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Strategy-proof Cardinal Decision Schemes
Bhaskar Dutta, Hans Peter and Arunava Sen
No 722
WARWICK ECONOMIC RESEARCH PAPERS
DEPARTMENT OF ECONOMICS
Strategy-proof Cardinal Decision Schemes
Bhaskar Dutta∗
, Hans Peters†and Arunava Sen‡
January 2005
∗Department of Economics, University of Warwick, Coventry, UK. Email :b.dutta@warwick.ac.uk
†Department of Quantitative Economics, University of Maastricht, Maastricht, The Netherlands. Email:
h.peters@ke.unimaas.nl
‡Indian Statistical Institute, New Delhi, India. Email:asen@isid.ac.in
1
1 Introduction
The classic results of Gibbard [6] and Satterthwaite [13] have shown that unless preferences
are restricted, the only decentralized mechanism which induces truth-telling behaviour by in-
dividual agents is the dictatorial one. This impossibility result has induced a huge literature
which analyzes the possibility of constructing strategy-proof mechanisms under various alter-
native frameworks. One variant, due to Gibbard [7,8], which is the main focus of this paper
is the extension of the original impossibility result to mechanisms which assign a probability
distribution over the set of feasible outcomes for each profile of preferences. Gibbard [7]
characterized the class of such strategy-proof probabilistic mechanisms or decision schemes.
He showed that a strategy-proof decision scheme must be a convex combination of duples
and unilaterals. A duple is a mechanism which assigns positive probability to at most two
alternatives, the pair of alternatives being independent of the profile of preferences, while
a unilateral is one where the preference ordering of a single individual dictates the social
lottery over feasible alternatives. 1
Such mechanisms need not satisfy even a weak form of efficiency. That is, even if all
individuals unanimously prefer an alternative ato all other alternatives, the mechanism need
not assign a probability of one to a. The only strategy-proof mechanisms satisfying even this
weak form of efficiency are random dictatorships, in which each individual is assigned a fixed
probability of being a dictator - fixed in the sense that these probabilities are independent
of the preference profile. Duggan [3] and Nandeibam [10] provide alternative proofs of the
random dictatorship result, while Dutta et al. [4] show that the random dictatorship result
holds even if the feasible set of alternatives is some convex set in <k(with k > 1), and
preferences are strictly convex and continuous with a unique peak.2
In the original Gibbard [7,8] framework, the decision scheme used only ordinal information
about individual preferences. However, Gibbard assumed that individual preferences were
represented by von Neumann-Morgenstern utility functions since these functions were used
to rank alternative probability distributions. Thus, the assumption that the decision scheme
can use only ordinal information about preferences imposed a strong invariance requirement
on the aggregation rule. In order to appreciate the strength of the invariance requirement, we
point out that strategy-proof ordinal decision schemes must satisfy a “local” property. That
is, suppose that a voter changes her preference by “switching” two contiguous alternatives. In
1See Barbera [1,2] for related characterizations of strategy-proof probabilistic mechanisms.
2See also Ehlers et al. [5].
2
the ordinal context, strategy-proofness will immediately imply that only the probabilities of
the two alternatives being switched are affected. This is a property with strong implications
and considerably simplifies the task of characterizing strategy-proof ordinal decision schemes.
In contrast, if a strategy-proof decision scheme utilizes cardinal information, then a change
in the utility of a single alternative for a voter could in principle, have a “global” impact,
that is, the probability of all alternatives could be affected. This makes the analysis in the
cardinal model far more difficult.
Despite this difficulty, Hylland [9], in an important and regrettably unpublished paper,
showed that the random dictatorship result holds even if the decision scheme is allowed to
use cardinal information. In this paper, we have two main objectives. First, we provide
an alternative and considerably simpler proof of Hylland’s theorem.3Second, we consider
a framework where essentially individuals cannot discern infinitesimally small differences in
utility. In particular, we assume that if an alternative ais strictly preferred to another
alternative b, then the utility difference between aand bis at least some fixed number which
we refer to as the grid size. We construct an example to show that the random dictatorship
result no longer holds when individual utility functions satisfy this additional restriction. We
then analyze the consequences of gradually reducing the grid size. That is, we consider an
arbitrary sequence of strategy-proof and unanimous decision schemes defined on a sequence
of decreasing grid sizes approaching zero. We obtain a ‘limit’ random dictatorship result in
the sense that the sequence of such decision schemes must converge to a random dictatorship
for all profiles for which the limit exists.
2 The Model
Let A={a1, a2, . . . , aM}be a finite set of alternatives, with M≥3. A lottery λis a
probability distribution over the set A, and can be identified with an M-vector whose jth
component λjdenotes the probability that λassigns to aj∈A. Clearly every component of
λis non-negative and the sum of the components is 1. The set of lotteries is denoted by L.
The set of voters will be denoted by I={1,2, . . . , N}. Each voter ihas a preference
ordering Riover the elements of the set A. The ordering Riis represented by an admissible
utility function ui, which is unique up to affine transformations. We normalize utility func-
tions by assuming that the utility of the maximal element, which is assumed to be unique,
3Nandeibam[11] has recently provided another proof of the Hylland result.
