal lotteries induces a vector of individual utilities. In this setting, by changing
person i’s personal outcome lottery, holding …xed the other people’s lotteries, we can induce a
rich range of these individual-utility vectors. With only the set of product lotteries, however, each
person faces the same outcome lottery (although their preferences over those lotteries may di¤er).
This limits the set of individual-utility vectors we can induce. If this set is not rich enough then
our axioms will lack bite.
The richness of this set depends on the degree to which di¤erent individuals di¤er in their
ranking of outcome lotteries, and the degree to which di¤erent outcome lotteries lead the impartial
observer to di¤erent welfare ranking of individuals. Most of the results below require relatively
mild richness conditions: either that individuals do not all agree in their preference for one outcome
over another or that the impartial observer does not always prefer to be one individual rather than
another.
10
For one result, however, –our second axiomatization of Harsanyi’s utilitarianism –we
use a stronger condition that requires there to be three or more agents.
Section 2 sets up the product-lottery framework. Section 3 shows that, if we adapt the Harsanyi
axioms to that framework, we obtain an axiomatization of generalized utilitarianism. Section 4
provides an additional axiom to accommodate Diamond’s concern for fairness. It shows that
forcing the impartial observer to ignore these concerns corresponds to being indi¤erent between
identity and outcome lotteries. This yields our …rst new axiomatization of Harsanyi’s utilitarian-
ism. Section 5 shows how to accommodate Paittanaik’s concerns for di¤erent attitudes toward
risk. It shows that forcing the impartial observer to ignore these concerns corresponds to imposing
9
Axioms of this form were also used by Safra & Weissengrin (2002).
10
But, in most cases, these conditions are essential to the results. We provide counter-examples in appendix A.
6

a weak form of independence axiom directly on the impartial observer. Going further and imposing
all axioms symmetrically on identity and outcome lotteries (given a rich environment) yields our
second new axiomatization of Harsanyi. There is a very large literature discussing Harsanyi, and
we do not attempt to summarize it, b ut 6 discusses s ome related technical papers. I n particular, it
considers what happens if we impose less structure and what happens if, like Harsanyi, we impose
more. Section 7 (following, for example, Weymark (1991)) introduces an explicit n otion of com-
parable welfare, and uses it to interpret some of our representation results. Appendix A provides
counter-examples to show that our axioms are essential. Appendix B provides those proofs not in
the text.
2 Set up and Notation
Let society consist of a …nite set of individuals I = f1; : : : ; Ig, I  2, with generic elements i and
j. The set of …nal outcomes or social states is den oted by X with generic element x. The set X
is assumed to have more than one element and to be a compact metrizable space and associated
with it is the set of events E, which is taken to be the Borel sigma-algebra of X. Let 4(X) (with
generic element `) denote the set of outcome lotteries; that is the set of probability measures on
(X; E) endowed with the weak convergence topology. We will sometimes refer to these lotteries
over outcomes as life chances: they represent the real risks f aced by each individual in their real
lives. With slight abuse of notation, we will let x or sometimes [x] denote the degenerate outcome
lottery that assigns probability weight 1 to social state x.
Each individual i in I, is endowed with a preference relation %
i
de…ned over the set of life-
chances 4(X). We assume throughout that for each i in I, the preference relation %
i
is a
complete, transitive and continuous binary relation on 4(X), and that its asymmetric part 
i
is non-empty. Henc e for each %
i
there exists a non-constant function V
i
: 4(X) ! R, satisfying
for any ` and `
0
in 4(X), V
i
(`)  V
i
(`
0
) if and only if ` %
i
`
0
. In summary, a society may be
characterized by the tuple

X; E; I; f%
i
g
i2I

.
In Harsanyi’s story, the impartial observer imagines herself behind a veil of ignorance, uncertain
about which identity she will assume in the given society. Let 4(I) denote the set of identity
7

lotteries on I. Let z denote the typical ele ment of 4(I), and let z
i
denote the probability assigned
by the identity lottery z to individual i. We will sometimes refer to these lotteries over identity
as accidents of birth: they represent the imaginary risks in the mind of the impartial observer of
being born as someone else. With slight abuse of notation, we will let i or sometimes [i] denote the
degenerate identity lottery that assigns probability weight 1 to th e impartial observer’s assuming
the identity of individual i.
As discussed above, we assume that the outcome and identity lotteries faced by the impartial
observer are independently distributed; that is, she faces a product lottery (z; `) 2 4(I) 4(X).
We shall sometimes refer to this as a product identity-outcome lottery or (where no confusion
arises) simply as an product lottery.
The impartial observer is endowed with a preference relation % de…ned ove r 4(I)4(X). We
assume throughout that % is c omplete, transitive and continuous, and that its asymmetric part 
is non-empty, and so it admits a (non-trivial) continuous representation V : 4(I)  4(X) ! R.
That is, for any pair of product lotteries, (z; `) and (z
0
; `
0
), (z; `) % (z
0
; `
0
) if and only if V (z; `) 
V (z
0
; `
0
).
3 Generalized Utilitarianism
In this section, we adapt the axioms from Harsanyi’s impartial observer theorem to apply to
the product-lottery framework, add a richness condition that there is some disagreement in the
underlying individual preferences over policies, and hence provide an axiomatization of generalized
utilitarianism.
The …rst axiom is Harsanyi’s acceptance principle. In degenerate product lotteries of the form
(i; `) or (i; `
0
), the impartial obse rver knows she will assum e identity i for sure. The acceptance
principle requires that, in this case, the impartial observer’s preferenc es % must coincide with that
individual’s preferences %
i
over life chances.
The Acceptance Principle. For all i in I and all `; `
0
2 4(X), ` %
i
`
0
if and only if (i; `) %
(i; `
0
).
Second, following Harsanyi, we assume that each individual i’s preferences satisfy the indepen-
8

dence axiom for the lotteries he faces, that is over the set of outcome lotteries 4(X). We state
this axiom in a slightly non-standard form.
Independence over Outcome Lotteries (for Individual i). Suppose `, `
0
2 4(X) are such
that ` 
i
`
0
. Then, for all
~
`,
~
`
0
2 4(X),
~
` %
i
~
`
0
if and only if 
~
`+(1  ) ` <
i

~
`
0
+(1  ) `
0
for all  in (0; 1].
Notice that the two outcome lotteries, ` and `
0
that are “mixed in” with weight (1  ) to
~
`
and
~
`
0
are themselves indi¤erent. The axiom states that ‘mixing in’two indi¤erent lotteries (with
equal weight) preserves the original preference order between
~
` and
~
`
0
prior to mixing.
The standard version of the independence axiom states that for all
~
`;
~
`
0
;
~
`
00
in 4(X),
~
` %
i
~
`
0
if and only if 
~
` + (1  )
~
`
00
%
i

~
`
0
+ (1  )
~
`
00
for all  in (0; 1]. That is, in its standard form,
the same outcome lottery
~
`
00
is ‘mixed-in”with weight (1  ) to
~
` and
~
`
0
. It is a simple exercise
to show that these two versions of independence are equivalent.
11
We use the form above to
emphasize the symmetry with the next axiom.
Third, following Harsanyi, we assume that the impartial observer’s preferences also satisfy
independence. Here, however, we need to be careful. First, the se t of product lotteries 4(I) 
4(X) is not a convex subset of 4(I X) and hence not all probability mixtures of product
lotteries are well de…ned. Second, the impartial observer faces two types of lottery, over outcomes
and over identities. The former risks are faced directly by real people, but are on ly faced indirectly
by the impartial observer once she assumes the identity of a real person. Once we impose the
independence axiom on each individual’s preferences, the acceptance principle already ensures
that the impartial observer res pects those individual preferences (and hence independence) over
outcome lotteries. Identity lotteries, however, are not faced by real people, but only faced by the
impartial obs erver in her thought experiment. Thus, to get independence over identity lotteries,
we need to impose it d irectly on the impartial observer’s preferences. The following axiom achieves
this.
12
11
In particu lar, the current form immediately implies the standard form. For the other direction, the standard
form implies that ` 
i
`
0
if and only if 
~
`
0
+ (1  ) ` 
i

~
`
0
+ (1  ) `
0
.
12
This axiom is based on Fishburn’s (1982, p.88) and Safra & Weissengrin’s (2003) substitution axioms for
9

Independence over Identity Lotteries (for the Impartial Observer). Suppose (z; `), (z
0
; `
0
) 2
4(I)  4(X) are such that (z; `)  (z
0
; `
0
). Then, for all ~z, ~z
0
2 4(I): (~z; `) % (~z
0
; `
0
) if
and only if (~z + (1  ) z; `) % (~z
0
+ (1  ) z
0
; `
0
) for all  in (0; 1].
To understand this axiom, …rst notice that the two mixtures on the right side of the implication
are identical to (~z; `)+(1  ) (z; `) and (~z
0
; `
0
)+(1  ) (z
0
; `
0
) respectively. These two mixtures
of product lotteries are well de…ned: they mix identity lotteries holding the outcome lottery …xed.
Second, notice that the two product lotteries, (z; `) and (z
0
; `
0
), that are ‘mixed in’with weight
(1  ) are themselves indi¤erent. The axiom states that ‘mixing in’ two indi¤erent lotteries
(with equal weight) preserves the the original preference order between (~z; `) and (~z
0
; `
0
) prior to
mixing.
13
Finally, notice that this axiom only applies to mixtures of identity lotteries holding
the outcome lotteries …xed, not to the opposite case: mixtures of outcome lotteries holding the
identity lotteries …xed. We will discuss this ‘opposite’axiom in section 5 below.
How do these axioms relate to the discussion in the introduction? Given acceptance, the
impartial ob server inherits her preferences over outcome lotteries from the preferences of the
individuals who will face those lotteries and whose identities she will assume. In particular,
the impartial observer inh erits independence over outcome lotteries indirectly from individuals’
preferences. By contrast, we can think of Harsanyi imposing such independence directly. We
will show in section 5 that this distinction allows us to accommodate Pattanaik’s concern about
di¤erent individuals’di¤erent attitudes toward risk. None the axioms above say anything about
how the impartial observer compares identity and outcome lotteries. In particular, unlike Harsanyi,
we do not implicitly assume that she is indi¤erent between accidents of birth and life chances. We
product lottery spaces. Their axioms, howeve r, apply whereever probability mixtures are well de…ned in this space.
For example, in our context, their axioms wo uld apply to mixtures of outcome lotteries. We only allow mixtures of
identity lotteries. In this re spect, our axiom is simil ar to Karni & Safra’s (2000) ‘constrained ind ependence’axiom,
but their axiom applies to all joint distributions over ide ntities and outcome s, not just to product lotteries.
13
One technical remark mi ght interest some readers. In the axiom, we allow the mixing of identity lotteries to
occur at two di¤erent outcome lotteries ; that is, we do not restrict ` to equal `
0
. We could de…ne a weaker axiom –
call in conditional independence – that simply imposes independence over identity l otteries at e ach …xed outcome
lottery

`. That is, for all

` 2 4(X ), if z; z
0
2 4 (I) are such that (z;

`)  (z
0
;

`) then for all ~z; ~z
0
2 4(I), (~z;

`) %
(~z
0
;

`) if and on ly if (~z + (1   ) z;

`) % (~z
0
+ (1  ) z
0
;

`) for all  in (0; 1]. Our stronger axiom is necessary for
the repre sentation results that follow. To sho w this, example 2 in a ppendix A shows that preferences can satisfy
the acceptance principle , independence over outcome lotteries for individuals, and conditional inde pendence over
identity lotteries for the impartial observer but not satisfy the (unconditional) independ ence axiom over identity
lotteries de…ned above.
10

will show in section 4 that this allows us accommodate Diamond’s concerns about fairness.
To obtain our representation results, we work with a richness condition on the domain of
individual preferences: we ass ume that none of the outcome lotteries under consideration are
Pareto dominated.
Absence of Unanimity For all `; `
0
2 4(X) if ` 
i
`
0
for some i in I then there exists j in
I such that `
0

j
`.
This condition is perhaps a natural restriction in the context of Harsanyi’s thought experiment.
That exercise is motivated by the need to make social choices when agents disagree. We d o not
need to imagine ourselves as an impartial observer facing a identity lottery to rule out social
alternatives that are Pareto dominated.
14
The following lemma does yet not impose independence over outcome lotteries on individuals
and hence yields a more general representation. The idea for this lemma comes from Karni & Safra
(2000) but they work with the full set of joint distributions 4(I X) whereas we are restricted
to the set of product lotteries 4(I)  4(X).
Lemma 1 Suppose absence of unanimity applies. Then the impartial observer satis…es the ac-
ceptance principle and independence over identity l otteries if and only if there exist a continuous
function V : 4(I)  4(X) ! R that represents %, and, for each individual i in I, a function
V
i
: 4(X) ! R, that represents %
i
, such that for all (z; `) in 4(I) 4(X),
V (z; `) =
I
X
i=1
z
i
V
i
(`):
Moreover the functions V
i
are unique up to common a¢ ne transformations.
The proof is in the appendix but a sketch is as follows. The …rst step follows Karni & Safra
(2000).
15
Fix some outcome lottery `
1
. Notice that, by independence, there exist two individual
14
For example, if absence of un animity fa ils but individuals satisfy independenc e, we could …rst discard all those
outcomes that are Pareto dominated by other outcome lotteries and then carry out the Harsanyi thought experiment
on the set of lot teries over the remaining undominated outcomes. Given independence, the set of undominated
lotteries over th e original outcome s is equal to the set of lotteries over the undominated outcomes, an d absence of
unanimity will ho ld for the new domain.
15
An alternate strategy would be to prove it as a special case of Theorem 8.
11

i
1
and i
1
such that

i
1
; `
1

%

z; `
1

%

i
1
; `
1

for all z; that is, i
1
is a best identity and i
1
is
a worst identity to assume given that the impartial observer will then face the outcome lottery
`
1
. Next, construct a representation for all product lotteries (z; `) such that

i
1
; `
1

% (z; `) %

i
1
; `
1

by …nding the weight  in [0; 1] such that the identity lottery 

i
1

+ (1  ) [i
1
] facing
the outcome lottery `
1
is indi¤erent to the ide ntity lottery z facing the outcome lottery `. Set
V (z; `) := . Independence over identity lotteries en su res that this representation is unique and
a¢ ne.
16
Up to this point the argument resembles a standard proof of the von Neumann-Morgenstern
theorem except that (so far) we have only constructed an a¢ ne representation for those identity-
outcome lotteries (z; `) such that

i
1
; `
1

% (z; `) %

i
1
; `
1

; that is, those (z; `) that are indi¤erent
to

z
0
; `
1

for some identity lottery z
0
at the particular …xed outcome lottery `
1
. Loosely speaking,
we have only represented an ‘interval’of the impartial observer’s preference s. To go further, Karni
& Safra exploit the fact that (for them) each individual faces a di¤erent outcome lottery. Instead,
we rely on our richness condition.
Lemma 9 in the appendix shows that, given absence of unanimity, we need at most two ‘…xed’
outcome lotteries (i.e., at most two ‘intervals’) to cover the entire range of the impartial observer’s
preferences. To keep the notation consistent with that in the appendix, let these two outcome
lotteries be denoted `
1
and `
2
. That is, th ere exists two outcome lotteries `
1
and `
2
such that for
all product lotteries (z; `) either (z; `) 

z
0
; `
1

for some z
0
, or (z; `)  (z
00
; `
2
) for some z
00
or
both. Moreover we can choose `
1
and `
2
such that their ‘intervals’overlap. With this step in hand,
standard arguments ensure that the a¢ ne representations are consistent on the two ‘intervals’,
and satisfy the usual uniqueness condition.
Finally, applying a¢ nity implies that the representation takes the form
P
i
z
i
V (i; `), and, by
acceptance, we can set V (i; ) := V
i
() to complete the proof.
In section 6, we sh ow that without absence of unanimity, we can still obtain a representation
similar to that in Lemma 1 but it will lack the uniqueness properties.
16
For uniqueness: strictly speaking, we need

i
1
; `
1



i
1
; `
1

. For a¢ nity, this step is w here the weaker condi-
tional independence discussed in footnote X wou ld not be su¢ cient: the product lotteries (z; `) we are representing
contain outcome l otteries other tha n just `
1
.
12

