Impossibility theorems for preference correspondences based on a new monotonicity concept arediscussed. Here monotonicity means that if preferences update in such a way that they get closerto an outcome then at the new situation this outcome remains chosen. Strong monotonicity requiresfurther that in those cases the outcome at the new profile is a subset of the outcome at the oldprofile. It is shown that only dictatorial preference correspondences are unanimous and stronglymonotone.microeconomics ;

Can Burak

Storcken Ton

English

Research Papers in Economics

Burak Can, Ton Storcken 
 
Impossibilities with Kemeny 
updating 
 
RM/11/049 
  
 
 Impossibilities with Kemeny updating
Burak Can and Ton Storcken
September 2011
Abstract
Impossibility theorems for preference correspondences based on a new
monotonicity concept are discussed. Here monotonicity means that if
preferences update in such a way that they get closer to an outcome then
at the new situation this outcome remains chosen. Strong monotonicity
requires further that in those cases the outcome at the new pro￿le is a
subset of the outcome at the old pro￿le. It is shown that only dictatorial
preference correspondences are unanimous and strongly monotone.
1 Introduction
In social choice theory, it is well-known that monotonicity (Maskin, 1985) leads
to some impossibility theorems. In fact Muller & Satterthwaite (1977) show
that under citizen sovereignty1, any monotone, single-valued social choice rule
is dictatorial. Satterthwaite (1975) relates these impossibilites to the Arrow￿ s
impossibility theorem for welfare functions.
In a recent study, Can & Storcken (2011) introduce ￿update monotonicity￿
for preference rules, i.e. social welfare functions/correspondences2 and shows
the existence of non-trivial welfare rules that satisfy this monotonicity condition.
In this companion paper, we analyze a similar monotonicity condition, Kemeny
update monotonicity, which requires that a rule preserves the outcome assigned
to a pro￿le whenever the pro￿le is changed such that each individual preference
is ￿closer￿ to the initial outcome. The closeness is measured, in particular,
by the Kemeny distance (Kemeny & Snell, 1960). A stronger version of this
distance based monotonicity is also analyzed.
Our study leads to three impossibility theorems regarding these monotonic-
ity conditions. Despite the fact that our ￿ndings are impossibility theorems
for welfare functions/correspondences, the proofs that we introduce di⁄er sub-
stantially from the standard Arrovian framework. The proofs are essentially
based on maximal con￿ icts, i.e., societies that are polarized into two completely
1Citizen sovereignty means for any alternative there exists a pro￿le wherein that alternative
is chosen. This condition is weaker than unanimity and Pareto optimality.
2A correspondence is a rule which is possibly not single-valued and assigns set-valued
outcomes to pro￿les.
1opposite preference. We ￿rst shortly introduce the model, and the properties.
Thereafter, we provide the lemmas which will be used to show three impossibil-
ity theorems later on concerning the Kemeny update monotonicity.
2 Model
2.1 Notation
Given a set of alternatives A; let L denote the set of all linear orders, i.e.,
complete, transitive and antisymmetric binary relations, over A. Let N be the
set of agents and let LN denote the set of all preference pro￿les p, which assign
to every agent i in N a preference p(i) in L. Given a pro￿le p and any subset
of agents S ￿ N, the set of all reported preferences from S at p is denoted
by p(S) = fp(i) : i 2 Sg. Given R 2 L we denote the inverse linear order
as :R = f(y;x) : (x;y) 2 Rg. For R in L and a coalition S ￿ N, a pro￿le