3
is one, while the utility of the worst element is zero. We do not require distinct alternatives
to have distinct utility levels (i.e. it is not required that Riis a strict ordering).
Let Udenote the set of admissible utility functions. We will use τ(ui) to refer to the
maximal element of utility function ui.
In section 4, we will impose an additional restriction on admissible utility functions - we
will assume that the minimal difference in utility levels of alternatives which have different
utilities is at least some η > 0. We refer to ηas the grid size.
A utility profile is an N-tuple (u1, u2, . . . , uN)∈ UN. Let udenote the utility profile
(u1, . . . , uN), and (u0
i, u−i) denote the profile (u1, . . . , ui−1, u0
i, ui+1, . . . , uN).
Definition 1 A Cardinal Decision Scheme (CDS) is a mapping φ:UN→ L.
A CDS utilizes cardinal information in individuals’ utility functions and specifies a prob-
ability distribution over the set of alternatives for each profile of utility functions. We let
φj(u), j= 1,2, . . . , M denote the probability on alternative ajin the lottery φ(u).
A CDS which only utilizes ordinal information about individual utility functions will be
called an Ordinal Decision Scheme.
Two admissible utility functions ui, u0
iare ordinally equivalent if for all ak, aj∈A,
ui(aj)≥ui(ak) iff u0
i(aj)≥u0
i(ak). Similarly, two utility profiles uand u0are ordinally
equivalent if each pair ui, u0
iis ordinally equivalent.
Definition 2 An Ordinal Decision Scheme (ODS) is a CDS φwith the property that φ(u) =
φ(u0)whenever uand u0are ordinally equivalent.
Different concepts of efficiency can be associated with decision schemes. One concept
which has been used is that of ex post efficiency.4
Definition 3 A CDS φis ex post efficient if for all aj, ak∈Aand for all admissible utility
profiles u,φk(u) = 0 if ui(aj)> ui(ak)for all i∈I.
An ex post efficient CDS ensures that a Pareto non-optimal alternative is never assigned
positive probability. A considerably weaker condition is that of Unanimity.
Definition 4 A CDS φsatisfies Unanimity if for all aj∈Aand for all admissible utility
profiles u,φj(u) = 1 if τ(ui) = ajfor all i∈I.
4See for instance Gibbard[7]), Duggan [3], Nandeibam [10].
4
Unanimity simply requires that if an alternative is best for all individuals, then it should be
assigned probability one.
Random dictatorships are an important class of ordinal decision schemes. These are rules
in which each individual has a fixed probability (that is, independent of the utility profile)
of being a dictator. More formally,
Definition 5 The CDS is a random dictatorship if there exist non-negative real numbers
β1, β2,· · · , βNwith Piβi= 1 such that for all u∈ UNand aj∈A,
φj(u) = X
{i|τ(ui)=aj}
βi
We assume that individuals rank alternative lotteries in terms of expected utility.
Definition 6 A CDS is manipulable by an individual i∈Iat u∈ UNvia u0
i∈ U if
M
P
j=1
u(aj)φj(u0
i, u−i)>
M
P
j=1
u(aj)φj(u).
Definition 7 A CDS is strategy-proof (SP) if it is not manipulable by any voter at any
profile.
Thus, a CDS is strategy-proof if no voter can strictly gain in terms of expected utility
by misrepresenting her true preferences.
3 The Hylland Theorem
An example of a strategy-proof decision scheme is the random dictatorship. If iis the
dictator, then the alternative which is first in i’s preference ordering is chosen with probability
one. Since the probability of any voter ibeing a dictator is independent of the profile of
preferences, it is easy to see that no individual has an incentive to misreveal preferences.
The random dictatorship in which each individual has an equal chance of being the dic-
tator is obviously anonymous and efficient - the voting scheme only puts positive weight on
alternatives which are Pareto optimal. This might seem to suggest that this random dic-
tatorship provides a positive resolution of the dilemma posed by the Gibbard-Satterthwaite
result- an equal distribution of power is consistent with efficiency and truthful revelation of
preferences. Unfortunately, random dictatorships possess an undesirable property, as shown
in the following example.
5
Example 1 Let |I|= 1000, and A={a1, . . . , a1001}. Consider a profile Psuch that for
each individual i,aiPia1001Piafor all a∈A\{ai, a1001 }. Although every individual considers
a1001 as the second most preferred alternative, and no two individuals agree on what is the
best alternative in A, a random dictatorship must assign zero probability to a1001.
This example provides a motivation to search for other strategy-proof decision schemes.
Unfortunately, Hylland [9] proved that random dictatorships constitute the only class of
unanimous and strategy-proof cardinal decision schemes. In this section, we provide a rela-
tively simple proof of Hylland’s theorem, which is stated below.
Theorem 1 A CDS satisfies strategy-proofness and unanimity if and only if it is a random
dictatorship.
Proof: It is clear that a random dictatorship satisfies unanimity and is also strategy-proof.
