Soc Choice Welf (2012) 39:599–614
DOI 10.1007/s00355-011-0641-3
ORIGINAL PAPER
Social preferences for the evaluation of procedures
Marc Fleurbaey
Received: 4 December 2011 / Accepted: 13 December 2011 / Published online: 21 December 2011
© Springer-Verlag 2011
Abstract The standard analysis of procedures (mechanisms), in the theory of implementation, focuses on the properties of the subset of possible outcomes. But since a
given procedure may yield very different outcomes in different circumstances (preference profile, information or rationality of players), it may be useful to rely on finegrained social preferences over outcomes in order to evaluate the procedure. This
paper introduces the notion of cross-profile social ordering, and illustrates how this
concept may be used for the assessment of procedures.
1 Introduction
Every procedure for conflict resolution amounts to making the parties with conflicting interests play a game. The equilibrium outcomes of the game, in various possible
profiles of the players, define a social correspondence. When one wants to evaluate
the procedure, this correspondence may naturally be assessed as is usually done in the
normative analysis of social correspondences, namely, by checking whether the correspondence satisfies some good ethical requirements. Such requirements often consist
merely in conditions bearing on the subset of outcomes for each profile. For instance,
an efficiency requirement may stipulate that every outcome must be Pareto-efficient;
an equity requirement may want every outcome to be envy-free, etc.
This paper has benefited from comments by two anonymous referees, as well as F. Maniquet, H. Moulin,
K. Tadenuma, W. Thomson, and reactions of the audience at the conference in honor of Maurice Salles in
Caen.
M. Fleurbaey (B)
Princeton University, Princeton, USA
e-mail: mfleurba@princeton.edu
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This approach would be satisfactory if one could always find good procedures satisfying all important requirements in all circumstances. But this is seldom the case.
For instance, strategy-proof correspondences, which are associated to direct revelation
games in which telling the truth is a dominant strategy, cannot generally be Pareto efficient and minimally impartial (by the Gibbard–Satterthwaite theorem).1 Procedures
that yield Pareto efficient and egalitarian-equivalent allocations (such as the games
proposed in Crawford (1979) and Demange (1984)) cannot always produce envy-free
allocations. Symmetrically, the classical divide-and-choose mechanism, which guarantees envy-freeness, does not generally make the chooser better off than at the per
capita bundle.
Now, when a procedure fails to yield good outcomes, one would like to know by
how much it fails. It would be regrettable to condemn a procedure which only slightly
fails to satisfy a particular requirement, but may be better than similar procedures
in satisfying other requirements. For such an assessment, a fine-grained ranking of
outcomes is needed, and this requires more than the definition of a good social correspondence. It requires a full-fledged ordering of outcomes, in the manner of social
choice.
In this article, the construction of social preferences over outcomes for various
profiles is defended as a useful approach for the evaluation of procedures. It appears
that the evaluation of procedures requires more than standard social preferences which
rank outcomes conditionally on profiles of individual preferences, and it is shown how
one can design “cross-profile” social preferences which compare outcomes obtained
in different profiles. An example of such a construction is provided, dealing with the
division of a bundle of divisible goods, and this example is pursued with the examination of several procedures in this context. The divide-and-choose mechanism, in
particular, is shown to perform very badly according to the social preferences proposed
in this example, and it is also shown that sophisticated mechanisms do not uniformly
dominate simpler procedures in all contexts. It is also shown how imperfections of the
players’ information may actually improve the performance of some mechanisms.
The article is organised as follows. The next section introduces the main concepts
and justifies the need for “cross-profile” social preferences. Then, in the example of the
sharing of divisible goods, social preferences are axiomatically studied in Sect. 3. Such
social preferences are applied to the evaluation of the divide-and-choose mechanism
in Sect. 4. Section 5 is devoted to other procedures. The last section concludes.
2 Evaluating procedures
Consider a population of players N = {1, . . . , n} facing a set of possible outcomes
X. Every player i has preferences over X which are formally defined as an ordering
(transitive and complete binary relation) Ri over X. Let R be the set of orderings
of X, and D the domain of possible preference profiles R N = (R1 , . . . , Rn ) in the
population.
1 Gibbard (1973); Satterthwaite (1975).
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A procedure may be described as a game form (S, g), such that S = S1 × · · · × Sn
is the set of permitted strategy profiles (s1 , . . . , sn ) and g : S → X is a mapping
which associates every strategy profile with an outcome in X. For any given procedure (S, g), an equilibrium concept yields a correspondence E : D → X which, for
every possible profile of preferences, describes the possible outcomes of the game. An
equilibrium concept incorporates data about the information available to the players
and about their rationality. There are therefore several relevant equilibrium concepts
for any procedure.
