We characterize binary decision rules which are independent and strongly paretian,or independent and almost strongly paretian when the individual preferences and the collective preference are weak orders.Binary decision rule, lexicographic dictatorship

JoÃ«l Moulen

Nicolas Gabriel Andjiga

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 3 
  
Arrow's theorem for weak orders 
  
 
 
Nicolas Gabriel Andjiga  
Ecole Normale Supérieure, UY I, Cameroun 
Joël Moulen  
Faculté des sciences, Université de Yaoundé I
Abstract 
We characterize binary decision rules which are independent and strongly paretian,or independent and almost strongly 
paretian when the individual preferences and the collective preference are weak orders.
We are grateful to all members of CREME (University of Yaounde I) for discussions and helpful remarks on previous drafts. 
Citation: Nicolas Gabriel Andjiga and Joël Moulen, (2009) ''Arrow's theorem for weak orders'', Economics Bulletin, Vol. 29 no.3 pp. 
2038-2044. 
Submitted: Jun 06 2009.   Published: August 20, 2009. 
 
     1 Introduction
Arrow￿ s Theorem (Arrow (1951)) states that, when the domain and the codomain of
a binary decision rule (BDR) are the set of weak orders, and when the BDR is paretian
and satis￿es the Independence of Irrelevant Alternatives (IIA) property, there exists a
player i0 such that, whenever i0 strictly prefers an option x to another option y, then the
collectivity also strictly prefers x to y. No information is given to the collective preference
when x and y are equivalent for i0 who is called the dictator. The aim of this note is to
characterize, as a consequence of the Arrow￿ s theorem, two subfamilies of arrovian BDRs
by strengthening the Pareto condition. The result obtained on strong paretian BDR is
also obtained by many other authors in di⁄erent frameworks [see Fishburn (1974), Craven
(1992), Hild (2001), Xu (2003)].
2 Notations and de￿nitions
Let N be a the ￿nite set of n voters and A a ￿nite set of m alternatives with n ￿ 2
and m ￿ 3. We will denote by B the set of binary relations on A; W (resp. WN)
the set of weak orders (transitive and complete binary relations ) on A (resp. the set
of pro￿les of weak orders on A). Given a binary relation R on A and a subset fx;yg
of A , we write Rjfx;yg = xy if x is strictly preferred to y and Rjfx;yg = (xy) if the
indi⁄erence holds between x and y. Moreover, given a subset fx;yg of A and two pro￿les
RN and QN, ￿(x;y;RN) is the set of all voters who strictly prefer x to y; the notation
RNjfx;yg = QNjfx;yg stands for ￿(x;y;RN) = ￿(x;y;QN) and ￿(y;x;RN) = ￿(y;x;QN);
and
￿
RS;QN￿S￿
is the pro￿le where preferences of voters in S are given by RS and
preferences of voters in N ￿S by QN￿S. As usual, 2N is the set of all non empty subsets
of N.
De￿nition 1 Let F be a BDR, that is a mapping F from WN to B.
(i) F satis￿es Independence of Irrelevant Alternatives (IIA) if
8RN;QN 2 WN, 8fx;yg ￿ A : RNjfx;yg = QNjfx;yg =) F(RN)jfx;yg = F(RN)jfx;yg:
(ii) F is paretian if
8RN 2 WN;8fx;yg ￿ A : ￿(x;y;RN) = N =) F(RN)jfx;yg = xy:
(iii) F is almost paretian if
8RN 2 WN;8fx;yg ￿ A : ￿(x;y;RN) = N =) F(RN)jfx;yg 2 fxy;(xy)g:
(iv) F is strongly paretian if 8RN 2 WN;8fx;yg ￿ A :
(iv-a)
￿
￿(x;y;RN) 6= ; and ￿(y;x;RN) = ;
￿
=) F(RN)jfx;yg = xy;
(iv-b) ￿(y;x;RN) = ; =) F(RN)jfx;yg 2 fxy;(xy)g:
(v) F is almost strongly paretian if
8RN 2 WN;8fx;yg ￿ A : ￿(y;x;RN) = ; =) F(RN)jfx;yg 2 fxy;(xy)g:
1De￿nition 2 Let F be a BDR.