p = (RS;:RN￿S), where p(S) = R and p(N ￿ S) = :R, is called a maximal
con￿ict between S and N ￿ S on R and :R. A preference correspondence or
rule is a function ’ from the set of preference pro￿les LN to the power set of
L, i.e. P(L), such that for all pro￿les p in LN the outcome ’(p) is a non-empty
subset of L. For R1 and R2 in L, the (Kemeny) distance between these two
linear orders is de￿ned by
￿(R1;R2) = #[(R1 ￿ R2) [ (R2 ￿ R1)]=2:
Two linear orders R and R0 are said to be elementary changes if ￿ (R;R0) = 1.
A sequence of linear orders (R0;R1;:::;Rk) is called a path from R0 to Rk if for
all i = 1 to k, Ri￿1 and Ri are elementary changes. A linear order R3 is said to
be between linear orders R1 and R2 if ￿(R1;R2) = ￿(R1;R3) + ￿(R3;R2): That
is if there is a shortest path via R3 from R1 to R2: A subset V of L is convex if
for any R1 and R2 in V , all linear orders between R1 and R2 are also in V:
2.2 Properties of preference rules
Unanimity: A rule ’ is unanimous if ’(RN) = fRg for all linear orders R.
Dictatoriality: A rule ’ is weakly dictatorial if there is a unique agent i, the
dictator, such that p(i) 2 ’(p) for all preference pro￿les p.
Convexity: A rule ’ is convex valued if ’(p) is a convex set for all preference
pro￿les p.
Single valuedness:A rule ’ is single valued if ’(p) is a singleton for all p in
LN:
(Kemeny-update) monotonicity: A rule ’ is Kemeny-update monotone if
for all R in ’(p) and all preference pro￿les q we have that R 2 ’(q); whenever
￿(q(i);R) ￿ ￿(p(i);R) for all i in N.
2A rule ’ is strongly Kemeny-update monotone if for all R in ’(p) and all pref-
erence pro￿les q we have that R 2 ’(q) ￿ ’(p); whenever
￿(q(i);R) ￿ ￿(p(i);R) for all i in N.
3 Results
From here on we refer to the (strong) Kemeny-update monotonicity simply as
(strong) monotonicity. Below are some implications of monotonicity together
with other conditions introduced above.
Lemma 1 Let ’ be monotone and unanimous.
a) Let p 2 LN. Then ’(p) ￿ p(N).
b) Let ’ be, in addition, convex valued. Let p 2 LN be such that for all i and j
in N, ￿ (p(i);p(j)) 6= 1. Then #’(p) = 1.
Proof. a) To the contrary let R 2 ’(p) ￿ p(N). Then there are R 2 L such
that ￿(R;R) = 1 and pro￿les q in LN such that q(i) = R for all i in N. As R 2
’(p) ￿ p(N) it follows that ￿(R;p(i)) ￿ 1 for all i 2 N. Monotonicity therefore
implies R 2 ’(q). Unanimity implies ’(q) = fRg. This yields the contradiction
R = R. b) Assume further that ’ is convex valued and ￿(p(i);p(j)) 6= 1 for
all agents i and j in N. Then, between two di⁄erent reported preferences p(i)
and p(j), there are linear orders which are not in p(N): As ’(p) is convex and
’(p) ￿ p(N) it follows that ’(p) can contain at most one element.
Coalition S ￿ N is winning (on maximal con￿icts) at rule ’ if ’(p) = fRg
for all R 2 L and all p = (RS;:RN￿S). Let W denote the set of coalitions
which are winning at ’:
Lemma 2 Let ’ be monotone and unanimous. Let S ￿ N. Then either S 2 W
or N ￿ S 2 W in each of the following two cases:
a) If ’ is convex valued,
b) If ’ is strongly monotone.
Proof. Let R1 and R2 be an elementary change. By Lemma 1 ’(RS
i ;:R
N￿S
i ) ￿
fRi;:Rig for i 2 f1;2g. Without loss of generality, let R1 2 ’(RS
1;:R
N￿S
1 ).
Because all maximal con￿ icts are connected by distance one preference de-
viations, it is su¢ cient to prove that :R2 = 2 ’(RS
2;:R
N￿S
2 ). Note that
￿(R1;:R2) ￿ 2: Therefore a) in case ’ is convex valued, by Lemma 1 and
monotonicity it follows from R1 2 ’(RS
1;:R
N￿S
1 ) that ’(RS
1;:R
N￿S
2 ) = fR1g:
Hence, :R2 = 2 ’(RS
1;:R
N￿S
2 ): Similarly, b) in case ’ is strongly monotone, the
latter follows because :R2 = 2 ’(RS
1;:R
N￿S
1 ). Now for both cases a) and b)
:R2 = 2 ’(RS
1;:R
N￿S
2 ), then monotonicity yields :R2 = 2 ’(RS
2;:R
N￿S
2 ).
Lemma 3 Let ’ be strongly monotone and unanimous. Then ’ is single valued.