The standard evaluation of a procedure relies on a social correspondence, that is, on
a correspondence ϕ : D → X which selects a subset of X for every possible preference
profile. This social correspondence is viewed as a normative criterion and is commonly
chosen on the basis of particular ethical requirements. Then, the evaluation consists
of checking whether, for a given equilibrium concept E, one has E(R N ) ⊆ ϕ(R N )
for every R N in D, or even whether one has E(R N ) = ϕ(R N ) for every R N in D.2
Sometimes, the evaluation more directly consists in checking whether E, viewed as
a social correspondence, satisfies various ethical requirements which are commonly
imposed on social correspondences.
This standard method of evaluation may be criticized as too coarse. Three complications are not satisfactorily dealth with.
First, it may happen that E(R N ) ⊆ ϕ(R N ) (or E(R N ) = ϕ(R N )) for some R N
in D, but not all. Declaring that a procedure is not good because it does not verify
E(R N ) ⊆ ϕ(R N ) for every R N may in some cases be too severe3 . One should actually
analyse the subset of profiles R N such that E(R N ) ⊆ ϕ(R N ) and examine whether it
contains, or fails to contain, the most relevant cases of possible profiles.
Second, even when E(R N ) ϕ(R N ) (or E(R N ) = ϕ(R N )) for some relevant R N
in D, it may be that E(R N ) is actually not so far from being contained in (or equal to)
ϕ(R N ). The difficulty here is that the distance between E(R N ) and ϕ(R N ) should not
be computed as a mathematical quantity, because what matters is the ethical distance.
The question is not so much how far, but, rather, how badly off the track E(R N ) lies.
A proper assessment of this requires a fine-grained criterion which synthesizes the
various ethical principles which matter in the case at hand.
Third, it may be that, for an equilibrium concept E, one has E(R N ) ⊆ ϕ(R N )
for every R N , but that for another concept E , one has E (R N ) ϕ(R N ) for some
relevant R N . This is a failure in “multiple” implementation. Then, one has to assess
the relative importance of the equilibrium concepts, and, again, the severity of failure
for E .
In order to assess the severity of failure, one would need a social ordering function (SOF) ψ : D → R, which for every profile R N defines an ordering over X
representing a normative ranking of all outcomes. Notice that, formally, the social
correspondence ϕ does already provide an ordering, but this is a very coarse ordering,
2 This distinguishes full implementation from partial implementation. Thomson (1996) argues that one
should be interested primarily in full implementation. Indeed, even if a correspondence ϕ satisfies many
good properties, a subcorrespondence of ϕ may fail to satisfy some of these properties.
3 The same can be said when ethical properties of E(R) are directly checked. It may be too severe to
declare E(R) bad when it fails to satisfy a particular requirement for some R.
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with only two indifference classes (the selected outcomes and the rejected outcomes).
With a finer ordering,4 one may hope to obtain a subtler evaluation of the procedure.
For instance, it might be quite valuable to learn that with procedure A, for every R N one
obtains an outcome which is better, according to ψ(R N ), than the outcome obtained
with procedure B for the same profile and equilibrium concept.
The possibility to devise a consistent and ethically appealing SOF on the sole
basis of individual ordinal and non interpersonally comparable preferences Ri has
been negated by a tradition dating back to Arrow’s theorem on the impossibility of
social choice (Arrow 1951) and culminating in Sen’s Nobel lecture (Sen 1999). It
has been argued, however, that Arrow’s conditions (more specifically, his axiom of
independence of irrelevant alternatives) were restrictive and that with more reasonable
requirements some quite good SOFs can indeed be constructed and even axiomatically
justified.5
But for the current purpose, it is not sure that a SOF is sufficient to answer the questions one would like to ask. Consider a case when E(R N ) ϕ(R N ) for some relevant
R N in D. One would like to know, for instance, how bad is the worst possible outcome
over all preference profiles. The identification of “the worst outcome over all profiles”
requires a ranking not only of outcomes conditional on a preference profile, but also on
pairs of outcome-cum-profile. Only then is one able to say whether outcome x under
profile R N is better or worse than outcome x under profile R N . This may afterward
be used in the comparison of procedures in the following way. It may be that the worst
outcome for procedure A is outcome x, which is obtained under profile R N , whereas
the worst outcome for procedure B is outcome x , which is obtained under profile
R N . If the pair (x , R N ) is deemed worse than the pair (x, R N ), then this provides an
argument in favor of procedure A.
