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Distribution of coalitional power under probabilistic voting
procedures
* Shasikanta Nandeibam
Department of Economics, University of Birmingham, Birmingham B15 2TT, 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)
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natives, ex-post Pareto optimality and regularity. 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.,
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1982; Barbera and Valenciano, 1983) studied the structure of coalitional power under
social welfare schemes. Social welfare schemes are stochastic social decision rules that
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map each preference profile to a lottery over social preference orderings. 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. McLen-nan (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
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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.
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citizens’ sovereignty. The Pareto principle, although widely accepted as very appealing, was derived from the primitive conditions. Our citizens’ sovereignty condition general-izes 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 mono-tonicity 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.
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 (5h1, . . . ,nj) and the universal set of social alternatives by X. Throughout, unless otherwise mentioned, we will maintain the assumption that ` .uNu5n$2 and ` .uXu5m$3. Also, we denote the set of all nonempty subsets of X by -and the set of all linear orderings on X by +.
The set of all possible preference profiles is the n-fold Cartesian product of+and is
N ˆ ˜ N
denoted by + . Preference profiles are denoted by R, R, R, . . . . For each R[+ , the ith coordinate of R, denoted by R , is the preference ordering of individual i in thei
preference profile R.
Definition 1. A probabilistic voting procedure (PVP) is a function K: X3-3
N N
+ →R such that: o K(x,B,R)5o K(x,B,R)51 for every (B,R)[-3+ . 1 x[B x[X
So K(x,B,R) is interpreted as the probability of society choosing x when the feasible set is B and preference profile is R. This means that, if x is not in the feasible set B, then
K assigns zero probability to it.
N ˆ
Definition 2. A PVP K satisfies regularity (R) if, for all B, B[- and for all R[+ :
ˆ ˆ
[x[B#B ]⇒[K(x,B,R)$K(x,B,R)].
which is known as property a. It is well known that R is a weaker condition than
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probabilistic rationalizability which requires the PVP to be induced by a lottery over+. N
Given any (B,R)[-3+ , for each i[N, we denote the restriction of R to B byi RiuB and the restriction of R to B by RuB5(R1uB, . . . ,RnuB ). 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, N
ˆ
for all B[- withuBu52 and for all R,R[+ :
ˆ ˆ
[RuB5RuB]⇒[K(x,B,R)5K(x,B,R ) for all x[B].
Definition 4. A PVP K satisfies independence of irrelevant alternatives (IIA) if, for all N
ˆ B[- and for all R,R[+ :
ˆ ˆ
[RuB5RuB]⇒[K(x,B,R)5K(x,B,R ) 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)50 whenever there exists some y[B\hxj such that yR xi
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
N
Thus,@(x,R,R) contains x and all the alternatives that are preferred to x according to R
i i
ˆ ˆ
but are preferred by x according to R and@(x,R,R) contains x and every alternative for i
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
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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 N
ˆ ˆ ˆ
R,R[+ , if (i) RuX\hxj5RuX\hxj and (ii) for each i[N and for all y[X, xR y ifi ˆ
xR y, then K(x,X,R )i $K(x,X,R).
Definition 8. A PVP K satisfies monotonicity (M) if, given any x[X and any N
ˆ ˆ
R,R[+ , if (i) RuX\hxj5RuX\hxj and (ii) for each i[N and for all y[X, xR y ifi
ˆ ˆ
xR y, then: (a) K(x,X,R )i $K(x,X,R) and (b) K( y,X,R )#K( y,X,R) for all y[ ˆ
@(x,R,R)\hxj.
Definition 9. A PVP K satisfies strong monotonicity (SM) if, given any x[X and any N
ˆ ˆ ˆ
R,R[+ , if (i) RuX\hxj5RuX\hxj and (ii) for each i[N and for all y[X, xR y ifi
ˆ ˆ ˆ
xR y, then: (a) K(x,X,R )#K(x,X,R), (b) K( y,X,R )#K( y,X,R) for all y[@(x,R,R)\hxj i
ˆ
and (c) oz[X \@(x,R,R )ˆ [K(z,X,R )2K(z,X,R)]#0.
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.
