Directory UMM :Data Elmu:jurnal:M:Mathematical Social Sciences:Vol40.Issue1.Jul2000:
Mathematical Social Sciences 40 (2000) 63–84
www.elsevier.nl / locate / econbase
Distribution of coalitional power under probabilistic voting
procedures
Shasikanta Nandeibam*
Department of Economics, University of Birmingham, Birmingham B15 2 TT, UK
Received September 1998; received in revised form February 1999; accepted March 1999
Abstract
Pattanaik and Peleg (1986) imposed regularity, ex-post Pareto optimality and independence of
irrelevant alternatives on a probabilistic voting procedure and showed that: (i) the distribution of
coalitional power for decisiveness in two-alternative feasible sets is subadditive in general, but
additive if the universal set has at least four alternatives; and (ii) the distribution of coalitional
power in an arbitrary feasible set is almost complete random dictatorship, and becomes complete
random dictatorship under certain additional conditions. This paper formulates the problem in
terms of citizens’ sovereignty and monotonicity conditions (which are in line with Arrow’s original
work) instead of ex-post Pareto optimality and proves that: (i) Pattanaik and Peleg’s coalitional
weights for decisiveness in two-alternative feasible sets become additive even with only three
alternatives in the universal set; and (ii) the distribution of coalitional power in an arbitrary
feasible set is completely characterized by random dictatorship without the additional conditions
of Pattanaik and Peleg. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Coalitional power; Probabilistic voting procedures; Weak monotonicity; Monotonicity; Strong
monotonicity
1. Introduction
Pattanaik and Peleg (1986) considered the distribution of power among coalitions to
influence the social choice probabilities corresponding to various possible feasible sets
under probabilistic voting procedures that satisfy independence of irrelevant alter-
*Tel.: 144-121-414-6221; fax: 144-121-414-7377.
E-mail address: [email protected] (S. Nandeibam)
0165-4896 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 99 )00039-6
64
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
natives, ex-post Pareto optimality and regularity.1 When individuals only have strict
preference orderings, Pattanaik and Peleg show the existence of a unique weight for each
coalition which can be interpreted as its power or chance to be decisive in every
two-alternative feasible set. These coalitional weights are subadditive in general but
additive if there are four or more alternatives in the universal set. This result can be seen
as the probabilistic analogue of the neutrality property for decisiveness of coalitions over
pairs of alternatives implied by the Arrow conditions. Using the probabilistic neutrality
result, Pattanaik and Peleg derive an almost complete characterization of random
dictatorship. Their characterization is not complete because, unlike the deterministic
framework, there must be four or more alternatives in the universal set. Also, for their
result to hold when the universal set is the feasible set, they need at least two more
alternatives in the universal set than the number of individuals in the society.
An alternative line of research initiated by Barbera and Sonnenschein (1978) and
subsequently pursued by other authors (e.g., McLennan, 1980; Bandyopadhyay et al.,
1982; Barbera and Valenciano, 1983)2 studied the structure of coalitional power under
social welfare schemes. Social welfare schemes are stochastic social decision rules that
map each preference profile to a lottery over social preference orderings.3 Barbera and
Sonnenschein impose probabilistic counterparts of binary independence and Pareto
conditions and derive a probabilistic neutrality result for decisiveness of coalitions over
pairs of alternatives. Their coalitional powers or weights are only subadditive. McLennan (1980) proves additivity when there are at least six alternatives in the universal set.
Although the probabilistic framework is still very similar to the deterministic
framework in the sense that the basic features of Arrow’s impossibility result persists,
albeit in the probabilistic form, imposing only the appropriate probabilistic counterparts
of Arrow’s conditions does not imply an additive coalitional power structure. To
generate additivity and a complete probabilistic counterpart of Arrow’s impossibility
result requires additional conditions. Thus, the earlier works suggest that there may be a
small hope of getting a more flexible distribution of power than complete additivity.
However, the purpose of this paper is to show that this may be a false hope. By resorting
to classical conditions of citizens’ sovereignty and monotonicity instead of the Pareto
condition which keeps us in line with Arrow’s framework, the distribution of coalitional
power becomes fully additive without imposing any of the additional conditions
mentioned in the earlier works. This indicates a much closer parallelism with Arrow’s
theorem than what the earlier works in probabilistic social choice theory seem to
suggest.
We consider the probabilistic voting procedures with linear individual preference
orderings of Pattanaik and Peleg (1986). In Arrow’s framework the primitive conditions
he imposed are collective rationality, binary independence, positive association and
1
A probabilistic voting procedure is the probabilistic counterpart of what is commonly known as a voting
procedure.
2
Although Barbera and Valenciano (1983) actually consider probabilistic voting procedures, their basic
framework is essentially the same as that of Barbera and Sonnenschein (1978) because they implicitly restrict
attention to two-alternative feasible sets.
3
So a social welfare scheme is the probabilistic counterpart of a social welfare function.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
65
citizens’ sovereignty. The Pareto principle, although widely accepted as very appealing,
was derived from the primitive conditions. Our citizens’ sovereignty condition generalizes Arrow’s condition of the same name in a straightforward manner. It says that there
is no pair of alternatives x and y such that the probability of choosing y when only x and
y are feasible is always positive no matter what the preference profile is.
Arrow’s positive association condition essentially requires that, if the only change
from an initial preference profile is to improve the relative position of an alternative x in
the preference profile, then x should remain socially chosen from any feasible set from
which it was chosen initially. In the presence of the other conditions, this is equivalent to
requiring that any alternative y distinct from x should remain socially unchosen if it was
not chosen initially whenever x was also feasible. Our probabilistic version of the former
condition which is called complete weak monotonicity says that, if the only change from
an initial preference profile is to improve the relative position of an alternative x in the
preference profile, then the probability of choosing x whenever it is feasible should not
go down. A straightforward generalization of the latter implied condition would require
that the probability assigned to any alternative y distinct from x should not rise whenever
x is also feasible. We formulate two conditions called complete monotonicity and
complete strong monotonicity which are weaker than this but stronger than complete
weak monotonicity. However, it turns out that, unlike the deterministic case, even these
weaker conditions are not equivalent to complete weak monotonicity in the presence of
the other conditions.
Complete weak monotonicity, complete monotonicity and complete strong monotonicity impose restrictions on the social choice probabilities for every feasible set which
contains the alternative whose relative position in the preference profile has improved.
We also formulate weaker counterparts of these three conditions by requiring the same
restrictions to hold when the universal set itself is the feasible set but not necessarily
when a proper subset of the universal set is the feasible set. These three weaker
conditions corresponding to complete weak monotonicity, complete monotonicity and
complete strong monotonicity are respectively called weak monotonicity, monotonicity
and strong monotonicity. In fact, most of the paper requires only these weaker
conditions.
The probabilistic neutrality result of this paper shows that, if the universal set has at
least three alternatives and the probabilistic voting procedure satisfies regularity,
citizens’ sovereignty, monotonicity and binary independence of irrelevant alternatives,
then there is a system of additive coalitional weights or powers for decisiveness in
two-alternative feasible sets. Using this restriction, our main theorem shows that the
distribution of coalitional power under a probabilistic voting procedure is completely
characterized by a random dictatorship if monotonicity is strengthened to complete
strong monotonicity. The main theorem also shows that our characterization remains
valid if we replace binary independence of irrelevant alternatives and complete strong
monotonicity by independence of irrelevant alternatives and strong monotonicity.
This paper also briefly looks at the strict social welfare schemes of Barbera and
Sonnenschein (1978). We reformulate our conditions in this framework by using the
probabilistic voting procedure induced by a strict social welfare scheme. As a corollary
of our results for probabilistic voting procedures, we show that the following hold even
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S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
with three or more alternatives in the universal set: (a) the system of coalitional weights
of Barbera and Sonnenschein will become additive if their Paretian condition is replaced
by our citizens’ sovereignty and monotonicity conditions; and (b) if the strict social
welfare scheme satisfies citizens’ sovereignty and either (i) binary independence of
irrelevant alternatives and complete strong monotonicity, or (ii) independence of
irrelevant alternatives and strong monotonicity, then the distribution of coalitional power
under it must be such that the induced probabilistic voting procedure is completely
characterized by a random dictatorship.
In Section 2 we introduce the basic framework and provide the definitions of most of
our conditions. Section 3 contains some of the existing relevant results as well as our
probabilistic neutrality result for decisiveness in two-alternative feasible sets. Our main
characterization of random dictatorship which was briefly outlined above is presented in
Section 4. In Section 5 we essentially reformulate our conditions and results in the strict
social welfare scheme framework. We conclude in Section 6.
2. Basic framework and definitions
There are n individuals in the society and m elements in the universal set of social
alternatives. We denote the society by N ( 5 h1, . . . ,nj) and the universal set of social
alternatives by X. Throughout, unless otherwise mentioned, we will maintain the
assumption that ` . uNu 5 n $ 2 and ` . uXu 5 m $ 3. Also, we denote the set of all
nonempty subsets of X by - and the set of all linear orderings on X by +.
The set of all possible preference profiles is the n-fold Cartesian product of + and is
˜ . . . . For each R [ + N , the
ˆ R,
denoted by + N . Preference profiles are denoted by R, R,
ith coordinate of R, denoted by R i , is the preference ordering of individual i in the
preference profile R.
Definition 1. A probabilistic voting procedure (PVP) is a function K: X 3 - 3
+ N → R 1 such that: ox [B K(x,B,R) 5 o x [X K(x,B,R) 5 1 for every (B,R) [ - 3 + N .
So K(x,B,R) is interpreted as the probability of society choosing x when the feasible
set is B and preference profile is R. This means that, if x is not in the feasible set B, then
K assigns zero probability to it.
Definition 2. A PVP K satisfies regularity (R) if, for all B, Bˆ [ - and for all R [ + N :
ˆ
[x [ B # Bˆ ] ⇒ [K(x,B,R) $ K(x,B,R)].
Regularity is a collective rationality property for PVP and it seems quite appealing. It
is a natural probabilistic analogue of a rationality condition due to Chernoff (1954)
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
67
which is known as property a. It is well known that R is a weaker condition than
probabilistic rationalizability which requires the PVP to be induced by a lottery over +.4
Given any (B,R) [ - 3 + N , for each i [ N, we denote the restriction of R i to B by
R i uB and the restriction of R to B by RuB 5 (R 1 uB, . . . ,R n uB). The following definitions
of binary independence of irrelevant alternatives and independence of irrelevant
alternatives are the appropriate counterparts of those in the deterministic framework.
Definition 3. A PVP K satisfies binary independence of irrelevant alternatives (BIIA) if,
for all B [ - with uBu 5 2 and for all R,Rˆ [ + N :
[RuB 5 Rˆ uB] ⇒ [K(x,B,R) 5 K(x,B,Rˆ ) for all x [ B].
Definition 4. A PVP K satisfies independence of irrelevant alternatives (IIA) if, for all
B [ - and for all R,Rˆ [ + N :
[RuB 5 Rˆ uB] ⇒ [K(x,B,R) 5 K(x,B,Rˆ ) for all x [ B].
Definition 5. A PVP K satisfies ex-post Pareto optimality (EPO) if, for all (B,R) [ - 3
+ N and each x [ B, K(x,B,R) 5 0 whenever there exists some y [ B\hxj such that yRi x
for all i [ N.
In the deterministic framework, a social choice rule is said to be imposed if, for some
pair of distinct alternatives x and y, society always chooses x over y no matter what the
social preference profile is. This suggests that we should also consider a probabilistic
social choice rule to be imposed if, for some pair of distinct alternatives x and y, society
always gives a positive chance to x being chosen over y no matter what the social
preference profile is. This consideration motivates the following citizens’ sovereignty
condition.
Definition 6. A PVP K satisfies citizens’ sovereignty (CS) if there does not exists any
distinct x,y [ X such that K(x,hx, yj,R) . 0 for all R [ + N .
Given any x [ X and any R,Rˆ [ + N , let
ˆ
@i (x,R,R)
5 hy [ X: yR i x and xRˆ i yj for each i [ N,
ˆ
ˆ
@ (x,R,R)
5 < @i (x,R,R).
i [N
ˆ
Thus, @i (x,R,R)
contains x and all the alternatives that are preferred to x according to R i
ˆ
but are preferred by x according to Rˆ i and @ (x,R,R)
contains x and every alternative for
ˆ
which there is someone who prefers it to x in R but prefers x to it in R.
In the deterministic social choice literature, the desirability of incorporating some
notion of positive association of social and individual values has motivated the
4
For example, see Lemma 3.13 in Pattanaik and Peleg (1986).
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
68
imposition of some form of monotonicity on social decision rules. Incorporating such
notions of positive association of social and individual values seems equally desirable in
the case of stochastic social decision rules. This has motivated our next three conditions,
namely weak monotonicity, monotonicity and strong monotonicity.
Definition 7. A PVP K satisfies weak monotonicity (WM) if, given any x [ X and any
R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then K(x,X,Rˆ ) $ K(x,X,R).
Definition 8. A
R,Rˆ [ + N , if (i)
xR i y, then: (a)
ˆ
@ (x,R,R)\hxj.