We prove the converse.
Step 1: We first show that for |N|= 2, a unanimous and strategy-proof CDS φis a random
dictatorship.
In the proof of this step, for k, j ∈ {1, . . . , M }with k6=jand a positive number η, we
frequently use the notation uη
jk for an admissible utility function that assigns 1 to aj, 1 −η
to ak, and strictly lower utilities to all other alternatives.
Pick aj∈A, and let u1be an admissible utility function such that τ(u1) = aj. Also
pick ak∈Aand η > 0, and consider uη
kj ∈ U. We now consider the consequences of letting
η→0.
Claim 1: limη→0(φj(u1, uη
kj ) + φk(u1, uη
kj )) = 1.
Proof: If voter 2 announces u0
2such that τ(u0
2) = aj, then φj(u1, u0
2) = 1 from unanimity.
So, in order to prevent voter 2 from manipulating at (u1, uη
kj ) by announcing u0
2, we must
have
φk(u1, uη
kj ) + (1 −η)φj(u1, uη
kj ) + (1 −φk(u1, uη
kj )−φj(u1, uη
kj ))α≥1−η,
where α:= max{uη
kj (as)|s6=k, j }<1−η < 1. Taking limits5as ηtends to 0, we obtain
lim
η→0φk(u1, uη
kj ) + φj(u1, uη
kj ) + (1 −φk(u1, uη
kj )−φj(u1, uη
kj ))α≥1,
which implies limη→0φk(u1, uη
kj ) + φj(u1, uη
kj )≥1. The reverse inequality is obviously true.
5Note that we can assume that these limits exist because all the probabilities lie in the unit simplex.
6
Claim 2: Let v1be an admissible utility function such that τ(v1) = aj. Then lim
η→0φk(u1, uη
kj ) =
lim
η→0φk(v1, uη
kj ).
Proof: Suppose that the claim is false. Assume w.l.o.g. that lim
η→0φk(u1, uη
kj ) = λk> λ0
k=
lim
η→0φk(v1, uη
kj ). Observe that Claim 1 implies that
lim
η→0Ptu1(at)φt(u1, uη
kj ) = 1 −λk+u1(ak)λkand lim
η→0Ptu1(at)φt(v1, uη
kj ) = 1 −λ0
k+
u1(ak)λ0
k. Therefore lim
η→0Ptu1(at)(φt(v1, uη
kj )−φt(u1, uη
kj )) = (1 −u1(ak))(λk−λ0
k). But
the RHS of this expression is strictly positive by assumption. Therefore there exists ηsmall
enough such that Ptu1(at)φt(v1, uη
kj )>Ptu1(at)φt(u1, uη
kj ). This implies that voter 1 can
manipulate φat (u1, uη
kj ) via v1which contradicts strategy-proofness of φ.
Let uη
jk ∈ U and let u2∈ U with τ(u2) = ak.
Claim 3: lim
η→0φj(uη
jk , u2) = 1 −lim
η→0φk(u1, uη
kj ).
Proof: Let lim
η1→0φj(uη1
jk , u2) = λjand let lim
η2→0φk(u1, uη2
kj ) = λ0
k. According to Claims 1 and
2, lim
η2→0φj(uη1
jk , uη2
kj ) = 1 −λ0
kfor all η1. Therefore
lim
η1,η2→0φj(uη1
jk , uη2
kj ) = lim
η1→0(1 −λ0
k) = 1 −λ0
k.
But Claim 2 also implies that lim
η1→0φj(uη1
jk , uη2
kj ) = λjfor all η2. Therefore
1−λ0
k= lim
η1,η2→0φj(uη1
jk , uη2
kj ) = lim
η2→0λj=λj
which is what we have to prove.
Let aj, ak, as, at∈Awith aj6=akand as6=at. Let u1and v1be admissible utility
functions such that τ(u1) = ajand τ(v1) = as.
Claim 4: lim
η→0φj(u1, uη
kj ) = lim
η→0φs(v1, uη
ts).
Proof: We know from Claim 2 that lim
η→0φ(u1, uη
kj ) does not depend on u1as long as the
first-ranked alternative in u1is aj. We can therefore denote this limit w.l.o.g. as λj(j, k). So
we have to prove that λj(j, k) = λs(s, t). We will first prove that λj(j, k) = λs(s, k).
Let δ,and γbe positive numbers and let v
1be an admissible utility function with
τ(v
1) = as,v
1(aj) = 1 −and v
1(al)≤for all al6=as, aj.
Now consider voter 1 in the profile (v
1, uγ
ks). Her maximal expected utility from truth-
telling is φs(v
1, uγ
ks) + (1 −)φj(v
1, uγ
ks) + (1 −φs(v
1, uγ
ks)−φj(v
1, uγ
ks)). If she announces
uδ
jk instead her minimal expected utility is φs(uδ
jk , uγ
ks) + (1 −)φj(uδ
jk , uγ
ks)). Since φis
strategy-proof, we have
φs(v
1, uγ
ks) + (1 −)φj(v
1, uγ
ks) + (1 −φs(v
1, uγ
ks)−φj(v
1, uγ
ks))
7
≥φs(uδ
jk , uγ
ks) + (1 −)φj(uδ
jk , uγ
ks).