This may still look a little crude. One may for instance perform this kind of comparison not only on all possible profiles but also on the most relevant ones, if such a subset
may be identified, for instance on grounds of realism (some preferences are exotic,
e.g., those that combine perfect substitutability for small quantities of commodities
and perfect complementarity for large quantities) or by referring to partial information
about the likely profile of a given population (if one knows that a population generally
prefers beer to wine, it may not be worth considering profiles in which most people
prefer wine). Even better, one would like to be able to say whether a given procedure
has an overall good or poor performance over the domain D. For instance, assume
that the ethical value of a pair (x, R N ) can be measured on the real line by a social
evaluation function W : X × R → R. One would then like to look at the distribution
of min W (E(R N ), R N ) for all R N in D, where
W (E(R N ), R N ) = {w ∈ R | ∃x ∈ E(R N ), w = W (x, R N )} .
4 Clearly, the idea is that there should be some relation between ϕ and ψ. The ϕ-acceptable allocations
should be ranked by ψ above the other allocations.
5 See in particular Hansson (1973); Pazner (1979); Kaneko and Nakamura (1979); Fleurbaey and Maniquet
(2008a,b); Dhillon and Mertens (1999).
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Looking at min W (E(R N ), R N ) amounts to focusing on the worst possible outcome
for every R N . If some probability measure μ was available on D, describing the likelihood of the various profiles, one would then be able to represent this distribution
conveniently by the inverse of the cumulative distribution function. This is the graph
of the function Z : [0, 1] → R defined by
μ ({R N ∈ D | min W (E(R N ), R N ) ≤ Z ( p)}) = p.
If two procedures A and B are such that Z A ( p) > Z B ( p) for all p, then this means
that A dominates B in the sense that it is unambiguously 6 more likely to yield a better
outcome, even if it may happen that for some R N procedure B has better outcomes
than procedure A. When the curves for Z A and Z B cross, it is more difficult to make a
judgment, and there are various possibilities (looking at the lowest part of the graphs,
or at the averages, etc.) which will not be explored in detail here.
3 The cross-profile Pazner–Schmeidler ordering
The rest of this article focuses on the example of the division of unproduced commodities. This is a simple context but it is of particular relevance in many conflicts.7 In
this section, the construction of a SOF which evaluates allocations across profiles of
individual preferences is studied. The application to the evaluation of procedures is
the topic of the next section.
In Fleurbaey (2005) an axiomatic defense of a particular SOF, the Pazner–Schmeidler ordering function, is provided, and one may build on this approach and seek an
extension of it to the evaluation of allocations for various profiles. This is of course
taken as an example and other criteria of a similar sort could be designed.
Let us consider allocations of goods ( ≥ 2) to n individuals (n ≥ 2). An allocation is a list of bundles, one for each agent: x = (x1 , . . . , xn ) ∈ Rn
+ . Individual
preferences over allocations are assumed to be self-centered, so that every individual i cares only about her own bundle xi . It is then convenient to define individual
preferences simply as bearing on individual bundles instead of general allocations.
Individual i’s preference ordering Ri is thus defined over R+ , and is assumed here to
be monotonic (xi ≥ yi implies xi Ri yi and xi yi implies xi Pi yi )8 , convex and
continuous. Let D denote the set of profiles R N = (R1 , . . . , Rn ) such that every Ri
satisfies these conditions.
A SOF is a mapping which, for every economy defined by a number n ≥ 2 and a
profile R N ∈ D, determines an ordering over the set of allocations Rn
+ . A cross-profile
social ordering function (CPSOF) is a mapping which, for every economy defined by a
population size n ≥ 2, determines an ordering over the set of allocations-cum-profiles
Rn
+ × D.
6 There is still, however, the possibility that the distribution of max W (E(R), R) is more favorable for B.
7 A lively description of many concrete examples is provided in Brams and Taylor (1999).
8 Vector inequalities are denoted ≥, >, .
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Pazner and Schmeidler (1978) proposed to select allocations which are Pareto-optimal, and such that every individual i is indifferent between his bundle and a particular
bundle proportional to a reference bundle ∈ R++ , that is, such that for some real
number λ, one has xi Ii λ for all i. As they themselves mention in their article, notably in the proof of the existence of ‘egalitarian-equivalent’ allocations, this solution
to the distribution problem can also be described by referring to the following SOF,
denoted RP S :
x RP S y ⇔ min vi (xi ) ≥ min vi (yi ),
i
i
where vi is a representation of i’s preferences defined by:
vi (xi ) = min{υ ∈ R+ | υ Ri xi }.