PVP K satisfies monotonicity (M) if, given any x [ X and any
Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xR i y if
K(x,X,Rˆ ) $ K(x,X,R) and (b) K( y,X,Rˆ ) # K( y,X,R) for all y [
Definition 9. A PVP K satisfies strong monotonicity (SM) if, given any x [ X and any
R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
ˆ
xR i y, then: (a) K(x,X,Rˆ ) # K(x,X,R), (b) K( y,X,Rˆ ) # K( y,X,R) for all y [ @ (x,R,R)\hxj
ˆ
and (c) o z [X \ @ (x,R,R)
[K(z,X,R ) 2 K(z,X,R)] # 0.
ˆ
It is quite clear that SM is stronger than M, which in turn is stronger than WM. It is
worth pointing out that WM, M and SM impose restrictions on the PVP only when the
universal set of alternatives X is the feasible set. WM is the appropriate probabilistic
version of the monotonicity or positive association property most widely used in the
deterministic literature. Although the deterministic versions of M and SM are not usually
imposed explicitly, it is straightforward to show that, in the presence of collective
rationalizability and binary independence of irrelevant alternatives or independence of
irrelevant alternatives, the deterministic versions of WM, M and SM are equivalent to
each other. It turns out that this is no longer true in the probabilistic social choice
framework. Thus, for some of our characterizations we need to impose the stronger
conditions M or SM which seems reasonable as they are quite appealing in their own
right.
Given any (B,R) [ - 3 + N , let
G(R i uB) 5 hx [ B: xR i y ;y [ Bj for each i [ N,
b (RuB) 5 < G(Ri uB)
i [N
and
L(x,RuB) 5 hi [ N: x [ G(R i uB)j for each x [ B.
G(R i uB) is the set containing the unique best alternative in B according to R i , b (RuB) is
the set of alternatives that are best in B according to the preferences in R and L(x,RuB) is
the set of individuals who have x as their best alternative in B according to their
respective preferences in R. So L(x,RuB) could be empty for some x [ B.
A PVP is said to be a random dictatorship if each individual has a chance to be the
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
69
dictator and this chance does not vary with the feasible set. To define this formally, let
us first define a dictatorial PVP. For each individual i [ N, the PVP in which i is the
dictator is a function di : X 3 - 3 + N → R 1 such that, for each (x,B,R) [ X 3 - 3 + N ,
di (x,B,R) 5
H10
if hxj 5 G(R i uB),
otherwise.
Thus, for each feasible set, di chooses for sure the best feasible alternative of individual
i.
Definition 10. A PVP K is a random dictatorship (RD) if there exists ai [ [0,1] for each
i [ N such that, for every (B,R) [ - 3 + N :
(a) o i [N ai 5 1 and
(b) K(x,B,R) 5 o i [N aidi (x,B,R) for all x [ B.
3. Probabilistic neutrality
In the deterministic case it is well known that citizens’ sovereignty and positive
association imply the Pareto principle in the presence of collective rationality and binary
independence.5 We begin this section by showing that a similar result holds in the
current framework, namely CS and WM imply EPO in the presence of R and BIIA.
Lemma 1. If a PVP K satisfies R, BIIA, CS and WM, then for all distinct x,y [ X and for
every R [ + N , K( y,hx,yj,R) 5 0 if L(x,Ruhx,yj) 5 N.
Proof. Let x [ X. Using CS, for each y [ X\hxj, let R y [ + N be such that
K( y,hx,yj,R y) 5 0. Let Rˆ [ + N be such that Rˆ uhx,yj 5 R y uhx,yj for all y [ X\hxj. Then
BIIA gives us K( y,hx,yj,Rˆ ) 5 0 for all y [ X\hxj, which together with R imply that
K( y,X,Rˆ ) 5 0 for all y [ X\hxj and K(x,X,Rˆ ) 5 1. Consider R˜ [ + N such that R˜ uX\hxj 5
R̂ uX\hxj and xR˜ i y for all y [ X\hxj and for each i [ N. Then WM implies that
K(x,X,Rˆ ) $ K(x,X,R˜ ) 5 1, which together with R gives us K(x,hx,yj,R˜ ) 5 1 for all
y [ X\hxj. Thus, K( y,hx,yj,R˜ ) 5 0 for all y [ X\hxj. Hence, as L(x,R˜ uhx,yj) 5 N for every
y [ X\hxj, BIIA implies that, for each y [ X\hxj and each R [ + N , K( y,hx,yj,R) 5 0 if
L(x,Ruhx,yj) 5 N. h
Corollary 1. If a PVP K satisfies R, BIIA, CS and WM, then K satisfies EPO.
Proof. Let (B,R) [ - 3 + N and let distinct x,y [ B be such that yR i x for all i [ N. Then
L( y,Ruhx,yj) 5 N and, hence, Lemma 1 implies that K(x,hx,yj,R) 5 0. Therefore, using R,
we have K(x,B,R) # K(x,hx,yj,R) 5 0, which implies that K(x,B,R) 5 0. h
5
For example, see Arrow (1963).
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
70
ˆ If R [ + N is
Lemma 2. Let the PVP K satisfy R and let B,Bˆ [ - be such that B # B.
ˆ
ˆ
such that o x [B K(x,B,R) 5 1, then K(x,B,R) 5 K(x,B,R) for all x [ B.
ˆ
Proof. Let B,Bˆ [ - and R [ + N be such that B # Bˆ and o x [B K(x,B,R)
5 1. We know
ˆ
that o x[B K(x,B,R) 5 1. By R, we also have K(x,B,R) $ K(x,B,R)
for all x [ B. Hence, it
ˆ
must be the case that K(x,B,R)
5 K(x,B,R) for all x [ B. h
When the Arrow conditions are imposed, a deterministic social choice procedure
satisfies a neutrality property. We know that analogues of this neutrality result are
satisfied in the probabilistic framework as well (e.g., Barbera and Sonnenschein, 1978;
McLennan, 1980; Pattanaik and Peleg, 1986). As the basic framework here is the same
as in Pattanaik and Peleg (1986), we can use their probabilistic neutrality result. So, we
summarize their key results on the existence and uniqueness of coalitional weights and
random dictatorship in the first two propositions. Pattanaik and Peleg (1986) actually
imposes IIA, however their proof only uses BIIA, except when they prove that the
coalitional weights are additive. Thus, in Proposition 1 we only require BIIA, except in
Proposition 1(f).
Proposition 1. If a PVP K satisfies R, BIIA and EPO, then:
(a) there exists a K (S) [ [0,1] for each S # N such that, for all distinct x,y [ X and
for all R [ + N , K(x,hx,yj,R) 5 a K (S) if xR i y for all i [ S and yR i x for all i [ N\S;
K
K
(b) for all S # T # N, a (T ) $ a (S);
K
K
(c) for all S # N, a (S) 1 a (N\S) 5 1;
(d) a K (N) 5 1 and a K (5) 5 0;
(e) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) $ a K (S < T );
( f ) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) 5 a K (S < T ) if m $ 4 and K
satisfies IIA.
Proof. For the proof of Proposition 1(a) and (c)–(f) see Pattanaik and Peleg (1986). So,
we only prove Proposition 1(b) here. Let S # T # N. Clearly, a K (T ) $ a K (S) follows
from Proposition 1(a) and (d) if S 5 T or T 5 N or S 5 5. So, suppose that 5 ± S ± T ±
N. Consider distinct x,y,z [ X and R [ + N such that xR i yR i z for all i [ S, yR i xR i z for
all i [ T \S and yR i zR i x for all i [ N\T. As yR i z for all i [ N, EPO implies that
K(x,hx,y,zj,R) 1 K( y,hx,y,zj,R) 5 1.
(1)
Thus, we have the following:
a K (T )
5 K(x,hx,zj,R)
$ K(x,hx,y,zj,R)
5 K(x,hx,yj,R)
5 a K (S)
[by Proposition 1(a)]
[by R]
[by (1) and Lemma 2]
[by Proposition 1(a)]. h
The following is the almost random dictatorship result of Pattanaik and Peleg (1986).6
6
See also McLennan (1980) and Nandeibam (1995) for alternative characterizations of random dictatorship
using strategy proofness.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
71
Proposition 2. If m $ 4 and the PVP K satisfies R, IIA and EPO, then:
(a) for all (B,R) [ - 3 + N with B ± X, K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all
x [ B;
(b) K(x,X,R) 5 o i [N a K (hij)di (x,X,R) for all x [ X if m $ n 1 2.
Proof. See Pattanaik and Peleg (1986). h
From Corollary 1 it is quite obvious that we can replace EPO by citizens’ sovereignty
and weak monotonicity in Propositions 1 and 2. For completeness, we will present here
only one of these results as our next corollary.
Corollary 2. If a PVP K satisfies R, BIIA, CS and WM, then:
(a) there exists a K (S) [ [0,1] for each S # N such that, for all distinct x,y [ X and
for all R [ + N , K(x,hx,yj,R) 5 a K (S) if xR i y for all i [ S and yR i x for all i [ N\S;
(b) for all S # T # N, a K (T ) $ a K (S);
(c) for all S # N, a K (S) 1 a K (N\S) 5 1;
(d) a K (N) 5 1 and a K (5) 5 0;
(e) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) $ a K (S < T );
( f ) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) 5 a K (S < T ) if m $ 4 and K
satisfies IIA.
Proof. Follows from Corollary 1 and Proposition 1. h
The two examples below show that neither m $ 4 can be dropped nor IIA relaxed to
BIIA in Propostion 1(f) and Corollary 2(f).
Example 1. Let m 5 3 and n 5 3 and define the PVP K as follows. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
K(x,B,R) 5
H
(1 / u b (RuB)u, if x [ b (RuB),
0,
otherwise.
K satisfies R, IIA, CS, WM and EPO, but the coalitional weights are not additive
because a K (S) 5 1 / 2 for all coalitions S containing one or two individuals. Thus, the
condition m $ 4 is required in Proposition 1(f) and Corollary 2(f).
Example 2. Let m 5 4 and n 5 3. Let K be the PVP defined by the following rule. For
each (x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if uBu 5 3, u b (RuB)u 5 3 and u b (RuX) > Bu 5 1, then
K(x,B,R) 5
H
1 / 2, if x [ b (RuX),
1 / 4, otherwise;
72
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
(b) if uBu ± 3 or u b (RuB)u , 3 or u b (RuX) > Bu ± 1, then
K(x,B,R) 5
H
(1 / u b (RuB)u, if x [ b (RuB),
0,
otherwise.
It can be checked that K satisfies R, BIIA, CS, WM and EPO. However, a K (S) 5 1 / 2 for
all coalitions S containing one or two individuals. Clearly, K violates IIA. Thus, IIA
cannot be relaxed to BIIA in Proposition 1(f) and Corollary 2(f).
In the probabilistic neutrality results presented so far, namely Proposition 1 and
Corollary 2, we can only interpret the weight of each coalition exactly as its probability
to be decisive in two-alternative feasible sets when the coalitional weights are additive.
Additional conditions have to be imposed to get additivity. This seems to suggest the
possibility of a gap in the parallelism with Arrow’s neutrality result. However, our next
result shows that this possible gap disappears if we impose monotonicity instead of weak
monotonicity.
Proposition 3. If a PVP K satisfies R, BIIA, CS and M, then:
(a) there exists a K (S) [ [0,1] for each S # N such that, for all distinct x,y [ X and
for all R [ + N , K(x,hx,yj,R) 5 a K (S) if xR i y for all i [ S and yR i x for all i [ N\S;
(b) for all S # T # N, a K (T ) $ a K (S);
(c) for all S # N, a K (S) 1 a K (N\S) 5 1;
(d) a K (N) 5 1 and a K (5) 5 0;
(e) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) 5 a K (S < T ).
Proof. As M is stronger than WM, Corollary 2(a)–(e) holds. So, we only need to prove
Proposition 3(e). Note that, by Corollary 1, EPO is satisfied. Let S,T # N be such that
S > T 5 5. If S 5 5 or T 5 5 or S < T 5 N, then a K (S) 1 a K (T ) 5 a K (S < T ) follows
from Proposition 3(c,d). So suppose that S ± 5, T ± 5 and S < T ± N. Consider distinct
x,y,z [ X and R [ + N such that, for every w [ X\hx,y,zj: (i) xR i yR i zR i w if i [ S, (ii)
yR i zR i xR i w if i [ T and (iii) zR i xR i yR i w if i [ N\(S < T ).
Let Rˆ [ + N be such that: (i) Rˆ i 5 R i if i [ S < T and (ii) Rˆ i uX\hyj 5 R i uX\hyj and
ˆ
G(Rˆ i uX) 5 hyj if i [ N\(S < T ). Then x [ @ ( y,R,R)
and M imply that
K(x,X,R) $ K(x,X,Rˆ ).
(2)
Clearly, for all w [ X\hx,yj, yRˆ i w for every i [ N. So, by EPO, we have K(x,X,Rˆ ) 1
K( y,X,Rˆ ) 5 1, which together with Lemma 2 gives us K(x,X,Rˆ ) 5 K(x,hx,yj,Rˆ ). However, by Proposition 3(a), K(x,hx,yj,Rˆ ) 5 a K (S) because xRˆ i y for all i [ S and yRˆ i x for
all i [ N\S. Thus, we have K(x,X,Rˆ ) 5 a K (S), which together with (2) imply that
K(x,X,R) $ a K (S).
Using similar reasoning, we can also conclude that K( y,X,R) $ a K (T ) and K(z,X,R) $
K
a (N\(S < T )). Thus, we have
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
1
$
$
$
5
73
K(x,X,R) 1 K( y,X,R) 1 K(z,X,R)
a K (S) 1 a K (T ) 1 a K (N\(S < T ))
a K (N)
[by Corollary 2(e)]
1
[by Proposition 3(d)].