Since the inequality above is true for all δ,and γ, we can take limits to obtain
lim
,γ,δ→0(φs(v
1, uγ
ks) + (1 −)φj(v
1, uγ
ks) + (1 −φs(v
1, uγ
ks)−φj(v
1, uγ
ks)) ≥
lim
,γ,δ→0(φs(uδ
jk , uγ
ks) + (1 −)φj(uδ
jk , uγ
ks)).
Observe that Claims 2 and 3 imply that lim
δ→0φj(uδ
jk , uγ
ks) = λj(j, k ) and
lim
γ→0φs(v
1, uγ
ks) = λs(s, k ). Also, Claim 1 implies that lim
δ→0φs(uδ
jk , uγ
ks) = 0 and lim
γ→0φj(v
1, uγ
ks) =
0. Therefore the inequality above reduces to
lim
→0(λs(s, k) + (1 −λs(s, k)) ≥lim
→0(1 −)λj(j, k)
So λs(s, k)≥λj(j, k). By reversing the roles of asand ajwe also have the reverse inequality,
and thus λs(s, k) = λj(j, k).
Define λk(s, k) := lim
η→0φk(v1, uη
ks), then by Claim 3 we have
λk(s, k) := lim
η→0φk(uη
ks, v2), where v2∈ U has τ(v2) = ak. By an argument symmetric to the
one in the first part of the proof Claim 4, we obtain λk(s, k) = λt(s, t). So altogether we
have
λj(j, k) = λs(s, k) = 1 −λk(s, k) = 1 −λt(s, t) = λs(s, t),
where the second and last equalities follow from Claim 1.
We now summarize the implication of Claims 1 through 4. There exists a real number
λlying between 0 and 1 with the following properties. Let ajand akbe two arbitrary
but distinct alternatives. Consider a utility profile where ajand akare first-ranked for
voters 1 and 2 respectively. Now consider a sequence of utility profiles where the utility
function of voter 1 is fixed but the utility function of voter 2 is changed in a way such
that akremains first-ranked and the utility of ajis increased to 1. Then the sequence of
probabilities associated with alternative ajconverges to λwhile that of akconverges to 1−λ.
Similarly, if we fix voter 2’s utility function and consider a sequence of utility functions for
voter 1 where akincreases to 1, then the sequence of probabilities associated with ajand ak
converges once again to λand 1 −λrespectively.
Claim 5: For all admissible utility profiles uand all j∈ {1, . . . , M}, if φj(u)>0, then
aj∈ {τ(u1), τ (u2)}.
Proof: Suppose that the Claim is false. Assume w.l.o.g. that there exist distinct alternatives
aj, akand asand an admissible utility profile uwhere ajand akare first-ranked by voters
8
1 and 2 respectively and φs(u)>0. Let ηand δbe positive numbers and let uη
js and uδ
ks
be admissible utility functions, and denote lim
η,δ→0φl(uη
js , uδ
ks) by λ0
lfor all al∈A. We first
prove that λ0
s>0.
In order to establish this, we start with a general observation. Let wbe a profile and at
be an alternative which is not first-ranked in w1. Let v1be an admissible utility function
with v1(at)> w1(at) and v1(al) = w1(al) for all al6=at. Then φt(v1, w2)≥φt(w). In
order to see this, observe that since φis strategy-proof, we must have Prw1(ar)φr(w)≥
Prw1(ar)φr(v1, w2) and Prv1(ar)φr(v1, w2)≥Prv1(ar)φr(w). Combining these two in-
equalities we have Pr(v1(ar)−w1(ar))(φr(v1, w2)−φr(w)) ≥0, which implies φt(v1, w2)≥
φt(w). Thus if we increase the utility of an alternative for a voter in a profile, the proba-
bility associated with that alternative cannot decline. Notice that this observation together
with our assumption that φs(u)>0 implies that for η,δsmall enough, φs(uη
js , uδ
ks)>0.
Moreover, this probability is non-increasing in ηand δ. Therefore λ0
s>0.
We now complete the proof of Claim 5.
For > 0 define admissible utility functions ¯u1and ¯u2such that
•τ(¯u1) = as,¯u1(aj) = 1 −, ¯u1(ak) = 0,¯u1(al) = 1 −(l+ 1)for all al6=as, aj, ak.
•τ(¯u2) = as,¯u2(ak) = 1 −, ¯u2(aj) = 0,¯u2(al) = 1 −(l+ 1)for all al6=as, aj, ak.
Then, by the summary of Claims 1–4 above,
lim
δ→0φs(¯u1, uδ
ks) = λ, lim
δ→0φk(¯u1, uδ
ks) = 1 −λ.
Suppose
λ<λ0
s+λ0
j+X
l6=s,j,k
λ0
l.