It is not very difficult to conceive an extension of this SOF into a CPSOF, also
denoted RP S for convenience, and defined by:
(x, R N ) RP S y, R N ⇔ min vi (xi ) ≥ min vi (yi ),
i
i
where
vi (xi ) = min{υ ∈ R+ | υ Ri xi },
vi (xi ) = min{υ ∈ R+ | υ Ri xi }.
Notice that the Pazner–Schmeidler SOF and CPSOF do not actually depend on a
definite bundle , but only of its direction. That is, for any , such that = λ
for λ ∈ R++ , one has
RP S = R P S .
Conversely, if RP S = R P S for some , , then there is λ ∈ R++ such that
= λ . In other words, they are uniquely and unambiguously defined by the direction of the vector .
We now focus on CPSOFs, and introduce some ethical requirements which appear
desirable for such objects.9 In order to minimize mathematical notations, quantifiers
are omitted whenever there is no ambiguity. The ordering defined by a CPSOF is
simply denoted (x, R N ) R (y, R N ) for weak preference, (x, R N ) P (y, R N ) for strict
preference, and (x, R N ) I (y, R N ) for indifference.
The first requirement is the Pareto criterion, in the following weak version.
Weak Pareto: If (x, R N ) and (y, R N ) are such that for all i, xi Pi yi , then
(x, R N ) P (y, R N ).
9 For a more extensive discussion of the ethical underpinnings of the axioms, see Fleurbaey (2005).
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The next axiom, inspired by the famous Pigou–Dalton transfer principle, says that
it is socially acceptable to transfer part of a bundle from one agent to another with
identical preferences when, both ex ante and ex post, the agent who gives has more of
all goods.
Transfer Principle for Equals: If (x, R N ) and (y, R N ) are two allocations with
one profile, and i and j are two agents with identical preferences, such that for some
δ 0,
yi − δ = xi x j = y j + δ,
whereas for all other agents k, xk = yk , then (x, R N ) R (y, R N ).
The next axiom extends the scope of egalitarianism to the case when agents may
have different preferences. But it is less demanding in terms of egalitarianism, since it
only says that it is socially acceptable to fully equalize an allocation which is proportional to by giving the average bundle to all agents. An allocation proportional to is an allocation such that every individual bundle is proportional to . This condition
can be related not only to the Pigou–Dalton principle, but also to a traditional notion of
fairness, according to which no individual should ever be given less than the average
consumption (Steinhaus 1948).
Minimal Egalitarianism for -Proportional Allocations: Let R N be any profile.
If x and y are two allocations proportional to , such that for all i,
xi =
1
yj,
n
j
then (x, R N ) R (y, R N ).
This axiom does not give by itself any guidance about how to choose —note that
what matters is not the precise value of , but the direction of this vector in R+ . It is
natural to take the available resources if one likes the Steinhaus principle referring to
average consumption. But there may be contexts in which other references might be
appealing. For instance, under uncertainty (commodities become contingent quantities
delivered in different states) it may be natural to think of certainty as the reference,
even if the total endowment is not itself certain.
The next axiom, inspired from Hansson (1973) and Pazner (1979), requires social
preferences on a pair of allocations to depend only on individual indifference curves
at these allocations. This condition is very appealing because it guarantees that social
preferences will not be sensitive to far-fetched details of the preferences. At the same
time, it makes it possible to take account of relevant features, such as marginal rates of
substitution, which are excluded by Arrow’s extremely restrictive condition of independence of irrelevant alternatives. It is extended here to a cross-profile scope, which
is rather natural. The comparison of two allocations-cum-profiles (x, R N ) and (y, R N )
should only depend on the indifference curves of the population at the two allocations
in the respective profiles.
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N , R
Cross-Profile Independence: Let x and y be two allocations, and R N , R N , R
N
be four profiles. If for all i, all q ∈ R+ ,
x i Ii q ⇔ x i Ii q
yi Ii q ⇔ yi Ii q,
then
N R y, R
N .
(x, R N ) R y, R N ⇔ x, R
We may now state the following result:
Theorem 1 Any CPSOF which satisfies Weak Pareto, Transfer Principle for Equals,
Minimal Egalitarianism for -Proportional Allocations and Cross-Profile Independence is such that for any x, y and R N , R N ,
(x, R N ) PP S y, R N ⇒ (x, R N ) P y, R N .