Therefore, we have
a K (S) 1 a K (T )
5 1 2 a K (N\(S < T ))
5 a K (S < T )
[by Proposition 3(c)]. h
We know that the PVP in Example 1 satisfies R, IIA, CS and WM and the PVP in
Example 2 satisfies R, BIIA, CS and WM. However, in both examples the coalitional
weights are not additive. It can be verified that in both examples the PVP K does not
satisfy M. Thus, Proposition 3(e) does not hold if M is relaxed to WM. We present more
examples below to establish that none of the other conditions in Proposition 3 can be
dropped either.
Example 3. Let m 5 3 and n 5 2. Fix a z [ X and define the PVP K as follows. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if B 5 hzj or z [
⁄ B, then
K(x,B,R) 5 uL(x,RuB)u / 2;
(b) if B ± hzj and z [ B, then
1, if 2 [ L(x,RuB) and uBu 5 2,
K(x,B,R) 5 1, if 1 [ L(x,RuB) and uBu 5 3,
0, otherwise.
5
K satisfies IIA, CS, M and EPO, but there are no coalitional weights. Obviously, K
violates R. Hence, R is essential for Propositions 1 and 3 and Corollary 2.
Example 4. Let m 5 3 and n 5 2. Given any (x,B,R) [ X 3 - 3 + N such that x [ B and
uBu $ 2, for each i [ N, let l(x,R i uB) be defined as follows:
(a) if uBu 5 2, then
l(x,R i uB) 5
H
2, if hxj 5 G(R i uB),
1, otherwise;
(b) if uBu 5 3, then
3, if hxj 5 G(R i uB),
l(x,R i uB) 5 1, if yR i x for all y [ B,
2, otherwise.
5
74
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
Then let K be the PVP defined by the following rule. For each (x,B,R) [ X 3 - 3 + N
such that x [ B:
(a) if uBu 5 1, then K(x,B,R) 5 1;
(b) if uBu 5 2, then K(x,B,R) 5 [ l(x,R 1 uB) 1 l(x,R 2 uB)] / 6;
(c) if uBu 5 3, then K(x,B,R) 5 [ l(x,R 1 uB) 1 l(x,R 2 uB)] / 12.
It can be checked that K satisfies R, IIA and SM, but violates CS and EPO. Clearly,
a K (5) 5 1 / 3, a K (h1j) 5 a K (h2j) 5 1 / 2 and a K (N) 5 2 / 3. Thus, Proposition 1 is no
longer true without EPO and CS is also necessary for Corollary 2 and Proposition 3.
Example 5. Let m 5 3 and n 5 2, and define the PVP K as follows. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if uBu 5 2 and b (RuX) > B ± 5, then
K(x,B,R) 5
H
1 / u b (RuX) > Bu, if x [ b (RuX),
0,
otherwise;
(b) if uBu ± 2 or b (RuX) > B 5 5, then
K(x,B,R) 5
H
1 / u b (RuB)u, if x [ b (RuB),
0,
otherwise.
In this example K satisfies R, CS and SM, but there are no coalitional weights.
Obviously, K violates BIIA. This shows that Propositions 1 and 3 and Corollary 2 do not
remain valid if we drop BIIA.
4. Random dictatorship
The probabilistic neutrality results derived in the previous section provide restrictions
on the distribution of coalitional power to influence the social choice probabilities in
binary choice situations. In this section we use one of them, namely Proposition 3, to
derive restrictions on the distribution of coalitional power to influence the social choice
probabilities when the feasible set is any subset of the universal set X.
Lemma 3. Let the PVP K satisfy R, IIA, CS and M and let (B,R) [ - 3 + N . If B ± X,
then K(x,B,R) $ a K (L(x,RuB)) for all x [ B.
Proof. Let (B,R) [ - 3 + N be such that B ± X and let x [ B. If L(x,RuB) 5 5, then
K(x,B,R) $ 0 5 a K (L(x,RuB)) follows from Proposition 3(d). If L(x,RuB) 5 N, then
Proposition 3(d) and EPO (which is satisfied because of Corollary 1) imply that
K(x,B,R) 5 1 5 a K (L(x,RuB)). So, suppose that 5 ± L(x,RuB) ± N. Since B ± X, let y [
X\B and R˜ [ + N be such that R˜ uB 5 RuB, G(R˜ i uX) 5 hxj for all i [ L(x,RuB) and there
exists j [ N\L(x,RuB) such that xR˜ j y. By R and IIA, we have
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
K(x,B,R) 5 K(x,B,R˜ ) $ K(x,X,R˜ ).
75
(3)
Consider Rˆ [ + N such that Rˆ uX\hyj 5 R˜ uX\hyj, G(Rˆ i uX) 5 hyj for all i [ N\L(x,RuB)
and xRˆ i yRˆ i z for all z [ X\hx,yj and for each i [ L(x,RuB). Clearly, for each z [ X\hx,yj,
yRˆ i z for all i [ N. So, EPO implies that K(x,X,Rˆ ) 1 K( y,X,Rˆ ) 5 1, which together with
Lemma 2 gives us K(x,X,Rˆ ) 5 K(x,hx,yj,Rˆ ). As xRˆ i y for all i [ L(x,RuB) and yRˆ i x for all
i [ N\L(x,RuB), Proposition 3(a) also implies that K(x,hx,yj,Rˆ ) 5 a K (L(x,RuB)). Thus, we
have
K(x,X,Rˆ ) 5 a K (L(x,RuB)).
(4)
ˆ ˜ ). So, by M, we also have K(x,X,R˜ ) $ K(x,X,Rˆ ).
It can be checked that x [ @ ( y,R,R
Hence, (3) and (4) imply that K(x,B,R) $ a K (L(x,RuB)). h
Proposition 4. Let the PVP K satisfy R, IIA, CS and M and let (B,R) [ - 3 + N . If
B ± X, then K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B.
N
Proof. Let (B,R) [ - 3 + be such that B ± X. By Lemma 3, we have K(x,B,R) $
a K (L(x,RuB)) for all x [ B. Clearly, hL(x,RuB): x [ Bj is a partition of N. Then, using
Proposition 3(d,e), we get 1 5 o x[B K(x,B,R) $ o x [B a K (L(x,RuB)) 5 a K (N) 5 1. So it
must be the case that K(x,B,R) 5 a K (L(x,RuB)) for all x [ B. We also know that
a K (L(x,RuB)) 5 o i [L(x,RuB ) a K (hij) 5 o i [N a K (hij)di (x,B,R) for all x [ B by Proposition
3(e). Therefore, K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B. h
It is clear from Examples 3–5 that the random dictatorship result in Proposition 4
does not hold without either R or IIA or CS. It can also be checked that the PVP in
Example 1 violates M. Thus, M cannot be relaxed to WM in Proposition 4. The PVP in
Example 5 violates both IIA and BIIA. So we can ask if IIA can be replaced by BIIA in
Proposition 4.
ˆ ¯ [+N
ˆ ˆ ˆ ˆ j and N 5 h1,2,3j. Also, let Bˆ 5 hx,y,z
ˆ ˆ ˆ j and let R,R
Example 6. Let X 5 hx,y,z,w
ˆ ˆ 1 yR
ˆ ˆ 1 zR
ˆ ˆ 1 w;
ˆ wR
ˆ ˆ 2 yR
ˆ ˆ 2 xR
ˆ ˆ 2 z;
ˆ wR
ˆ ˆ 3 zR
ˆ ˆ 3 xR
ˆ ˆ 3 y;
ˆ xR
ˆ ¯ 1 zR
ˆ ¯ 1 yR
ˆ ¯ 1 w;
ˆ R¯ 2 5 Rˆ 2 and
be such that: xR
R¯ 3 5 Rˆ 3 . Fix any 0 , e , 1 / 3 and define the PVP K by the following rule. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
ˆ ¯ j, then
(a) if B 5 Bˆ and R [ hR,R
K(x,B,R) 5
H
ˆ
(1 1 2e ) / 3, if x 5 x,
(1 2 e ) / 3,
otherwise;
ˆ ¯ j, then
(b) if B ± Bˆ or R [
⁄ hR,R
K(x,B,R) 5 uL(x,RuB)u / 3.
ˆ
It can be verified that K satisfies R, BIIA, CS and SM, but K(x,B,R)
± 1/3 5
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S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
ˆ or R.
¯ Clearly, K violates IIA. Thus, Proposition
a K (L(x,RuBˆ )) for every x [ Bˆ if R is R
4 no longer holds if we relax IIA to BIIA.
Just to confirm that the feasible set cannot be the universal set in Proposition 4, we
provide the following example.
Example 7. Let m 5 3 and n 5 3. Also, for each R [ + N , let
PO(R) 5 hx [ X: there does not exist y [ X\hxj such that yR i x for all i [ Nj.
Define the PVP K as follows. For each (x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if B 5 X and PO(R) 5 X, then K(x,B,R) 5 1 / 3;
(b) if B ± X or PO(R) ± X, then K(x,B,R) 5 uL(x,RuB)u / 3.
In this example K satisfies R, IIA, CS and M. However, it can be checked that, for
R [ + N such that b (RuX) ± PO(R) 5 X, K(x,X,R) 5 1 / 3 and L(x,RuX) 5 5 if x [
X\b (RuX). Thus, we cannot drop the condition B ± X in Proposition 4.
In the deterministic framework, the Arrow conditions imply the neutrality property of
decisiveness over pairs of alternatives which in turn generates the dictatorship theorem.
Similarly, Proposition 3 shows that imposing what looks like the probabilistic counterparts of the Arrow conditions generates the probabilistic neutrality property of
decisiveness over pairs of alternatives. However, unlike in the deterministic framework,
Proposition 4 and Example 7 show that the probabilistic neutrality property derived in
Proposition 3 only implies almost, but not complete, random dictatorship. Thus, it is
natural to investigate the cause of this discrepancy between the deterministic and the
probabilistic frameworks.
It can be shown that the PVP in Example 7 satisfies probabilistic rationalizability. So
we still need the feasible set to be different from the universal set even if we strengthen
regularity to probabilistic rationalizability in Proposition 4. Thus, imposing R rather than
probabilistic rationalizability cannot be the cause of the above-mentioned difference.
We know that WM is widely used in the classical deterministic framework but M and
SM are seldom explicitly imposed. However, we already pointed out that in the
deterministic case the three conditions are equivalent to each other in the presence of
collective rationality and binary independence or independence conditions. The PVP in
Example 1 satisfies R, IIA and WM but not M. The PVP in Example 7 satisfies R, IIA
and M, but it can be verified that it violates SM.7 Thus, WM, M and SM are no longer
equivalent to each other in the probabilistic framework even when R or probabilistic
rationalizability and IIA are imposed. This suggests that, when R, IIA and CS are
imposed, although M is sufficient for the coalitional weights to be additive, it is only
strong enough to derive the almost, but not complete, random dictatorship result of
Proposition 4. Hence, the discrepancy between the two frameworks that we observed
7
The PVP in Example 1 also satisfies probabilistic rationalizability.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
77
above may be caused by the equivalence between WM, M and SM in the deterministic
framework and their non-equivalence in the probabilistic framework in the presence of
the other relevant conditions. Our next result confirms that this is indeed the case,
because the feasible set does not have to be different from the universal set in
Proposition 4 if we replace M by the stronger condition SM.
Proposition 5. If a PVP K satisfies R, IIA, CS and SM and (B,R) [ - 3 + N , then
K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B.
Proof. Let R [ + N . Given Proposition 4, we only need to show that, for each x [ X,
K(x,X,R) 5 o i [N a K (hij)di (x,X,R). As EPO is satisfied by Corollary 1, if x [ X is such
that L(x,RuX) 5 N, then EPO together with Proposition 3(d,e) gives us K(x,X,R) 5 1 5
a K (N) 5 o i [N a K (hij)di (x,X,R). So suppose that L(x,RuX) ± N for every x [ X. Consider
any z [ X. Then it can be verified that there exists y [ X\hzj and Rˆ [ + N such that: (i)
Rˆ uX\hyj 5 RuX\hyj, (ii) G(Rˆ i uX) 5 hyj for all i [ N\L(z,RuX), (iii) zRˆ i yRˆ i w for all
ˆ
ˆ
w [ X\hy,zj and each i [ L(z,RuX) and (iv) either z [ @ ( y,R,R)
or hzj 5 X\@ ( y,R,R).
Thus, by SM, we must have
K(z,X,R) $ K(z,X,Rˆ ).
(5)
It can be checked that yRˆ i w for all i [ N if w [ X\hy,zj. So, EPO implies that
K( y,X,Rˆ ) 1 K(z,X,Rˆ ) 5 1, which together with Lemma 2 gives us
K(z,X,Rˆ ) 5 K(z,hy,zj,Rˆ ).