Then, for small enough, 1 can manipulate φat (¯u1, uδ
ks) via uη
js as δ→0. Hence,
λ≥λ0
j+X
l6=j,k
λ0
l.(1)
We similarly have
lim
η→0φj(uη
js ,¯u2) = λ, lim
η→0φs(uη
js ,¯u2) = 1 −λ.
In order to prevent 2 from manipulating φat (uη
js ,¯u2) for small values of as η→0, we
need
1−λ≥λ0
k+X
l6=j,k
λ0
l.(2)
9
Combining inequalities (1) and (2) we obtain
1≥1 + X
l6=j,k
λ0
l.
This implies that λ0
l= 0 for each l6=j, k. This contradicts λ0
s>0, and hence completes the
proof of Claim 5.
Combining Claims 1–5, we see that for any profile with unequal top alternatives all prob-
ability is assigned to the top alternatives (Claim 5), and that agents 1 and 2 can guarantee
probabilities as close to λand 1 −λas desired on their respective top alternatives (Claims
1–4). Hence, φis a random dictatorship with weights λand 1 −λ. This completes the proof
of Step 1.
Step 2: We now show that a unanimous and strategy-proof CDS is a random dictatorship
for arbitrary N. We assume that the statement is true for all Iwith N−1 or fewer agents,
and we now establish it for N. So let φbe an N-agent CDS satisfying unanimity and
strategy-proofness.
Define a CDS g:UN−1→ L for an N−1 agent society, as follows:
for all u1, u3, . . . , uN∈ UN−1,g(u1, u3, . . . , uN) = φ(u1, u1, u3, . . . , uN)
Then ginherits unanimity from φ. We first show that gis strategy-proof. Clearly, if i∈
{3, . . . , N }manipulates gat (u1, u3, . . . , uN), then imanipulates φat (u1, u1, u3, . . . , uN).
This contradicts the assumption that φis strategy-proof.
Since 1 cannot manipulate φat u= (u1, u1, . . . , uN) via u2,
M
X
k=1
u1(ak)φk(u)≥
M
X
k=1
u1(ak)φk(u2, u1, . . . , uN).
Similarly, since 2 cannot manipulate (u2, u1, . . . , uN) via u2, we have
M
X
k=1
u1(ak)φk(u2, u1, . . . , uN)≥
M
X
k=1
u1(ak)φk(u2, u2, . . . , uN).
Putting these inequalities together,
M
X
k=1
u1(ak)φk(u)≥
M
X
k=1
u1(ak)φk(u2, u2, . . . , uN).
Hence, 1 cannot manipulate gat uvia u2. This shows that gis strategy-proof.
The induction hypothesis establishes that gmust be a random dictatorship. Let βbe the
weight of the “coalesced” individual 1 in the random dictatorship g, while βiis the weight
for i= 3, . . . , N .
10
Fix an arbitrary (N−2)-tuple of utilities (u3, . . . , uN), and with some abuse of notation,
write φ(u1, u2)≡φ(u1, u2, u3, . . . , uN) for any pair u1, u2.
Step 2.1: Suppose β= 0.
We want to show that for all (u1, u2), φ(u1, u2) = φ(u1, u1).
Suppose not. Then, there are u1, u2and aksuch that
φk(u1, u2)> φk(u1, u1).(3)
Now, for > 0, choose usuch that
τ(u) = ak, u(aj) = for all aj6=ak.
Note that
φ(u, u) = φ(u1, u1) (4)
since the coalesced individual has zero weight in the random dictatorship g. From equations
(3) and (4) and the specification of u, it follows that
lim
→0
M
X
j=1
u(aj)φj(u1, u2)>lim
→0
M
X
j=1
u(aj)φj(u, u).(5)
In order to prevent individual 1 from manipulating φat (u, u, u3, . . . , uN), we need
M
X
j=1
u(aj)φj(u, u)≥
M
X
j=1
u(aj)φj(u1, u).
In order to prevent individual 2 from manipulating φat (u1, u, u3, . . . , uN), we need
M
X
j=1
u(aj)φj(u1, u)≥
M
X
j=1
u(aj)φj(u1, u2).
Putting these inequalities together, we need
M
X
j=1
u(aj)φj(u, u)≥
M
X
j=1
u(aj)φj(u1, u2).(6)
But, equation (5) shows that this cannot be satisfied for all values of , a contradiction.
Hence, in this case, φis a random dictatorship with weights (0,0, β3, . . . , βN).
Step 2.2: Suppose β > 0.
Let I0={3, . . . , N }. Define a function h:U{1,2}→ L as follows:
for all u1, u2, aj:hj(u1, u2) = 1
β[φj(u1, u2)−X
{i∈I0|τ(ui)=aj}
βi].
11
We want to show that his a 2-person CDS satisfying strategy-proofness and unanimity.
First, we show that his a CDS. That is, hj(u1, u2)≥0 for all aj∈A, and Pjhj(u1, u2) =
1.
Note that Pjhj(u1, u2) = 1 follows from the definition of hitself. So, we only need to
show that each hj(u1, u2) is non-negative.