Proof Let (x, R N ) PP S (y, R N ). This means that
min vi (xi ) > min vi (yi ).
i
i
Let a, b, c, d, e, f, g be such that
min vi (xi ) > g > e > d > b > a > min vi (yi )
i
i
and
c=
b + (n − 1)d
,
n
f =
(n − 1)e + g
.
n
Let x ,y , x , y and y be defined by:
x1 = g; ∀i = 1, xi = e.
∀i, xi = f .
y1 = b; ∀i = 1, yi = d.
∀i, yi = c.
∀i, yi = a.
Let R 1N , R 2N , R 3N , R 4N be such that for all q ∈ R+ ,
∀i, yi Ii1 q ⇔ yi Ii q,
y1 I11 q ⇔ y1 I1 q,
∀i = 1, q Ri1 yi ⇔ q ≥ yi ;
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∀i, yi Ii2 q ⇔ yi Ii1 q,
y1 I12 q ⇔ y1 I1 q,
∀i = 1, yi Ii2 q ⇔ yi Ii q;
∀i, yi Ii3 q ⇔ yi Ii2 q,
q R13 x1 ⇔ q ≥ x1 ,
∀i = 1, xi Ii3 q ⇔ xi Ii q;
∀i, xi Ii4 q ⇔ xi Ii3 q,
∀i, xi Ii4 q ⇔ xi Ii q.
By Weak Pareto, Transfer Principle for Equals, Minimal Egalitarianism for -Proportional Allocations and Cross-Profile Independence, one has, from Theorem 4 in
Fleurbaey (2005),
y , R N
P y, R N ,
because mini vi (yi ) = a > mini v (yi ). By Cross-Profile Independence,
y , R 1N
P y, R N ,
and by Weak Pareto,
y , R 1N
P
y , R 1N .
By Cross-Profile Independence,
y , R 2N P y , R 1N .
By Minimal Egalitarianism,
y , R 2N
P
y , R 2N ,
and by Cross-Profile Independence,
y , R 3N P y , R 2N .
By Weak Pareto,
x , R 3N
P
y , R 3N ,
and by Cross-Profile Independence,
x , R 4N P y , R 3N .
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M. Fleurbaey
By Minimal Egalitarianism,
x , R 4N
P
x , R 4N
and by Cross-Profile Independence,
x , R N
P
x , R 4N .
By Weak Pareto,
(x, R N ) P x , R N .
Summarizing, by transitivity one obtains (x, R N ) P (y, R N ), the desired conclusion.
From the proof of the theorem one can also deduce that, if a CPSOF is such that,
for any x, y and R N ,
(x, R N ) PP S (y, R N ) ⇒ (x, R N ) P (y, R N ),
and if in addition it satisfies Cross-Profile Independence, then for any x, y and R N , R N ,
(x, R N ) PP S y, R N ⇒ (x, R N ) P y, R N .
Indeed the property
(x, R N ) PP S (y, R N ) ⇒ (x, R N ) P (y, R N )
can be used in the proof in place of Minimal Egalitarianism and Weak Pareto.
On the basis of this observation, one can extend any axiomatic justification of the
Pazner–Schmeidler SOF into a justification of the Pazner–Schmeidler CPSOF, simply
by adding Cross-Profile Independence to the list of axioms.
4 Evaluating divide-and-choose
As an illustration of how a CPSOF may be used for the evaluation of a procedure,
we may consider the divide-and-choose mechanism, and take the Pazner–Schmeidler
CPSOF as the normative criterion. It may look strange to use the Pazner–Schmeidler
CPSOF, which is based on the idea of egalitarian-equivalence, for the assessment of a
mechanism which is closely related to the no-envy criterion. Indeed, egalitarian-equivalence and no-envy have been shown in the fair allocation literature to represent two
polar equity principles which fall apart in many contexts. But any CPSOF may be used
to evaluate any mechanism, and if one may expect the divide-and-choose to behave
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poorly according to the Pazner–Schmeidler CPSOF, it remains quite interesting to see
how bad the assessment actually is. 10 It is convenient to define
WP S (x, R N ) = min vi (xi ).
i
For simplicity, let us focus on the case with two agents. Let ∈ R++ be the total
amount of resources to be divided, and assume to begin with that the agents may have
any monotonic, convex and continuous preferences (D is the domain of such profiles).
Let agent 1 divide and agent 2 choose.