(6)
It can also be checked that zRˆ i y for all i [ L(z,RuX) and yRˆ i z for all i [ N\L(z,RuX),
which together with Proposition 3(a) imply that K(z,hy,zj,Rˆ ) 5 a K (L(z,RuX)). Thus, (5)
and (6) imply that K(z,X,R) $ a K (L(z,RuX)). Then, as z was arbitrarily chosen from X,
we have K(x,X,R) $ a K (L(x,RuX)) for every x [ X. Since hL(x,RuX): x [ Xj is a partition
of N, Proposition 3(d,e) then imply that 1 5 o x [X K(x,X,R) $ o x[X a K (L(x,RuX)) 5
a K (N) 5 1. Hence, we must have K(x,X,R) 5 a K (L(x,RuX)) for all x [ X. By Proposition
3(e), we also know that a K (L(x,RuX)) 5 o i [L(x, Rux) a K (hij) 5 o i [N a K (hij)di (x,X,R) for all
x [ X. Therefore, K(x,X,R) 5 o i [N a K (hij)di (x,X,R) for all x [ X. h
In deriving our main probabilistic neutrality result, Proposition 3, we only required
BIIA, but we imposed IIA in the random dictatorship results presented so far in this
section. Furthermore, we know from Example 6 that IIA cannot be relaxed to BIIA in
Propositions 4 and 5. However, in the deterministic framework, given the other Arrow
conditions, binary independence is sufficient to derive the dictatorship theorem. Arrow’s
positive association axiom puts restriction on the social choices induced by the social
preference relation for any feasible set that contains the alternative which has moved up
in individual preferences and not necessarily just the universal set. In contrast, WM, M
and SM impose restrictions on the PVP only when the universal set X is the feasible set.
This is the possible reason why BIIA (in conjunction with the other relevant conditions)
is not sufficient to derive the random dictatorship result. In order to verify this, we
strengthen WM, M and SM in an obvious manner that puts restriction on the PVP for
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S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
every feasible set that contains the alternative which has moved up in the individual
preferences.
Definition 11. A PVP K satisfies complete weak monotonicity (CWM) if, given x [ X
and R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then K(x,B,Rˆ ) $ K(x,B,R) for every B [ - with x [ B.
Definition 12. A PVP K satisfies complete monotonicity (CM) if, given x [ X and
R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then for all B [ - with x [ B: (a) K(x,B,Rˆ ) $ K(x,B,R) and (b) K( y,B,Rˆ ) #
ˆ
K( y,B,R) for all y [ B > @ (x,R,R)\hxj.
Definition 13. A PVP K satisfies complete strong monotonicity (CSM) if, given x [ X
and R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then for all B [ - with x [ B: (a) K(x,B,Rˆ ) $ K(x,B,R), (b) K( y,B,Rˆ ) # K( y,B,R)
ˆ
for all y [ B > @ (x,R,R)\hxj
and (c) o z [B >(X \ @ (x,R,R))
[K(z,B,Rˆ ) 2 K(z,B,R)] # 0.
ˆ
Clearly, CSM is stronger than CM, CWM, SM, M and WM, CM is stronger than
CWM, M and WM, and CWM is stronger than WM. We know that the PVP in Example
6 satisfies R, BIIA, CS and SM but violates IIA. It can be verified that this PVP also
satisfies CM but not CSM. This shows that, even if we impose CM, the random
dictatorship results of Propositions 4 and 5 do not remain valid if we relax IIA to BIIA.
This naturally leaves us with only one more possible option, namely strengthen M and
SM to CSM and replace IIA by BIIA and check whether Propositions 4 and 5 remain
valid.
Proposition 6. If a PVP K satisfies R, BIIA, CS and CSM and (B,R) [ - 3 + N , then
K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B.
Proof. Let (B,R) [ - 3 + N . As EPO is satisfied by Corollary 1, if x [ B is such that
L(x,RuB) 5 N, then EPO together with Proposition 3(d,e) imply that K(x,B,R) 5 1 5
a K (N) 5 o i [N a K (hij)di (x,B,R). So suppose that L(x,RuB) ± N for every x [ B. Consider
any z [ B. Then it can be checked that there exists y [ B\hzj and Rˆ [ + N such that: (i)
Rˆ uX\hyj 5 RuX\hyj, (ii) G(Rˆ i uB) 5 hyj for all i [ N\L(z,RuB), (iii) zRˆ i yRˆ i w for all
ˆ
w [ B\hy,zj and each i [ L(z,RuB) and (iv) either z [ @ ( y,R,R)
or hzj 5 B >
ˆ
(X\@ ( y,R,R)).
Thus, using CSM, we get
K(z,B,R) $ K(z,B,Rˆ ).
(7)
It can be checked that yRˆ i w for all i [ N if w [ B\hy, zj. So EPO implies that
K( y,B,Rˆ ) 1 K(z,B,Rˆ ) 5 1, which together with Lemma 2 gives us
K(z,B,Rˆ ) 5 K(z,hy,zj,Rˆ ).
(8)
As zRˆ i y for all i [ L(z,RuB) and yRˆ i z for all i [ N\L(z,RuB), Proposition 3(a) also implies
that K(z,hy,zj,Rˆ ) 5 a K (L(z,RuB)). Combining (7) and (8), we then get K(z,B,R) $
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
79
a K (L(z,RuB)). Thus, as z is any alternative from B, we have K(x,B,R) $ a K (L(x,RuB)) for
all x [ B. Obviously, hL(x,RuB): x [ Bj is a partition of N. Then, using Proposition
3(d,e), we get 1 5 o x[B K(x,B,R) $ o x [B a K (L(x,RuB)) 5 a K (N) 5 1. Hence, we must
have K(x,B,R) 5 a K (L(x,RuB)) for all x [ B. By Proposition 3(e), we also know that
a K (L(x,RuB)) 5 o i [L(x,RuB ) a K (hij) 5 o i [N a K (hij)di (x,B,R) for all x [ B. Therefore, we
have K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B. h
As in the case of WM, M and SM, it is straightforward to show that, in the presence
of collective rationality and binary independence, CWM, CM and CSM are equivalent to
each other in the deterministic framework but not in the probabilistic framework. Thus,
Proposition 6 is not surprising because it shows that, if we want to impose BIIA rather
than IIA and derive the exact probabilistic analogue of the result in the deterministic
framework, then we must explicitly impose CSM.
To formally state our main characterization, we now prove the following straightforward result which establishes the converse of Propositions 5 and 6.
Proposition 7. If a PVP K satisfies RD, then K satisfies R, IIA, CS and CSM.8
Proof. Let the PVP K satisfy RD with individual weights ai [ [0,1] for each i [ N such
that o i [N ai 5 1.
ˆ Clearly, for each i [ N and for all
(R): Let R [ + N and let B,Bˆ [ - with B # B.
ˆ
x [ B, x [
⁄ G(Ri uBˆ ) if x [
⁄ G(Ri uB). So, for each i [ N, di (x,B,R) $ di (x,B,R)
for all x [ B.
ˆ
ˆ
Hence, K(x,B,R) 5 o i [N aidi (x,B,R) $ o i [N aidi (x,B,R) 5 K(x,B,R) for all x [ B. Thus, K
satisfies R.
ˆ ˜ [ + N be such that R˜ uB 5 Rˆ uB. Then it is obvious that
(IIA): Let B [ - and let R,R
di (x,B,Rˆ ) 5 di (x,B,R˜ ) for all x [ B and each i [ N. So, K(x,B,Rˆ ) 5 o i [N aidi (x,B,Rˆ ) 5
o i [N aidi (x,B,R˜ ) 5 K(x,B,R˜ ) for all x [ B. Therefore, K satisfies IIA.
(CS): Given any distinct x,y [ X and any R [ + N such that yR i x for all i [ N, we
have di (x,hx,yj,R) 5 0 for all i [ N and, hence, K(x,hx,yj,R) 5 o i [N aidi (x,hx,yj,R) 5 0.
Thus, K satisfies CS.
(CSM): Let (B,R) [ - 3 + N . Also, let x [ B and Rˆ [ + N be such that Rˆ uX\hxj 5
RuX\hxj and for each y [ X and each i [ N, xRˆ i y if xR i y. Then it can be checked that
L( y,Rˆ uB) # L( y,RuB) for all y [ B\hxj. Thus, K( y,B,Rˆ ) 5 o i [N aidi ( y,B,Rˆ ) 5
o i [L( y,Rˆ uB ) ai # o i [L( y,RuB ) ai 5 o i [N aidi ( y,B,R) 5 K( y,B,R) for all y [ B\hxj, which implies that
K(x,B,Rˆ ) $ K(x,B,R),
ˆ
K( y,B,Rˆ ) # K( y,B,R) for all y [ B > @ (x,R,R)\hxj
and
O
ˆ ))
z [ @ >(X \ @ (x,R,R
8
K(z,B,Rˆ ) #
O
K(z,B,R).
ˆ ))
z [ @ >(X \ @ (x,R,R
In fact, RD implies probabilistic rationalizability as well.
80
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
Hence, K also satisfies CSM. h
Given Proposition 7, it is quite obvious that a randomly dictatorial PVP also satisfies
R, BIIA, CM, CWM, SM, M, WM and EPO. We are now ready to present the main
theorem of this paper which provides alternative characterizations of random dictatorship to those already existing in the literature (e.g., McLennan, 1980; Nandeibam, 1995;
Pattanaik and Peleg, 1986).
Theorem. Let K be a PVP.
(a) K satisfies R, IIA, CS and SM if and only if K satisfies RD.
(b) K satisfies R, BIIA, CS and CSM if and only if K satisfies RD.
Proof. The theorem readily follows from Propositions 5, 6 and 7. h
It is worth emphasizing that our theorem completely characterizes random dictatorship
by using only conditions that are in the spirit of Arrow’s conditions in the classical
deterministic framework. Thus, our theorem can be viewed as filling the gap left in
Pattanaik and Peleg (1986) in the sense that it provides an almost exact probabilistic
analogue of Arrow’s impossibility theorem for voting procedures with strict individual
preference orderings.
5. Randomized social preference
Barbera and Sonnenschein (1978) call the probabilistic analogue of a strict social
welfare function (where both the individual and social preference orderings are strict) a
strict social welfare scheme (SSWS). It maps each strict preference profile to a lottery
over strict social preference orderings. In this section we will briefly examine SSWS and
show that, using our results on PVP, we can derive an almost exact probabilistic
analogue of Arrow’s impossibility theorem for strict social welfare functions. The
coalitional weights for decisiveness in pairwise comparison of alternatives derived by
Barbera and Sonnenschein are only subadditive. McLennan (1980) showed that for it to
be additive there must be at least six alternatives in the universal set. As a corollary of
our Proposition 3, we will show that the coalitional weights of Barbera and Sonnenschein will become additive with three or more alternatives in the universal set if we
add the appropriate stochastic positive association condition.
Definition 14. A strict social welfare scheme (SSWS) is a function g : + N → D( + ),
where D( + ) is the set of all lotteries over +.
For each SSWS g, let Kg : X 3 - 3 + N → R 1 be the PVP induced by g, i.e. Kg is
such that, for every (B,R) [ - 3 + N : Kg (x,B,R) 5 p( g(R),x,B) for all x [ B, where
p( g(R),x,B) is the sum of the probabilities assigned by the lottery g(R) to all those
linear orderings in + that rank x as the best alternative in B.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
81
Given any SSWS g, we can use the induced PVP Kg to translate the conditions from
the PVP framework to the SSWS framework in the obvious manner. For simplicity and
without creating much confusion, we will give the same names to these conditions in the
SSWS framework as their respective counterparts in the PVP framework. Thus, we will
say that a SSWS g satisfies BIIA if Kg satisfies BIIA, g satisfies IIA if Kg satisfies IIA,
and so on.9
We must point out that there is an important caveat in interpreting the random
dictatorship result to be presented in this section as an almost exact probabilistic
analogue of Arrow’s impossibility theorem. Given the above convention, when we say
that a SSWS g satisfies RD, it is not necessarily the case that there exists ai [ [0,1] for
each i [ N such that: (i) o i [N ai 5 1 and (ii) for each R [ + N , the probability assigned
by g(R) to R i is equal to ai for every i [ N. The example below confirms this important
observation.
ˆ ˆ ˆ ˆ j and N 5 h1,2j. Also, let: (i) Rˆ [ + N be such that
Example 8. Let X 5 hx,y,z,w
ˆ ˆ 1 yR
ˆ ˆ 1 zR
ˆ ˆ 1 wˆ and wR
ˆ ˆ 2 zR
ˆ ˆ 2 yR
ˆ ˆ 2 x,
ˆ (ii) R9 [ + be such that xR9zR9yR9w
ˆ ˆ ˆ ˆ and (iii) R¯ [ + be
xR
ˆ ¯ ˆ ¯ ˆ ¯ ˆ Define the SSWS g as follows. For each R [ + N :
such that wRyRzRx.
ˆ then g(R) assigns a probability of 1 / 2 to each of R 1 and R 2 ;
(a) if R ± R,
ˆ then g(R) assigns a probability of 1 / 4 to each of Rˆ 1 , Rˆ 2 , R9 and R.
¯
(b) if R 5 R,
It can be checked that Kg satisfies RD with individual weights a1 5 a2 5 1 / 2 and, hence,
g satisfies RD. However, g(Rˆ ) assigns only a probability of 1 / 4 to each of Rˆ 1 and Rˆ 2 .
