In this note we define generalized monotonicity which is a generalized version of monotonicity due to Muller and Satterthwaite (1979) for a social choice function under individual preferences which permit indifference, and shall show that generalized monotonicity and strategy-proofness are equivalent.generalized monotonicity

Yasuhito Tanaka

English

Research Papers in Economics

Generalized monotonicity and strategy−proofness: A note 
Yasuhito Tanaka
Faculty of Law, Chuo University, Japan
Abstract
In this note we define generalized monotonicity which is a generalized version of
monotonicity due to Muller and Satterthwaite (1979) for a social choice function under
individual preferences which permit indifference, and shall show that generalized
monotonicity and strategy−proofness are equivalent.
I would like to thank Bhaskar Dutta and the anonymous referee for helpful comments.
Citation: Tanaka, Yasuhito, (2001) "Generalized monotonicity and strategy−proofness: A note." Economics Bulletin, Vol. 4,
No. 11 pp. 1−6
Submitted: June 25, 2001.  Accepted: October 10, 2001.
URL: http://www.economicsbulletin.com/2001/volume4/EB−01D70005A.pdf1 Introduction
In this note we deﬁne generalized monotonicity which is a generalized version
of monotonicity in Muller and Satterthwaite (1979) for a social choice function
under individual preferences which permit indiﬀerence, and we shall show that
generalized monotonicity and strategy-proofness are equivalent.
2 Notation, deﬁnitions and preliminary results
There are a set of individuals N, and a set of alternatives A for a social problem.
The number of individuals n is a ﬁnite positive integer which is larger than or
equal to 2. The number of alternatives is also a ﬁnite positive integer which is
larger than or equal to 3. The individuals are represented by individual i, j and
so on, and the alternatives are represented by x, y, z and so on. The preference
of individual i about the alternatives is represented by a weak order Ri, which is
reﬂexive, complete (connected) and transitive. The asymmetric part (strict pref-
erence) and the symmetric part (indiﬀerence) of Ri are denoted by Pi and Ii. We
allow indiﬀerence in individual preferences.
A social choice function (or voting rule) is a mapping from an n-tuple of re-
ported preferences of the individuals to an alternative. An n-tuple of individual
preferences is called a proﬁle of individual preferences (or an individual prefer-
ence proﬁle). Each proﬁle is denoted by a, b, c and so on. At a proﬁle a, for
example, individual i’s preference is denoted by Ra
i, Pa
i and Ia
i . When a social
choice function chooses x at a proﬁle a, we denote C(a) = x. We call the alter-
native which is chosen by a social choice function the winner of the social choice
function. We consider a resolute social choice function, which chooses only one
of the alternatives at any proﬁle. Further we assume that social choice functions
are non-imposed or onto. It means that for any alternative and any social choice
function there is a proﬁle of individual preferences at which the alternative is cho-
sen by the social choice function.
We deﬁne strategic manipulability and strategy-proofness of a social choice
function.
Strategic manipulability There are two individual preference proﬁles a and b
such as a social choice function chooses x at a and y at b. Between a and
b only the preference of one individual (denoted by i) is diﬀerent (b is an i-
variant of a). Ifindividual i has a preference xPb
iy, thesocial choice function
is strategically manipulable by him at b because he can make the social
1choice function choose x by reporting falsely his preference Ra
i when his
true preference is Rb
i. Similarly, if he has a preference yPa
i x, the social
choice function is strategically manipulable by him at a.
Strategy-proofness If a social choice function is not strategically manipulable
by any individual at any individual preference proﬁle, it is strategy-proof.
Next, we deﬁne generalized monotonicity which is a generalized version of
monotonicity due to Muller and Satterthwaite (1979).
generalized monotonicity There is a proﬁle of individual preferences a such as
for alternatives x and y
(1) individuals in a group V (V ⊂ N): xPa
iy
(2) individuals in a group V0 (V0 ⊂ N, V0 ∩ V = ∅): xIa
i y
(3) others (group V00): yPa
i x
and a social choice function chooses x (C(a) = x). We do not assume any
speciﬁcation of individual preferences about alternatives other than x and y.
There is another proﬁle b such as
(1) individuals in V: xPb
iy, other preferences are not speciﬁed
(2) individuals in V0: xPb
iy or their preferences are the same as those at a
(3) V00: not speciﬁed
Then, the social choice function does not choose y at b (C(b) , y).
Now we show the following lemma.
Lemma 1. Strategy-proofness implies generalized monotonicity.
In the following proof we use notation in the above deﬁnition of generalized
monotonicity.
Proof. Let individuals 1 to m (0 ≤ m ≤ n) belong to V, individuals m + 1 to m0
(m ≤ m0 ≤ n) belong to V0, and individuals m0 + 1 to n belong to V00. Consider
a preference proﬁle c other than a and b such as individuals in V and V0 have a
preference xPc
iyPc
iz, and individuals in V00 have a preference yPc
i xPc
iz, where z is
an arbitrary alternative other than x and y.
2Let a1 be a preference proﬁle such as only the preference of individual 1 has
changed from Ra
1 to Rc
1, and suppose that at a1 the social choice function chooses
an alternative other than x. Then, individual 1 has an incentive to report falsely
his preference Ra
1 when his true preference is Rc
1, and so we have C(a1) = x. By
the same logic we ﬁnd that when the preferences of individuals 1 to m0 change
from Ra
i to Rc
i, the social choice function chooses x (C(am0
) = x). Next, let am0+1