The representation in Lemma 1 puts no restriction on the V
i
-functions. But, if we now add
the assumption that each individual satis…es independence over outcome lotteries then it follows
immediately that each V
i
-function must be a strictly increasing transformation of a von Neumann-
Morgenstern expected-utility representation. Thus, we obtain a generalized utilitarian represen-
tation.
Theorem 2 (Generalized Utilitarianism) Suppose that absence of unanimity applies. Then
the following are equivalent:
(a) The impartial observer satis…es the acceptance principle and independence over identity lot-
teries, and each individual satis…es independence over outcome lotteries
(b) There exist a continuous function V : 4(I)  4(X) ! R that represents %, and, for each
individual i in I, a von Neumann-Morgenstern function U
i
: 4(X) ! R that represents %
i
and a continuous, strictly increasing function 
i
: R ! R, such that, for all (z; `) in 4(I) 
4(X),
V (z; `) =
I
X
i=1
z
i

i
[U
i
(`)] .
where, for each i, U
i
(`) =
Z
X
U
i
(x) ` (dx). Moreover the functions U
i
are unique up to a¢ ne
transformations, and the composite functions 
i
 U
i
are unique up to a common a¢ ne
transformation.
Notice that, while the representation of each individual’s pref erenc es U
i
is a¢ ne in outcome
lotteries, in general, the representation of the impartial observer’s preferences V is n ot.
4 Accommodating Fairness: Harsanyi vs. Diamond.
In this section, we …rst introduce a new axiom on the impartial observer’s preferences to ensure
that the generalized utilitarian representation is concave and hence accommodates Diamond. We
then show that tightening this axiom yields Harsanyi’s utilitarianism.
So far we have placed no restriction on the shape of the 
i
-functions excep t that they are
increasing. An analogy may help to see why we want concavity. In a standard utilitarian so-
cial welfare function, each u
i
-function maps individual i’s income to an individual utility. These
13

incomes di¤er across people, and concavity is associated with income egalitarianism. In a general-
ized utilitarian social welfare function, each 
i
-function maps individual i’s expected utility U
i
(`)
to a utility of the impartial observer. These expected utilities di¤e r across people, and concavity
is associated with expected-utility egalitarianism.
17
It is easy to show that, if the 
i
-functions are concave then the impartial observer will respect
Diamond’s preferences in ou r initial e xample.
18
But having preferences respect Diamond’s choice
in this particular example is not enough to ensure in general that the 
i
-functions are concave:
for example, the underlying social choice problem may not contain two outcomes and two people
with (i; x
i
)  (j; x
j
) and (i; x
j
)  (j; x
i
). We need to generalize the idea of the example.
The example involved two indi¤erence sets of the impartial observer, that containing (i; x
i
)
and (j; x
j
) and that containing (i; x
j
) and (j; x
i
). Diamond preferred a randomization between
these indi¤erenc e sets in outcome lotteries (i.e., real life chances) to a randomization in identity
lotteries (i.e., imaginary accidents of birth). To generalize, supp ose the impartial observer is
indi¤erent between (z; `
0
) and (z
0
; `), and consider the product lottery (z; `) that (in general) lies
in a di¤erent indi¤erence set. There are two ways to randomize betwe en these indi¤erence sets
while remaining in the set of product lotteries. The product lottery (z; ` + (1  ) `
0
) randomizes
between these indi¤erence sets in outcome lotteries (i.e., real life chances); while the product lottery
(z + (1  ) z
0
; `) randomizes between these indi¤erence sets in identity lotteries (i.e., imaginary
accidents of birth). The example suggests that Diamond prefers the former.
Preference for Life Chances. For any pair of identity lotteries z and z
0
in 4(I), and any
pair of outcome lotteries ` and `
0
in 4(X), (z; `
0
)  (z
0
; `) then (z; ` + (1  ) `
0
) %
(z + (1  ) z
0
; `) for all  in (0; 1).
If the pref erenc e sign is reversed in the above axiom, we say that the impartial observer exhibits
preference for accidents of birth. If both apply, we say that the impartial observer is indi¤erent
between accidents of birth and life chances.
17
This is sometimes called ‘ex ante egalitarianism’. See for exa mple, Broome (1 984), Myerso n (19 81), Hammond
(1981, 1982) and Meyer (1991). In our context, it is perhaps be tter to call this ‘interim’ egalitarianism since it
refers to distributions ‘after’the resolution of the identity lottery but ‘before’the resolution of the outcome lottery.
18
See footnote 6. To get Diamond’s strict preference, we require strict concav ity.
14

If we add this axiom to the conditions of Theorem 2, then we obtain concave generalized
utilitarianism.
Proposition 3 (Concavity) Suppose that absence of unanimity and all the axioms of Theorem
2 apply, so that V (z; `) =
P
I
i=1
z
i

i
[U
i
(`)] is a generalized utilitarian representation. Then the
impartial observer exhibits preference for life chances if and only if each of the 
i
-functions is
concave.
To show that concavity is su¢ cient, recall that V is a¢ ne in identity lotteries and each U
i
is
a¢ ne in outcome lotteries. Thus, if we set V
i
(`) := 
i
[U
i
(`)] for all `, then V
i
is concave if and only
if 
i
is concave. Imposing concavity, we obtain V (z; ` + (1  ) `
0
) =
P
I
i=1
z
i
V
i
(`+(1  ) `
0
) 
P
I
i=1
z
i
[V
i
(`) + (1  ) V
i
(`
0
)] = V (z; `) + (1  ) V (z; `
0
). Using the fact that (z; `
0
)  (z
0
; `),
the last expression is equal to V (z; `) + (1  ) V (z
0
; `) = V (z + (1   ) z
0
; `). Hence the
impartial observer exhibits a preference for life chances.
The proof that preference for life chances implies concavity is in the appendix but a discussion
follows. At …rst glance, the representation in Lemma 1 resembles a recursive expected-utility
model such as
P
i
z
i
v (`
i
) in which z
i
is the probability of being faced by the outcome lottery `
i
.
In that setting, an analog of preference for life chances implies that the fu nc tion v is concave.
19
But there are two ways in which the current model di¤ers from this recursive one. First, in place
of a single v, each V
i
represents a di¤erent individual’s preferences. Sec ond, in place of a vector
of `
i
’s, each individual faces the same outcome lottery `, thus the set of product lotteries (our
objects of choice) are not isomorphic to the set of compound lotteries. In a sense, however, the
…rst problem alleviates the second.
The role of absence of unanimity in the proof is to ensure that there is enough variation in the
individual preferences to make up for the lack of variation in the outcome lottery. Consider the
map that takes each outcome lottery ` to its corresponding vector of utilities (V
1
(`) ; : : : ; V
I
(`)).
Absence of unanimity ensures that the range of this map is rich enough for our axioms to bite.
And absence of unanimity is essential: example 3 in appendix A shows that without this richness
19
See, for exampl e, Kreps & Porteus (1979) or Grant, Kajii & Polak (1998) .
15

condition on the underlying individual preferences, the V
i
’s (and hence the 
i
’s) need not be
concave.
In contrast to Diamond, Harsanyi implicitly imposes indi¤erence between life chances and
accidents of birth. If we impose this indi¤erence as an explicit axiom then, as a corollary of
Proposition 3, we obtain that each 
i
-function must be a¢ ne. In this case, if we let
^
U
i
:= 
i
U
i
,
then
^
U
i
is itself a von Neumann-Morgenstern expected-utility representation of %
i
. Thus, we
immediately obtain our …rst new axiomatization of Harsanyi’s utilitarian representation.
Theorem 4 (Utilitarianism I) Suppose that absence of unanimity applies. Then the following
are equivalent:
(a) The impartial observer satis…es the acceptance principle and independence over identity lot-
teries, each individual satis…es independence over outcome lotteries, and the impartial ob-
server is indi¤erent between life chances and accidents of birth.
(b) There exist a continuous function V : 4(I)  4(X) ! R that represents %, and, for
each individual i in I, a function
^
U
i
: 4(X) ! R that is a von Neumann-Morgenstern
expected-utility representation of %
i
, such that for all (z; `) in 4(I) 4(X),
V (z; `) =
I
X
i=1
z
i
^
U
i
(`)
where, for each i,
^
U
i
(`) =
Z
X
^
U
i
(x) ` (dx). Moreover the functions
^
U
i
are unique up to
common a¢ ne transformation.
To summarize: if we start from the axioms that gave us a generalized utilitarianism and add
Diamond’s preference for life chances then we obtain concave generalized utilitarianism. But if we
assume that life chances and accidents of birth are equivalent in the sense of indi¤erenc e we are
forced back to Harsanyi’s utilitarianism.
5 Di¤erent risk attitudes: Harsanyi vs. Pattanaik
In this section, we …rst sh ow how generalized utilitarianism can accommodate di¤erent risk at-
titudes. More interestingly, we then show that if we impose a weak form of independence over
16

outcome lotteries directly on the impartial observer (rather than just allow her to in herit outcome-
lottery independence via the acceptance principle), th en she is forced to ignore di¤erent risk atti-
tudes. Finally, we will show that strengthening this axiom –so that identity and ou tcome lotteries
are treated symmetrically in terms of the axioms – (almost) forces us again back to Harsanyi’s
utilitarianism.
Recall that Pattanaik’s critique concerned di¤erent risk attitudes of di¤erent individuals. The
impartial observer’s interpersonal welfare comparisons might rank (i; x
i
)  (j; x
j
) and (i; x
j
) 
(j; x
i
), but if person i is more comfortable facing risk than person j, she might rank

i;
1
2
[x
i
] +
1
2
[x
j
]



j;
1
2
[x
i
] +
1
2
[x
j
]

. Harsanyi’s utilitarianism rules this out. An analogy might be useful. In the
standard representative-agent model of consump tion over time, each time period is assigned one
utility function. This utility fu nction must re‡ect both risk aversion in that period and substi-
tutions between periods. Once utilities are scaled for inter-temporal welfare comparisons, there
is limited scope to accommodate di¤erent risk attitudes across periods . Harsanyi’s utilitarian
impartial observer assigns one utility function per person. This utility function must re‡ect both
risk aversion of that person and substitutions between people. Once utilities are scaled for in-
terpersonal welfare comparisons, there is limited scope to accommodate di¤erent risk attitudes
across people.
Given the analogy, it is not surprising that generalized utilitarianism can accommodate Pat-
tanaik. Each person is now assigned two functions, 
i
and u
i
, so we can separate interpersonal
welfare comparison from risk aversion. To be more speci…c, we generalize Pattanaik’s example.
Suppose the impartial observer ranks (i; `)  (j; `
0
) and (i;
~
`)  (j;
~
`
0
). Then, for all  in (0; 1), we
say that the two outcome lotteries 
~
`+(1   ) ` and 
~
`
0
+(1  ) `
0
are similar risks for individu-
als i and j respectively. Suppose that the generalized utilitarian impartial observer always prefers
to face similar risks as person i than as person j. In this case, loosely spe aking, we require the
function 
i
to be a concave transformation of 
j
on the ‘relevant domain’. The next proposition
makes this precise.
Proposition 5 (Di¤erent Risk Attitudes.) Suppose that absence of unanimity and all the ax-
ioms of Theorem 2 apply, so that V (z; `) =
P
I
i=1
z
i

i
[U
i
(`)] is a generalized utilitarian represen-
17

tation. Then the impartial observer always (weakly) prefers to face similar risks as individual i
than as individual j if and only if the composite function 
1
i
 
j
is convex on the the domain
U
ji
:= fu 2 R : there exists `; `
0
2 4(X) with (i; `)  (j; `
0
) and U
j
(`
0
) = ug.
The proof is in the appendix but a discussion follows. Recall that we say that agent j is more
(income) risk averse than agent i if the function that maps income to agent j’s von Neumann-
Morgenstern utility is a concave transformation of that for agent i. For each i, the function 
i
maps agent i’s von Neumann-Morgenstern utility to utilities of the impartial observer. Thus, to
say that 
1
i
 
j
is convex is to say that the function that maps the impartial observer’s utility
to agent j’s von Neumann -Morgens tern utility (i.e., 
1
j
) is a concave transformation of that of
agent i. We return to this discussion in section 7 when we introduce a cardinally measurable and
comparable welfare.
In contrast to Pattanaik, Harsanyi imp licitly imposes indi¤erence as to which person should
face similar risks; that is, he ignores di¤erent risk attitudes.
20
Harsanyi makes this assumption
when he imposes independence over outcome lotteries directly on the impartial observer, rather
than just allowing such inde pendence to be inherited from individual preferences via the acceptance
principle. In fact, even the following weak independence su¢ ces.
Weak Independence over Outcome Lotteries (for the impartial observer). Suppose (i; `),
(j; `
0
) 2 I  4(X) are such that (i; `)  (j; `
0
). Then, for all
~
`,
~
`
0
2 4(X): (i;
~
`) % (j;
~
`
0
) if
and only if (i; 
~
` + (1   ) `) % (j; 
~
`
0
+ (1  ) `
0
) for all  in (0; 1]
Notice …rst that this axiom is almost symmetric to independence over identity lotteries for the
impartial observer: there the mixing involves identity lotteries holding the outcome lotteries …xed;
here the mixing involves outcome lotteries holding the identity lotteries …xed. But this axiom is
weak in that it restricts the (…xed) identity lotteries to be degenerate. Second, given acceptance,
imposing weak independence over outcome lotteries directly on the impartial observer implies
independence over outcome lotteries for each individual i, but the converse is not true. Third, if
20
We know from sectio n 4 t hat indi¤erence as to which agent should face similar risks is implied by indi¤erence
between ac cidents of birth and life chances. But the converse is not true: i ndi¤erence as to which agent should face
similar risks is weaker.
18