Consider u1, u2such that τ(u1) = aj6=ak=τ(u2).
Claim 1:φl(u1, u2)≥φl(u1, u1) for all al6=aj.
Proof: Suppose there is al6=ajsuch that φl(u1, u2)< φl(u1, u1). Choose usuch that
τ(u) = aj,u(ai)≥1−for all ai6=aj, al, and u(al) = 0. Then, since φ(u1, u1) = φ(u, u),
lim
→0
M
X
i=1
u(ai)φi(u1, u2)>lim
→0
M
X
i=1
u(ai)φi(u, u).(7)
But, this shows that equation (6) is not satisfied for some value of , and hence contradicts
the assumption that φis strategy-proof.
Claim 1 establishes that for all l6=j,hl(u1, u2)≥0. We still need to show that
hj(u1, u2)≥0. But, notice that we could have “started” from u2, and proved that φl(u1, u2)≥
φl(u2, u2) for all l6=k. This shows that hj(u1, u2)≥0.
We now want to show that hsatisfies unanimity. Choose any u1, u2such that τ(u1) =
τ(u2) = ajfor some aj∈A. Take any ak∈A, and let the upper contour set of u1for akbe
B(k, u1) = {l∈ {1, . . . , M } | u1(al)> u1(ak)}.
Claim 2:φ(u1, u2) = φ(u1, u1).
Proof: Suppose there is some aksuch that
X
l∈B(k,u1)
[φl(u1, u1)−φl(u1, u2)] <0.(8)
For small > 0 choose usuch that
(i) u1and uare ordinally equivalent.
(ii) u(al)≥1−for all l∈B(k, u1).
(iii) u(al)≤for all l6∈ B(k, u1).
Now, strategy-proofness of φimplies that equation (6) also holds for the new specification
of u.
Noting that φ(u, u) = φ(u1, u1), equations (8) and (6) cannot hold simultaneously as →0.
12
Hence, either the claim is true or φ(u1, u1) stochastically dominates φ(u1, u2), i.e., all sums in
the LHS of (8) are non-negative and at least one sum is positive. But, note that if φ(u1, u1)
stochastically dominates φ(u1, u2), then it is well known6that
X
al∈A
u1(al)φl(u1, u1)>X
al∈A
u1(al)φl(u1, u2).(9)
Noting that φ(u1, u1) = φ(u2, u2), equation (9) shows that 1 manipulates φat (u1, u2) via
u2. Hence, Claim 2 is true.
Claim 2 immediately establishes that hsatisfies unanimity.
We now show that his strategy-proof. Pick any utility functions u1, u2, u0
1. Then
M
X
j=1
u1(aj)hj(u1, u2) =
M
X
j=1
u1(aj)1
β
φj(u1, u2, u3, . . . , uN)−X
{i∈I0|τ(ui)=aj}
βi
≥X
j
u1(aj)1
β
φj(u0
1, u2, u3, . . . , uN)−X
{i∈I0|τ(ui)=aj}
βi
=
M
X
j=1
u1(aj)hj(u0
1, u2).
Therefore voter 1 cannot manipulate in h. An identical argument establishes that 2 cannot
manipulate heither.
Hence, hmust be a random dictatorship with weights α1and α2.
Let β1=α1βand β2=α2β. We want to show that φis a random dictatorship with
weights β1, . . . , βN. Notice that we would have proved this if we can show that the weights
of the 2-agent hconstructed earlier do not depend on the choice of (u3, . . . , uN) used in the
construction of h. In fact, it is sufficient to show that the weights do not change when (say)
u3changes to u0
3, because we can change the profile from (u3, . . . , uN) to (u0
3, . . . , u0
N) by
changing utility functions one at a time.
Suppose that the weights change to α0
1and α0
2with α0
1> α1when u3changes to u0
3. We
show that this violates strategy-proofness of φ.
First, suppose τ(u3) = τ(u0
3) = aj. Consider u1, u2such that τ(u1) = ajand τ(u2) = al
where u3(al) = 0, that is, alis the worst element in terms of u3. Then, it is easy to check that
1 manipulates φat (u1, u2, u3, . . . , uN) since there is a probability transfer of (βα0
1−βα1)
from alto aj(with probabilities on all other elements remaining the same) when 3 states u0
3
rather than u3. Hence, the weights cannot change if the top elements of u3and u0
3are the
same.
6See, for instance Quirk and Saposnik [12].
13
Now, suppose τ(u3) = ajand τ(u0
3) = ak6=aj. Using arguments of the previous
paragraph, we can assume that u3(ak) = 1−and u3(al) = 0. Again, assume that τ(u1) = aj
and τ(u2) = al. Then,
M
X
i=1
u3(ai) [φi(u1, u2, u3, . . . , uN)−φi(u1, u2, u0
3, . . . , uN)] = β(α0
1−α1)−β3.
This difference can be made positive by choosing small enough. So, φviolates SP.
This concludes the proof of the induction step, and thus of Theorem 1.