We first consider the case when agent 1 knows exactly agent 2’s preferences. Then
agent 1 selects her best point x1∗ on agent 2’s Kolm curve 11 , defined by
K 2 (R2 ) = q ∈ R+ | q I2 − q .
More precisely, agent 1 proposes x1∗ and −x1∗ , expecting agent 2 to choose −x1∗ (to
make sure that agent 2 makes this choice, any small ε added to − x1∗ and subtracted
from x1∗ will do).12
Since /2 belongs to K 2 (R2 ), necessarily x1∗ R1 /2. In contrast, since x1∗ I2 −
∗
x1 , and
1
1
= x1∗ +
− x1∗ ,
2
2
2
one has /2 R2 − x1∗ . As a consequence,
WP S
x1∗ , − x1∗ , R N = v2 − x1∗ .
The following result determines the best and the worst possible results over the
∗
∗
∗
∗
domain D. Let WP
S (R N ) denote the value of WP S ((x 1 , − x 1 ), R N ) when x 1 is
obtained as above for the profile R N .
Theorem 2 For any profile R N ∈ D,
∗
0 ≤ WP
S (R N ) ≤ 1/2,
∗
0
∗
1
and there exist profiles R 0N and R 1N in D such that WP
S (R N ) = 0, WP S (R N ) =
1/2.
10 The divide-and-choose mechanism is also criticized as a rather poor implementation of the envy-free
and efficient correspondence, because it selects the best allocations for the divider, in blatant violation of
anonymity (Thomson 1996).
11 See Kolm (1972) for a thorough construction of the Kolm curve and an introduction to the divide-andchoose mechanism.
12 The Kolm curve is such that whenever q is above it, − q is below it and q P − q (and conversely).
2
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M. Fleurbaey
∗
Proof The fact that 0 ≤ WP
S (R N ) ≤ 1/n is an immediate consequence of
/2 R2 − x1∗ R2 0,
which is equivalent to 0 ≤ v2 ( − x1∗ ) ≤ 1/2.
Let R 0N be any profile such that
(1 , 0, . . . , 0) P1 (0, 2 , . . . , ),
(1 , 0, . . . , 0) I2 (0, 2 , . . . , ) I2 0.
In such a profile, agent 1 proposes (1 , 0, . . . , 0) or (0, 2 , . . . , ), agent 2 chooses
∗
0
the latter and then WP
S (R N ) = v2 (0, 2 , . . . , ) = v2 (0) = 0.
Let R 1N be any profile such that R11 = R21 . Then a best strategy for agent 1 is to
∗
1
propose /2, leading to WP
S (R N ) = v2 (/2) = 1/2.
As a supplement to this result, one can show that whenever R1 and R2 have different
∗
marginal rates of substitution at /2, then WP
S (R N ) < 1/2. All this is devastating for the divide-and-choose mechanism. It performs definitely worse, according to
the Pazner-Schmeidler CPSOF, than the crude mechanism which simply gives /2 to
each agent independently of their preferences. With this “naive” mechanism, the value
∗
of WP
S is invariably 1/2, and is therefore almost always better, and never worse, than
the outcome of divide-and-choose.
Let us now consider the case when agent 1 is unsure about agent 2’s preferences.
This uncertainty may be crudely described by a set K 2 of possible Kolm curves for
agent 2. Let the union of all these curves be denoted
2 =
K
K2
K 2 ∈K 2
Assume that agent 1 adopts a maximin strategy, considering the worst possible case
for herself. Then she must focus on the envelope curve of all the Kolm curves:
2 , q ≮ q
2 | ∀q ∈ K
q∈K
and select her best bundle x1∗ in this set. The argument proving this fact may be briefly
described as follows. First, the best bundle must be the best bundle in one of the Kolm
curves in K 2 . If agent 1 picks a Kolm curve which is not on the envelope, there is a
risk that the bundle x1 she selects will lie above the true Kolm curve, in which case,
by construction of the Kolm curve (see footnote 8), agent 2 will choose x1 and leave
x1 R1 /2 and therefore, by convexity of pref −
x1 to agent 1. Now, necessarily x1 . Similarly, x1∗ R1 /2, since /2 belongs to the envelope
erences, /2 R1 − ∗
x1 . In addition, since x1∗ lies below (or on)
curve, and as a consequence x 1 R1 − ∗
x1∗ to agent 1. This
the true Kolm curve, agent 2 accepts to take − x1 and to leave ∗
shows that the maximin strategy is to select x1 (i.e., on the envelope curve).