N
Given any (B,R)[-3+ , let
G(RiuB )5hx[B: xR yi ;y[Bjfor each i[N,
b(RuB )5<G(RiuB ) i[N
and
L(x,RuB )5hi[N: x[G(RiuB )jfor each x[B.
G(RiuB ) 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.
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
N N
Thus, for each feasible set,dichooses for sure the best feasible alternative of individual
i.
Definition 10. A PVP K is a random dictatorship (RD) if there existsai[[0,1] for each
N i[N such that, for every (B,R)[-3+ :
(a)oi[Na 5i 1 and
(b) K(x,B,R)5oi[Na di i(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
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independence. 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
Corollary 1. If a PVP K satisfies R, BIIA, CS and WM, then K satisfies EPO.
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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:
K
Proof. For the proof of Proposition 1(a) and (c)–(f) see Pattanaik and Peleg (1986). So,
K K
The following is the almost random dictatorship result of Pattanaik and Peleg (1986).
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Proposition 2. If m$4 and the PVP K satisfies R, IIA and EPO, then:
N K
(a) for all (B,R)[-3+ with B±X, K(x,B,R)5oi[Na (hij)di(x,B,R) for all
x[B;
K
(b) K(x,X,R)5oi[Na (hij)di(x,X,R) for all x[X if m$n12.
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:
K
(a) there existsa (S )[[0,1] for each S#N such that, for all distinct x, y[X and
N K
for all R[+ , K(x,hx, yj,R)5a (S ) if xR y for all ii [S and yR x for all ii [N\S;
K K
(b) for all S#T#N,a (T )$a (S );
K K
(c) for all S#N,a (S )1a (N\S )51;
K K
(d ) a (N )51 anda (5)50;
K K K
(e) for all S,T#N such that S>T55,a (S )1a (T )$a (S<T );
K K K
( f ) for all S,T#N such that S>T55,a (S )1a (T )5a (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 m53 and n53 and define the PVP K as follows. For each
N
(x,B,R)[X3-3+ such that x[B:
(1 /ub(RuB )u, if x[b(RuB ),
K(x,B,R)5
H
0, otherwise.
K satisfies R, IIA, CS, WM and EPO, but the coalitional weights are not additive K
because a (S )51 / 2 for all coalitions S containing one or two individuals. Thus, the condition m$4 is required in Proposition 1(f) and Corollary 2(f).
Example 2. Let m54 and n53. Let K be the PVP defined by the following rule. For
N
each (x,B,R)[X3-3+ such that x[B:
(a) if uBu53, ub(RuB )u53 and ub(RuX )>Bu51, then
1 / 2, if x[b(RuX ),
K(x,B,R)5
H
(b) ifuBu±3 or ub(RuB )u,3 or ub(RuX )>Bu±1, then all coalitions S containing one or two individuals. Clearly, K violates IIA. Thus, IIA cannot be relaxed to BIIA in Proposition 1(f) and Corollary 2(f).
In the probabilistic neutrality results presented so far, namely Proposition 1 and Corollary 2, we can only interpret the weight of each coalition exactly as its probability to be decisive in two-alternative feasible sets when the coalitional weights are additive. Additional conditions have to be imposed to get additivity. This seems to suggest the possibility of a gap in the parallelism with Arrow’s neutrality result. However, our next result shows that this possible gap disappears if we impose monotonicity instead of weak monotonicity.
Proposition 3. If a PVP K satisfies R, BIIA, CS and M, then:
K
Proof. As M is stronger than WM, Corollary 2(a)–(e) holds. So, we only need to prove
Proposition 3(e). Note that, by Corollary 1, EPO is satisfied. Let S,T#N be such that
K K K
Using similar reasoning, we can also conclude that K( y,X,R)$a (T ) and K(z,X,R)$ K
1 $ K(x,X,R)1K( y,X,R)1K(z,X,R)
K K K
$ a (S )1a (T )1a (N\(S<T )) K
$ a (N ) [by Corollary 2(e)]
5 1 [by Proposition 3(d)].
Therefore, we have
K K K
a (S )1a (T ) 5 12a (N\(S<T )) K
5 a (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 m53 and n52. Fix a z[X and define the PVP K as follows. For each N
(x,B,R)[X3-3+ such that x[B:
(a) if B5hzjor z[⁄ B, then
K(x,B,R)5uL(x,RuB )u/ 2;
(b) if B±hzj and z[B, then
1, if 2[L(x,RuB ) anduBu52,
K(x,B,R)5
50,
1, if 1[L(x,RuB ) anduBu53, otherwise.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.