We now present the probabilistic neutrality result for SSWS derived in Barbera and
Sonnenschein (1978) and sharpened in McLennan (1980). This shows the existence of
subadditive coalitional weights that become additive if there are at
www.elsevier.nl / locate / econbase
Distribution of coalitional power under probabilistic voting
procedures
Shasikanta Nandeibam*
Department of Economics, University of Birmingham, Birmingham B15 2 TT, UK
Received September 1998; received in revised form February 1999; accepted March 1999
Abstract
Pattanaik and Peleg (1986) imposed regularity, ex-post Pareto optimality and independence of
irrelevant alternatives on a probabilistic voting procedure and showed that: (i) the distribution of
coalitional power for decisiveness in two-alternative feasible sets is subadditive in general, but
additive if the universal set has at least four alternatives; and (ii) the distribution of coalitional
power in an arbitrary feasible set is almost complete random dictatorship, and becomes complete
random dictatorship under certain additional conditions. This paper formulates the problem in
terms of citizens’ sovereignty and monotonicity conditions (which are in line with Arrow’s original
work) instead of ex-post Pareto optimality and proves that: (i) Pattanaik and Peleg’s coalitional
weights for decisiveness in two-alternative feasible sets become additive even with only three
alternatives in the universal set; and (ii) the distribution of coalitional power in an arbitrary
feasible set is completely characterized by random dictatorship without the additional conditions
of Pattanaik and Peleg. 2000 Elsevier Science B.V. All rights reserved.
Keywords: Coalitional power; Probabilistic voting procedures; Weak monotonicity; Monotonicity; Strong
monotonicity
1. Introduction
Pattanaik and Peleg (1986) considered the distribution of power among coalitions to
influence the social choice probabilities corresponding to various possible feasible sets
under probabilistic voting procedures that satisfy independence of irrelevant alter-
*Tel.: 144-121-414-6221; fax: 144-121-414-7377.
E-mail address: [email protected] (S. Nandeibam)
0165-4896 / 00 / $ – see front matter 2000 Elsevier Science B.V. All rights reserved.
PII: S0165-4896( 99 )00039-6
64
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
natives, ex-post Pareto optimality and regularity.1 When individuals only have strict
preference orderings, Pattanaik and Peleg show the existence of a unique weight for each
coalition which can be interpreted as its power or chance to be decisive in every
two-alternative feasible set. These coalitional weights are subadditive in general but
additive if there are four or more alternatives in the universal set. This result can be seen
as the probabilistic analogue of the neutrality property for decisiveness of coalitions over
pairs of alternatives implied by the Arrow conditions. Using the probabilistic neutrality
result, Pattanaik and Peleg derive an almost complete characterization of random
dictatorship. Their characterization is not complete because, unlike the deterministic
framework, there must be four or more alternatives in the universal set. Also, for their
result to hold when the universal set is the feasible set, they need at least two more
alternatives in the universal set than the number of individuals in the society.
An alternative line of research initiated by Barbera and Sonnenschein (1978) and
subsequently pursued by other authors (e.g., McLennan, 1980; Bandyopadhyay et al.,
1982; Barbera and Valenciano, 1983)2 studied the structure of coalitional power under
social welfare schemes. Social welfare schemes are stochastic social decision rules that
map each preference profile to a lottery over social preference orderings.3 Barbera and
Sonnenschein impose probabilistic counterparts of binary independence and Pareto
conditions and derive a probabilistic neutrality result for decisiveness of coalitions over
pairs of alternatives. Their coalitional powers or weights are only subadditive. McLennan (1980) proves additivity when there are at least six alternatives in the universal set.
Although the probabilistic framework is still very similar to the deterministic
framework in the sense that the basic features of Arrow’s impossibility result persists,
albeit in the probabilistic form, imposing only the appropriate probabilistic counterparts
of Arrow’s conditions does not imply an additive coalitional power structure. To
generate additivity and a complete probabilistic counterpart of Arrow’s impossibility
result requires additional conditions. Thus, the earlier works suggest that there may be a
small hope of getting a more flexible distribution of power than complete additivity.
However, the purpose of this paper is to show that this may be a false hope. By resorting
to classical conditions of citizens’ sovereignty and monotonicity instead of the Pareto
condition which keeps us in line with Arrow’s framework, the distribution of coalitional
power becomes fully additive without imposing any of the additional conditions
mentioned in the earlier works. This indicates a much closer parallelism with Arrow’s
theorem than what the earlier works in probabilistic social choice theory seem to
suggest.
We consider the probabilistic voting procedures with linear individual preference
orderings of Pattanaik and Peleg (1986). In Arrow’s framework the primitive conditions
he imposed are collective rationality, binary independence, positive association and
1
A probabilistic voting procedure is the probabilistic counterpart of what is commonly known as a voting
procedure.
2
Although Barbera and Valenciano (1983) actually consider probabilistic voting procedures, their basic
framework is essentially the same as that of Barbera and Sonnenschein (1978) because they implicitly restrict
attention to two-alternative feasible sets.
3
So a social welfare scheme is the probabilistic counterpart of a social welfare function.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
65
citizens’ sovereignty. The Pareto principle, although widely accepted as very appealing,
was derived from the primitive conditions. Our citizens’ sovereignty condition generalizes Arrow’s condition of the same name in a straightforward manner. It says that there
is no pair of alternatives x and y such that the probability of choosing y when only x and
y are feasible is always positive no matter what the preference profile is.
Arrow’s positive association condition essentially requires that, if the only change
from an initial preference profile is to improve the relative position of an alternative x in
the preference profile, then x should remain socially chosen from any feasible set from
which it was chosen initially. In the presence of the other conditions, this is equivalent to
requiring that any alternative y distinct from x should remain socially unchosen if it was
not chosen initially whenever x was also feasible. Our probabilistic version of the former
condition which is called complete weak monotonicity says that, if the only change from
an initial preference profile is to improve the relative position of an alternative x in the
preference profile, then the probability of choosing x whenever it is feasible should not
go down. A straightforward generalization of the latter implied condition would require
that the probability assigned to any alternative y distinct from x should not rise whenever
x is also feasible. We formulate two conditions called complete monotonicity and
complete strong monotonicity which are weaker than this but stronger than complete
weak monotonicity. However, it turns out that, unlike the deterministic case, even these
weaker conditions are not equivalent to complete weak monotonicity in the presence of
the other conditions.
Complete weak monotonicity, complete monotonicity and complete strong monotonicity impose restrictions on the social choice probabilities for every feasible set which
contains the alternative whose relative position in the preference profile has improved.
We also formulate weaker counterparts of these three conditions by requiring the same
restrictions to hold when the universal set itself is the feasible set but not necessarily
when a proper subset of the universal set is the feasible set. These three weaker
conditions corresponding to complete weak monotonicity, complete monotonicity and
complete strong monotonicity are respectively called weak monotonicity, monotonicity
and strong monotonicity. In fact, most of the paper requires only these weaker
conditions.
The probabilistic neutrality result of this paper shows that, if the universal set has at
least three alternatives and the probabilistic voting procedure satisfies regularity,
citizens’ sovereignty, monotonicity and binary independence of irrelevant alternatives,
then there is a system of additive coalitional weights or powers for decisiveness in
two-alternative feasible sets. Using this restriction, our main theorem shows that the
distribution of coalitional power under a probabilistic voting procedure is completely
characterized by a random dictatorship if monotonicity is strengthened to complete
strong monotonicity. The main theorem also shows that our characterization remains
valid if we replace binary independence of irrelevant alternatives and complete strong
monotonicity by independence of irrelevant alternatives and strong monotonicity.
This paper also briefly looks at the strict social welfare schemes of Barbera and
Sonnenschein (1978). We reformulate our conditions in this framework by using the
probabilistic voting procedure induced by a strict social welfare scheme. As a corollary
of our results for probabilistic voting procedures, we show that the following hold even
66
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
with three or more alternatives in the universal set: (a) the system of coalitional weights
of Barbera and Sonnenschein will become additive if their Paretian condition is replaced
by our citizens’ sovereignty and monotonicity conditions; and (b) if the strict social
welfare scheme satisfies citizens’ sovereignty and either (i) binary independence of
irrelevant alternatives and complete strong monotonicity, or (ii) independence of
irrelevant alternatives and strong monotonicity, then the distribution of coalitional power
under it must be such that the induced probabilistic voting procedure is completely
characterized by a random dictatorship.
In Section 2 we introduce the basic framework and provide the definitions of most of
our conditions. Section 3 contains some of the existing relevant results as well as our
probabilistic neutrality result for decisiveness in two-alternative feasible sets. Our main
characterization of random dictatorship which was briefly outlined above is presented in
Section 4. In Section 5 we essentially reformulate our conditions and results in the strict
social welfare scheme framework. We conclude in Section 6.
2. Basic framework and definitions
There are n individuals in the society and m elements in the universal set of social
alternatives. We denote the society by N ( 5 h1, . . . ,nj) and the universal set of social
alternatives by X. Throughout, unless otherwise mentioned, we will maintain the
assumption that ` . uNu 5 n $ 2 and ` . uXu 5 m $ 3. Also, we denote the set of all
nonempty subsets of X by - and the set of all linear orderings on X by +.
The set of all possible preference profiles is the n-fold Cartesian product of + and is
˜ . . . . For each R [ + N , the
ˆ R,
denoted by + N . Preference profiles are denoted by R, R,
ith coordinate of R, denoted by R i , is the preference ordering of individual i in the
preference profile R.
Definition 1. A probabilistic voting procedure (PVP) is a function K: X 3 - 3
+ N → R 1 such that: ox [B K(x,B,R) 5 o x [X K(x,B,R) 5 1 for every (B,R) [ - 3 + N .
So K(x,B,R) is interpreted as the probability of society choosing x when the feasible
set is B and preference profile is R. This means that, if x is not in the feasible set B, then
K assigns zero probability to it.
Definition 2. A PVP K satisfies regularity (R) if, for all B, Bˆ [ - and for all R [ + N :
ˆ
[x [ B # Bˆ ] ⇒ [K(x,B,R) $ K(x,B,R)].
Regularity is a collective rationality property for PVP and it seems quite appealing. It
is a natural probabilistic analogue of a rationality condition due to Chernoff (1954)
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
67
which is known as property a. It is well known that R is a weaker condition than
probabilistic rationalizability which requires the PVP to be induced by a lottery over +.4
Given any (B,R) [ - 3 + N , for each i [ N, we denote the restriction of R i to B by
R i uB and the restriction of R to B by RuB 5 (R 1 uB, . . . ,R n uB). The following definitions
of binary independence of irrelevant alternatives and independence of irrelevant
alternatives are the appropriate counterparts of those in the deterministic framework.
Definition 3. A PVP K satisfies binary independence of irrelevant alternatives (BIIA) if,
for all B [ - with uBu 5 2 and for all R,Rˆ [ + N :
[RuB 5 Rˆ uB] ⇒ [K(x,B,R) 5 K(x,B,Rˆ ) for all x [ B].
Definition 4. A PVP K satisfies independence of irrelevant alternatives (IIA) if, for all
B [ - and for all R,Rˆ [ + N :
[RuB 5 Rˆ uB] ⇒ [K(x,B,R) 5 K(x,B,Rˆ ) for all x [ B].
Definition 5. A PVP K satisfies ex-post Pareto optimality (EPO) if, for all (B,R) [ - 3
+ N and each x [ B, K(x,B,R) 5 0 whenever there exists some y [ B\hxj such that yRi x
for all i [ N.
In the deterministic framework, a social choice rule is said to be imposed if, for some
pair of distinct alternatives x and y, society always chooses x over y no matter what the
social preference profile is. This suggests that we should also consider a probabilistic
social choice rule to be imposed if, for some pair of distinct alternatives x and y, society
always gives a positive chance to x being chosen over y no matter what the social
preference profile is. This consideration motivates the following citizens’ sovereignty
condition.
Definition 6. A PVP K satisfies citizens’ sovereignty (CS) if there does not exists any
distinct x,y [ X such that K(x,hx, yj,R) . 0 for all R [ + N .
Given any x [ X and any R,Rˆ [ + N , let
ˆ
@i (x,R,R)
5 hy [ X: yR i x and xRˆ i yj for each i [ N,
ˆ
ˆ
@ (x,R,R)
5 < @i (x,R,R).
i [N
ˆ
Thus, @i (x,R,R)
contains x and all the alternatives that are preferred to x according to R i
ˆ
but are preferred by x according to Rˆ i and @ (x,R,R)
contains x and every alternative for
ˆ
which there is someone who prefers it to x in R but prefers x to it in R.
In the deterministic social choice literature, the desirability of incorporating some
notion of positive association of social and individual values has motivated the
4
For example, see Lemma 3.13 in Pattanaik and Peleg (1986).
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
68
imposition of some form of monotonicity on social decision rules. Incorporating such
notions of positive association of social and individual values seems equally desirable in
the case of stochastic social decision rules. This has motivated our next three conditions,
namely weak monotonicity, monotonicity and strong monotonicity.
Definition 7. A PVP K satisfies weak monotonicity (WM) if, given any x [ X and any
R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then K(x,X,Rˆ ) $ K(x,X,R).
Definition 8. A
R,Rˆ [ + N , if (i)
xR i y, then: (a)
ˆ
@ (x,R,R)\hxj.
PVP K satisfies monotonicity (M) if, given any x [ X and any
Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xR i y if
K(x,X,Rˆ ) $ K(x,X,R) and (b) K( y,X,Rˆ ) # K( y,X,R) for all y [
Definition 9. A PVP K satisfies strong monotonicity (SM) if, given any x [ X and any
R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
ˆ
xR i y, then: (a) K(x,X,Rˆ ) # K(x,X,R), (b) K( y,X,Rˆ ) # K( y,X,R) for all y [ @ (x,R,R)\hxj
ˆ
and (c) o z [X \ @ (x,R,R)
[K(z,X,R ) 2 K(z,X,R)] # 0.