we look at the case where both (i; `)  (j; `
0
) and (i;
~
`)  (j;
~
`
0
), the new axiom immediately forces
the impartial observer to be in di¤erent between facing similar risks as person i and person j. Thus
(given Proposition 5), the 
i
and 
j
-functions must be identical up to a¢ ne transformations.
To obtain a tighter result, we introduce a second richness condition that is the symmetric
analog of absence of unanimity.
Redistributive Scope For all z; z
0
in 4(I), if (z; x)  (z
0
; x) for some x in X then there exists
x
0
in X such that (z
0
; x
0
)  (z; x
0
).
Given acceptance, absence of unanimity implies: for all `; `
0
in 4(X), if (i; `)  (i; `
0
) for some
i in I then there exists j in I su ch that (j; `
0
)  (j; `). With absence of unanimity, there are no
Pareto dominated outcome lotteries. With redistributive scope, there are no ‘dominated’identity
lotteries. In particular, for each pair of individuals i and j, if there is an ou tcome at which the
impartial observer would prefer to be individual i then there is another outcome at which she
would prefer to be ind ividual j. Intuitively, there is some other policy outcome in which either
person i has been made su¢ ciently worse o¤ or person j has been made su¢ ciently better o¤
(or both) such that their ranking has been reversed. Despite the formal symmetry between these
two richness conditions, redistributive scope is perhaps more restrictive in practice: for example,
there may be policy settings in which, under every policy under consideration, one agent is always
better o¤ than an other. It will apply however, in standard private-go od allocation problems with
(ex ante) symmetric agents.
With redistributive scope, imposing weak independence over outcome lotteries directly on
the impartial observer yields a generalized u tilitarian representation in which there is a common
-function.
Proposition 6 (Common -Function) Suppose that absence of unanimity and redistributive
scope both apply. Then the impartial observer satis…es the acceptance principle, independence
over identity lotteries and weak independence over outcome lott eries if and only if there exist a
continuous function V : 4(I)  4(X) ! R that represents %, for each individual i in I, a von
Neumann-Morgenstern function
^
U
i
: 4(X) ! R that represents %
i
, and a ( common) continuous,
19

strictly increasing function  : R ! R, such that, for all (z; `) in 4(I) 4(X),
V (z; `) =
I
X
i=1
z
i

h
^
U
i
(`)
i
.
where, for each i,
^
U
i
(`) =
Z
X
^
U
i
(x) ` (dx). Moreover the functions
^
U
i
are unique up to a common
a¢ ne transformations, and the composite functions  
^
U
i
are unique up to a common a¢ ne
transformation.
The proof is in the appendix but a sketch follows. Given acceptance, weak independence
over outcome lotteries for the impartial observer implies independence over outcome lotteries
for each individual. Hence, theorem 2 implies that preferences admit a generalized utilitarian
representation. Weak independence over outcome lotteries also implies that the impartial observer
is indi¤erent between facing similar risks as person i or person j. Proposition 5 then tells us that

1
i
 
j
is a¢ ne on the relevant interval, U
ji
.
Our redistributive scope condition ensures that there exist two individuals, call them i
1
and
i
2
, such that for all individuals j either U
ji
1
or U
ji
2
is not trivial. Thus, loosely speaking, all the

j
-functions are a¢ ne transformations of one another. The proof then con structs a common -
function (applying appropriate a¢ ne transformations to each von Neumann-Morgenstern utility
function U
i
to form
^
U
i
). Without the redistributive scope condition we could s till construct a
representation with a common -function –a remark in the appendix gives an example –but we
would lose the tight uniqueness conditions.
Recall that when we assumed that the impartial observer was indi¤erent between identity and
outcome lotteries –that is, she ignored Diamond’s concerns –we were forced back to Harsanyi’s
utilitarianism. If we assume that the impartial observer is indi¤erent as to who should face similar
risks –that is, she ignores Pattanaik’s c once rns — we are not forced back to Harsanyi, but only to
a common -function. Nevertheless, once we introduce weak independence over outcome lotteries
directly on the impartial observer, it seems natural to ask what happens if we impose strong
independence; that is, if we treat identity and outcome lotteries symmetrically in terms of the
axioms.
20

Strong Independence over Outcome Lotteries (for the Impartial Observer). Suppose (z; `),
(z
0
; `
0
) 2 4(I)  4(X) are such that (z; `)  (z
0
; `
0
). Then for all
~
`,
~
`
0
2 4(X): (z;
~
`) %
(z
0
;
~
`
0
) if and only if (z; 
~
` + (1  a) `) % (z
0
; 
~
`
0
+ (1  a) `
0
) for all  in (0; 1].
This axiom is the symmetric analog of our independence over identity lotteries for the impartial
observer reversing the roles of identity lotteries and outcome lotteries. It is stronger than weak
independence over outcome lotteries in that it allows the (…xed) identity lotteries z and z
0
to be
non-degenerate.
One might conjecture that treating identity and outcome lotteries symmetrically in terms of
the axioms would force us back to Harsanyi: more precisely, given the acceptance principle, if we
symmetrically impose absence of unanimity and redistributive scope, and we symmetrically impose
(strong) ind ependence both over outcome lotteries and over identity lotteries on the impartial
observer then we would again obtain Harsanyi’s utilitarianism. After all, we know from lemma
1 that abse nce of unanimity and independence over identity lotteries gives us a representation
that is a¢ ne in identity lotteries. Symmetrically, redistributive scope and (strong) independence
over outcome lotteries give us a representation that is a¢ ne in outcome lotteries. This suggests
that having both sets of properties gives us a represe ntation that is a¢ ne in both identity and
outcome lotteries; i.e., utilitarianism. But there is a ‡aw in this argument: the representation
that is a¢ ne in identity lotteries need not be the same representation as that which is a¢ ne in
outcome lotteries. The following example illustrates what can go wrong.
For the purpose of the example, let I = f1; 2g and X= fx
1
; x
2
g. To simplify notation, for each
z 2 4(I), let q = z
2
; and for each ` 2 4(X) let p := `(x
2
). With slight abuse of notation, we
will write (q; p) % (q
0
; p
0
) for (z; `) % (z
0
; `
0
), and write V (q; p) for V (z; `).
Example 1 Let agent 1’s preferences be given by U
1
(p) = (1  2p), and let agent 2’s preferences
be given by U
2
(p) = (2p  1). Let the impartial observer’s preferences be given by V (q; p) :=
(1  q)  [U
1
(p)] + q [U
2
(p)], where the (common) -function is given by:
 [u] =

u
k
for u  0
(u)
k
for u < 0
, for some k > 0
21

These preferen ces are not utilitarian unles s k = 1. Nevertheless, it is clear that absence
of unanimity and redistributive scope both apply in this example. And, by proposition 6 (the
common  result), this impartial observer satis…es acceptance and independence over identity
lotteries. It remains to show that she also satis…es strong independence over outcome lotteries.
Consider the inverse function 
1
(u) = u
1=k
for u  0 and 
1
(u) = (u)
1=k
for u < 0.
This is a strictly increasing function. Therefore, the function 
1
[V (; )] represents the same
preferences as V (; ). Some simple algebra (provided in the appendix) shows that we can write

1
[V (q; p)] = (1  p) 
1
[(1  2q)] + p
1
[(2q  1)] :
This alternative representation is symmetric to the original representation V (; ) with the p’s and
q’s reversed and 
1
replacing . Since the alternative representation is a¢ ne in p, preferences
must satisfy strong independence over outcome lotteries.
Notice that the underlying preferences in this example resemble those in the example in the
introduction. We could think of x
1
as the outcome in which person 1 gets some indivisible good,
and x
2
as the outcome in which person 2 gets it. As advocated by Harsanyi, if the impartial
observer thinks she is equally likely to be either person, she is indi¤erent as to whom is given the
good. And if the impartial obse rver thinks the good is equally likely to be given to either person,
she is indi¤erent as to whom she is when she faces that risk. Nevertheless, her preferences are not
utilitarian for more complicated randomizations.
Although this example is special, some aspects of it are quite general. For any generalized
utilitarian representation V with common , the associated function 
1
 V will always be an
alternative representation of the same preferences. Moreover, lemma 11 in the appendix shows
that, if we start from the conditions of proposition 6 (the common  result) but replace weak with
strong independence over outcome lotteries then this alternative representation 
1
V is always
a¢ ne in outcome lotteries.
Our alternative representation takes the form:

1
[V (z; `)] := 
1
I
X
i=1
z
i

i
h
^
U
i
(`)
i
!
22

We can think of this as a “generalized mean” of the distribution of utilities induced by (z; `).
21
In particular, given each individual’s (expected) u tility function
^
U
i
, we can again think of each
outcome lottery ` as inducing a vector of individual (expected) utilities

^
U
1
(`) ; : : : ;
^
U
I
(`)

. Thus,
each product lottery (z; `) induces a distribution of individual utilities where each
^
U
i
(`) is assigned
probability z
i
. Since 
1
V and each
^
U
i
function are a¢ ne on outcome lotteries, the generalized
mean must be a¢ ne on the induced set of utility vectors. If the generalized mean were a¢ ne
over all utility vectors (or over a su¢ ciently rich set) then it would have to be the ordinary
arithmetic mean; that is,  would have to be a¢ ne and our representation would reduce to
Harsanyi’s utilitarianism.
22
But sinc e, in our pro du ct-lottery setting, every individual f aces the
same outcome lottery, the set of utility vectors that are induced by outcome lotteries nee d not be
rich. In example 1, the induced utility vectors all lie in the line segment from (1; 1) to (1; 1).
Once again, we need a rich set of underlying outcomes and/or preferences to induce a su¢ ciently
rich set of utility lotteries to be forced to utilitarianism.
The example shows that the two richness c onditions we have used so far, absence of unanimity
and redistributive scope, are not enough. But they are close. The -function in the example is
a homogenous function. Again, this is general: lemma 12 in the appendix shows that if start
from the conditions of proposition 6 but replace weak with strong independence over outcome
lotteries then the common -function is always homogenous; that is homogeneity is necessary.
23
But homogeneity is not su¢ cient except in very special cases. Notice in the example that the
point of in‡ection of the homogenous -function (the “zero”) occurs exactly at outcome lottery
p = 1=2 where the impartial observer is indi¤erent over which identity lottery she faces. This is
a very “knife-edge” property, and it can be ruled out in a numb er of ways. The following extra
richness condition su¢ ces.
Three-Player Richness For all outcomes x; y in X, and all  in [0; 1], there exist individuals i
21
See, for exampl e, Hardy, Littlewood and Polya (1934) ch.3.
22
Strictly spea king, an a¢ ne transformation of the arithmetic mean. For a proof, see Hardy, Littlewood and
Polya (1934) p.86.
23
Strictly speaking, it must be an a ¢ ne transformation of a ho mogenou s function since our representation is
only unique up to a¢ ne transformat ions.
23

and j in I such that (i;  [x] + (1  ) [y])  (j ;  [x] + (1  ) [y]).
Given redistributive scope, three-player richness implies that there must be at least three individ-
uals. In words, it says that there is no outcome lottery involving just two outcomes at which the
impartial observer is indi¤erent over all th e possible identities she could assume. In example 1,
the condition was violated at p = 1=2, sinc e the impartial observer was indi¤erent between being
either person there. If we add a third person to the example, then the condition would be met
provided that either the outcome lottery at which the impartial observer is indi¤erent betwe en
being person 1 or person 3 or th e outcome lottery at which she is indi¤erent between being person
2 or person 3 is not exactly equal to 1=2.
24
With this extra condition in place (and hence troublesome examples like example 1 ruled out),
the symmetric richness conditions and symmetric independence axioms over identity and outcome
lotteries yield Harsanyi’s utilitarianism.
Theorem 7 (Utilitarianism II) Suppose that absence of unanimity, redistributive scope and
three-player richness all apply. Then the following are equivalent:
(a) The impartial observer satis…es the acceptance principle, independence over identity lotteries
and (strong) independence over outcome lotteries
(b) There exist a continuous function V : 4(I)  4(X) ! R that represents % and, for each
i in I, a function
^
U
i
: 4(X) ! R that is a von Neumann-Morgenstern expected-utility
representation of %
i
, such that for all (z; `) in 4(I) 4(X),
V (z; `) =
I
X
i=1
z
i
^
U
i
(`)
where, for each i,
^
U
i
(`) =
Z
X
^
U
i
(x) ` (dx). Moreover the functions
^
U
i
are unique up to
common a¢ ne transformation.
24
Notice that, if there are three or mo re possible out comes, the condition still o nly places rest rictions for lotterie s
invol ving just two. In particular, there still could be some lottery on the interi or of t he simplex where the impartial
observer is indi¤erent as to identity. The condition would, however, be violated if there were di visible and disposable
pri vate goods and (hence) some outcome that equa lized the welfare of all individuals. In that setting, however,
since the outcome set is itself very rich, we can anyway induce a su¢ ciently rich set of utility lotteries.
24

The proof is in the appendix. Example 1 shows that three-player richness is essential. Example
4 in appendix A shows that redistributive scope is also essential.
To summarize: if we start from the axioms that gave us generalized utilitarianism then it is easy
to accommodate Pattanaik’s concerns about di¤erent attitudes toward risk. Forcing the impartial
observer to be indi¤erent as to who faces risk, does not force us to Harsanyi’s utilitarianism
but only to a common -function. Such indi¤erence h owever, is equivalent to imposing a weak
form of independence over outcome lotteries directly on the impartial observer. If we go further
and assume that identity and outcome lotteries are equivalent in the sense that all axioms are
symmetric across the two types of lottery, then (provided the underlying problem is rich) we are
again forced to Harsanyi’s utilitarianism.
6 Assuming less and assuming more: related literature.
In this section, we …rst ask what happens to our representations if we assu me less. In particular,
we consider dropping our richness considerations altogether. Then we switch around and compare
our results to those in the literature that assume more. In particular, we show how Harsanyi’s
axioms (imposed on preferences over all joint distributions over identities and outcomes) imply
all of the axioms (imposed just on preferences over pro du ct lotteries) of each of our utilitarian
theorems.
Assuming less. If we are not worried about uniqueness, we can obtain a representation of
the form
P
I
i=1
z
i
V
i
(`) without imposing absence of unanimity. Recall our sketch proof of lemma
1. Absence of unanimity ensured there exists two outcome lotteries `
1
and `
2
such that for all
product lotteries (z; `) either (z; `) 

z
0
; `
1

for some z
0
, or (z; `)  (z
00
; `
2
) for some z
00
or both.
Moreover we can choose `
1
and `
2
such that the ‘intervals’of such indi¤erent lotteries ‘overlap’. If
we do not impose absence of unanimity then two complications arise. First we may require many
more than two ‘intervals’to cover the indi¤erence sets of the impartial obse rver. The main step
in the proof of theorem 8 is to show that we can always …nd a countable number of ‘intervals’
that cover these indi¤erence s ets, and to construct a representation using these intervals. The
second complication is that the intervals may not overlap. Without such overlapping, we do not
25

obtain uniqueness. Absence of unanimity is su¢ cient but not necessary to obtain uniqueness. The
precise condition is given in case one of the proof of theorem 8. This theorem provides the most
general representation result in the paper.
Theorem 8 The following are equivalent:
(a) The impartial observer’s preferences % satisfy the acceptance principle and independence
over identity lotteries.
(b) There exist a continuous function V : 4(I)  4(X) ! R and, for each i in I, a function
V
i
: 4(X) ! R, such that V represents %; for each i, V
i
represents %
i
; and for all (z; `) in
4(I) 4(X),
V (z; `) =
I
X
i=1
z
i
V
i
(`):
Assuming more. At a technical level, the papers closest to ours are Karni & Safra (2000),
Fishburn (1982, ch 7) and Safra & Weissengrin (2002). Karni & Safra p roduce a representation
similar to lemma 1. The key di¤erence is that their axioms apply to the full set of joint distributions
so they can apply recursive arguments. Both Fishburn and Safra & Weissengrin work with product
lottery spaces like ours. Fishburn provides axioms on product spaces of mixture sets to obtain
multi-linear representations. His context was games in which opponents’ mixed strategies are
independent. Safra & Weissengrin adapt this approach to derive Harsanyi’s utilitarianism in a
setting where the impartial observer faces only product lotteries. The key di¤erence between their
result and our two utilitarianism theorems (theorem 4 and theorem 7) is that Safra & Weissengrin
directly impose independence on the impartial observer for all mixtures that are well de…ned in
the space of product lotteries. Implicitly, therefore, they not only impose both independence over
identity lotteries and strong independence over outcome lotteries, but also a th ird independence
axiom over hybrids of the other two:
Independence over Hybrid Lotteries (for the Impartial Observer). Suppose (z; `), (z
0
; `
0
) 2
4(I)4(X) are such that (z; `)  (z
0
; `
0
). Then for all 2 4(I) and all
~
`
0
24(X): (~z; `) %
26