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Example 4. Let m53 and n52. Given any (x,B,R)[X3-3+ such that x[B and uBu$2, for each i[N, letl(x,RiuB ) be defined as follows:
(a) if uBu52, then
2, ifhxj5G(RiuB ), l(x,RiuB )5
H
1, otherwise;
(b) ifuBu53, then
3, ifhxj5G(RiuB ),
1, if yR x for all y[B, l(x,RiuB )5
52,
iN
Then let K be the PVP defined by the following rule. For each (x,B,R)[X3-3+
such that x[B:
(a) if uBu51, then K(x,B,R)51;
(b) ifuBu52, then K(x,B,R)5[l(x,R1uB )1l(x,R2uB )] / 6;
(c) if uBu53, then K(x,B,R)5[l(x,R1uB )1l(x,R2uB )] / 12.
It can be checked that K satisfies R, IIA and SM, but violates CS and EPO. Clearly,
K K K K
a (5)51 / 3, a (h1j)5a (h2j)51 / 2 and a (N )52 / 3. Thus, Proposition 1 is no longer true without EPO and CS is also necessary for Corollary 2 and Proposition 3.
Example 5. Let m53 and n52, and define the PVP K as follows. For each
N
(x,B,R)[X3-3+ such that x[B:
(a) if uBu52 and b(RuX )>B±5, then
1 /ub(RuX )>Bu, if x[b(RuX ),
K(x,B,R)5
H
0, otherwise;
(b) ifuBu±2 or b(RuX )>B55, then
1 /ub(RuB )u, if x[b(RuB ),
K(x,B,R)5
H
0, otherwise.
In this example K satisfies R, CS and SM, but there are no coalitional weights. Obviously, K violates BIIA. This shows that Propositions 1 and 3 and Corollary 2 do not remain valid if we drop BIIA.
4. Random dictatorship
The probabilistic neutrality results derived in the previous section provide restrictions on the distribution of coalitional power to influence the social choice probabilities in binary choice situations. In this section we use one of them, namely Proposition 3, to derive restrictions on the distribution of coalitional power to influence the social choice probabilities when the feasible set is any subset of the universal set X.
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Lemma 3. Let the PVP K satisfy R, IIA, CS and M and let (B,R)[-3+ . If B±X,
K
then K(x,B,R)$a (L(x,RuB )) for all x[B.
N
Proof. Let (B,R)[-3+ be such that B±X and let x[B. If L(x,RuB )55, then
K
K(x,B,R)$05a (L(x,RuB )) follows from Proposition 3(d). If L(x,RuB )5N, then
Proposition 3(d) and EPO (which is satisfied because of Corollary 1) imply that
K
K(x,B,R)515a (L(x,RuB )). So, suppose that 5±L(x,RuB )±N. Since B±X, let y[ N
˜ ˜ ˜
X\B and R[+ be such that RuB5RuB, G(RiuX )5hxj for all i[L(x,RuB ) and there ˜
˜ ˜
K ˆ ˆ ˆ ¯
a (L(x,RuB )) for every x[B if R is R or R. Clearly, K violates IIA. Thus, Proposition
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.
N Example 7. Let m53 and n53. Also, for each R[+ , let
PO(R)5hx[X: there does not exist y[X\hxjsuch that yR x for all ii [Nj.
N
Define the PVP K as follows. For each (x,B,R)[X3-3+ such that x[B:
(a) if B5X and PO(R)5X, then K(x,B,R)51 / 3; (b) if B±X or PO(R)±X, then K(x,B,R)5uL(x,RuB )u/ 3.
In this example K satisfies R, IIA, CS and M. However, it can be checked that, for
N
R[+ such that b(RuX )±PO(R)5X, K(x,X,R)51 / 3 and L(x,RuX )55 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 counter-parts 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
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and M, but it can be verified that it violates SM. 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
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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.