4 Strategy-proofness with utility grids
Our proof technique suggests an interesting extension of the basic framework. In particular,
our proof relies heavily on the fact that we can specify utility profiles where the utility of
some alternative is arbitrarily close to 1 although it is not maximal. How essential is this in
generating the random dictatorship result? In order to answer this question, we now assume
that an admissible utility function has the property that the minimal difference in utility
levels of alternatives which have different utilities is at least some η > 0.
More formally, let η=ηk= (M−1)−kwhere kis a positive integer. For every such η,
an admissible utility function is a mapping ui:A→ {0, η, 2η, . . . , 1−η, 1}, satisfying the
restrictions that there exists a unique element aj∈Asuch that ui(aj) = 1 and that there
exists some ak∈Asuch that ui(ak) = 0.
For every grid size η, we shall let Uηdenote the set of admissible utility functions. A utility
profile is an N-tuple (u1, u2, . . . , uN)∈[Uη]N. Note that if η > η0, then [Uη]N⊂[Uη0]N.
We shall let u∈[Uη]Nand (u0
i, u−i)∈[Uη]Ndenote the utility profiles (u1, . . . , uN), and
(u1, . . . , ui−1, u0
i, ui+1, . . . , uN) respectively.
4.1 An Example
The random dictatorship result no longer holds in this framework. The following counter-
example demonstrates that non-maximal elements can get positive probability for some
utility profiles.
Example 2 Let I={1,2},|A|= 3. As before, the best alternative has utility 1, the worst
has utility 0, while the maximum utility that the middle alternative can get is 1−η.
Consider the following rule φ∗.
14
(i) If τ(u1) = τ(u2), then φ∗assigns probability 1to the unanimous top alternative.
(ii) If there are only two Pareto optimal alternatives at a profile, then φ∗assigns 0.5to
each of these.
(iii) If there are three Pareto optimal alternatives at the profile u, but ui(ak)<0.5for
some iwhere akis the middle alternative, then φ∗assigns probability 0.5to each top
alternative.
(iv) Otherwise, φ∗assigns 0.5−dto each top alternative and 2dto the middle alternative,
where dis independent of the profile and d≤η
2(1+η).
Clearly, φ∗is unanimous. To see that φ∗is strategy-proof, suppose the true profile uis
such that either cases (ii) or (iii) apply. Without loss of generality, let u1(a1)> u1(a2)≥
u1(a3). Clearly, 1 cannot increase the probability weight on a1. If u1(a2)<0.5, then 1 does
not gain by increasing the probability weight on a2since at least half of any such increase
comes from a reduction in the probability weight on a1, 1’s most-preferred alternative. If
u1(a2)≥0.5, then either u2(a2)<0.5 in which case 1 cannot increase the probability weight
on a2, or u2(a2) = 1 in which case 1 can only increase the weight on a2to 1.
In case (iv), both individuals have (say) ui(a2)≥0.5. Neither wants to decrease the
weight on a2to 0 since this will mean an increase of 1
2din the probability weight on the worst
alternative. Neither individual gains by declaring a2to be the most-preferred alternative.
Finally, note that the weight on the middle alternative cannot be greater than η
2(1+η). For
suppose, u2(a2)> u2(a3)≥0.5> a1, and u1(a3)> u1(a1) = 1 −η > u1(a2). If 1 declares his
true utility function, then his expected utility (when 2 also declares his true utility function)
is 0.5. If instead 1 declares u0
1(a1)> u0
1(a3)>0.5> u0
1(a2), then the probability weights
will be
φ∗
1(u0
1, u2) = 0.5−d, φ∗
3(u0
1, u2) = 2d, φ∗
2(u0
1, u2) = 0.5−d
In order to prevent this lottery from giving 1 an expected utility greater than 0.5, we need
the upper bound on d.
The example suggests the following related lines of inquiry. First, notice that there
is an upper bound on the probability on the middle alternative. Moreover, this upper
bound is an increasing function of the grid size. So, is it generally true that if a CDS is
strategy-proof and unanimous, then the maximum probability on non-maximal elements
is an increasing function of grid size? The question is interesting because the maximum
15
possible probability on non-maximal alternatives is a “crude” measure of the distance from
some random dictatorship since the latter assigns zero probability to such alternatives.
Second, the CDS constructed in the example approaches a random dictatorship in the
limit as the grid size approaches zero. Again, it is of considerable interest to see whether such
a ‘limit’ random dictatorship result is true. We turn to these questions in the subsequent
sections.
4.2 A Limit Result
In this section, we first prove a ‘limit’ random dictatorship result, thus answering the second
question at the end of the preceding subsection. We then turn briefly to the first question
concerning the maximal probability on non-maximal elements.
Consider the following situation. For k= 1,2, . . . let {φηk}be a sequence of strategy-
proof and unanimous CDS’s, each one defined on Uk:= (Uηk)N. Note that for any u∈ U N
there is a minimal number kusuch that u∈Ukfor all k≥ku. With some abuse of notation
we can therefore define
lim
k→∞ φηk(u) = lim
k→∞, k≥ku
φηk(u).