This reasoning also shows that uncertainty is favorable to agent 2 in the sense that
the choice he is proposed is now such that −
x1∗ may lie strictly above his true Kolm
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curve and is never below. This usually entails that − x1∗ P2 − x1∗ , that is, he
may end up better off than when agent 1 knows his true preferences. But this is not
x1∗ ,
always the case. It is not difficult to construct examples where − x1∗ P2 − in which case uncertainty is harmful to both agents, and worsens the performance of
divide-and-choose for the Pazner–Schmeidler CPSOF.
There is an extreme case which suggests, however, that uncertainty is likely to
be beneficial to the performance of divide-and-choose as assessed by the Pazner–
Schmeidler CPSOF. Assume that uncertainty is extreme, so that agent 1 has no idea
of agent 2’s preferences. Then the maximin strategy consists of proposing /2. In
other words, divide-and-choose under complete uncertainty coincides with the naive
egalitarian mechanism, and performs uniformly better than under certainty.
5 Other procedures
Pursuing on the same simple example with two agents, it is interesting to examine
other mechanisms. Crawford (1979) describes a variant of divide-and-choose, dubbed
the “equal division divide-and-choose” in which the divider must propose a choice
between two bundles, one of which must be /2.
Under full information, agent 1 (the divider) will select her best bundle x1e on the
set
C2 (R2 ) = q ∈ R+ | − q I2 /2 ,
and give agent 2 the choice between /2 and −x1e . Since /2 ∈ C2 (R2 ), necessarily
x1e R1 /2, whereas − x1e I2 /2. This means that for any R N ,
1
v1 x1e ≥ = v2 − x1e .
2
e
e
e
e
Let WP
S (R N ) denote the value of WP S ((x 1 , − x 1 ), R N ) when x 1 is obtained as
e
above for the profile R N . This mechanism, then, always yields WP S (R N ) = 1/2,
and is therefore better than divide-and-choose, although not better than the naive equal
split. It is better than equal split, however, in virtue of the fact that v1 (x1e ) > 1/2 is often
obtained. But this asymmetry in favor of the divider may also be seen as a drawback
of the mechanism.
Interestingly, uncertainty can never worsen the performance of the mechanism as
e
measured by WP
S (R N ), since this always yields at least 1/2 (even when agent 1
does not play maximin). Indeed, agent 2 is guaranteed v2 ≥ 1/2 by being offered
/2. And agent 1, by proposing /2 or − x1e , is offering herself the prospect of
e
x1e such that /2 P1 x1e . In this
obtaining /2 or x1 , and will therefore never select uncertainty context, the outcome may actually happen to be better than 1/2. In other
words, uncertainty can only improve the performance of this mechanism.
Crawford proposed a refined mechanism which removes the asymmetry and, under
full information and in a subgame perfect equilibrium, always selects the Paretoefficient egalitarian-equivalent allocation, that is, the allocation which maximizes
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M. Fleurbaey
WP S (x, R N ) over all feasible allocations. Under such circumstances this mechanism is therefore the best according to the Pazner-Schmeidler criterion. The mechanism works as follows.13 First, the agents simultaneously announce a real number λ.
Second, agent i who announced λi > λ j selects some xic and gives agent j the choice
between − xic and λi .
This mechanism has the typical drawback of mechanisms involving backward
induction, in relation to what happens out of equilibrium. At the subgame perfect
equilibrium, the two agents normally announce the same λ∗ (equal to the maximum
value of WP S (x, R N ) over all feasible allocations). If one of the agents announces a
lower λ, the other, who becomes the divider, is facing an enigma. Either one of them
is ill-informed, or the agent who announces a lower λ is irrational, and in either case
what will happen at the next step becomes quite uncertain.
It is possible to remedy this problem by altering the timing of the moves slightly.
Let the mechanism proceed as follows. Every agent i selects a bundle xic and a number
λi , and gives the other a choice between − xic and λi . Then only the agent who
proposed the lower λi actually makes a choice.
Let us see why this mechanism performs well under full information. For a given λi ,
every agent will select xic such that − xic I j λi . Whenever λi < λ∗ , i may pick xic
such that xic Pi λi . Therefore, if player j proposes λ j < λ∗ , then agent i, by proposing λ j < λi < λ∗ , offers herself access to xic such that xic Pi λi Pi λ j Ii − x cj .
In other words, it is in every agent’s interest to compete for the role of divider, which
gives her an advantage whenever λi < λ∗ . At the equilibrium, both agents propose
λ∗ , which is then equal to the value of WP S (x, R N ) at the resulting allocation. For
any R N , then, this mechanism yields WP S (x c , R N ) = λ∗ .