ˆ
It is quite clear that SM is stronger than M, which in turn is stronger than WM. It is
worth pointing out that WM, M and SM impose restrictions on the PVP only when the
universal set of alternatives X is the feasible set. WM is the appropriate probabilistic
version of the monotonicity or positive association property most widely used in the
deterministic literature. Although the deterministic versions of M and SM are not usually
imposed explicitly, it is straightforward to show that, in the presence of collective
rationalizability and binary independence of irrelevant alternatives or independence of
irrelevant alternatives, the deterministic versions of WM, M and SM are equivalent to
each other. It turns out that this is no longer true in the probabilistic social choice
framework. Thus, for some of our characterizations we need to impose the stronger
conditions M or SM which seems reasonable as they are quite appealing in their own
right.
Given any (B,R) [ - 3 + N , let
G(R i uB) 5 hx [ B: xR i y ;y [ Bj for each i [ N,
b (RuB) 5 < G(Ri uB)
i [N
and
L(x,RuB) 5 hi [ N: x [ G(R i uB)j for each x [ B.
G(R i uB) is the set containing the unique best alternative in B according to R i , b (RuB) is
the set of alternatives that are best in B according to the preferences in R and L(x,RuB) is
the set of individuals who have x as their best alternative in B according to their
respective preferences in R. So L(x,RuB) could be empty for some x [ B.
A PVP is said to be a random dictatorship if each individual has a chance to be the
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
69
dictator and this chance does not vary with the feasible set. To define this formally, let
us first define a dictatorial PVP. For each individual i [ N, the PVP in which i is the
dictator is a function di : X 3 - 3 + N → R 1 such that, for each (x,B,R) [ X 3 - 3 + N ,
di (x,B,R) 5
H10
if hxj 5 G(R i uB),
otherwise.
Thus, for each feasible set, di chooses for sure the best feasible alternative of individual
i.
Definition 10. A PVP K is a random dictatorship (RD) if there exists ai [ [0,1] for each
i [ N such that, for every (B,R) [ - 3 + N :
(a) o i [N ai 5 1 and
(b) K(x,B,R) 5 o i [N aidi (x,B,R) for all x [ B.
3. Probabilistic neutrality
In the deterministic case it is well known that citizens’ sovereignty and positive
association imply the Pareto principle in the presence of collective rationality and binary
independence.5 We begin this section by showing that a similar result holds in the
current framework, namely CS and WM imply EPO in the presence of R and BIIA.
Lemma 1. If a PVP K satisfies R, BIIA, CS and WM, then for all distinct x,y [ X and for
every R [ + N , K( y,hx,yj,R) 5 0 if L(x,Ruhx,yj) 5 N.
Proof. Let x [ X. Using CS, for each y [ X\hxj, let R y [ + N be such that
K( y,hx,yj,R y) 5 0. Let Rˆ [ + N be such that Rˆ uhx,yj 5 R y uhx,yj for all y [ X\hxj. Then
BIIA gives us K( y,hx,yj,Rˆ ) 5 0 for all y [ X\hxj, which together with R imply that
K( y,X,Rˆ ) 5 0 for all y [ X\hxj and K(x,X,Rˆ ) 5 1. Consider R˜ [ + N such that R˜ uX\hxj 5
R̂ uX\hxj and xR˜ i y for all y [ X\hxj and for each i [ N. Then WM implies that
K(x,X,Rˆ ) $ K(x,X,R˜ ) 5 1, which together with R gives us K(x,hx,yj,R˜ ) 5 1 for all
y [ X\hxj. Thus, K( y,hx,yj,R˜ ) 5 0 for all y [ X\hxj. Hence, as L(x,R˜ uhx,yj) 5 N for every
y [ X\hxj, BIIA implies that, for each y [ X\hxj and each R [ + N , K( y,hx,yj,R) 5 0 if
L(x,Ruhx,yj) 5 N. h
Corollary 1. If a PVP K satisfies R, BIIA, CS and WM, then K satisfies EPO.
Proof. Let (B,R) [ - 3 + N and let distinct x,y [ B be such that yR i x for all i [ N. Then
L( y,Ruhx,yj) 5 N and, hence, Lemma 1 implies that K(x,hx,yj,R) 5 0. Therefore, using R,
we have K(x,B,R) # K(x,hx,yj,R) 5 0, which implies that K(x,B,R) 5 0. h
5
For example, see Arrow (1963).
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
70
ˆ If R [ + N is
Lemma 2. Let the PVP K satisfy R and let B,Bˆ [ - be such that B # B.
ˆ
ˆ
such that o x [B K(x,B,R) 5 1, then K(x,B,R) 5 K(x,B,R) for all x [ B.
ˆ
Proof. Let B,Bˆ [ - and R [ + N be such that B # Bˆ and o x [B K(x,B,R)
5 1. We know
ˆ
that o x[B K(x,B,R) 5 1. By R, we also have K(x,B,R) $ K(x,B,R)
for all x [ B. Hence, it
ˆ
must be the case that K(x,B,R)
5 K(x,B,R) for all x [ B. h
When the Arrow conditions are imposed, a deterministic social choice procedure
satisfies a neutrality property. We know that analogues of this neutrality result are
satisfied in the probabilistic framework as well (e.g., Barbera and Sonnenschein, 1978;
McLennan, 1980; Pattanaik and Peleg, 1986). As the basic framework here is the same
as in Pattanaik and Peleg (1986), we can use their probabilistic neutrality result. So, we
summarize their key results on the existence and uniqueness of coalitional weights and
random dictatorship in the first two propositions. Pattanaik and Peleg (1986) actually
imposes IIA, however their proof only uses BIIA, except when they prove that the
coalitional weights are additive. Thus, in Proposition 1 we only require BIIA, except in
Proposition 1(f).
Proposition 1. If a PVP K satisfies R, BIIA and EPO, then:
(a) there exists a K (S) [ [0,1] for each S # N such that, for all distinct x,y [ X and
for all R [ + N , K(x,hx,yj,R) 5 a K (S) if xR i y for all i [ S and yR i x for all i [ N\S;
K
K
(b) for all S # T # N, a (T ) $ a (S);
K
K
(c) for all S # N, a (S) 1 a (N\S) 5 1;
(d) a K (N) 5 1 and a K (5) 5 0;
(e) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) $ a K (S < T );
( f ) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) 5 a K (S < T ) if m $ 4 and K
satisfies IIA.
Proof. For the proof of Proposition 1(a) and (c)–(f) see Pattanaik and Peleg (1986). So,
we only prove Proposition 1(b) here. Let S # T # N. Clearly, a K (T ) $ a K (S) follows
from Proposition 1(a) and (d) if S 5 T or T 5 N or S 5 5. So, suppose that 5 ± S ± T ±
N. Consider distinct x,y,z [ X and R [ + N such that xR i yR i z for all i [ S, yR i xR i z for
all i [ T \S and yR i zR i x for all i [ N\T. As yR i z for all i [ N, EPO implies that
K(x,hx,y,zj,R) 1 K( y,hx,y,zj,R) 5 1.
(1)
Thus, we have the following:
a K (T )
5 K(x,hx,zj,R)
$ K(x,hx,y,zj,R)
5 K(x,hx,yj,R)
5 a K (S)
[by Proposition 1(a)]
[by R]
[by (1) and Lemma 2]
[by Proposition 1(a)]. h
The following is the almost random dictatorship result of Pattanaik and Peleg (1986).6
6
See also McLennan (1980) and Nandeibam (1995) for alternative characterizations of random dictatorship
using strategy proofness.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
71
Proposition 2. If m $ 4 and the PVP K satisfies R, IIA and EPO, then:
(a) for all (B,R) [ - 3 + N with B ± X, K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all
x [ B;
(b) K(x,X,R) 5 o i [N a K (hij)di (x,X,R) for all x [ X if m $ n 1 2.
Proof. See Pattanaik and Peleg (1986). h
From Corollary 1 it is quite obvious that we can replace EPO by citizens’ sovereignty
and weak monotonicity in Propositions 1 and 2. For completeness, we will present here
only one of these results as our next corollary.
Corollary 2. If a PVP K satisfies R, BIIA, CS and WM, then:
(a) there exists a K (S) [ [0,1] for each S # N such that, for all distinct x,y [ X and
for all R [ + N , K(x,hx,yj,R) 5 a K (S) if xR i y for all i [ S and yR i x for all i [ N\S;
(b) for all S # T # N, a K (T ) $ a K (S);
(c) for all S # N, a K (S) 1 a K (N\S) 5 1;
(d) a K (N) 5 1 and a K (5) 5 0;
(e) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) $ a K (S < T );
( f ) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) 5 a K (S < T ) if m $ 4 and K
satisfies IIA.
Proof. Follows from Corollary 1 and Proposition 1. h
The two examples below show that neither m $ 4 can be dropped nor IIA relaxed to
BIIA in Propostion 1(f) and Corollary 2(f).
Example 1. Let m 5 3 and n 5 3 and define the PVP K as follows. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
K(x,B,R) 5
H
(1 / u b (RuB)u, if x [ b (RuB),
0,
otherwise.
K satisfies R, IIA, CS, WM and EPO, but the coalitional weights are not additive
because a K (S) 5 1 / 2 for all coalitions S containing one or two individuals. Thus, the
condition m $ 4 is required in Proposition 1(f) and Corollary 2(f).
Example 2. Let m 5 4 and n 5 3. Let K be the PVP defined by the following rule. For
each (x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if uBu 5 3, u b (RuB)u 5 3 and u b (RuX) > Bu 5 1, then
K(x,B,R) 5
H
1 / 2, if x [ b (RuX),
1 / 4, otherwise;
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S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
(b) if uBu ± 3 or u b (RuB)u , 3 or u b (RuX) > Bu ± 1, then
K(x,B,R) 5
H
(1 / u b (RuB)u, if x [ b (RuB),
0,
otherwise.
It can be checked that K satisfies R, BIIA, CS, WM and EPO. However, a K (S) 5 1 / 2 for
all coalitions S containing one or two individuals. Clearly, K violates IIA. Thus, IIA
cannot be relaxed to BIIA in Proposition 1(f) and Corollary 2(f).
In the probabilistic neutrality results presented so far, namely Proposition 1 and
Corollary 2, we can only interpret the weight of each coalition exactly as its probability
to be decisive in two-alternative feasible sets when the coalitional weights are additive.
Additional conditions have to be imposed to get additivity. This seems to suggest the
possibility of a gap in the parallelism with Arrow’s neutrality result. However, our next
result shows that this possible gap disappears if we impose monotonicity instead of weak
monotonicity.
Proposition 3. If a PVP K satisfies R, BIIA, CS and M, then:
(a) there exists a K (S) [ [0,1] for each S # N such that, for all distinct x,y [ X and
for all R [ + N , K(x,hx,yj,R) 5 a K (S) if xR i y for all i [ S and yR i x for all i [ N\S;
(b) for all S # T # N, a K (T ) $ a K (S);
(c) for all S # N, a K (S) 1 a K (N\S) 5 1;
(d) a K (N) 5 1 and a K (5) 5 0;
(e) for all S,T # N such that S > T 5 5, a K (S) 1 a K (T ) 5 a K (S < T ).
Proof. As M is stronger than WM, Corollary 2(a)–(e) holds. So, we only need to prove
Proposition 3(e). Note that, by Corollary 1, EPO is satisfied. Let S,T # N be such that
S > T 5 5. If S 5 5 or T 5 5 or S < T 5 N, then a K (S) 1 a K (T ) 5 a K (S < T ) follows
from Proposition 3(c,d). So suppose that S ± 5, T ± 5 and S < T ± N. Consider distinct
x,y,z [ X and R [ + N such that, for every w [ X\hx,y,zj: (i) xR i yR i zR i w if i [ S, (ii)
yR i zR i xR i w if i [ T and (iii) zR i xR i yR i w if i [ N\(S < T ).
Let Rˆ [ + N be such that: (i) Rˆ i 5 R i if i [ S < T and (ii) Rˆ i uX\hyj 5 R i uX\hyj and
ˆ
G(Rˆ i uX) 5 hyj if i [ N\(S < T ). Then x [ @ ( y,R,R)
and M imply that
K(x,X,R) $ K(x,X,Rˆ ).
(2)
Clearly, for all w [ X\hx,yj, yRˆ i w for every i [ N. So, by EPO, we have K(x,X,Rˆ ) 1
K( y,X,Rˆ ) 5 1, which together with Lemma 2 gives us K(x,X,Rˆ ) 5 K(x,hx,yj,Rˆ ). However, by Proposition 3(a), K(x,hx,yj,Rˆ ) 5 a K (S) because xRˆ i y for all i [ S and yRˆ i x for
all i [ N\S. Thus, we have K(x,X,Rˆ ) 5 a K (S), which together with (2) imply that
K(x,X,R) $ a K (S).
Using similar reasoning, we can also conclude that K( y,X,R) $ a K (T ) and K(z,X,R) $
K
a (N\(S < T )). Thus, we have
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
1
$
$
$
5
73
K(x,X,R) 1 K( y,X,R) 1 K(z,X,R)
a K (S) 1 a K (T ) 1 a K (N\(S < T ))
a K (N)
[by Corollary 2(e)]
1
[by Proposition 3(d)].