Obviously, this limit does not have to exist for every u. For instance, take different recurring
random dictatorships in the sequence of CDS’s. We will show, however, that there exists a
random dictatorship ¯
φsuch that
¯
φ(u) = lim
k→∞ φηk(u)
for all u∈ U Nfor which the limit exists.
We first establish a ‘local’ version of this result in the following theorem. We use the
same notation as in the preceding paragraph.
Theorem 2 Let ˆu∈U1such that lim
k→∞ φηk(ˆu)exists.7Then, lim
k→∞ φηk
j(ˆu)>0implies that
τ(ˆui) = ajfor some i∈I.
Proof: Throughout the proof, we will use the fact that Uk⊂Uk+1 for all k≥1.
Since all subsequences of {φηk}converge on ˆuand U1is finite, we can construct a
subsequence of the given sequence of CDSs which converges on every u∈U1. So we have a
subsequence φ1,k such that φ1(u)≡lim
k→∞ φ1,k(u) exists for every u∈U1.
7If the limit does not exist, then the theorem holds for any convergent subsequence. Note that since each
φk(ˆu) lies in the unit simplex, every such sequence must have a convergent subsequence.
16
We want to show that φ1is a strategy-proof and unanimous CDS on U1. To check
unanimity, pick any u∈U1such that for some aj∈A,τ(ui) = ajfor all i∈I. Then, for
all ηk,φ1,k
j(u) = 1. Hence, φ1
j(u) = limk→∞ φ1,k
j(u) = 1.
We now check that φ1is strategy-proof. Suppose to the contrary that φ1is not strategy-
proof. Then, there are u∈U1, i ∈Iand u0
i∈Uη1such that
M
X
j=1
ui(aj)φ1
j(u0
i, u−i)>
M
X
j=1
ui(aj)φ1
j(u).
But, this contradicts the fact that for each k,
M
X
j=1
ui(aj)φ1,k
j(u0
i, u−i)≤
M
X
j=1
ui(aj)φ1,k
j(u).
Next, since U2is finite, we may construct a subsequence of the sequence φ1,k which
converges on every u∈U2. So we have a subsequence φ2,k such that φ2(u)≡limkφ2,k(u)
exists for every u∈U2. Then, it follows from previous arguments that φ2is a strategy-proof
and unanimous CDS on U2. Also, by construction, φ1and φ2coincide on U1.
Continuing in this way, we construct an infinite sequence φ1, φ2, . . . of CDS’s such that
each φkis a strategy-proof and unanimous CDS on Uk, and coincides with φ`on U`for each
` < k .
Let u∈SkUk. Then u∈Tk≥kuUk, and therefore limk≥kuφk(u) exists, and is in fact
equal to φku(u). Denote this limit by ¯
φ(u). Then it follows that ¯
φis a strategy-proof and
unanimous CDS on SkUk, and therefore from Theorem 1 is a random dictatorship. It follows
in particular that limk→∞ φηk(ˆu) = φ1(ˆu) = ¯
φ(ˆu) has zero probability on the non-maximal
elements.
Now let u∈ UNbe an arbitrary profile such that limk→∞ φηk(u) exists. Since u∈
Tk≥kuUk, it follows from the last paragraph of the proof of Theorem 2 that this limit is
equal to ¯
φ(u) where ¯
φis the random dictatorship constructed there. Hence, we have the
following consequence of (the proof of) Theorem 2.
Corollary 1 For k= 1,2, . . . let {φηk}be a sequence of strategy-proof and unanimous
CDS’s, each one defined on (Uηk)N. Then there exists a random dictatorship ¯
φon UN
such that
¯
φ(u) = lim
k→∞ φηk(u)
for all u∈ U Nfor which this limit exists.
17
Finally, the corollary implies that, when applied to a fixed utility profile, the probabilities
put on non-maximal elements by a converging sequence of unanimous and strategy-proof
CDS’s must converge to zero as the grid size converges to zero. So this provides a partial
answer to the first question raised at the end of the preceding subsection.
5 Conclusion
We have investigated the structure of strategy-proof, cardinal-valued decision schemes sat-
isfying unanimity. One of our contributions is to provide a new and independent proof of
Hylland’s Random Dictatorship Theorem. The other is to establish a limit random dicta-
torship result as the size of the utility grid tends to zero. We believe that it is important to
analyze strategy-proof cardinal schemes in the finite utility grid model because it sheds light
on the role of cardinalization in generating various possibility results. For instance, we would
like to be able to determine the maximum probability that can be placed (by a strategy-proof
cardinal decision scheme) on non-maximal alternatives for any profile, as a function of the
size of the utility grid. It is easy to obtain upper bounds for these probabilities (which vanish
in the limit) by extending the arguments that we have used in the proof of Theorem 1 if we
make the additional assumption that the decison schemes satisfy ex-post efficiency; however
we are unable to show that these bounds are attained. In fact, the class of such cardinal
decision schemes appears to fairly “thin” if there are at least four alternatives. We hope to
able to address these issues in future research.
18
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