(i) F is complete if 8RN 2 WN, F(RN) is complete.
(ii) F is transitive if 8RN 2 WN, F(RN) is transitive.
(iii) F is dictatorial if
9i0 2 N= 8RN 2 WN;8fx;yg ￿ A : Ri0jfx;yg = xy =) F(RN)jfx;yg = xy
(iv) F is strongly dictatorial if 9i0 2 N= 8RN 2 WN;F(RN) = Ri0:
(v) F is a lexicographic dictatorship of order q if there exists q distinct voters
i1;i2;:::;iq such that 8RN 2 WN;8fx;yg ￿ A, 8k 2 f1;2;::;qg :
a)
￿
Rikjfx;yg = xy and 8t < k;Ritjfx;yg = (xy)
￿
=) F(RN)jfx;yg = xy;
b) Rikjfx;yg = (xy); 8k < q =) F(RN)jfx;yg = Rqjfx;yg.
For k 2 f1;2;::;qg, ik will then be called the dictator of order k for F and the decisive
dictator for F given S 2 2N is the dictator for F in S with the smallest order.
(vi) F is a lexicographic dictatorship if F is a lexicographic dictatorship of order n.
(vii) F is null if 8RN 2 WN;8fx;yg ￿ A : F(RN)jfx;yg = (xy).
When F is null, F will be called a lexicographic dictatorship of order q = 0.
3 The main result
Let us recall the well-known Arrow￿ s Theorem :
Theorem 1 (Arrow 1951) If F is paretian, IIA, transitive and complete then F is
dictatorial.
We use theorem 1 to establish the following :
Theorem 2 F is strongly paretian, IIA, transitive and complete if and only if F is a
lexicographic dictatorship.
Proof. (a) Su¢ ciency. Let F be a lexicographic dictatorship.
(a1) Completeness. Let RN 2 LN and fx;yg ￿ A. Suppose that ￿(x;y;RN) [
￿(y;x;RN) = ;. Then Rijfx;yg = (xy) for all i 2 N. By de￿nition of a lexicographic
dictatorship, F
￿
RN￿
jfx;yg = (xy). Now suppose that ￿(x;y;RN)[￿(y;x;RN) 6= ; and let
ip be the decisive dictator for F given ￿(x;y;RN)[￿(y;x;RN). Then Fjfx;yg = Ripjfx;yg.
Since Rip 2 W, F is complete.
(a2) IIA. Let RN;QN 2 LN and fx;yg ￿ A such that RNjfx;yg = QNjfx;yg. Sup-
pose that ￿(x;y;RN) [ ￿(y;x;RN) = ;. Then F
￿
RN￿
jfx;yg = (xy). Since RNjfx;yg =
QNjfx;yg, ￿(x;y;QN) [ ￿(y;x;QN) = ;. Hence F
￿
QN￿
jfx;yg = (xy). Now suppose that
￿(x;y;RN)[￿(y;x;RN) 6= ; and let ip be the decisive dictator for F given ￿(x;y;RN)[
￿(y;x;RN). Then F
￿
RN￿
jfx;yg = Ripjfx;yg. Since RNjfx;yg = QNjfx;yg, ￿(x;y;QN) [
2￿(y;x;QN) 6= ; and ip is also the decisive dictator for F given ￿(x;y;QN) [ ￿(y;x;QN).
Hence F
￿
QN￿
jfx;yg = Qipjfx;yg. Therefore F
￿
RN￿
jfx;yg = F
￿
QN￿
jfx;yg.
(a3) Transitivity. Let RN 2 LN and fx;y;zg ￿ A.
(a3-1) Suppose that F
￿
RN￿
jfx;yg = (xy) and F
￿
RN￿
jfy;zg = (yz). Then by the
de￿nition of a lexicographic dictatorship, Rijfx;yg = (xy) and Rijfy;zg = (yz) for all
i 2 N. By transitivity of individual preferences, Rijfx;zg = (xz) for all i 2 N. Hence
F
￿
RN￿
jfx;zg = (xz).