(z
0
;
~
`
0
) if and only if (~z + (1   ) z; `) % (z
0
; 
~
`
0
+ (1  a) `
0
) for all  in (0; 1].
This axiom is similar to the other independence axioms for the impartial observer except that
the lotteries being mixed on the left are identity lotteries (holding outcome lotteries …xed ), while
the lotteries being mixed on the right are outcome lotteries (holding identity lotteries …xed).
This hybrid independence axiom is quite strong: in particular, it can be shown that it implies
indi¤erence between life chances and acc idents of birth. Thus, we can think of Safra & Weissengrin
as assuming the union of our axioms from theorem 4 and theorem 7, our two utilitarianism results.
The Safra & Weissengrin theorem helps explain how Harsanyi comes implicitly to assume both
indi¤erence between life chances and accidents of birth and indi¤erence over who should face
similar risks; and hence con‡ict with Diamond and Pattanaik. Recall that Harsanyi works with
the full set of joint distributions 4(I  X). He imposes independence directly on the impartial
observer for all mixtures de…ned on that space. If we then restrict these preferences from the larger
set 4(I X) to j ust the product lotteries 4(I)4(X) then independence applies to any mixture
that is still well de…ned. That is, all three of Safra & Weissengrin’s independence axioms apply.
But, as we have just argued, the third of the Safra & Weissengrin axioms (independence over
hybrid lotteries for the impartial observer) implies indi¤erence between life chances and accidents
of birth; in con‡ict with Diamond. And, as we argued in section 5, the second of the Safra
& Weissengrin axioms (strong independence over outcome lotteries for the impartial observer)
implies indi¤erence over who should face similar risks; in con‡ict with Pattanaik.
Notice that these con‡icts are not over the idea of independence per se. We can ass ume that
each individual satis…es independence over the lotteries he faces, namely outcome lotteries; that
the impartial observer respects these individual preferences over outcome lotteries; and that she
satis…es independence over the lottery she faces, namely identity lotteries, but this does not imply
Harsanyi’s utilitarianism: it only implies generalized utilitarianism.
7 Welfare Inequality and Risk
In this section, we introduce an explicit notion of comparable welfare, and use it to interpret
some of our representation results. Let w
i
: 4(X) ! R be agent i’s welfare function, and let
27

w : 4(I)  4(X) ! R be the impartial observer’s welfare function. These w
i
’s are functions of
life chances rather than …nal outcomes, so think of them as interim welfares. Similarly, we can
think of the impartial observer’s welfare function w as ex ante welfare. Let us assume that these
welfare functions guide choice. That is,
Congruence For each individual i in I and f or `; `
0
in 4(X), ` %
i
`
0
if and only if w
i
(`) 
w
i
(`
0
). For the impartial observer, for all (z; `) ; (z
0
; `
0
) in 4(I)  4(X), (z; `) % (z
0
; `
0
) if
and only if w (z; `)  w (z
0
; `
0
).
Following Weymark (1991), let us further assume that the impartial ob server adopts the welfare
of agent i when she puts herself in the shoes of agent i.
Principle of Welfare Identity For each individual i in I and for ` in 4(X), w
i
(`) = w (i; `).
For the remainder of this section, we assume that both of these axioms apply. As Weymark
(1991) notes, taken together, congruence and the principle of welfare identity imply acceptance.
Furthermore, they entail that for any pair of individuals i and j, and any pair of life-chances ` and
`
0
, the ranking between (i; `) and (j; `
0
) is completely determined by the ranking b e tween w
i
(`)
and w
j
(`
0
). That is, the welfare functions (w
1
(:) ; : : : ; w
I
(:)) are at least ordinally measurable and
fully comparable.
To relate these welfare measures to the generalized utilitarian representation obtained in the-
orem 2, de…ne for each ind ividual i, the function g
i
: R ! R that maps individual i’s interim
welfare to his von Neumann-Morgenstern expected utility. That is, for each individual i, and for
all ` in 4(X):
g
i
(w
i
(`)) = U
i
(`)
Similarly, let g : R ! R denote the mapping from the impartial observer’s ex ante welfare to her
von Neumann-Morgenstern utility. Thus, if the conditions of the orem 2 apply, then for all (z; `)
in 4(I)  4(X):
g (w (z; `)) =
I
X
i=1
z
i

i
 U
i
(`)
28

Given this, we can now re-interpret th e functions 
i
in terms of welfare. Applying the principle
of welfare identity, we get 
i
 U
i
(`) = g [w(i; `)] = g [w
i
(`)] = g

g
1
i
(U
i
(`))

. Thus, for each
individual i, the function 
i
is given by the function g  g
1
i
.
We can then re-express the generalized utilitarian social welfare function from theorem 2 in
terms of our g-functions and welfare to yield
w (z; `) = g
1
I
X
i=1
z
i
g
i
(w
i
(`))
!
:
With only ordinally measurable welfares, the shape and hence degree of curvature of g can
vary as one considers di¤erent (common) monotonic transformations of (w
1
(:) ; : : : ; w
I
(:)). In this
case (following Sen (1977)), we can no more interpret the shape of g than can Harsanyi interpret
his social welfare function as being linear in welfare. We know from theorem 2, however, that 
i
(which is equal to g  g
1
i
) is invariant to any common increasing transformations of the welfare
functions (w
1
() ; : : : ; w
I
()). That is, if we take ( ^w
1
() ; : : : ; ^w
I
()), where ^w
i
= h  w
i
and h : R
! R is an increasing function, then the functions that map the transf ormed welfares to the von
Neumann-Morgenstern utilities are now given by ^g
i
= g
i
 h
1
and ^g = g  h
1
And so,
^

i

^g ^g
1
i
= g h
1
h g
1
i
= g g
1
i
. This provides some intuition why the 
i
-functions are unique
up to positive a¢ ne transformations.
Suppose we go further and however assume that welfares are cardinally measurable. In this
case, we can give more interpretation to the g-functions. In particular, for a …xed identity lottery z
(for example, a uniform lottery), we can associate the representation w (z; `) above with a Bergson-
Samuelson social welfare fun ction that maps the induced vectors of individual interim welfares,
(w
1
(`) ; : : : ; w
I
(`)), to ‘social welfare’(that is, the impartial observer’s ex ante welfare, before she
knows whom she will become).
25
In such an interpretation, since we have imposed cardinal measurability, g is uniquely-de…ned
up to positive a¢ ne transformations. In this case, the degree of concavity of g may be interpreted
as measuring the degree of the impartial observer’s aversion to interim welfare inequality or h er
attitudes toward the risks embodied in accidents of birth.
25
Analogous to the discussion in sections 4 and 5, we need a rich set of underlying preferences to induce a rich
set of individual interim welfares.
29

This is exactly what we should expect. Had we started from the viewpoint of a Bergson-
Samuelson social welfare function, then we would immediately have interpreted Diamond’s notion
of fairness as aversion to interim welfare inequality. This in turn would have led us to a social
welfare function which is concave in individual interim utilities. Instead we started from the
viewpoint of representing an impartial observe r’s preferences, and replaced Harsanyi’s implicit
assumption that the impartial observer is indi¤erent between life chances and accidents of birth
with Diamond’s preference for life chances. If we now impose cardinally measurable welfare, we
arrive at the same point.
An explicit notion of welfare also helps us interpret the di¤erent attitudes to risk in Proposition
5. Recall that the impartial observer is more willing to take on similar risks in the identity of
person i than that of person j if and only if 
i
is a concave transformation of 
j
. If welfare
is cardinally measurable then 
i
is a concave transformation of 
j
if and only if g
i
is a convex
transformation of g
j
. This corresponds to our usual notion of income risk aversion except that
instead of being risk averse over income, our individuals are risk averse over welfares: individual i
is less welfare risk averse than individual j. In other words, each function g
i
captures individual
i’s attitudes toward the welfare risk embo d ied in her life chances. In this setting, imposing either
that the impartial observer is indi¤erent between life chances and accidents of birth or imposing
directly that she respect even weak independence over outcome lotteries forces all people to have
the same welfare risk aversion.
Once we allow our social welfare function to take into account that di¤erent agents may have
di¤erent degrees of welfare risk aversion, we may in fact no longer wish to accept Diamond’s
fairness axiom. There may be case s where the impartial observer may actually prefer accidents
of birth to life chances. Suppose society contains people who are extremely welfare risk averse,
but suppose that our impartial observer is only mildly (interim) welfare inequality averse. In
this case, the functions g
i
might be more concave than the function g, and hence the functions

i
= g  g
1
i
would be convex. The impartial observer, anticipating the discomfort that real-life
uncertainty would cause real people, prefers to absorb the risk in the imaginary world of her
thought experiment.
30

Although the case of convex 
i
functions may seem odd, it corresponds to an argument some-
times used by conservatives to defe nd caste-like societies. Prefe rring accidents of birth to life
chances corresponds to preferring risks that resolve early. In a di¤erent context, Grant, Kajii &
Polak (1998) argue that a preference for early resolution corresponds to an intrinsic preference for
information: anxious agents may prefer to know their fate soon. In the context of the impartial
observer, if individuals are highly risk averse over their welfares, they may prefer for uncertainty
to have been resolved by the time they are born. They might prefer “to know their place”.
Appendix A: Examples
For each of the following examples, let I = f1; 2g and X= fx
1
; x
2
g. To simplify notation, for each
z 2 4(I), let q = z
2
; and for each ` 2 4(X) let p := `(x
2
). Then, with slight ab use of notation,
we write (q; p) % (q
0
; p
0
) for (z; `) % (z
0
; `
0
), and write V (q; p) for V (z; `).
Example 1 is introduced and discussed in the text. It shows that absence of unanimity and
redistributive scope can apply and the impartial ob server can satisfy acceptance and (strong)
independence over both identity lotteries and outcome lotteries but that the impartial observer
need not be utilitarian in Harsanyi’s sense; in particular, the -function need not be a¢ ne. That
is, three-person richness is essential for theorem 7 (Utilitarianism II).
Here we just complete the argument to show we can write:

1
[V (q; p)] = (1  p) 
1
[(1  2q)] + p
1
[(2q  1)] .
31

To see this, it is instructive to rewrite V (q; p) as follows:
V (q; p) =
8
>
>
<
>
>
:
(1  2q) (1  2p)
k
for p < 1=2
(2q  1) (2p  1)
k
for p > 1=2
=
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(1  2q) (1  2p)
k
for q < 1 =2, p < 1=2 (and V (q; p) > 0)
(2q  1) (1  2p)
k
for q > 1 =2, p < 1=2 (and V (q; p) < 0)
0 for (2q  1) (2p  1) = 0
(1  2q) (2p  1)
k
for q < 1 =2, p > 1=2 (and V (q; p) < 0)
(2q  1) (2p  1)
k
for q > 1 =2, p > 1=2 (and V (q; p) > 0)
.
Hence,

1
 V (q; p) =
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
(1  2q)
1=k
(1  2p) for q < 1=2, p < 1=2
(2q  1)
1=k
(1  2p) for q > 1=2, p < 1=2
0 for (2q  1) (2p  1) = 0
(1  2q)
1=k
(2p  1) for q < 1=2, p > 1=2
(2q  1)
1=k
(2p  1) for q > 1=2, p > 1=2
=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
(1  p)
h
(1  2q)
1=k
i
+ p
h
[(2q  1)]
1=k
i
for q < 1 =2
0 for q = 1=2
(1  p)
h
[(1  2q)]
1=k
i
+ p
h
(2q  1)
1=k
i
for q > 1 =2
= (1  p) 
1
[(1  2q)] + p
1
[(2q  1)]
which equals (1  p) 
1
[(1  2q)] + p
1
[(2q  1)] as desired. 
Example 2 shows that preferences can satisfy the acceptance principle, inde pendence (over
outcome lotteries) for individuals, and conditional independence over identity lotteries for the
impartial observer (as de…ned in footnote 13 but not satisfy the (unconditional) independence
axiom over identity lotteries.
26
26
Karni & Sa fra (2000, p.324) provide an example of preferences de…ned on 4 (I)  X that satisfy the analog
of conditional independence but not the analog of unconditional independence. This example extends the idea to
4 (I)  4 (X) :
32

Example 2 Let agent 1’s preferences be given by U
1
(p) = (2p  1) =4, and let agent 2’s pref-
erences be given by U
2
(p) = 3 (2p  1) =4. Notice that these individual preferences satisfy inde-
pendence. Let the impartial observer’s preferences be given by V (q; p) = (2p  1)

q
2
 1=4

.
By construction, these preferences satisfy the acceptance principle. To show that they satisfy
conditional independence over identity lotteries, notice that for each …xed p, the function V (q; p)
is monotone in q. If p = 1=2, then the impartial observer is indi¤erent over all q and conditional
independence follows trivially. If p > 1=2, then (~q; p) % (~q
0
; p) if and only if ~q  ~q
0
. Thus, for all
 2 (0; 1], (~q + (1  ) q; p) % (~q
0
+ (1  ) q; p) if and only if ~q  ~q
0
. The case f or p < 1=2 is
similar.
To show that these preferences violate (unconditional) independence over identity lotteries,
let p := 0 and let p
0
:= 1. Let q = q
0
:= 1=2 so that V (q; p) = V (q
0
; p
0
) = 0. Let ~q := 0 and
let ~q
0
= 1=
p
2 so that V (~q; p) = V (~q
0
; p
0
) = 1=4. Let  := 1=2. Then V (~q + (1  ) q; p) =


(1=4)
2
 1=4

= 3=16. But V (~q
0
+ (1  ) q
0
; p
0
) =

1=4 + 1=

2
p
2

2
1=4 = (1=16 + 1=8 +
1=

4
p
2

 1=4) =

3=16 +

1 
p
2

=

4
p
2

< 3=16, violating (unconditional) independence. 
Example 3 shows that the impartial observer’s preferences can satisfy all the conditions of
proposition 3 (the concavity result) except absence of unanimity and yet the functions 
i
need
not be concave. That is, absence of unanimity is essential.
Example 3 Let the individual’s preferences be given by U
1
(p) = U
2
(p) = p, and let the impartial
observer’s preferences be given by V (q; p) := (1  q) 
1
[U
1
(p)] + q
2
[U
2
(p)] where