Under incomplete information, the mechanism can only perform less well, since
λ∗ is the upper bound for all mechanisms. If the agents play maximin, however,
x c, RN ) ≥
leading to some allocation x c , then the mechanism guarantees WP S (
1/2. If the agents do not play maximin, any agent i may in some cases end up with
x c , R N ) < 1/2. This mechanism may therefore
(1 − λi ) < /2, and then WP S (
perform less well, in some cases of incomplete information, than the equal division
divide-and-choose. Therefore Crawford’s mechanism does not uniformly dominate
the less ambitious equal division divide-and-choose.
Finally, let us conclude this analysis by looking at the egalitarian competitive equilibrium, which can be viewed as a procedure as well. It is defined as the competitive
equilibrium in which the two agents have the same endowment /2. To make the
analysis more realistic as far as the strategic setting is concerned, consider the situation in which our two agents i = 1, 2 are actually two groups containing a continuum of agents, each group having mass one. Then the worst situation that can
obtain for the WP S function is when the agents do not trade. In this case one has
WP S (x, R N ) = 1/2. As soon as the agents trade and strictly prefer their consumption
to their initial endowment, one has WP S (x, R N ) > 1/2. In summary, this procedure
dominates the simple equal split, but of course cannot do as well as the specially
designed Crawford procedure.
13 For more than two agents, Crawford’s mechanism may involve non-feasible allocations out of equilib-
rium. This problem is solved by a variant of the mechanism introduced by Demange (1984).
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Social preferences for the evaluation
613
6 Conclusion
This article has introduced a series of questions about how to evaluate procedures
with the help of a CPSOF. Not all of the suggestions have been illustrated. It has been
shown how a CPSOF could be conceived and one may perhaps emphasize how benign
and natural the extension appears, once the initial SOF has been designed. It has then
been shown how to use such a tool to make some evaluations of arbitrary procedures.
In the simple applications of the previous two sections, however, we have ignored the
possibility to exclude some implausible preference profiles, or to rely on a probability
measure describing the likelihood of various profiles, as suggested in Sect. 2. It is,
obviously, rather hard to make use of such refinements in a general, abstract context,
and they may be more useful in more concrete applications, where specific information
is available about the types of individuals involved.
Other issues would deserve to be considered in future research. A fact which has
been set aside is the possibility for some players to have externalities in their preferences. Altruistic or spiteful preferences actually abound in real-life cases. This entails
difficulties at every step of the analysis. First, one may wonder how such externalities
should be taken into account in the definition of social preferences. It has become
customary to exclude them in order to avoid introducing biases against the altruist and
in favor of the malicious. It does make sense to rely only on self-centered individuals
preferences in the construction of social preferences. The idea is that social preferences, if based on proper ethical principles, do take account of the various interests at
stake at least as well as any altruist would do.14 A second difficulty occurs when the
individuals play the game, and fail to pursue their own interest in order to promote, or
dampen, their partner’s interests. Then it becomes very hard to forecast the outcome
of the game, unless one has a precise idea of the particular form of externalities in the
case at hand. It may very well be that a given procedure is very good in the absence
of externalities but is more vulnerable than another in their presence.
Yet another point would deserve further research. Procedures should not only be
evaluated according to the outcomes they produce, but also according to their ability to
make the players learn about their (opponent’s, and possibly their own) preferences and
move toward better outcomes thanks to the learning process. This does not disqualify
social preferences as introduced here, but requires looking at how good the outcomes
are when starting with different kinds of imperfect information. In other words, one
then needs a finer description of the various degrees of imperfection of information in
order to assess the performance of procedures which modify the players’ information
along the game. It may happen that one procedure yields good outcomes when starting
from one kind of initial information, and is less good when the initial information is
different. We have observed this phenomenon above, when comparing full information to imperfect information, but a more precise analysis of the subcases of imperfect
information would be quite interesting.
Another interesting direction of research involves ethical properties which are not
simply described as features of a particular allocation, but involve several allocations.
14 Often, the altruist herself does not want her own altruism to be counted as her own interest, but would
like social preferences to espouse her impartial view of the social good.
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Monotonicity properties (with respect to resources or population) are a case in point.
When a procedure (e.g., the egalitarian competitive equilibrium) violates monotonicity, this cannot be evaluated by a CPSOF. A measure of the degree of violation of
monotonicity would require a measure of how bad a pair of allocations is.
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