(a3-2) Suppose that F
￿
RN￿
jfx;yg = (xy) and F
￿
RN￿
jfy;zg = yz. Since F is a
lexicographic dictatorship, Rijfx;yg = (xy) for all i 2 N and ￿(y;z;RN) 6= ;. By transi-
tivity, ￿(y;z;RN) = ￿(x;z;RN) and ￿(z;y;RN) = ￿(z;x;RN). Since F
￿
RN￿
jfy;zg = yz,
the decisive dictator for F given ￿(y;z;RN) [ ￿(z;y;RN) belongs to ￿(y;z;RN). Hence
the decisive dictator for F given ￿(x;z;RN) [ ￿(z;x;RN) belongs to ￿(x;z;RN). Thus
F
￿
RN￿
jfx;zg = xz.
(a3-3) Suppose that F
￿
RN￿
jfx;yg = xy and F
￿
RN￿
jfy;zg = (yz). Since F is a lexi-
cographic dictatorship, Rijfy;zg = (yz) for all i 2 N and ￿(x;y;RN) 6= ;. By transitivity,
￿(x;z;RN) = ￿(x;y;RN) and ￿(z;x;RN) = ￿(y;x;RN). Since F
￿
RN￿
jfx;yg = xy, the
decisive dictator for F given ￿(x;y;RN) [ ￿(y;x;RN) belongs to ￿(x;y;RN). Hence
the decisive dictator for F given ￿(x;z;RN) [ ￿(z;x;RN) belongs to ￿(x;z;RN). Thus
F
￿
RN￿
jfx;zg = xz.
(a3-4) Suppose that F
￿
RN￿
jfx;yg = xy and F
￿
RN￿
jfy;zg = yz. Let ip and iq
be respectively the decisive dictator for F given ￿(x;y;RN) [ ￿(y;x;RN) and given
￿(y;z;RN) [ ￿(z;y;RN). Then ip 2 ￿(x;y;RN) and iq 2 ￿(y;z;RN). Consider ik a dic-
tator for F the order of which is k. Suppose that k < p and k < q. Then Rikjfx;yg = (xy)
and Rikjfy;zg = (yz). By transitivity Rikjfx;zg = (xz). Now suppose that p ￿ q. Then
Ripjfy;zg = yz for p = q or Ripjfy;zg = (yz) for p < q. Hence Ripjfx;zg = xz and ip is the
dictator for F given ￿(x;z;RN)[￿(z;x;RN). Similarly if p > q then Riqjfy;zg = yz and iq
is the decisive dictator given ￿(x;z;RN)[￿(z;x;RN). In both cases, F
￿
RN￿
jfx;zg = xz.
Clearly, F
￿
RN￿
is transitive.
(a4) Strongly Paretian. Consider a pro￿le RN and fx;yg ￿ A.
(a4-1) Suppose that there exists S 2 2N such that ￿(x;y;RN) = S and ￿(y;x;RN) =
;. Then the decisive dictator for F given ￿(x;y;RN)[￿(y;x;RN) belongs to ￿(x;y;RN).
Therefore F
￿
RN￿
jfy;zg = xy.
(a4-2) Suppose that F
￿
RN￿
jfx;yg = xy. By the de￿nition of F the decisive dictator
for F given ￿(x;y;RN) [ ￿(y;x;RN) belongs to ￿(x;y;RN). Therefore ￿(x;y;RN) 6= ;.
(b) Necessity. Conversely let F be a strongly paretian, IIA, transitive and complete BDR.
(b1) Since F is strongly paretian, F is paretian and thus F is paretian, IIA, transitive
and complete. By Theorem 1, F is dictatorial. Hereafter the dictator of F is denoted
by i. Let N1 = N ￿ fi1g and de￿ne the BDR F1 on WN1 by F1
￿
RN1￿
= F
￿
Ifi1g;RN1￿
.
There is no di¢ culty to observe that F1 is strongly paretian, IIA, transitive and complete.
3F1 is therefore paretian, IIA, transitive and complete. By Theorem 1, F1 is dictatorial
and the dictator for F2 is denoted i2.