1
(u) :=

1=4 + u=2 for u  1=2
u for u > 1=2

2
(u) :=

u for u  1=2
2u  1=2 for u > 1=2
Since U
1
= U
2
, both individuals have the same ranking over outcome lotteries and so the
impartial observer’s preferenc es violate absence of unanimity. Clearly, the functions 
1
(:) and

2
(:) are not concave. To see that the impartial observer satis…es preference for life chances,
without loss of generality let p  p
0
and notice that (q; p
0
)  (q
0
; p) implies either p  p
0
 1=2
or p
0
 p  1=2. But in either case, the functions 
1
and 
2
are concave (in f act, a¢ ne) on the
domain [p; p
0
] and hence V (q + (1  ) q
0
; p)  (in fact, =) V (q
0
; p + (1  ) p
0
), as desired.
33

Notice that redistributive scope cannot replace absence of unanimity in this proposition. In-
deed, the preferences above satisfy redistributive scope. 
Example 4 shows that preferences can satisfy all the conditions of Theorem 7 except redistributive
scope yet the 
i
’s are not a¢ ne. That is, redistributive scope is essential.
Example 4 Let agent 1’s preferences be given by
^
U
1
(p) = 1 bp, and let agent 2’s preferences be
given by
^
U
2
(p) = 1+bp = 
^
U
1
(p), for some b 2 (0; 1). Fix k > 0, k 6= 1 and set  (u) := u
k
, for
u in [0; 1] and set  (u) = (u)
k
, for u in [1; 0). Suppose the impartial observer’s preference
relation % is represented by the following function:
V (q; p) = (1  q) 

^
U
1
(p)

+ q

^
U
2
(p)

= (1  2q) (1  bp)
k
The impartial observer’s preference relation, %, generated by V (q; p) clearly fails to satisfy
redistributive scope, since for all p in [0; 1),


^
U
1
(p)

 

^
U
2
(p)

= 2 (1  bp)
k
> 0.
That is, (1; `)  (2; `) for all `. For the same reason, three-player richness holds trivially: V is
decreasing in q.
Absence of unanimity holds, since
h
^
U
1
(p) 
^
U
1
(p
0
)
ih
^
U
2
(p) 
^
U
2
(p
0
)
i
= b
2
(p  p
0
)
2
< 0
for all p 6= p
0
.
To see that strong independence over outcome lotteries holds, notice that

1
 V (q; p) =
8
>
>
<
>
>
:
(1  2q)
1=k
(1  bp) if q  1=2
(2q  1)
1=k
(1 + bp) if q > 1=2
which for given q is a¢ ne in p. 
Appendix B: Proofs
Proof of Lemma 1. Since the represe ntation is a¢ ne in identity lotteries, it is immediate that the
represented preferences satisfy the axioms. We will show that the axioms imply the representation.
34

Let outcome lotteries `
1
; `
2
(not necessarily distinct) and identity lotteries z
1
; z
2
(not necessar-
ily distinct) be such that

z
1
; `
1

 (z
2
; `
2
) and such that

z
1
; `
1

% (z; `) % (z
2
; `
2
) for all product
lotteries (z; `). That is, the product lottery

z
1
; `
1

is weakly better than all other product lotter-
ies, and the product lottery (z
2
; `
2
) is weakly worse than all other product lotteries. And let the
identity lotteries z
1
and z
2
(not necessarily distinct) be such that

z
1
; `
1

%

z; `
1

%

z
1
; `
1

for
all product lotteries

z; `
1

, and

z
2
; `
2

% (z; `
2
) % (z
2
; `
2
) for all pro d uc t lotteries (z; `
2
). That
is, given outcome lottery `
1
, the identity lottery z
2
is (weakly) worse than all other identity lotter-
ies; and, given outcome lottery `
2
, the identity lottery z
2
is (weakly) better than all other identity
lotteries. Their existence of these special lotteries follows from continuity of %, non-emptyness of
; and the compactness of  (I)   (X). Moreover, by independence over identity lotteries, we
can take z
1
; z
1
; z
2
; and z
2
each to be a degenerate identity lottery. Let these be i
1
; i
1
; i
2
; and i
2
respectively.
The following lemma is helpful.
Lemma 9 Assume absence of unanimity applies and that the impartial observer satis…es accep-
tance and independence over identity lotteries. Let i
1
; i
1
; i
2
; i
2
; `
1
; and `
2
be de…ned as above. Then
(a) either

i
1
; `
1

 (i
2
; `
2
), or

i
2
; `
2



i
1
; `
1

, or

i
2
; `
2



i
1
; `
1

. And (b), for all product
lotteries (z; `), either

i
1
; `
1

% (z; `) %

i
1
; `
1

or

i
2
; `
2

% (z; `) % (i
2
; `
2
) or both.
Proof. (a) If `
1
= `
2
, then the …rst two cases both hold. Otherwise, suppose that the …rst two
cases do not hold; that is,

i
1
; `
1

 (i
2
; `
2
) and

i
1
; `
1



i
2
; `
2

. By the de…nition of i
1
, we know
that

i
2
; `
1

%

i
1
; `
1

, and hence

i
2
; `
1

 (i
2
; `
2
). Using absence of unanimity and acceptance ,
there must exist another individual ^{ 6= i
2
such that (^{; `
2
) 

^{; `
1

. Again by the de…nition of i
1
,
we know that

^{; `
1

%

i
1
; `
1

, and henc e (^{; `
2
) 

i
1
; `
1

. By the de…nition of i
2
, we know that

i
2
; `
2

% (^{; `
2
), and hence

i
2
; `
2



i
1
; `
1

, as desired. Part (b) follows immediately from (a).

Given continuity, an immediate consequence of the lemma is that there exists two outcome
lotteries `
1
and `
2
such that for all product lotteries (z; `) either (z; `) 

z
0
; `
1

for some z
0
, or
(z; `)  (z
00
; `
2
) for some z
00
or both. Moreover, we can choose the z
0
such that its support only
35

contains individuals i
1
and i
1
. And similarly for z
00
with respect to i
2
and i
2
.
The proof of lemma now proc eed s with two cases.
Case (1)
27
The easiest case to consider is where the lotteries `
1
and `
2
, de…ned above are equal.
In this case, for the individuals i
1
and i
1
de…ned in lemma 9,

i
1
; `
1



i
1
; `
1

, and

i
1
; `
1

%
(z; `) %

i
1
; `
1

, for all (z; `). Then, for each (z; `), let V (z; `) be de…ned by

V (z; `)

i
1

+ (1  V (z; `)) [i
1
] ; `
1

 (z; `) :
By continuity and independence over identity lotteries, such a V (z; `) exists and is unique.
To show that this representation is a¢ ne, notice that if

V (z; `)

i
1

+ (1  V (z; `)) [i
1
] ; `
1


(z; `) and

V (z
0
; `)

i
1

+ (1  V (z
0
; `)) [i
1
] ; `
1

 (z
0
; `) then independence over identity lotteries
implies ([V (z; `) + (1  ) V (z
0
; `)]

i
1

+[1V (z; `)(1  ) V (z
0
; `)] [i
1
] ; `
1
)  (z + (1  ) z
0
; `).
Hence V (z; `) + (1  ) V (z
0
; `) = V (z + (1  ) z
0
; `).
Since any identity lottery z in  (I) can be written as z =
P
i
z
i
[i], proceeding sequentially
on I, a¢ nity implies V (z; `) =
P
i
z
i
V (i; `). Finally, by acceptance, V (i; ) agrees with %
i
on
 (X) Hence, if we de…ne V
i
:  (X) ! R by V
i
(`) = V (i; `), then V
i
represents individual i’s
preferences. The uniqueness argument is standard: see for example, Karni & Safra (2000, p.321).
Case (2). Let outcome lotteries `
1
; `
2
be de…ned as above, and let individuals i
1
; i
1
; i
2
; i
2
be
de…ned as in lemma 9 and its proof. If

i
1
; `
1

 (i
2
; `
2
) then

i
1
; `
1

% (z; `) %

i
1
; `
1

for all
(z; `) and hence case (1) applies. Similarly, if

i
2
; `
2



i
1
; `
1

then

i
2
; `
2

% (z; `) % (i
2
; `
2
) for
all (z; `), and again case (1) applies (with `
2
in place of `
1
). Hence suppose that

i
1
; `
1



i
2
; `
2

and that

i
1
; `
1

 (i
2
; `
2
). Then, by lemma 9,

i
1
; `
1



i
2
; `
2



i
1
; `
1

 (i
2
; `
2
); that is, we
have two overlapping intervals that ‘span’the entire range of the impartial observer’s preferences.
Then, just as in case (1), we can construct an a¢ ne function V
1
(; ) to represent the impartial
observer’s preferences % restricted to those (z; `) such that

i
1
; `
1

% (z; `) %

i
1
; `
1

, and we can
construct an a¢ ne function V
2
(; ) to represent % restricted to those (z; `) such that

i
2
; `
2

%
27
This case is similar to case (1) in Safra & Weisengrin (2003, p.184). This case is also analogous to case (1) of
Karni & Safra (2000, p.320) except that, in their settin g, the analog of `
1
is a vector of ou tcome lotteries, with a
di¤erent outcome lottery for each agent. One implication of this is that, in their setting, if case (1) does not apply,
then there must exi st an agent i and two (vectors of) outcome lotteries `
0
and `
00
such that (i; `
0
) % (z; `) % (i; `
00
)
for all (z; `). This is not true her e.
36

(z; `) % (i
2
; `
2
). We can then app ly an a¢ ne re-normalization of either V
1
or V
2
such the (re-
normalized) representations agree on the ‘ove rlap’

i
2
; `
2

% (z; `) %

i
1
; `
1

. Since V
1
(; ) and
V
2
(; ) are a¢ ne, the re-normalized representation is a¢ ne, and induction on I (plus acceptance)
gives us V (z; `) =
P
i
z
i
V
i
(`) as before. Again, uniquenes s f ollows from stand ard arguments. 
Proof of Theorem 2 (Generalized Utilitarianism): In the text.
Proof of Proposition 3 (Concavity) The proof of su¢ ciency is in the text. For necessity,
again de…ne V
i
(`) := 
i
 U
i
(`). We need to show that for all i and all `, `
0
2 4(X), V
i
(` +
(1  ) `
0
)  V
i
(`) + (1  ) V
i
(`
0
) for all  in [0; 1]. By acceptance, it is enough to show that
V (i; ` + (1  ) `
0
) V (i; `) + (1  ) V (i; `
0
) for all  in (0; 1). So let % exhibit preference
for life chances, …x i and consider `, `
0
2 4(X). Assume …rst that ` 
i
`
0
By acceptance,
V (i; `) = V (i; `
0
). Hence, by preference for life chances,
V (i; ` + (1  ) `
0
)
 V ( [i] + (1  ) [i] ; `) (by preference for life chances)
= V (i; `)
= V (i; `) + (1  ) V (i; `
0
) (since V (i; `) = V (i; `
0
)),
as desired.
Assume henceforth that ` 
i
`
0
(and, by acceptance, V (i; `) > V (i; `
0
)). By ab sen ce of una-
nimity, there must exist a j su ch that V (j; `) < V (j; `
0
). There are three cases to consider.
(a) If V (i; `
0
)  V (j; `) then, by the representation in lemma 1, there exists z
0
(of the form
 [i] + (1  ) [j]) such that V (z
0
; `) = V (i; `
0
). Thus, for all  in (0; 1),
V (i; ` + (1  ) `
0
)
 V ( [i] + (1  ) z
0
; `) (by preference for life chances)
= V (i; `) + (1  ) V (z
0
; `)
= V (i; `) + (1  ) V (i; `
0
) (since V (z
0
; `) = V (i; `
0
)),
as desired.
37

Assume henceforth that V (j; `) > V (i; `
0
) (which implies V (j; `
0
) > V (i; `
0
)).
(b) If V (j; `
0
)  V (i; `) then, by the representation in lemma 1, there exists z (of the form
 [i] + (1  ) [j]) such that V (z; `
0
) = V (i; `). Thus, f or all  in (0; 1),
V (i; `
0
+ (1  ) `)
 V ( [i] + (1  ) z; `
0
) (by preference for life chances)
= V (i; `
0
) + (1   ) V (z; `
0
)
= V (i; `
0
) + (1   ) V (i; `) (since V (z; `
0
) = V (i; `) ),
as desired.
(c) Finally, let V (i; `) > V (j; `
0
) > V (j; `) > V (i; `
0
). By the continuity of V , there exist 
0
; 
0
in (0; 1) such that 
0
> 
0
, and such that V (i; 
0
` +

1  
0

`
0
) = V (j; `
0
) and V (i; 
0
` +
(1  
0
) `
0
) = V (j; `). Denote `
0
= 
0
` + (1  
0
) `
0
. Then, similarly to part (a),
V
i
(` + (1  ) `
0
)  V
i
(`) + (1  ) V
i
(`
0
)
for all  2 (0; 1). Next, denote `
0
= 
0
` +

1  
0

`
0
. Then, similarly to part (b),
V
i
(`
0
+ (1  ) `
0
)  V
i
(`
0
) + (1  ) V
i

`
0

for all  2 (0; 1). Therefore, restricted to the line segment [`
0
; `], the graph of V
i
lies weakly
above the line connecting (`
0
; V
i
(`
0
)) and

`
0
; V
i

`
0

(as does the point (`
0
; V
i
(`
0
))) and weakly
above the line connecting (`
0
; V
i
(`
0
)) and (`; V
i
(`)) (as does the point

`
0
; V
i

`
0

). Hence,
V
i
(` + (1  ) `
0
)  V
i
(`) + (1  ) V
i
(`
0
) for all  2 (0; 1). 
Proof of Theorem 4 (Utilitarianism I): In the text.
Proof of Proposition 5 (Di¤erent R isk Attitudes). First, notice that if U
ji
is not empty
then it is a closed interval. If U
ji
has an empty interior then the proposition holds trivially true.
Therefore, assume that U
ji
= [u
ji
; u
ji
] where u
ji
< u
ji
.
To prove that 
1
i
 
j
convex is su¢ cient, …x `; `
0
;
~
` and
~
`
0
such that V (i; `) = V (j; `
0
) and
V

i;
~
`

= V

j;
~
`
0

. We want to show that V

i; 
~
` + (1   ) `

 V

j; 
~
`
0
+ (1  ) `
0

. By
38

construction, both U
j
(`
0
) and U
j

~
`
0

lie in U
ji
. Moreover, we have U
i
(`) = 
1
i

j
[U
j
(`
0
)] and
U
i

~
`

= 
1
i
 
j
h
U
j

~
`
0
i
Applying the representation we obtain,
V

i; 
~
` + (1   ) `

= 
i
h
U
i


~
` + (1   ) `
i
(by the representation)
= 
i
h
U
i

~
`

+ (1  ) U
i
(`)
i
(by a¢ nity of U
i
)
= 
i
h

1
i
 
j
h
U
j

~
`
0
i
+ (1  ) 
1
i
 
j
[U
j
(`
0
)]
i
(by the representation)
 
i
h

1
i
 
j
h
U
j

~
`
0

+ (1  ) U
j
(`
0
)
ii
(by convexity of 
1
i
 
j
)
= 
j
h
U
j


~
`
0
+ (1  ) `
0
i
(by a¢ nity of U
j
)
= V

j; 
~
`
0
+ (1  ) `
0

(by the representation)

To prove that 
1
i
 
j
convex is necessary, …x v,w in U
ji
. By the de…nition of U
ji
, there exists
outcome lotteries `; `
0
2 4(X) such that U
j
(`
0
) = v and U
i
(`) = 
1
i
 
j
(v); and there exists
outcome lotteries
~
`;
~
`
0
2 4(X) such that U
j

~
`
0

= w and U
i

~
`

= 
1
i

j
(w). By construction,
we have V (i; `) = V (j; `
0
) and V

i;
~
`

= V

j;
~
`
0

. Therefore, for all  in (0; 1)

i
h
U
i


~
` + (1   ) `
i
 
j
h
U
j


~
`
0
+ (1  ) `
0
i
)
U
i

~
`

+ (1  ) U
i
(`)  
1
i
 
j
h
U
j

~
`
0

+ (1  ) U
j
(`
0
)
i
)

1
i
 
j
(w) + (1  ) 
1
i
 
j
(v)  
1
i
 
j
(w + (1  ) v)
Since v and w were arbitrarily, the last inequality corresponds to the convexity of 
1
i

j
on U
ji
.