be a preference proﬁle such as the preference of individual m0 + 1, as well as the
preferences of the ﬁrst m0 individuals, has changed from Ra
m0+1 to Rc
m0+1, and sup-
pose that at am0+1 the social choice function chooses y. Then, individual m0 + 1
has an incentive to report falsely his preference Rc
m0+1 when his true preference is
Ra
m0+1 because yPa
m0+1x. On the other hand, if the social choice function chooses an
alternative other than x and y at am0+1, individual m0 + 1 has an incentive to report
falsely his preference Ra
m0+1 when his true preference is Rc
m0+1 because xPc
m0+1z.
Therefore, we have C(am0+1) = x. By the same logic we ﬁnd that when the pref-
erences of all individuals have changed from Ra
i to Rc
i, the social choice function
must choose x (C(c) = x).
Now, suppose that from c to b the individual preferences change one by one
from Rc
i to Rb
i. Then, when the preference of some individual changes, the winner
of the social choice function can not change directly from x to y. If the social
choicefunctionchoosesywhenthepreferenceofanindividualinV orV0 (denoted
by j) changes from Rc
j to Rb
j, individual j has an incentive to report falsely his
preference Rc
j when his true preference is Rb
j because xPb
jy. On the other hand,
if the social choice function chooses y when the preference of an individual in
V00 (denoted by k) changes from Rc
k to Rb
k, individual k has an incentive to report
falsely his preference Rb
k when his true preference is Rc
k because yPc
kx.
It remains the possibility, however, that the winner of the social choice func-
tion changes from x through z(, x,y) to y. Suppose that when the preferences of
someindividualshavechangedfromRc
i toRb
i, thewinnerofthesocialchoicefunc-
tion is z(, x, y), and further when the preference of individual l has changed from
Rc
l to Rb
l, the winner of the social choice function becomes y. Since he prefers y to
z at c, he can get y by misrepresenting his preference Rb
l when his true preference
is Rc
l. Therefore, if the social choice function is strategy-proof, in the sequence
of changes of individual preferences the winner of the social choice function does
not change from x through z to y. Hence, we must have C(b) , y. 
A group V in this lemma may be the set of all individuals, or may be a set
consisting of only one individual.
33 Equivalence of generalized monotonicity and strategy-proofness
In this section we shall show the equivalence of generalized monotonicity and
strategy-proofness.
Theorem 1. Generalized monotonicity and strategy-proofness are equivalent.
Proof. Lemma 1 has shown that strategy-proofness implies generalized mono-
tonicity so that only the converse needs to be proved.
Suppose that at a proﬁle of individual preferences a a social choice function
chooses x (C(a) = x), and assume that the social choice function which satis-
ﬁes generalized monotonicity is strategically manipulable. Then, there is a case
where, when the preference of one individual (denoted by i) changes from Ra
i to
Rb
i (denote such a proﬁle by b), the winner of the social choice function changes
from x to y, and individual i has a preference yPa
i x.
Consider another proﬁle of individual preferences c at which individual i has
a preference yPc
i xPc
iz where z is a arbitrary alternative other than x and y, and the
preferences of the other individuals are the same as those at a. If the social choice
function chooses y at c, since individual i prefers y to x at a and c, generalized
monotonicity implies that the social choice function does not choose x at a. This
contradicts with the assumption, and so y is not chosen at c. Comparing a and
c about x and z, the preferences of individuals other than individual i have not
changed, and individual i has a preference xPc
iz at c and his preference at a about
x and z is not speciﬁed. Therefore, from generalized monotonicity z is not chosen
at c, and so the social choice function must choose x at c.
On the other hand, comparing b and c about x and y, the preferences of in-
dividuals other than individual i have not changed, individual i has a preference
yPc
i x at c, and his preference at b is not speciﬁed. Therefore, from generalized
monotonicity x is not chosen at c. This contradicts with the above result. Hence,
the social choice function must be strategy-proof. 
4 Concluding remarks
The equivalence of strategy-proofness and generalized monotonicity presented in
this paper does not require all preference orderings to exist like as the proof of
the Gibbard-Satterthwaite theorem by Sen(2000) in the case of linear individual
preferences. All that is required is that for all pairs of alternatives x and y there
exists an admissible ordering where x is ranked ﬁrst uniquely and y is ranked
second uniquely.
4We can show the Gibbard-Satterthwaite theorem (Gibbard(1973) and Satterth-
waite(1975)) in the case of individual preferences which permit indiﬀerence using
generalized monotonicity.
References
[1] A. Gibbard, Manipulation of voting schemes: A general result, Economet-
rica 41 (1973), 587-602.
[2] E. Muller and M. A. Satterthwaite, The equivalence of strong positive as-
sociation and strategy-proofness, Journal of Economic Theory 14 (1977),
412-418.
[3] M. A. Satterthwaite, Strategy-proofness and Arrow’s conditions: Existence
and correspondence theorems for voting procedures and social welfare func-
tions, Journal of Economic Theory 10 (1975), 187-217.
5

Manipulation of voting schemes: A general result,

Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions,

The equivalence of strong positive association and strategy-proofness,

Generalized monotonicity and strategy-proofness: A note

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Generalized monotonicity and strategy-proofness: A note

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