Therefore, we have
a K (S) 1 a K (T )
5 1 2 a K (N\(S < T ))
5 a K (S < T )
[by Proposition 3(c)]. h
We know that the PVP in Example 1 satisfies R, IIA, CS and WM and the PVP in
Example 2 satisfies R, BIIA, CS and WM. However, in both examples the coalitional
weights are not additive. It can be verified that in both examples the PVP K does not
satisfy M. Thus, Proposition 3(e) does not hold if M is relaxed to WM. We present more
examples below to establish that none of the other conditions in Proposition 3 can be
dropped either.
Example 3. Let m 5 3 and n 5 2. Fix a z [ X and define the PVP K as follows. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if B 5 hzj or z [
⁄ B, then
K(x,B,R) 5 uL(x,RuB)u / 2;
(b) if B ± hzj and z [ B, then
1, if 2 [ L(x,RuB) and uBu 5 2,
K(x,B,R) 5 1, if 1 [ L(x,RuB) and uBu 5 3,
0, otherwise.
5
K satisfies IIA, CS, M and EPO, but there are no coalitional weights. Obviously, K
violates R. Hence, R is essential for Propositions 1 and 3 and Corollary 2.
Example 4. Let m 5 3 and n 5 2. Given any (x,B,R) [ X 3 - 3 + N such that x [ B and
uBu $ 2, for each i [ N, let l(x,R i uB) be defined as follows:
(a) if uBu 5 2, then
l(x,R i uB) 5
H
2, if hxj 5 G(R i uB),
1, otherwise;
(b) if uBu 5 3, then
3, if hxj 5 G(R i uB),
l(x,R i uB) 5 1, if yR i x for all y [ B,
2, otherwise.
5
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S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
Then let K be the PVP defined by the following rule. For each (x,B,R) [ X 3 - 3 + N
such that x [ B:
(a) if uBu 5 1, then K(x,B,R) 5 1;
(b) if uBu 5 2, then K(x,B,R) 5 [ l(x,R 1 uB) 1 l(x,R 2 uB)] / 6;
(c) if uBu 5 3, then K(x,B,R) 5 [ l(x,R 1 uB) 1 l(x,R 2 uB)] / 12.
It can be checked that K satisfies R, IIA and SM, but violates CS and EPO. Clearly,
a K (5) 5 1 / 3, a K (h1j) 5 a K (h2j) 5 1 / 2 and a K (N) 5 2 / 3. Thus, Proposition 1 is no
longer true without EPO and CS is also necessary for Corollary 2 and Proposition 3.
Example 5. Let m 5 3 and n 5 2, and define the PVP K as follows. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if uBu 5 2 and b (RuX) > B ± 5, then
K(x,B,R) 5
H
1 / u b (RuX) > Bu, if x [ b (RuX),
0,
otherwise;
(b) if uBu ± 2 or b (RuX) > B 5 5, then
K(x,B,R) 5
H
1 / u b (RuB)u, if x [ b (RuB),
0,
otherwise.
In this example K satisfies R, CS and SM, but there are no coalitional weights.
Obviously, K violates BIIA. This shows that Propositions 1 and 3 and Corollary 2 do not
remain valid if we drop BIIA.
4. Random dictatorship
The probabilistic neutrality results derived in the previous section provide restrictions
on the distribution of coalitional power to influence the social choice probabilities in
binary choice situations. In this section we use one of them, namely Proposition 3, to
derive restrictions on the distribution of coalitional power to influence the social choice
probabilities when the feasible set is any subset of the universal set X.
Lemma 3. Let the PVP K satisfy R, IIA, CS and M and let (B,R) [ - 3 + N . If B ± X,
then K(x,B,R) $ a K (L(x,RuB)) for all x [ B.
Proof. Let (B,R) [ - 3 + N be such that B ± X and let x [ B. If L(x,RuB) 5 5, then
K(x,B,R) $ 0 5 a K (L(x,RuB)) follows from Proposition 3(d). If L(x,RuB) 5 N, then
Proposition 3(d) and EPO (which is satisfied because of Corollary 1) imply that
K(x,B,R) 5 1 5 a K (L(x,RuB)). So, suppose that 5 ± L(x,RuB) ± N. Since B ± X, let y [
X\B and R˜ [ + N be such that R˜ uB 5 RuB, G(R˜ i uX) 5 hxj for all i [ L(x,RuB) and there
exists j [ N\L(x,RuB) such that xR˜ j y. By R and IIA, we have
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
K(x,B,R) 5 K(x,B,R˜ ) $ K(x,X,R˜ ).
75
(3)
Consider Rˆ [ + N such that Rˆ uX\hyj 5 R˜ uX\hyj, G(Rˆ i uX) 5 hyj for all i [ N\L(x,RuB)
and xRˆ i yRˆ i z for all z [ X\hx,yj and for each i [ L(x,RuB). Clearly, for each z [ X\hx,yj,
yRˆ i z for all i [ N. So, EPO implies that K(x,X,Rˆ ) 1 K( y,X,Rˆ ) 5 1, which together with
Lemma 2 gives us K(x,X,Rˆ ) 5 K(x,hx,yj,Rˆ ). As xRˆ i y for all i [ L(x,RuB) and yRˆ i x for all
i [ N\L(x,RuB), Proposition 3(a) also implies that K(x,hx,yj,Rˆ ) 5 a K (L(x,RuB)). Thus, we
have
K(x,X,Rˆ ) 5 a K (L(x,RuB)).
(4)
ˆ ˜ ). So, by M, we also have K(x,X,R˜ ) $ K(x,X,Rˆ ).
It can be checked that x [ @ ( y,R,R
Hence, (3) and (4) imply that K(x,B,R) $ a K (L(x,RuB)). h
Proposition 4. Let the PVP K satisfy R, IIA, CS and M and let (B,R) [ - 3 + N . If
B ± X, then K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B.
N
Proof. Let (B,R) [ - 3 + be such that B ± X. By Lemma 3, we have K(x,B,R) $
a K (L(x,RuB)) for all x [ B. Clearly, hL(x,RuB): x [ Bj is a partition of N. Then, using
Proposition 3(d,e), we get 1 5 o x[B K(x,B,R) $ o x [B a K (L(x,RuB)) 5 a K (N) 5 1. So it
must be the case that K(x,B,R) 5 a K (L(x,RuB)) for all x [ B. We also know that
a K (L(x,RuB)) 5 o i [L(x,RuB ) a K (hij) 5 o i [N a K (hij)di (x,B,R) for all x [ B by Proposition
3(e). Therefore, K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B. h
It is clear from Examples 3–5 that the random dictatorship result in Proposition 4
does not hold without either R or IIA or CS. It can also be checked that the PVP in
Example 1 violates M. Thus, M cannot be relaxed to WM in Proposition 4. The PVP in
Example 5 violates both IIA and BIIA. So we can ask if IIA can be replaced by BIIA in
Proposition 4.
ˆ ¯ [+N
ˆ ˆ ˆ ˆ j and N 5 h1,2,3j. Also, let Bˆ 5 hx,y,z
ˆ ˆ ˆ j and let R,R
Example 6. Let X 5 hx,y,z,w
ˆ ˆ 1 yR
ˆ ˆ 1 zR
ˆ ˆ 1 w;
ˆ wR
ˆ ˆ 2 yR
ˆ ˆ 2 xR
ˆ ˆ 2 z;
ˆ wR
ˆ ˆ 3 zR
ˆ ˆ 3 xR
ˆ ˆ 3 y;
ˆ xR
ˆ ¯ 1 zR
ˆ ¯ 1 yR
ˆ ¯ 1 w;
ˆ R¯ 2 5 Rˆ 2 and
be such that: xR
R¯ 3 5 Rˆ 3 . Fix any 0 , e , 1 / 3 and define the PVP K by the following rule. For each
(x,B,R) [ X 3 - 3 + N such that x [ B:
ˆ ¯ j, then
(a) if B 5 Bˆ and R [ hR,R
K(x,B,R) 5
H
ˆ
(1 1 2e ) / 3, if x 5 x,
(1 2 e ) / 3,
otherwise;
ˆ ¯ j, then
(b) if B ± Bˆ or R [
⁄ hR,R
K(x,B,R) 5 uL(x,RuB)u / 3.
ˆ
It can be verified that K satisfies R, BIIA, CS and SM, but K(x,B,R)
± 1/3 5
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S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
ˆ or R.
¯ Clearly, K violates IIA. Thus, Proposition
a K (L(x,RuBˆ )) for every x [ Bˆ if R is R
4 no longer holds if we relax IIA to BIIA.
Just to confirm that the feasible set cannot be the universal set in Proposition 4, we
provide the following example.
Example 7. Let m 5 3 and n 5 3. Also, for each R [ + N , let
PO(R) 5 hx [ X: there does not exist y [ X\hxj such that yR i x for all i [ Nj.
Define the PVP K as follows. For each (x,B,R) [ X 3 - 3 + N such that x [ B:
(a) if B 5 X and PO(R) 5 X, then K(x,B,R) 5 1 / 3;
(b) if B ± X or PO(R) ± X, then K(x,B,R) 5 uL(x,RuB)u / 3.
In this example K satisfies R, IIA, CS and M. However, it can be checked that, for
R [ + N such that b (RuX) ± PO(R) 5 X, K(x,X,R) 5 1 / 3 and L(x,RuX) 5 5 if x [
X\b (RuX). Thus, we cannot drop the condition B ± X in Proposition 4.
In the deterministic framework, the Arrow conditions imply the neutrality property of
decisiveness over pairs of alternatives which in turn generates the dictatorship theorem.
Similarly, Proposition 3 shows that imposing what looks like the probabilistic counterparts of the Arrow conditions generates the probabilistic neutrality property of
decisiveness over pairs of alternatives. However, unlike in the deterministic framework,
Proposition 4 and Example 7 show that the probabilistic neutrality property derived in
Proposition 3 only implies almost, but not complete, random dictatorship. Thus, it is
natural to investigate the cause of this discrepancy between the deterministic and the
probabilistic frameworks.
It can be shown that the PVP in Example 7 satisfies probabilistic rationalizability. So
we still need the feasible set to be different from the universal set even if we strengthen
regularity to probabilistic rationalizability in Proposition 4. Thus, imposing R rather than
probabilistic rationalizability cannot be the cause of the above-mentioned difference.
We know that WM is widely used in the classical deterministic framework but M and
SM are seldom explicitly imposed. However, we already pointed out that in the
deterministic case the three conditions are equivalent to each other in the presence of
collective rationality and binary independence or independence conditions. The PVP in
Example 1 satisfies R, IIA and WM but not M. The PVP in Example 7 satisfies R, IIA
and M, but it can be verified that it violates SM.7 Thus, WM, M and SM are no longer
equivalent to each other in the probabilistic framework even when R or probabilistic
rationalizability and IIA are imposed. This suggests that, when R, IIA and CS are
imposed, although M is sufficient for the coalitional weights to be additive, it is only
strong enough to derive the almost, but not complete, random dictatorship result of
Proposition 4. Hence, the discrepancy between the two frameworks that we observed
7
The PVP in Example 1 also satisfies probabilistic rationalizability.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
77
above may be caused by the equivalence between WM, M and SM in the deterministic
framework and their non-equivalence in the probabilistic framework in the presence of
the other relevant conditions. Our next result confirms that this is indeed the case,
because the feasible set does not have to be different from the universal set in
Proposition 4 if we replace M by the stronger condition SM.
Proposition 5. If a PVP K satisfies R, IIA, CS and SM and (B,R) [ - 3 + N , then
K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B.
Proof. Let R [ + N . Given Proposition 4, we only need to show that, for each x [ X,
K(x,X,R) 5 o i [N a K (hij)di (x,X,R). As EPO is satisfied by Corollary 1, if x [ X is such
that L(x,RuX) 5 N, then EPO together with Proposition 3(d,e) gives us K(x,X,R) 5 1 5
a K (N) 5 o i [N a K (hij)di (x,X,R). So suppose that L(x,RuX) ± N for every x [ X. Consider
any z [ X. Then it can be verified that there exists y [ X\hzj and Rˆ [ + N such that: (i)
Rˆ uX\hyj 5 RuX\hyj, (ii) G(Rˆ i uX) 5 hyj for all i [ N\L(z,RuX), (iii) zRˆ i yRˆ i w for all
ˆ
ˆ
w [ X\hy,zj and each i [ L(z,RuX) and (iv) either z [ @ ( y,R,R)
or hzj 5 X\@ ( y,R,R).
Thus, by SM, we must have
K(z,X,R) $ K(z,X,Rˆ ).
(5)
It can be checked that yRˆ i w for all i [ N if w [ X\hy,zj. So, EPO implies that
K( y,X,Rˆ ) 1 K(z,X,Rˆ ) 5 1, which together with Lemma 2 gives us
K(z,X,Rˆ ) 5 K(z,hy,zj,Rˆ ).