3Proof. Let p be a preference pro￿le and R 2 ’(p): It is su¢ cient to prove
that ’(p) = fRg: Let S = fi 2 N : p(i) = Rg: Take q = (RS; b RN￿S) and
r = (RS;R
N￿S
); where ￿
￿
R;R
￿
= 1, ￿(R; b R) = 1, and R 6= b R. By Lemma
1 R is not in ’(q): As monotonicity implies that R is in both ’(q) and ’(r)
and therewith strong monotonicity implies that ’(q) ￿ ’(r): It follows that
’(r) = fRg and R is not in ’(r): So, monotonicity implies that R is not in
’(:R
S
;R
N￿S
): In view of Lemma 2 S wins on maximal con￿ icts. But then
strong monotonicity implies ’(p) = fRg:
Lemma 4 Let ’ be monotone, unanimous and convex valued. Let S;T 2 W:
Then S \ T 2 W:
Proof. Let R1; R2 and R3 be three linear orders in L such that 2 ￿ ￿(R1;R2) ￿
￿(R1;R3) ￿ ￿(R2;R3) and take fX;Y;Zg = fS \T;N ￿S;S ￿Tg: Take pro￿le
p = (RX
1 ;RY
2 ;RZ
3 ): Suppose S\T = 2 W: Then monotonicity and Lemma 2 imply
that X[Y; X[Z and Y [Z are in W: Lemma 1 yields that ’(p) = fRtg for some
t 2 f1;2;3g: Suppose ’(p) = fR1g. Then Lemma 1 and monotonicity imply
that ’(RX
1 ;RY [Z
2 ) = fR1g: So, R2 = 2 ’(RX
1 ;RY [Z
2 ) and therefore monotonicity
yields that R2 = 2 ’(RY [Z
2 ;:RX
2 ): So, Lemma 1 implies ’(RY [Z
2 ;:RX
2 ) = f:R2g
contradicting Y [ Z is in W. So, ’(p) 6= fR1g: Similarly it follows that ’(p) 6=
fR2g and ’(p) 6= fR3g. This however contradicts ’(p) = fRtg for some t in
f1;2;3g: So, S \ T 2 W:
Theorem 1 A unanimous, convex valued and monotone rule is weakly dicta-
torial.
Proof. Let R = p(j) = 2 ’(p) for some j 2 N and preference pro￿le p: Then
monotonicity implies that R = 2 ’(Rfjg;:RN￿fjg): So, Lemma 2 N ￿ fjg 2 W:
As ; = 2 W; in view of Lemma 4, it follows that for some agent i at all preference
pro￿les q we have that q(i) 2 ’(q): Because rule ’ is convex valued there is at
most one such agent. Therefore, ’ is weakly dictatorial
Theorem 2 A single valued, unanimous and monotone rule is dictatorial.
Proof. Note that a single valued rule is convex valued. So, by Theorem 1
the rule is weakly dictatorial. Let i be such that p(i) 2 ’(p) for all preference
pro￿les p: Then because of single valuedness ’(p) = fp(i)g for all preference
pro￿les and ’ is dictatorial with dictator i:
Theorem 3 A strongly monotone and unanimous rule is dictatorial.
Proof. Follows from Theorem 2 and Lemma 3.
Example 1 Monotone but not strongly monotone rules
Fix a linear order R in L: For arbitrary preference pro￿les p de￿ne rule
’R(p) by ’R(p) = p(N)￿fRg if p(N) 6= fRg and ’R(p) = fRg if p(N) = fRg:
It is straight forward to show that ’R is monotone, unanimous and that it is
neither convex valued nor strongly monotone. At rule ’R agents play the same
role therefore it is not weakly dictatorial and hence also not dictatorial.
44 References
Arrow, K. J. (1963): ￿Social Choice and Individual Values￿ , 2nd ed. New York:
Wiley.
Can, B., Storcken, T. (2011): ￿Update Monotone Preference Rules￿(Work-
ing paper)
Fishburn, P. C. (1982): ￿Monotonicity paradoxes in the theory of elections￿ ,
Discrete Applied Mathematics, 4, Issue 2, 119-134,
Kemeny, J., Snell, L. (1960): ￿Mathematical Models in the Social Sciences￿
Ginn, Boston
Maskin, E. (1985): ￿The Theory of Implementation in Nash Equilibrium,￿in
L. Hurwicz, D. Schmeidler, and H. Sonnenschein, eds., Social Goals and Social
Organization: Essays in Honor of Elisha A. Panzer, Cambridge: Cambridge
University Press.
Muller, E. and M. A. Satterthwaite (1977): ￿The Equivalence of Strong
Positive Association and Strategy-Proofness,￿Journal of Economic Theory, 14,
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Existence and Correspondence Theorems for Voting Procedures and Social Wel-
fare Functions,￿Journal of Economic Theory, 10, 187-217.
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