Proof of Proposition 6 (Common -function) . To show that a common  function is
su¢ cient, it is enough to show that the representation V (; ) (as de…ned in the proposition)
satis…es weak independence over outcome lotteries. If (i; `)  (j; `
0
) then 
h
^
U
i
(`)
i
= 
h
^
U
j
(`
0
)
i
,
hence
^
U
i
(`) =
^
U
j
(`
0
). Similarly, (i;
~
`) % (j;
~
`
0
) implies
^
U
i

~
`


^
U
j

~
`
0

. By a¢ nity of
^
U
i
and
^
U
j
,
39

we have
V (i; 
~
` + (1   ) `) = 
h

^
U
i

~
`

+ (1  )
^
U
i
(`)
i
and
V (j; 
~
`
0
+ (1  ) `
0
) = 
h

^
U
j

~
`
0

+ (1  )
^
U
j
(`
0
)
i
:
Hence (i; 
~
` + (1   ) `) % (j; 
~
`
0
+ (1  ) `
0
). 
To show that weak independence implies a common , …rst notice that weak independence over
outcome lotteries and the acceptance principle imply that all individuals satisfy independence over
outcome lotteries. Thus the conditions of theorem 2 are met. Let V (z; `) =
P
I
i=1
z
i

i
[U
i
(`)] be
the corresponding represe ntation.
Suppose individuals i and j are such that the interval U
ji
as de…ned in Proposition 5 has a
non-empty interior. Clearly, the corresponding interval U
ij
must also have non-empty interior.
We argued in the text that weak independence implies that the impartial observer is indi¤erent
between facing similar risks as person i or person j. By Proposition 5, it follows that 
1
i
 
j
is
a¢ ne on U
ji
. Since U
ji
has a non-empty interior, 
1
i
 
j
has a unique extens ion on R. De…ne
a new von Neuman n-M orgenstern utility fu nc tion
^
U
j
for agent j by the a¢ ne transformation,
^
U
j
(`) := 
1
i
 
j
[U
j
(`)] for all ` in 4(X). De…ne a new transformation function
^

j
for agent
j by setting
^

j

^
U
j
(`)

:= 
j
(U
j
(`)). Thus, in particular, if 
j
[U
j
(`
0
)] = 
i
[U
i
(`)] (and hence
U
i
(`) 2 U
ij
), then
^
U
j
(`
0
) = U
i
(`), and hence
^

j
(u) = 
i
(u) for all u in U
ij
. By construction, the
new s ocial welfare function with U
j
replaced by
^
U
j
and 
j
replaced by
^

j
still has a generalized
utilitarian form and represents the same preferences. With slight abuse of notation we can write
^

j
= 
i
, even if this extends the domain of 
i
.
To complete the proof, it is su¢ cient to show that any two individuals (call them j
1
and j
N
)
can be ‘conne cted’by a sequence of ‘intermediary’individuals (call them j
2
through j
N1
) such
that U
j
n
j
n+1
has non-empty interior for all n = 1; : : : ; N  1. This is where we use our second
richness condition, redistributive scope. In fact, we never need more than two such intermediaries.
As in lemma 9, let outcome lotteries `
1
; `
2
(not necessarily d istinct) and identity lotteries z
1
; z
2
(not necessarily d istinct) be such that

z
1
; `
1

 (z
2
; `
2
) and such that

z
1
; `
1

% (z; `) % (z
2
; `
2
)
for all identity-outcome lotteries (z; `). That is, the product lottery

z
1
; `
1

is weakly better than
40

all other product lotteries, and the product lottery (z
2
; `
2
) is weakly worse than all other product
lotteries. And (symmetric to lemma 9), let `
1
and `
2
(not necessarily distinct) be such that

z
1
; `
1

%

z
1
; `

%

z
1
; `
1

and

z
2
; `
2

% (z
2
; `) % (z
2
; `
2
) for all outcome lotteries `. That is,
given identity lottery z
1
, th e outcome lottery `
1
is (weakly) worse than all other outcome lotteries;
and, given identity lottery z
2
, the outcome lottery `
2
is (weakly) better than all other outcome
lotteries. The existence of these special lotteries follows from continuity of %, non-emptyness of
; and the compactness of  (I)   (X).
By independence over identity lotteries, we can take z
1
and z
2
each to be a degenerate identity
lottery. Let these be i
1
and i
2
respectively. But then, by weak independence over outcome lotteries,
we can take `
1
; `
1
; `
2
; and `
2
each to be a degenerate identity lottery. Let these be x
1
; x
1
; x
2
; and
x
2
respectively.
The following lemma is symmetric to lemma 9. The proof is essentially the same with redis-
tributive scope playing the role of absence of unanimity.
Lemma 10 Assume redistributive scope holds and that the impartial observer satis…es indepen-
dence over identity lotteries and weak independence over outcome lotteries. Let i
1
; i
1
; x
1
; x
1
; x
2
;
and x
2
be de…ned as above. Then (a) either

i
1
; x
1

 (i
2
; x
2
), or

i
2
; x
2



i
1
; x
1

, or

i
2
; x
2



i
1
; x
1

. And (b), for all product lotteries (z; `), either

i
1
; x
1

% (z; `) %

i
1
; x
1

or

i
2
; x
2

% (z; `) % (i
2
; x
2
) or both.
Given acce ptance and the fact that individual preferences are not degenerate,

i
1
; x
1



i
1
; x
1

and

i
2
; x
2

 (i
2
; x
2
). Hence an immediate consequence of lemma 10 is that U
i
1
i
1
has non-empty
interior and that, for all individuals j in I, either U
ji
1
has non empty interior or U
ji
2
has non
empty interior (or both). Thus all individuals are con nec ted as desired.
The uniqueness of  U
i
up to common a¢ ne transformations follows from Lemma 1. For the
uniqueness of the
^
U
i
functions, notice that (i; `)  (j; `
0
) implies
^
U
i
(`) =
^
U
j
(`
0
), and that (by the
redistributive scope), for each i there exists a j such that
b
U
ji
has a non-empty interior. 
Remark. If we drop redistributive scope, we can still c onstruct the representation but we would
lose the uniqueness result. For example, consider a two person society in which (i; `) % (j; `
0
)
41

for all `; `
0
in 4(X). In this case, let
^
U
i
be an a¢ ne transformation of U
i
such that
^
U
i
(`) >
U
j
(`
0
) if (i; `)  (j; `
0
) and such that
^
U
i
(`) = U
j
(`
0
) if (i; `)  (j; `
0
). And let
^

i
be such that
^

i

^
U
i
 
i
 U
i
. Then simply set  :=
^

i
on range of
^
U
i
and  := 
j
on the range of U
j
. These
ranges have at most one point in common.
Proof of Theorem 7 (Utilitarianism II). It is clear that (b) implies (a). We will show
(a))(b). Clearly, strong indep e nde nce implies weak independence over outcome lotteries, hence
proposition 6 applies. Let V (z; `) =
P
I
i=1
z
i

h
^
U
i
(`)
i
be as d e…ned there. It is enough to show
that the common -function is a¢ ne. Since the proof is long, we will break it into 6 steps, and
we will signpost some parts.
Step 1 of the proof consists of the following lemma showing that the function 
1
 V is a¢ ne
on 4(X).
Lemma 11 Suppose that absence of unanimity and redistributive scope both apply, and that the
impartial observer satis…es the acceptance principle, independence over identity lotteries and strong
independence over outcome lotteries. Let V and  be de…ned as in proposition 6; that is, V (z; `) =
P
I
i=1
z
i

h
^
U
i
(`)
i
. Then for each z in 4(I) the function 
1
 V (z; ) : 4(X) ! R is a¢ ne.
Proof. Fix an identity lottery z and an individual i. Similar to the notation in proposition 5,
let
b
U
i
 R be the interval such that u 2
b
U
i
implies that there exists an ` such that
^
U
i
(`) = u. We
will …rst show that 
1
V (z; ) is a¢ ne on the inverse image of
b
U
i
; that is, on the subset of outcome
lotteries f` 2 4(X) : 
1
 V (z; `) 2
b
U
i
g. If this inverse image is empty then a¢ nity is trivial.
Hence consider two outcome lotteries `; `
0
(not necessarily distinct) such that 
1
 V (z; `) 2
b
U
i
and 
1
 V (z; `
0
) 2
b
U
i
. By the de…nition of
b
U
i
, there exists two outcome lotteries

` and

`
0
such
that 
1
V (z; `) =
^
U
i


`

and 
1
V (z; `
0
) =
^
U
i


`
0

; that is, (z; `) 

i;

`

and (z; `
0
) 

i;

`
0

.
Applying independence over outcome lotteries, yields
(z; ` + (1  ) `
0
) 

i; 

` + (1   )

`
0

42

for all  in [0; 1] (hence 
1
 V (z; ` + (1  ) `
0
) 2
b
U
i
). Applying the representation yields:

1
 V (z; ` + (1  ) `
0
) =
^
U
i



` + (1   )

`
0

= 
^
U
i


`

+ (1  )
^
U
i


`
0

(by a¢ nity of
^
U
i
)
= 
1
 V (z; `) + (1  ) 
1
 V (z; `
0
) .
where the third line is by de…nition of

` and

`
0
. This argument holds for all i.
An immediate consequence of the lemma 10 is that there exist two individuals i
1
and i
2
such
that range


1
 V (z; )


b
U
i
1
[
b
U
i
2
and the inverse image of
b
U
i
1
[
b
U
i
2
is  (X). We know that

1
V (z; ) is a¢ ne on the inverse image of
b
U
i
1
and
b
U
i
2
. Moreover, by lemma 10, the interior of
b
U
i
1
i
2
(= interior
b
U
i
1
\
b
U
i
2
) is not empty. Hence 
1
 V (z; ) is a¢ ne on  (X). This argument
holds for all z. 
Let the individuals i
1
; i
1
; i
2
and i
2
(not necessarily distinct) and outcome lotteries `
1
and
`
2
(not necessarily distinct) be de…ned as in lemma 9. By strong independence over outcome
lotteries, we can take `
1
and `
2
each to b e a degenerate identity lottery. Let these be x
1
and x
2
respectively. Recall that, given our representation with a common -function, (i; `) 
(j; `
0
) implies
^
U
i
(`) =
^
U
j
(`
0
). Hence, by lemma 9(a) either
^
U
i
1
(x
1
) =
^
U
i
2
(x
2
), or
^
U
i
2
(x
2
) =
^
U
i
1
(x
1
), or
^
U
i
2
(x
2
) >
^
U
i
1
(x
1
). And, by lemma 9(b), for all product lotteries (z; `), we have

1

P
i
z
i

h
^
U
i
(`)
i
2
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
[
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
. We will …rst concentrate on
the interval
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
, but we will return to the interval
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
in step 5.
If
^
U
i
1

x
1

=
^
U
i
1

x
1

then a¢ nity is trivial.
28
Hence assume
^
U
i
1

x
1

<
^
U
i
1

x
1

.
De…ning ^x and u. Since

i
1
; x
1



i
1
; x
1

, by redistributive scope, there exists an outcome ^x
such that (i
1
; ^x) 

i
1
; ^x

. Consider the outcome lotteries `
[]
de…ned by `
[]
:=  [^x]+(1  )

x
1

.
By continuity of both
^
U
i
1
and
^
U
i
1
, there must exist an outcome lottery

` (:= `
[


]
) such th at

i
1
;

`



i
1
;

`

. Let u be given by
u := 
1

V

i
1
;

`

= 
1

V

i
1
;

`

(1)
The level of utility u is going to be important in the argument below. By the de…nition of x
1
, if
28
In this case, lemma 9 implies
^
U
i
2
(x
2
) =
^
U
i
1
(x
1
) and hence showing  is a¢ ne on [
^
U
i
2
(x
2
);
^
U
i
2
(x
2
)] would be
enough.
43

u does not lie in the interval
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
then u <
^
U
i
1

x
1

.
Step 2 of the proof is to show that, for all u
0
and u
00
2
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
and all  and  in
[0; 1],

1
[ [u
0
+ (1  ) u] + (1  )  [u
00
+ (1  ) u]]
= 
1
[ (u
0
) + (1  )  (u
00
)] + (1   ) u. (2)
To show this, …x u
0
and u
00
2
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
such that u
0
< u
00
. Denote by z
0
= 
0

i
1

+

1  
0

[i
1
] and z
00
= 
00

i
1

+

1  
00

[i
1
], the identity lotteries with support just on i
1
and i
1
for which 
1

V

z
0
; x
1

= u
0
, and 
1

V

z
00
; x
1

= u
00
. Also …x  and  in [0; 1], and de…ne
u

by:
u

:= 
1

V

z
0
+ (1  ) z
00
; x
1

(3)
By the de…nition of u and the fact that V is a¢ ne in id entity lotteries, we have
u = 
1

V

z
0
+ (1  ) z
00
;

`

(4)
By lemma 11, the function 
1
 V (z
0
+ (1  ) z
00
; ) is a¢ ne on  (X), hence combining ex-
pressions (3) and (4), we get

1

V

z
0
+ (1  ) z
00
; 

x
1

+ (1  )

`

= u

+ (1  ) u (5)
Our two a¢ nity properties allows us to expand the left side of expression . First, by the a¢ nity
of V

; 

x
1

+ (1  )

`

on  (I), we get
V

z
0
+ (1  ) z
00
; 

x
1

+ (1  )

`

= V

z
0
; 

x
1

+ (1  )

`

+ (1  ) V

z
00
; 

x
1

+ (1  )

`

= 


1
 V

z
0
; 

x
1

+ (1  )

`

+ (1  ) 


1
 V

z
00
; 

x
1

+ (1  )