N
that L(x,RuX )5N, then EPO together with Proposition 3(d,e) gives us K(x,X,R)515
K K
which together with Proposition 3(a) imply that K(z,hy,zj,R )5a (L(z,RuX )). Thus, (5)
K
and (6) imply that K(z,X,R)$a (L(z,RuX )). Then, as z was arbitrarily chosen from X,
K
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 N
ˆ ˆ ˆ
and R,R[+ , if (i) RuX\hxj5RuX\hxjand (ii) for each i[N and for all y[X, xR y ifi ˆ
xR y, then K(x,B,R )i $K(x,B,R) for every B[- with x[B.
Definition 12. A PVP K satisfies complete monotonicity (CM) if, given x[X and N
Definition 13. A PVP K satisfies complete strong monotonicity (CSM) if, given x[X N
Clearly, CSM is stronger than CM, CWM, SM, M and WM, CM is stronger than CWM, M and WM, and CWM is stronger than WM. We know that the PVP in Example 6 satisfies R, BIIA, CS and SM but violates IIA. It can be verified that this PVP also satisfies CM but not CSM. This shows that, even if we impose CM, the random dictatorship results of Propositions 4 and 5 do not remain valid if we relax IIA to BIIA. This naturally leaves us with only one more possible option, namely strengthen M and SM to CSM and replace IIA by BIIA and check whether Propositions 4 and 5 remain valid. L(x,RuB )5N, then EPO together with Proposition 3(d,e) imply that K(x,B,R)515
K K
a (L(z,RuB )). Thus, as z is any alternative from B, we have K(x,B,R)$a (L(x,RuB )) for all x[B. Obviously, hL(x,RuB ): x[Bj is a partition of N. Then, using Proposition
K K
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 straight-forward result which establishes the converse of Propositions 5 and 6.
8
Proposition 7. If a PVP K satisfies RD, then K satisfies R, IIA, CS and CSM.
Proof. Let the PVP K satisfy RD with individual weightsai[[0,1] for each i[N such
that oi[Na 5i 1.
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 dictator-ship 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 Son-nenschein will become additive with three or more alternatives in the universal set if we add the appropriate stochastic positive association condition.
N
Definition 14. A strict social welfare scheme (SSWS) is a function g :+ →D(+),
whereD(+) is the set of all lotteries over +.
N
For each SSWS g, let K : X3-3+ →R be the PVP induced by g, i.e. K is
g 1 g
N
such that, for every (B,R)[-3+ : K (x,B,R)g 5p( 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
Given any SSWS g, we can use the induced PVP K to translate the conditions fromg
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 K satisfies BIIA, g satisfies IIA if K satisfies IIA,g g
9
and so on.
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 existsai[[0,1] for
N
each i[N such that: (i) o a 51 and (ii) for each R[+ , the probability assigned i[N i
by g(R) to R is equal toi ai for every i[N. The example below confirms this important
observation.
such that wRyRzRx. Define the SSWS g as follows. For each R[+ :
ˆ
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 least six alternatives in the universal set.
Proposition 8. If a SSWS g satisfies BIIA and EPO, then there is a function N
Barbera and Sonnenschein’s (1978) Paretian condition only requires that, given any distinct x, y[X and any N
Proof. See Barbera and Sonnenschein (1978) for the proof of Proposition 8(a)–(e) and
McLennan (1980) for the proof of Proposition 8(f). h
Kg
It is clear thatmg(S )5a (S ) for every S#N. So Proposition 8(a)–(e) can be viewed
as corollaries of Proposition 1(a)–(e). Proposition 1(f) also implies an additive mg if there are at least four alternatives in X and BIIA is replaced by IIA in Proposition 8. Furthermore, we can replace EPO by CS and WM in Proposition 8 because of Corollary 1.
Corollary 3. If a SSWS g satisfies BIIA, CS and M, then there is a function N
m : 2 →R such that:
g 1
N
(a) for all distinct x, y[X and for all R[+ , mg(L(x,Ruhx, yj))5K (x,hx, yjg ,R); (b) for all S#T#N, mg(T )$mg(S );
(c) for all S#N, mg(S )1mg(N\S )51; (d ) mg(N )51 and mg(5)50;
(e) for all S,T#N such that S>T55, mg(S )1mg(T )5mg(S<T ).
Kg
Proof. Clearly, K satisfies R, BIIA, CS and M, andg mg(S )5a (S ) for all S#N.