(b2) Suppose that there exists q voters i1, i2, ..., iq with q < n such that i1 is
the dictator for F and for all k 2 f2;:::;qg, ik is the dictator for Fk￿1 where for all
k 2 f1;2;:::;qg, Fk is the BDR de￿ned on WNk by Fk
￿
RNk￿
= F
￿
Ifi1;i2;:::;ikg;RNk￿
with
Nk = N ￿ fi1;i2;:::;ikg. Since F is strongly paretian, IIA, transitive and complete on
WN, it is easy to observe that Fq is strongly paretian, IIA, transitive and complete on
WNq. By Theorem 1, Fq is dictatorial and the dictator for Fq is denoted by iq+1.
(b3) After n iterations described at (b1) and (b2), F is a lexicographic dictatorship.
Theorem 3 (Arrow bis) If F is almost paretian, IIA, transitive and complete then F
is either dictatorial or null.
Proof. Let F be an almost paretian, IIA, transitive and complete BDR.
Case 1 : Suppose that F is paretian. Then by theorem 1 F is dictatorial.
Case 2 : Suppose that F is not paretian. Then there exists RN 2 LN and fx;yg ￿ A
such that ￿
￿
x;y;RN￿
= N and F
￿
RN￿
jfx;yg 6= xy. Since F is complete and al-
most paretian, F
￿
RN￿
jfx;yg = (xy). Now given QN and z 2 Anfxg, let prove that
F
￿
QN￿
jfx;zg = (xz).
(a) Let prove that for all pro￿le QN and z 2 Anfxg such that ￿
￿
x;z;RN￿
= N,
F
￿
QN￿
jfx;yg = (xy) holds. Suppose that z = y. Then RNjfx;yg = QNjfx;yg and by
IIA, F
￿
QN￿
jfx;yg = (xy). Now suppose that z 2 Anfx;yg. Consider a pro￿le RN
1 at
which for all voters, x is the most preferred alternative, z is the second best and y is
the third best. Then by transitivity of individual preferences, x is strictly preferred to
y by all voters. Therefore RNjfx;yg = RN
1 jfx;yg and by IIA, F
￿
RN
1
￿
jfx;yg = (xy). Also
observe that F is almost paretian, ￿
￿
x;z;RN
1
￿
= N and ￿
￿
z;y;RN
1
￿
= N. Consequently
F
￿
RN
1
￿
jfx;zg 6= zx and F
￿
RN
1
￿
jfz;yg 6= yz. By transitivity of F, F
￿
RN
1
￿
jfz;yg = (yz) and
F
￿
RN
1
￿
jfx;zg = (zx). Since QNjfx;zg = RN
1 jfx;zg and F is IIA, then F
￿
QN￿
jfx;zg = (zx).
(b) Since at (a) above, ￿
￿
z;y;RN
1
￿
= N and F
￿
RN
1
￿
jfz;yg = (yz), then by IIA, for
all pro￿le QN and z 2 Anfx;yg such that ￿
￿
z;y;RN￿
= N, F
￿
QN￿
jfz;yg = (zy) holds.
(c) Let prove that for all pro￿le QN and z 2 Anfx;yg such that ￿
￿
y;z;RN￿
= N,
F
￿
QN￿
jfz;yg = (zy) holds. Consider a pro￿le RN
2 at which for all voters, x is the most
preferred alternative and y is the second best alternative. Then RNjfx;yg = RN
2 jfx;yg
and by IIA, F
￿
RN
2
￿
jfx;yg = (xy). Also observe that RN
1 jfx;zg = RN
2 jfx;zg. Thus by IIA,
F
￿
RN
2
￿
jfz;xg = (xz). Since F is transitive, F
￿
RN
2
￿
jfz;yg = (yz). But QNjfy;zg = RN
2 jfy;zg.
Therefore by IIA, F
￿
QN￿
jfy;zg = (yz).
(d) Let prove that for all pro￿le QN and z 2 Anfxg, F
￿
QN￿
jfz;xg = (zx) holds. Just
consider a pro￿le RN
3 at which for all voter i : x and z are ranked according to Qi and are
strictly preferred to any other alternative in Anfx;zg.
4(d-1) Suppose that z 2 Anfx;yg. Then note that ￿
￿
x;y;RN
3
￿
= N and ￿
￿
z;y;RN
3
￿
=
N. Then by stages (a) and (b) above, F
￿
RN
3
￿
jfy;xg = (yx) and F
￿
RN
3
￿
jfz;yg = (zy).