(6)
It can also be checked that zRˆ i y for all i [ L(z,RuX) and yRˆ i z for all i [ N\L(z,RuX),
which together with Proposition 3(a) imply that K(z,hy,zj,Rˆ ) 5 a K (L(z,RuX)). Thus, (5)
and (6) imply that K(z,X,R) $ a K (L(z,RuX)). Then, as z was arbitrarily chosen from X,
we have K(x,X,R) $ a K (L(x,RuX)) for every x [ X. Since hL(x,RuX): x [ Xj is a partition
of N, Proposition 3(d,e) then imply that 1 5 o x [X K(x,X,R) $ o x[X a K (L(x,RuX)) 5
a K (N) 5 1. Hence, we must have K(x,X,R) 5 a K (L(x,RuX)) for all x [ X. By Proposition
3(e), we also know that a K (L(x,RuX)) 5 o i [L(x, Rux) a K (hij) 5 o i [N a K (hij)di (x,X,R) for all
x [ X. Therefore, K(x,X,R) 5 o i [N a K (hij)di (x,X,R) for all x [ X. h
In deriving our main probabilistic neutrality result, Proposition 3, we only required
BIIA, but we imposed IIA in the random dictatorship results presented so far in this
section. Furthermore, we know from Example 6 that IIA cannot be relaxed to BIIA in
Propositions 4 and 5. However, in the deterministic framework, given the other Arrow
conditions, binary independence is sufficient to derive the dictatorship theorem. Arrow’s
positive association axiom puts restriction on the social choices induced by the social
preference relation for any feasible set that contains the alternative which has moved up
in individual preferences and not necessarily just the universal set. In contrast, WM, M
and SM impose restrictions on the PVP only when the universal set X is the feasible set.
This is the possible reason why BIIA (in conjunction with the other relevant conditions)
is not sufficient to derive the random dictatorship result. In order to verify this, we
strengthen WM, M and SM in an obvious manner that puts restriction on the PVP for
78
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
every feasible set that contains the alternative which has moved up in the individual
preferences.
Definition 11. A PVP K satisfies complete weak monotonicity (CWM) if, given x [ X
and R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then K(x,B,Rˆ ) $ K(x,B,R) for every B [ - with x [ B.
Definition 12. A PVP K satisfies complete monotonicity (CM) if, given x [ X and
R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then for all B [ - with x [ B: (a) K(x,B,Rˆ ) $ K(x,B,R) and (b) K( y,B,Rˆ ) #
ˆ
K( y,B,R) for all y [ B > @ (x,R,R)\hxj.
Definition 13. A PVP K satisfies complete strong monotonicity (CSM) if, given x [ X
and R,Rˆ [ + N , if (i) Rˆ uX\hxj 5 RuX\hxj and (ii) for each i [ N and for all y [ X, xRˆ i y if
xR i y, then for all B [ - with x [ B: (a) K(x,B,Rˆ ) $ K(x,B,R), (b) K( y,B,Rˆ ) # K( y,B,R)
ˆ
for all y [ B > @ (x,R,R)\hxj
and (c) o z [B >(X \ @ (x,R,R))
[K(z,B,Rˆ ) 2 K(z,B,R)] # 0.
ˆ
Clearly, CSM is stronger than CM, CWM, SM, M and WM, CM is stronger than
CWM, M and WM, and CWM is stronger than WM. We know that the PVP in Example
6 satisfies R, BIIA, CS and SM but violates IIA. It can be verified that this PVP also
satisfies CM but not CSM. This shows that, even if we impose CM, the random
dictatorship results of Propositions 4 and 5 do not remain valid if we relax IIA to BIIA.
This naturally leaves us with only one more possible option, namely strengthen M and
SM to CSM and replace IIA by BIIA and check whether Propositions 4 and 5 remain
valid.
Proposition 6. If a PVP K satisfies R, BIIA, CS and CSM and (B,R) [ - 3 + N , then
K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B.
Proof. Let (B,R) [ - 3 + N . As EPO is satisfied by Corollary 1, if x [ B is such that
L(x,RuB) 5 N, then EPO together with Proposition 3(d,e) imply that K(x,B,R) 5 1 5
a K (N) 5 o i [N a K (hij)di (x,B,R). So suppose that L(x,RuB) ± N for every x [ B. Consider
any z [ B. Then it can be checked that there exists y [ B\hzj and Rˆ [ + N such that: (i)
Rˆ uX\hyj 5 RuX\hyj, (ii) G(Rˆ i uB) 5 hyj for all i [ N\L(z,RuB), (iii) zRˆ i yRˆ i w for all
ˆ
w [ B\hy,zj and each i [ L(z,RuB) and (iv) either z [ @ ( y,R,R)
or hzj 5 B >
ˆ
(X\@ ( y,R,R)).
Thus, using CSM, we get
K(z,B,R) $ K(z,B,Rˆ ).
(7)
It can be checked that yRˆ i w for all i [ N if w [ B\hy, zj. So EPO implies that
K( y,B,Rˆ ) 1 K(z,B,Rˆ ) 5 1, which together with Lemma 2 gives us
K(z,B,Rˆ ) 5 K(z,hy,zj,Rˆ ).
(8)
As zRˆ i y for all i [ L(z,RuB) and yRˆ i z for all i [ N\L(z,RuB), Proposition 3(a) also implies
that K(z,hy,zj,Rˆ ) 5 a K (L(z,RuB)). Combining (7) and (8), we then get K(z,B,R) $
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
79
a K (L(z,RuB)). Thus, as z is any alternative from B, we have K(x,B,R) $ a K (L(x,RuB)) for
all x [ B. Obviously, hL(x,RuB): x [ Bj is a partition of N. Then, using Proposition
3(d,e), we get 1 5 o x[B K(x,B,R) $ o x [B a K (L(x,RuB)) 5 a K (N) 5 1. Hence, we must
have K(x,B,R) 5 a K (L(x,RuB)) for all x [ B. By Proposition 3(e), we also know that
a K (L(x,RuB)) 5 o i [L(x,RuB ) a K (hij) 5 o i [N a K (hij)di (x,B,R) for all x [ B. Therefore, we
have K(x,B,R) 5 o i [N a K (hij)di (x,B,R) for all x [ B. h
As in the case of WM, M and SM, it is straightforward to show that, in the presence
of collective rationality and binary independence, CWM, CM and CSM are equivalent to
each other in the deterministic framework but not in the probabilistic framework. Thus,
Proposition 6 is not surprising because it shows that, if we want to impose BIIA rather
than IIA and derive the exact probabilistic analogue of the result in the deterministic
framework, then we must explicitly impose CSM.
To formally state our main characterization, we now prove the following straightforward result which establishes the converse of Propositions 5 and 6.
Proposition 7. If a PVP K satisfies RD, then K satisfies R, IIA, CS and CSM.8
Proof. Let the PVP K satisfy RD with individual weights ai [ [0,1] for each i [ N such
that o i [N ai 5 1.
ˆ Clearly, for each i [ N and for all
(R): Let R [ + N and let B,Bˆ [ - with B # B.
ˆ
x [ B, x [
⁄ G(Ri uBˆ ) if x [
⁄ G(Ri uB). So, for each i [ N, di (x,B,R) $ di (x,B,R)
for all x [ B.
ˆ
ˆ
Hence, K(x,B,R) 5 o i [N aidi (x,B,R) $ o i [N aidi (x,B,R) 5 K(x,B,R) for all x [ B. Thus, K
satisfies R.
ˆ ˜ [ + N be such that R˜ uB 5 Rˆ uB. Then it is obvious that
(IIA): Let B [ - and let R,R
di (x,B,Rˆ ) 5 di (x,B,R˜ ) for all x [ B and each i [ N. So, K(x,B,Rˆ ) 5 o i [N aidi (x,B,Rˆ ) 5
o i [N aidi (x,B,R˜ ) 5 K(x,B,R˜ ) for all x [ B. Therefore, K satisfies IIA.
(CS): Given any distinct x,y [ X and any R [ + N such that yR i x for all i [ N, we
have di (x,hx,yj,R) 5 0 for all i [ N and, hence, K(x,hx,yj,R) 5 o i [N aidi (x,hx,yj,R) 5 0.
Thus, K satisfies CS.
(CSM): Let (B,R) [ - 3 + N . Also, let x [ B and Rˆ [ + N be such that Rˆ uX\hxj 5
RuX\hxj and for each y [ X and each i [ N, xRˆ i y if xR i y. Then it can be checked that
L( y,Rˆ uB) # L( y,RuB) for all y [ B\hxj. Thus, K( y,B,Rˆ ) 5 o i [N aidi ( y,B,Rˆ ) 5
o i [L( y,Rˆ uB ) ai # o i [L( y,RuB ) ai 5 o i [N aidi ( y,B,R) 5 K( y,B,R) for all y [ B\hxj, which implies that
K(x,B,Rˆ ) $ K(x,B,R),
ˆ
K( y,B,Rˆ ) # K( y,B,R) for all y [ B > @ (x,R,R)\hxj
and
O
ˆ ))
z [ @ >(X \ @ (x,R,R
8
K(z,B,Rˆ ) #
O
K(z,B,R).
ˆ ))
z [ @ >(X \ @ (x,R,R
In fact, RD implies probabilistic rationalizability as well.
80
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
Hence, K also satisfies CSM. h
Given Proposition 7, it is quite obvious that a randomly dictatorial PVP also satisfies
R, BIIA, CM, CWM, SM, M, WM and EPO. We are now ready to present the main
theorem of this paper which provides alternative characterizations of random dictatorship to those already existing in the literature (e.g., McLennan, 1980; Nandeibam, 1995;
Pattanaik and Peleg, 1986).
Theorem. Let K be a PVP.
(a) K satisfies R, IIA, CS and SM if and only if K satisfies RD.
(b) K satisfies R, BIIA, CS and CSM if and only if K satisfies RD.
Proof. The theorem readily follows from Propositions 5, 6 and 7. h
It is worth emphasizing that our theorem completely characterizes random dictatorship
by using only conditions that are in the spirit of Arrow’s conditions in the classical
deterministic framework. Thus, our theorem can be viewed as filling the gap left in
Pattanaik and Peleg (1986) in the sense that it provides an almost exact probabilistic
analogue of Arrow’s impossibility theorem for voting procedures with strict individual
preference orderings.
5. Randomized social preference
Barbera and Sonnenschein (1978) call the probabilistic analogue of a strict social
welfare function (where both the individual and social preference orderings are strict) a
strict social welfare scheme (SSWS). It maps each strict preference profile to a lottery
over strict social preference orderings. In this section we will briefly examine SSWS and
show that, using our results on PVP, we can derive an almost exact probabilistic
analogue of Arrow’s impossibility theorem for strict social welfare functions. The
coalitional weights for decisiveness in pairwise comparison of alternatives derived by
Barbera and Sonnenschein are only subadditive. McLennan (1980) showed that for it to
be additive there must be at least six alternatives in the universal set. As a corollary of
our Proposition 3, we will show that the coalitional weights of Barbera and Sonnenschein will become additive with three or more alternatives in the universal set if we
add the appropriate stochastic positive association condition.
Definition 14. A strict social welfare scheme (SSWS) is a function g : + N → D( + ),
where D( + ) is the set of all lotteries over +.
For each SSWS g, let Kg : X 3 - 3 + N → R 1 be the PVP induced by g, i.e. Kg is
such that, for every (B,R) [ - 3 + N : Kg (x,B,R) 5 p( g(R),x,B) for all x [ B, where
p( g(R),x,B) is the sum of the probabilities assigned by the lottery g(R) to all those
linear orderings in + that rank x as the best alternative in B.
S. Nandeibam / Mathematical Social Sciences 40 (2000) 63 – 84
81
Given any SSWS g, we can use the induced PVP Kg to translate the conditions from
the PVP framework to the SSWS framework in the obvious manner. For simplicity and
without creating much confusion, we will give the same names to these conditions in the
SSWS framework as their respective counterparts in the PVP framework. Thus, we will
say that a SSWS g satisfies BIIA if Kg satisfies BIIA, g satisfies IIA if Kg satisfies IIA,
and so on.9
We must point out that there is an important caveat in interpreting the random
dictatorship result to be presented in this section as an almost exact probabilistic
analogue of Arrow’s impossibility theorem. Given the above convention, when we say
that a SSWS g satisfies RD, it is not necessarily the case that there exists ai [ [0,1] for
each i [ N such that: (i) o i [N ai 5 1 and (ii) for each R [ + N , the probability assigned
by g(R) to R i is equal to ai for every i [ N. The example below confirms this important
observation.
ˆ ˆ ˆ ˆ j and N 5 h1,2j. Also, let: (i) Rˆ [ + N be such that
Example 8. Let X 5 hx,y,z,w
ˆ ˆ 1 yR
ˆ ˆ 1 zR
ˆ ˆ 1 wˆ and wR
ˆ ˆ 2 zR
ˆ ˆ 2 yR
ˆ ˆ 2 x,
ˆ (ii) R9 [ + be such that xR9zR9yR9w
ˆ ˆ ˆ ˆ and (iii) R¯ [ + be
xR
ˆ ¯ ˆ ¯ ˆ ¯ ˆ Define the SSWS g as follows. For each R [ + N :
such that wRyRzRx.
ˆ then g(R) assigns a probability of 1 / 2 to each of R 1 and R 2 ;
(a) if R ± R,
ˆ then g(R) assigns a probability of 1 / 4 to each of Rˆ 1 , Rˆ 2 , R9 and R.
¯
(b) if R 5 R,
It can be checked that Kg satisfies RD with individual weights a1 5 a2 5 1 / 2 and, hence,
g satisfies RD. However, g(Rˆ ) assigns only a probability of 1 / 4 to each of Rˆ 1 and Rˆ 2 .
We now present the probabilistic neutrality result for SSWS derived in Barbera and
Sonnenschein (1978) and sharpened in McLennan (1980). This shows the existence of
subadditive coalitional weights that become additive if there are at