Therefore, the corollary readily follows from Proposition 3. h
Needless to say, as Example 1 satisfies probabilistic rationalizability, it can easily be converted into the SSWS framework to show that M cannot be relaxed to WM in Corollary 3 if we want to get additive coalitional weights.
Finally, we have the following characterization of random dictatorship for SSWS which is almost like the stochastic version of Arrow’s impossibility theorem for strict social welfare functions with the caveat already mentioned about our notion of randomly dictatorial SSWS.
Corollary 4. Let g be a SSWS.
(a) g satisfies IIA, CS and SM if and only if g satisfies RD; (b) g satisfies BIIA, CS and CSM if and only if g satisfies RD.
Kg
Proof. Since mg(S )5a (S ) for every S#N the corollary is a straightforward
consequence of our Theorem. h
6. Conclusion
different groups of individuals in the probabilistic social decision process. The existing literature first derives the restrictions implied by these conditions for social choice or comparison situations involving exactly two alternatives. In particular, the existence of a unique weight for each coalition is established. This can be viewed as a probabilistic version of the neutrality property for decisiveness over pairs of alternatives. However, to interpret the weight of a coalition exactly as its probability to be decisive over pairs of alternatives, the coalitional weights need to be additive, but this requires additional conditions. Our neutrality result proved the existence of additive coalitional weights without the additional conditions if we impose monotonicity which seems stronger than the stochastic version of Arrow’s positive association condition. The reason we need the stronger condition but the weaker condition is enough for the deterministic case is because, in the presence of the other relevant conditions, the two are equivalent to each other in the deterministic framework but not in the probabilistic framework. Thus, although we use a stronger condition than the stochastic analogue of positive association, to a certain extent, our neutrality result is an almost exact probabilistic version of Arrow’s neutrality result.
In the case of PVP, Pattanaik and Peleg (1986) proved that the restrictions implied by the probabilistic counterparts of collective rationality, independence and Pareto con-ditions for arbitrary feasible sets is almost random dictatorship if there are at least four alternatives in the universal set. Their result left two gaps: (i) in contrast to the deterministic framework they needed four or more alternatives in the universal set, and (ii) for their random dictatorship result to hold when the universal set itself is the feasible set they needed at least two more alternatives in the universal set than the number of individuals in the society. Our main result has shown that a randomly dictatorial PVP is completely characterized without the additional conditions of Pattanaik and Peleg (1986) by R, CS and either IIA and SM or BIIA and CSM. Conditions SM and CSM seem stronger than the probabilistic counterpart of Arrow’s positive association condition. However, just as in the neutrality result, we need the seemingly stronger conditions because, in the presence of the other relevant conditions, they are equivalent to each other in the deterministic framework but not in the probabilistic framework. Furthermore, in our random dictatorship result, we require IIA with SM, but BIIA is sufficient with CSM. The reason for this is that SM puts restriction only when the universal set is the feasible set whereas CSM, like Arrow’s positive association condition, imposes restrictions for all feasible sets that contain the alternative which has moved up in the preference profile. Thus, our characterization of randomly dictatorial PVP is also more or less the exact probabilistic analogue of Arrow’s impossibility theorem for voting procedures with strict preference orderings.
Acknowledgements
I would like to thank an anonymous referee for helpful comments and suggestions. I take full responsibility for all remaining errors.
References
Arrow, K.J., 1963. Social Choice and Individual Values, 2nd ed., Wiley, New York.
Bandyopadhyay, T., Deb, R., Pattanaik, P.K., 1982. The structure of coalitional power under probabilistic group decision rules. J. Econ. Theor. 27, 366–375.
Barbera, S., Sonnenschein, H., 1978. Preference aggregation with randomized social orderings. J. Econ. Theor. 18, 244–254.
Barbera, S., Valenciano, F., 1983. Collective probability judgements. Econometrica 51, 1033–1046. Chernoff, H., 1954. Rational selections of decision functions. Econometrica 22, 422–443.
McLennan, A., 1980. Randomized preference aggregation: additivity of power and strategy proofness. J. Econ. Theor. 22, 1–11.
Nandeibam, S., 1995. A note on randomized social choice and random dictatorships. J. Econ. Theor. 66, 581–589.