`

(6)
Second, by the a¢ nity of 
1
 V (z
0
; ) and 
1
 V (z
00
; ) on  (X), we have


1
 V

z
0
; 

x
1

+ (1  )

`

=


1
 V

z
0
; x
1

+ (1  ) 
1
 V

z
0
;

`

and (7)


1
 V

z
00
; 

x
1

+ (1  )

`

=


1
 V

z
00
; x
1

+ (1  ) 
1
 V

z
00
;

`

. (8)
44

Substituting u
0
= 
1
 V

z
0
; x
1

, and u
00
= 
1
 V

z
00
; x
1

, and u = 
1
 V

z
0
;

`

= 
1

V

z
00
;

`

, expressions (7) and (8) become
[u
0
+ (1  ) u] and [u
00
+ (1  ) u]
respectively. Substituting these back into expression (6) and then substituting back into the left
side of expression (5), yields

1
[ [u
0
+ (1  ) u] + (1  )  [u
00
+ (1  ) u]] = u

+ (1  ) u. (9)
Using the de…nition of u

in expression (3) and the a¢ nity of V

; x
1

on  (I), we have
u

= 
1

V

z
0
+ (1  ) z
00
; x
1

= 
1

V

z
0
; x
1

+ (1  ) V

z
00
; x
1

= 
1




1

V

z
0
; x
1

+ (1  ) 


1

V

z
00
; x
1

= 
1
[ (u
0
) + (1  )  (u
00
)] (10)
where the last line follows from the de…nition of u
0
and u
00
. Substituting expression (10) back into
expression (9) yields expression (2) as desired. Our choice of u
0
; u
00
;  and  was arbitrary, so this
completes step 2. 
Re-normalization. Recall that functions
h
^
U
i
i
i2I
are unique only up to a common a¢ ne trans-
formation and that the composite functions
h
 
^
U
i
i
i2I
are also unique only up to a common
a¢ ne transformation. Hence we can re-normalize such that the utility level u = 0. With slight
abuse of notation, we will continue to use  and
h
^
U
i
i
i2I
to denote these re-normalized functions.
With this re-normalization, expression (2) becomes

1
[ (u
0
) + (1  )  (u
00
)] = 
1
[ (u
0
) + (1  )  (u
00
)] (11)
Since u
0
< u
00
were arbitrary, expression (11) holds (for all  and  in [0; 1]) for all utility pairs in
the (re-normalized) interval
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
. Recall that 0 need not lie in this interval.
Step 3 is to show that express ion (11) also holds (for all  and  in [0; 1]) for all utility pairs
u
0
< u
00
in
h
0;
^
U
i
1

x
1

i
even if 0 <
^
U
i
1

x
1

; that is, even if u does not lie in
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
.
45

To show this, we will establish that expression (11) holds in each interval in a sequence of intervals
I
0
; I
1
,. . . , with (i) I
0
:=
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
; (ii) I
n
\ I
n+1
an interval with positive length (and
so having a non-empty interior), for all n = 0; 1; : : :; and (iii)
[
1
n=0
I
n
=

0;
^
U
i
1

x
1

i
.
Fix an ~ 2 (0; 1), for which ~
^
U
i
1

x
1

>
^
U
i
1

x
1

(> ~
^
U
i
1

x
1

). Set I
n
:=
h
~
n
^
U
i
1

x
1

; ~
n
^
U
i
1

x
1

i
.
By construction I
n
\ I
n+1
is an interval with positive length and
[
1
n=0
I
n
=

0;
^
U
i
1

x
1

i
. To
see that I
n
satis…es (11), consider a pair of utilities u
0
and u
00
in I
n
and …x ;  in [0; 1]. By
construction both ^u
0
:= u
0
=~
n
and ^u
00
:= u
00
=~
n
are in
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
. Since ~
n
and ~
n
are
in [0; 1], expression (11) implies

1
[ (~
n
^u
0
) + (1  )  (~
n
^u
00
)] = ~
n

1
[ (^u
0
) + (1  )  (^u
00
)] (12)
and

1
[ (~
n
^u
0
) + (1  )  (~
n
^u
00
)] = ~
n

1
[ (^u
0
) + (1  )  (^u
00
)] . (13)
Substituting u
0
for ~
n
^u
0
and u
00
for ~
n
^u
00
and then combining expressions (12) and (13), we obtain

1
[ (u
0
) + (1  )  (u
00
)] = 
1
[ (u
0
) + (1  )  (u
00
)] ,
as required. 
Step 4 consists of the following lemma showing that  must be an a¢ ne transformation of a
homogenous function.
Lemma 12 Suppose  (:) satis…es equation (11) for all u
0
; u
00
in
h
min
n
0;
^
U
i
1

x
1

o
;
^
U
i
1

x
1

i
and
all  and  in [0; 1], then
 (u) =

Cu
k
+ D u  0
C (u)
k
+ D u < 0
for some C; k in R
++
and some D in R.
Case 1.
29
^
U
i
1

x
1

 0. We shall show that
 (u
00
)   (u
0
) =  () [ (u
00
)   (u
0
)] . (14)
29
This dr a ws on Moulin’s [1988 p45] proof of Robert’s [1980] theorem that a s ocial welfare ordering that i s
additively separable and ind ependent of common utility scale admits a generalized utilitarian repres entation with
a power fun ction.
46

for all u
0
; u
00
2
h
0;
^
U
i
1

x
1

i
and all  2 (0; 1).
Consider four positive numbers u
1
, ^u
1
, u
2
, ^u
2
in
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
, such that u
1
< ^u
1
,
u
2
> ^u
2
and suppose (contra-hypothesis) that for some  2 (0; 1),
 (u
1
)   (^u
1
)
 (u
2
)   (^u
2
)
6=
 (u
1
)   (^u
1
)
 (u
2
)   (^u
2
)
=: r. (15)
Then we have,

1

1
1 + r
 (u
1
) +
r
1 + r
 (u
2
)

= 
1

1
1 + r
 (^u
1
) +
r
1 + r
 (^u
2
)

And (15) implies that

1

1
1 + r
 (u
1
) +
r
1 + r
 (u
2
)

6= 
1

1
1 + r
 (^u
1
) +
r
1 + r
 (^u
2
)

.
But (11) implies (setting  = 1= (1 + r))

1

1
1 + r
 (^u
1
) +
r
1 + r
 (^u
2
)

= 
1

1
1 + r
 (^u
1
) +
r
1 + r
 (^u
2
)

,
and 
1

1
1 + r
 (u
1
) +
r
1 + r
 (u
2
)

= 
1

1
1 + r
 (u
1
) +
r
1 + r
 (u
2
)

,
leading to a contradiction. Hence, (14) obtains.
The continuous solutions of (14) are known (Aczel [1966]) to be
 (u) = C
+
u
k
+
+ D
+
for some C
+
; k
+
in R
++
and some D
+
in R.
Case 2. 0 2

^
U
i
1

x
1

;
^
U
i
1

x
1


. By an analogous argument to the one employed in case 1, for
u in the sub-interval
h
0;
^
U
i
1

x
1

i
, we obtain  (u) = C
+
u
k
+
+ D
+
; and for u in the sub-interval
h
^
U
i
1

x
1

; 0

 R

, we obtain  (u) = C

(u)
k

+ D

, for some C

; k

in R
++
and some D

in R. Continuity of  implies D
+
= D

=: D. Thus we obtain:
 (u) =
8
>
>
<
>
>
:
C
+
u
k
+
+ D u  0
C

(u)
k

+ D u < 0
. (16)
It remains to show k
+
= k

and C
+
= C

.
47

To show k
+
= k

, we again exploit expression (11). Consider u
0
; u
00
2

^
U
i
1

x
1

;
^
U
i
1

x
1


such that u
0
< 0, u
00
> 0. Then, for any ;  in (0; 1)
 (u
0
) + (1  )  (u
00
) = C

(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
+ D
and  (u
0
) + (1  )  (u
00
) = C

(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
+ D.
Choose ;  in (0; 1), such that
C

(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
> 0 and  C

(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
> 0.
Therefore, on the left side of (11) we have

1
( (u
0
) + (1  )  (u
00
)) =
C


k

(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
C
+
!
1=k
+
and on the right side of (11) we have

1
( (u
0
) + (1  )  (u
00
)) = 
C

(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
C
+
!
1=k
+
=
C


k
+
(u
0
)
k

+ (1  ) C
+
(u
00
)
k
+
C
+
!
1=k
+
.
This is possible only if k
+
= k

=: k.
It only remains to show that C
+
= C

. Recall that, by redistributive scope and strong
independence ove r outcome lotteries, there exists an outcome ^x such that
^
U
i
1
(^x) >
^
U
i
1
(^x). We
used this fact to construct u. Case 2 (i.e., u = 0 2

^
U
i
1

x
1

;
^
U
i
1

x
1


) corresponds to the
situation in which
^
U
i
1
(^x) > 0 >
^
U
i
1

x
1

.
Recalling the notation we used to de…ne u , let `
[]
:=  [^x] + (1  )

x
1

. By strong inde-
pendence over outcome lotteries and our construction,
^
U
i
1

`
[]

is linear and decreasing in ,
and is positive at  = 0 and negative at  = 1; and
^
U
i
1

`
[]

is linear and increasing in ,
and is negative at  = 0 and positive at  = 1. Let

 correspond to u; that is, `
[


]
=

` and
^
U
i
1

`
[


]

=
^
U
i
1

`
[


]

= 0. By outcome independence,

 is implicitly given by


^
U
i
1
(^x) +

1 



^
U
i
1

x
1

= 0 =


^
U
i
1
(^x) +

1 



^
U
i
1

x
1

. (17)
48

Using

 we can write for  >

,
^
U
i
1

`
[]

=
^
U
i
1
(^x)

 



=

1 



and
^
U
i
1

`
[]

=
^
U
i
1
(^x)

 



=

1 



,
and so,
^
U
i
1

`
[]

^
U
i
1

`
[]

=
^
U
i
1
(^x)
^
U
i
1
(^x)
.
Similarly, for  <

,
^
U
i
1

`
[]

=
^
U
i
1

x
1


  

=

 and
^
U
i
1

`
[]

=
^
U
i
1

x
1


  

=

, and so,
^
U
i
1

`
[]

^
U
i
1

`
[]

=
^
U
i
1

x
1

^
U
i
1
(x
1
)
.
Furthermore, again from equation (17), since 
^
U
i
1

x
1

=
^
U
i
1
(^x) =

=

1 



= 
^
U
i
1

x
1

=
^
U
i
1
(^x),
we have
^
U
i
1
(^x) =
h

^
U
i
1
(^x)
i
=
h

^
U
i
1

x
1

i
=
^
U
i
1

x
1

. Hence
^
U
i
1

`
[]

^
U
i
1

`
[]

=
^
U
i
1

x
1

^
U
i
1
(x
1
)
(18)
for all  6=

.
Also to simplify notation, let z
[]
:=  [i
1
] + (1  )

i
1

. Using this notation, case 2 implies
V

z
[0]
; `
[0]

 V

z
[1]
; `
[1]

> V

z
[1]
; `
[0]

and V

z
[1]
; `
[1]

> V

z
[0]
; `
[1]

. Let v := V

z
[1]
; `
[


]

=
 (0). By independence over identity lotteries, V

z
[]
; `
[


]

= v for all .
By construction, for all  <

, V

z
[0]
; `
[]

> V

z
[0]
; `
[


]

> V

z
[1]
; `
[]

; and for all  >

,
V

z
[0]
; `
[]

< V

z
[0]
; `
[


]

< V

z
[1]
; `
[]

. Thus, by the a¢ nity of V (; `) on  (I), for all ,
V

z
[]
; `
[]

is a¢ ne in . Thus there exists a u nique  2 (0; 1) such that V

z
[]
; `
[0]

= v. That
is,

z
[]
; `
[0]



z
[]
; `
[


]

. An immediate implication of independence over outcome lotteries, is
that V

z
[]
; `
[]

= v for all  

. We claim that V

z
[]
; `
[]

= v for all . Suppose not: that is,
without loss of generality, there exists a  >

 such that V

z
[]
; `
[]

> v. Then, by independence
over outcome lotteries, by mixing with

z
[]
; `
[0]

, we would have V

z
[]
; `
[]

> v for all  > 0, a
contradiction. Thus V

z
[]
; `
[]

= v for all .
We can solve for  using the de…nition of V and the fact that


^
U
i
1

x
1


+ (1  ) 

^
U
i
1

x
1


= 

^
U
i
1
(^x)

+ (1  ) 

^
U
i
1
(^x)

Hence
 =


^
U
i
1

x
1


 

^
U
i
1
(^x)




^
U
i
1
(^x)

 

^
U
i
1
(x
1
)

+



^
U
i
1
(x
1
)

 

^
U
i
1
(^x)

49

By the de…nition of , we have

h
^
U
i
1

`
[]

i
+ (1  ) 
h
^
U
i
1

`
[]

i
= v (19)
for all . By the a¢ nity of
^
U, we have
^
U
i
1

`
[]

= 
^
U
i
1
(^x) + (1  )
^
U
i
1

x
1

and
^
U
i
1

`
[]

=

^
U
i
1
(^x) + (1  )
^
U
i
1

x
1

. Since this holds for all , and since  is di¤erentiable almost every-
where, we have

0
h
^
U
i
1

`
[]

i
^
U
i
1
(^x) 
^
U
i
1

x
1


+ (1  ) 
0
h
^
U
i
1

`
[]

i
^
U
i
1
(^x) 
^
U
i
1

x
1


= 0 (20)
at almost all .
Indeed, for all  6=

 such that
^
U
i
1

`
[]

and
^
U
i
1

`
[]

lie in

^
U
i
1

x
1

;
^
U
i
1

x
1


, we can use
our homogenous expression for  and obtain:

0
h
^
U
i
1

`
[]

i
= K
0
h
^
U
i
1

`
[]

i
, (21)
where K := (1  )

^
U
i
1

x
1


^
U
i
1
(^x)

=
h


^
U
i
1
(^x) 
^
U
i
1

x
1

i
is a constant (that is, does not
depend on ).
Since  is a power function we have by plugging in to expression (21), for  >

,
kC
+

^
U
i
1

`
[]


k1
= KkC



^
U
i
1

`
[]


k1
.
This reduces to
"
^
U
i
1

`
[]


^
U
i
1

`
[]

#
k1
= K
C

C
+
.
Similarly for for  <

,
kC



^
U
i
1

`
[]


k1
= KkC
+

^
U
i
1

`
[]


k1
.
This reduces to
"

^
U
i
1

`
[]

^
U
i
1

`
[]