Since F is transitive, F
￿
RN
3
￿
jfx;zg = (xz). But QNjfx;zg = RN
3 jfz;xg. Therefore by IIA,
F
￿
QN￿
jfz;xg = (zx).
(d-2) Suppose that z = y and consider t 2 Anfx;yg. Then observe that ￿
￿
x;t;RN
3
￿
=
N and ￿
￿
y;t;RN
3
￿
= N. Then by stages (a) and (c) in the present proof, F
￿
RN
3
￿
jft;xg =
(tx) and F
￿
RN
3
￿
jft;yg = (ty). Since F is transitive, F
￿
RN
3
￿
jfx;zg = (xz). But QNjfx;zg =
RN
3 jfz;xg. Therefore by IIA, F
￿
QN￿
jfz;xg = (zx).
To conclude, consider a pro￿le QN and fa;bg ￿ A. First suppose that x 2 fa;bg. Then
by stage (d), F
￿
QN￿
jfa;bg = (ab). Now suppose that x = 2 fa;bg. Then by stage (d),
F
￿
QN￿
jfa;xg = (ax) and F
￿
QN￿
jfb;xg = (bx). By transitivity of F, F
￿
QN￿
jfb;ag = (ba).
As conslusion in case 2, F is null.
Theorem 4 F is almost strongly paretian, IIA, transitive and complete if and only if F
is either null or there exists q ￿ n such that F is a lexicographic dictatorship of order q.
Proof. (i) Su¢ ciency. It is obvious that lexicographic dictatorships of any order q are
almost strongly paretian, IIA, transitive and complete BDRs.
(ii) Necessity. Conversely let F be an almost strongly paretian, IIA, transitive and
complete BDR.
(ii1) Since F is almost strongly paretian , F is almost paretian and thus F is almost
paretian, IIA, transitive and complete. By the Theorem 3, F is either null or F is
dictatorial.
If F is null, the proof ends and F is lexicographic dictatorship of order q = 0. Oth-
erwise F is dictatorial and its dictator is denoted by i1. Let N1 = N ￿ fi1g and de￿ne
the BDR F1 on W N1 by F1
￿
RN1￿
= F
￿
Ifi1g;RN1￿
. It is easy to prove that F1 is almost
strongly paretian, IIA, transitive and complete. F1 is therefore almost paretian, IIA, tran-
sitive and complete. By Theorem 3, F1 is either null or F1 is dictatorial and its dictator
is denoted i2.
If F1 is null, F is lexicographic dictatorship of order 1 and the proof ends. Otherwise
F2 is dictatorial and its dictator is denoted by i2.
(ii2) Suppose that there exists q voters i1, i2, ..., iq with q < n such that i1 is the
dictator of F and for all k 2 f2;:::;qg, ik is the dictator of Fk￿1 where for all k 2
f1;2;:::;qg, Fk is the BDR de￿ned on WNk by Fk
￿
RNk￿
= F
￿
Ifi1;i2;:::;ikg;RNk￿
with
Nk = N ￿fi1;i2;:::;ikg. Since F is almost strongly paretian, IIA, transitive and complete
on WN, it is easy to observe that Fq is almost strongly paretian, IIA, transitive and
complete on WNq. By Theorem 3, Fq is either null or dictatorial.
If Fq is null, F is a lexicographic dictatorship of order q and the proof ends. Otherwise
Fq is dictatorial and its dictator is denoted by iq+1.
5(ii3) By at most n iterations described at (ii1) and (ii2), F is null or is a lexicographic
dictatorship of some order q with 0 ￿ q ￿ n.
References
Arrow, K. J. (1951) Social Choice and Individual Values, Wiley.
Craven, J. (1992) Social Choice: A Framework for Collective Decisions and
Individual Judgments, Cambridge University Press.
Fishburn, P. C. (1974) ￿Lexicographic orders, utilities and decision rules: a survey￿ ,
Management science 20,1442-1471.
Hild, M. (2001) ￿A note on impossibility theorems and seniority rules￿Working paper
SSWP1123, California Institute of Technology.
XU, Y. (2003) ￿On Hierarchical Dictators￿ , Working paper, Georgia State University.
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Arrow's theorem for weak orders

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