#
k1
= K
C
+
C

.
But, by expression (18), the ratio in the left side of both these expressions is equal to 
^
U
i
1

x
1

=
^
U
i
1

x
1

for all  6=

. Thus we have shown that C
+
= C

. 
Step 5 extends the argument to cover the interval
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
. So far we have shown
that  must have the form given in Lemma 12 –that is, an a¢ ne transformation of a homogenous
50

function – on the interval
h
^
U
i
1

x
1

; U
i
1

x
1

i
. We next show that the same function extends over
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
.
We can repeat step 2 through step 4 above focussing on the interval
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
. The
argument is the same except that we need to be careful about the normalization that set u = 0
prior to step 3. Since we are re-normalizing a second time, we have to keep track of how this
second re-normalization is related to the …rst.
In particular, to be consistent with our notational convention above, let
h
^
U
i
i
i2I
and  be the
individual levels and -function given the normalization that set u = 0 above. In these utility
units, let the utility level that is analogous to u (see expression (1) for the de…nition) for our
analysis of the interval
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
be u
2
. Denote our re-normalized utility function for
each individual i by
~
U
i
(`) :=
^
U
i
(`)  u
2
(so that the utility level u
2
is re-normalized to zero as
before). For each utility level u, let ~u denote the corresponding re-normalized individual utility
level and let
~
 denote the correspondingly re-normalized -function. Then we can re-normalize
~

such that for all u in R,
~
 [~u] =
~
 [(u  u
2
)] =  [u].
By repeating steps 2 to 4, we know that
~
 [~u] must have a form analogous to that in Lemma
12 on the interval
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
. With slight abuse of notation, we can keep track of th e
re-normalization by writing
~
 (u) =
8
>
>
<
>
>
:
~
C [u  u
2
]
~
k
+
~
D u  u
2
 0

~
C ((u  u
2
))
~
k
+
~
D u  u
2
< 0
.
By lemma 9, we know that
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
\
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
has a non-empty interior.
Thus, the
~
 (u) and  (u) must coincide on this interval. Clearly if either function were a¢ ne then
both functions must be a¢ ne and we would be done. Suppose then that k 6= 1 and
~
k 6= 1. We
will show that this implies u
2
= 0; that is, the two normalizations must be the same.
Suppose …rst that the overlap
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
\
h
^
U
i
1

x
1

;
^
U
i
1

x
1

i
contains a subinterval
in which both u > 0 and u  u
2
> 0. Then we know that
~
C [u  u
2
]
~
k
+
~
D = Cu
k
+ D (22)
51

for all u in that subinterval. Di¤erentiating yields
~
k
~
C [u  u
2
]
~
k1
= kCu
k1
:
Notice that, if k =
~
k = 2, then we would have [u  u
2
] =u = C=
~
C and, since the right side is
constant, this implies u
2
= 0. Therefore assume k 6= 2 or
~
k 6= 2 . Di¤erentiating again, dividing
the second derivative by the …rst, and rearranging yields
[u  u
2
]
u
=
~
k  1
k  1
.
But again the right side is constant, implying u
2
= 0. The argument on subintervals where either
u < 0 or u  u
2
< 0 is similar.
Since u
2
= 0 (if k 6= 1), the …rst derivative red uce s to u
k
~
k
=
~
k
~
C=kC but again the right side
is a constant hence k =

k so that
~
k
~
C=kC = u
k
~
k
= 1, and hence C =
~
C. Finally, using expression
(22), we obtain D =
~
D. In other words, the two functions  and
~
 must be the same. 
Step 6 completes the proof by showing that k = 1. To do this, we invoke our third richness
condition, three-player richness.
Recall from the proof of Lemma 12, for the outcome lottery `
[


]
:=

 [^x] +

1 



x
1

we
had
^
U
i
1

`
[


]

=
^
U
i
1

`
[


]

. That is,

i
1
; `
[


]



i
1
; `
[


]

. Hence by three-player richness, there
exists another individual ^{ , such that

^{; `
[


]



i
1
; `
[


]

. That is,
^
U
^{

`
[


]

6= 0. Consider the
graphs of
^
U
i
1

`
[]

and
^
U
^{

`
[]

as functions of  2[0; 1] to R. Both are lines. The …rst passes
through the point


; 0

, while the second does not. And, by the de…nition of x
1
, ^x and i
1
, the line
^
U
i
1

`
[]

is strictly decreasing. Suppose
^
U
^{

`
[


]

> 0 – the argument for the case
^
U
^{

`
[


]

< 0
is similar. Then we can …nd  and 
0
such that 0 <  < 
0
<

 and s uch that the vectors

^
U
i
1

`
[]

;
^
U
^{

`
[]


 0 and

^
U
i
1

`
[
0
]

;
^
U
^{

`
[
0
]


 0. Moreover, since
^
U
^{

`
[


]

6= 0, these
vectors are not colinear.
By the a¢ nity of 
1
 V (lemma 11), for all z; `; `
0
and all ,

1
"
X
i
z
i

h

^
U
i
(`) + (1  )
^
U
i
(`
0
)
i
#
= 
1
"
X
i
z
i

h
^
U
i
(`)
i
#
+ (1  ) 
1
"
X
i
z
i

h
^
U
i
(`
0
)
i
#
(23)
52

In particular, this must hold for z = (1=2)

i
1

+(1=2) [^{], `
[]
and `
[
0
]
. Substituting in these values
along with our homogenous functional forms  (u) = Cu
k
+ D and 
1
(v) = [(v  D) =C]
1=k
, The
left side of expression (23) becomes:

1

1
2



^
U
i
1

`
[]

+ (1  )
^
U
i
1

`
[
0
]


+
1
2



^
U
^{

`
[]

+ (1  )
^
U
^{

`
[
0
]



= 
1

1
2
C



^
U
i
1

`
[]

+ (1  )
^
U
i
1

`
[
0
]


k
+


^
U
^{

`
[]

+ (1  )
^
U
^{

`
[
0
]


k

+ D

=
1
2
1=k



^
U
i
1

`
[]

+ (1  )
^
U
i
1

`
[
0
]


k
+


^
U
^{

`
[]

+ (1  )
^
U
^{

`
[
0
]


k

1=k
And the right side of expression (23) becomes:
1
2
1=k



^
U
i
1

`
[]


k
+

^
U
^{

`
[]


k

1=k
+ (1  )


^
U
i
1

`
[
0
]


k
+

^
U
^{

`
[
0
]


k

1=k
!
Combining these yields:



^
U
i
1

`
[]

+ (1  )
^
U
i
1

`
[
0
]


k
+


^
U
^{

`
[]

+ (1  )
^
U
^{

`
[
0
]


k

1=k
= 


^
U
i
1

`
[]


k
+

^
U
^{

`
[]


k

1=k
+ (1  )


^
U
i
1

`
[
0
]


k
+

^
U
^{

`
[
0
]


k

1=k
(24)
Notice that if “=”were replaced by “”then expression (24) becomes the Minkowski inequality.
Recall that if the Minkowski inequality holds with equality (and the vectors involved are not
colinear) then k = 1. Since the vectors we chose were not colinear, we have k = 1, completing the
proof. 
Remark. Notice that our third richness condition, three-player richness, was only used in the last
step (step 6) of the proof. Speci…cally, it allowed us to construct vectors that were not colinear,
and hence to apply the Minkowski inequality.
30
The previous step (step 5) illustrates how our counterexample (example 1) relies on their
being two outcomes and two agents. Recall the construction of u. Starting from the interval
h
^
U
i
1

x
1

; U
i
1

x
1

i
, redistributive scope ensured there existed an outcome ^x such that (i
1
; ^x) 

i
1
; ^x

. Continuity then ensures there exits an outcome lottery `
[


]
between x
1
and ^x such that
30
Eve n here, we o nly need this condition if
^
U
i
1
(^x)  0. If
^
U
i
1
(^x) > 0, then we have

i
1
; x
1

 (i
1
; ^x) and

i
1
; x
1



i
1
; ^x

. In this case, our …rst richness cond ition (absence of unanimity) already implies there exists an ^{
such tha t (^{; ^x) 

^{; x
1

%

i
1
; x
1

, hence
^
U
^{

`
[


]

> 0.
53


i
1
; `
[


]



i
1
; `
[


]

, and u corresponded to the utility level at that lottery. Similarly, starting
from the interval
h
^
U
i
2
(x
2
) ;
^
U
i
2
(x
2
)
i
, redistributive scope ensured there exists an outcome ^x
2
such
that (i
1
; ^x
2
) 

i
1
; ^x
2

and continuity ensures there exits an outcome lottery `
[


2
]
between x
2
and
^x
2
such that

i
2
; `
[


2
]



i
2
; `
[


2
]

, and u
2
corresponded to the u tility level at that lottery. An
implication of step 5 is that if u 6= u
2
then  is a¢ ne. In the example, there are only two outcomes
and two agents, hence `
[


]
and `
[


2
]
must be the same lottery, and therefore u and u
2
are trivially
equal. But in a world with three agents or three outcomes, such a coincidence is knife edge.
Proof of Theorem 8. Remark: in the proof of lemma 1, for each product lottery (z; `) except
the best and the worst, we found some lottery `
0
and two individuals i and j such that (i; `
0
) 
(z; `)  (j; `
0
). Absence of unanimity ensured u s that such a lottery and pair of individuals existed.
We then constructed a ‘local’representation V (z; `) that solves
(V (z; `) [i] + (1  V (z; `)) [j] ; `
0
)  (z; `) .
When we attempt to generalize this idea without absence of unanimity, there might exist “prob-
lem” product lotteries (z; `), other than j us t the best or the worst, such that no lottery `
0
and
individuals i and j exist with the property above. If (z; `) is a such a “problem”product lottery
then all product lotteries in its indi¤erence set have the same problem.
More formally, let
~
V be a continuous utility function representing % that we can use as a
benchmark to label indi¤erence sets. Without loss of generality, assume that
~
V ’s image is equal
to [0; 1]. Let us de…ne the set of “problem”indi¤erence levels as follows
# =
n
v 2 (0; 1) : @i; j 2 I and ` 2 4(X) s.t.
~
V (i; `) > v >
~
V (j; `)
o
.
We claim that the set # is closed (relative to (0; 1)). Suppose not. That is, let v
n
! v be a
sequence in # (where v 2 (0; 1)) and assume there exists i; j 2 I and ` 2 4(X) such that
~
V (i; `) >
v >
~
V (j; `). The n f or su¢ c iently large n,
~
V (i; `) > v
n
>
~
V (j; `): a contradiction.
For all v in #, continuity of
~
V implies
~
V (z; `) = v for some product lottery (z; `). By the
de…nition of #, if
~
V (z; `) = v 2 # then either min
i
~
V (i; `)  v or max
i
~
V (i; `)  v. By indepen-
dence over outcome lotteries, max
i
~
V (i; `) 
~
V (z; `)  min
i
~
V (i; `). Hence,
~
V (z; `) = v implies
there exists at least one individual j such that
~
V (j; `) = v. And, using indepe nd enc e over identity
54

lotteries again,
~
V (z; `) = v implies V (i; `) = v for all individuals i in the support of the identity
lottery z.
Moreover, we claim that, for all v in the interior of #, if
~
V (j; `) = v for some individual j then
~
V (i; `) = v for all individuals i. Suppose not. That is, without loss of generality, let
~
V (i; `) < v.
Then for all v
0
such that
~
V (i; `) < v
0
< v, we have
~
V (i; `) < v
0
<
~
V (j; `). But this contradicts v
being interior.
The proof proceeds in three cases.
Case 1: the set # is empty. Lemma 1 is a special case of this case, and this case yields the
same uniqueness conditions as lemma 1. Fix " > 0 and denote
BL
"
=
n
(z; `) 2 4(I)  4(X) : 1  " 
~
V (z; `)  "
o
Since # is empty, for each t in (0; 1) we can …nd an outcome lottery `
t
for which there exist
individuals i and j such that
~
V (i; `
t
) > t >
~
V (j; `
t
). Let
BL
t
=
n
(z; `) 2 4(I)  4(X) :
~
V

i; `
t

>
~
V (z; `) >
~
V

j; `
t

o
.
As in the proof of lemma 1, construct a function V
t
that represents % on the closure of BL
t
and which is a¢ ne in identity lotteries. Since fBL
t
g
"t1"
is an open cover of th e compact set
BL
"
, we can …nd a …nite cover fBL
t
1
; : : : ; BL
t
K
g. The intersection of any two adjacent sets is
non-empty. Therefore, we can therefore re-normalize these ‘local’representations to …nd an a¢ ne
function V
"
that represents % on BL
"
. For "
0
in (0; "), the a¢ ne function V
"
0
can be chosen
to agree with V
"
on BL
"
. By continuity, the limit function V = lim
"!0
V
"
is well de…ned. By
a¢ nity, as in the proof of lemma 1, we can write V (z; `) = 
i
z
i
V
i
(`) where, for each i in I, by
acceptance, the function V
i
(`) := V (i; `) represents %
i
on 4(X).
Case 2: the set # is …nite. The n we can write # = fv
1
; : : : ; v
K1
g where k
0
> k implies
v
k
0
> v
k
. Let v
0
:= 0 and v
K
:= 1. Fix an interval of the form [v
k1
; v
k
], k = 1; : : : ; K. By
independence over identity lotteries, if
~
V (z; `) 2 (v
k1
; v
k
) then min
i
~
V (i; `) < v
k
. Hence, by the
de…nition of #, max
i
~
V (i; `)  v
k
. Similarly, max
i
~
V (i; `) > v
k1
and hence min
i
~
V (i; `)  v
k1
.
That is, if
~
V (z; `) 2 (v
k1
; v
k
) then
~
V (i; `) 2 [v
k1
; v
k
] for all i. Moreover, if
~
V (z; `) = v
k1
(resp.
55

v
k
) then
~
V (i; `) = v
k1
(resp. v
k
) for all i in the support of z. Therefore, following the method of
case 1, we can construct a function V
k
(z; `) = 
i
z
i
V
k
i
(`) that represents % on f(z; `) 2 4(I) 
4(X) :
~
V (z; `) 2 [v
k1
; v
k
]g. To complete the representation, we simply re-normalize these K
functions such that they agree on the produc t lotteries (z; `) such that
~
V (z; `) = v
k
for k = 1;
: : : ; K. For example, we can re-normalize s uch that the range of V
k
is [k  1; k]. Notice that, as
this construction suggests, we do not have uniqueness in this case.
Case 3: the set # is in…nite. Choose  2 (0; 1) n# De…ne

+
= min f1; min fv 2 # : v >  gg


= max f0; max fv 2 # : v <  gg
Clearly 

<  < 
+
(recall # is a closed set). As in cases 1 and 2, de…ne functions V and V
i
on [

; 
+
]. The set (0; 1) n# is covered by a countable number of disjoint intervals of the form
(

; 
+
), and hence the functions V and V
i
can be constructed (inductively and continuously)
on their closed union. In this way the functions are also de…ned for #
0
, the set of # ’s boundary
points.
Let v be an interior point of # and let
v
+
= min

1; min

v
0
2 #
0
: v
0
> v

v

= max

0; max

v
0
2 #
0
: v
0
< v

Clearly, v

< v < v
+
and all %
i
agree on f` 2 4(X) :
~
V (z; `) 2 (v

; v
+
) for some zg.
Choose V that (with continuity) agrees with V of the former step at the indi¤erence sets that are
associated with v

and v
+
, such that it represents % on this set. To conclude, de…ne
V
i
: f` 2 4(X) :
~
V (z; `) 2

v

; v
+

for some zg ! R
by V
i
(`) = V (z; `) and note that V (z; `) = 
i
z
i
V
i
(`) is trivially satis…ed.
Finally, as the numb er of (non trivially) open components of # is also countable, the construc-
tion of the desired functions can be carried out easily. 
56

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