A procedure to construct a social welfare function from a social choice function is suggested and it is shown that the dictatorial are the only unanimous social welfare functions that can be reconstructed from a social choice function that does not change the social choice when a defeated alternative is moved to the last position in all the individual preferences.

Antonio Quesada

English

Research Papers in Economics

From social choice functions to dictatorial social welfare
functions 
Antonio Quesada
Universidad de Murcia
Abstract
A procedure to construct a social welfare function from a social choice function is suggested
and it is shown that the dictatorial are the only unanimous social welfare functions that can
be reconstructed from a social choice function that does not change the social choice when a
defeated alternative is moved to the last position in all the individual preferences.
The difference between night and day can be illustrated by comparing the original version of this paper (night) with the final
version that incorporates the two referees’ suggestions and corrections (day). I am very grateful to them for taking the trouble to
make this transformation (and recommending the publication of this paper) and to Jordi Massó, the associate editor, for
accepting the publication of this paper.
Citation: Quesada, Antonio, (2002) "From social choice functions to dictatorial social welfare functions." Economics
Bulletin, Vol. 4, No. 16 pp. 1−7
Submitted: July 19, 2002.  Accepted: July 19, 2002.
URL: http://www.economicsbulletin.com/2002/volume4/EB−02D70006A.pdf− 1−  
1. Introduction 
 
Arrow’s (1963) theorem, on the one hand, expresses an impossibility result: one cannot 
have a non-dictatorial preference aggregation rule (social welfare function) that satisfies 
unanimity and a condition of independence of irrelevant alternatives (IIA); see Feldman 
(1980) or Saari (1998, 2001). But, on the other, Arrow’s theorem makes social choice 
theory challenging (and, therefore, interesting) because it merely makes apparent the 
inadequacy of a specific approach to construct reasonable social welfare functions. 
 
For strict preferences, this note suggests the alternative approach consisting of 
constructing social welfare functions from social choice functions, which are rules 
transforming preference profiles not into preferences but into alternatives. The 
motivating idea is to define aggregation rules (social welfare functions) from simpler 
aggregation rules (social choice functions), so that one can use the solution to a certain 
aggregation problem to solve a more complex aggregation problem. 
 
To see how just one social choice function can be used to construct a social welfare 
function, let P be a preference profile and g a social choice function defined on the 
universal domain and satisfying the weak Pareto principle
1 (see A4: if all individuals 
prefer alternative x to alternative y then g cannot select y). The first alternative in the 
social preference associated with P is g(P). The problem is next to determine the second 
alternative in the social preference associated with P using g. One possible solution 
consists of replacing P with some preference profile P' that is “very close” to P and 
such that g does not associate with P' the previously selected alternative g(P). 
 
Since g satisfies the weak Pareto principle, moving g(P) to the last position in each 
individual preference from profile P ensures that g(P) will not be chosen. The resulting 
preference profile P'  is in a sense “very close” to P, because the ordering of the 
“relevant” alternatives (all but g(P)) is the same in both profiles. So g(P') could be 
chosen as the second alternative in the social preference associated with P. 
 
The same logic could now be applied to P' to determine the third alternative in the 
social preference: just define P'' to be the preference profile obtained from P' by ranking 
                                                 
1 I am very grateful to Referee 1 for calling to my attention that the iterative construction defined next is 
essentially the same one used in the proof of Lemma 2 in Ehlers and Weymark (2001) and to Referee 2 
for noticing that, without the weak Pareto principle, the aggregation rule to be defined next need not be a 
social welfare function. − 2−  
g(P') last in each individual preference from preference profile P' and declare that g(P'') 
is the third alternative in the social preference that corresponds to P. And so on. 
 
The interest of this procedure is that the “instructions” needed to run a social welfare 
function could be “coded” by a social choice function, which is a structure simpler than 
a social welfare function. With always strict preferences, the main result presented in 
this note (Proposition 2.2) characterizes dictatorial social welfare functions by means of 
the weak Pareto principle (see A1: if all individuals prefer alternative x to alternative y 
then x is socially preferred to y) and the condition that, using the previous procedure, the 
social welfare function can be reconstructed from a social choice function that, in 
addition, satisfies the following property: if preference profile P' is obtained from P by 
ranking alternative x ≠  g(P) last in each individual preference then g(P') = g(P)
2. In 
words, if all individuals give a defeated alternative the minimum support with 
everything else the same, then the winning alternative remains winning. 
 
Since Proposition 2.2 is proved by showing that a social welfare function satisfying the 
preceding condition also satisfies IIA it follows that IIA can be subsumed under the 
approach of trying to reduce social welfare functions to social choice functions. 
 
 
2. Definitions, assumptions and result 
 
Let N = {1, 2, …  , n} be a finite set whose members designate individuals, A a finite set 
containing m elements representing alternatives and L the set of complete, transitive and 
asymmetric binary relations that can be defined on A. Each member of L expresses a 
strict preference relation on A (no two alternatives are indifferent) and can be 
represented by a vector π  = (x1 , … , xm) of the m alternatives, with xp preferred to xq if, 
and only if, p < q. Intuitively, π  consists of a ranking of m positions. For 1 ≤  k ≤  m, let 
kπ  designate the alternative filling the kth position in ranking π . 
 
For π  ∈  L, P ∈  Ln, x ∈  A, y ∈  A\{x} and i ∈  N: (i) Pi denotes individual i’s preference in 
preference profile P; (ii) π {x,y} denotes the restriction of π  to the set {x, y}; and (iii) 
                                                 
2 Postulating the property that a social welfare function can be reconstructed from a social choice function 
makes the requirement that g satisfies the weak Pareto principle unnecessary, because the property 
presumes that the outcome of the procedure must be a social welfare function. It is this presumption that 
automatically restricts the domain of social choice functions that can be used to construct a social welfare 
function. Without this presumption, the domain restriction must be explicit; for instance, the social choice 
function must satisfy the weak Pareto principle. − 3−  
P {x,y} abbreviates (P1  {x,y} , … , Pn {x,y}). The preference profile P↓ x obtained from P 
∈  Ln by moving x ∈  A to the last position in each Pi is the profile P' in which, for all i ∈  
N, P'i is defined as follows. Let 1 ≤  k ≤  m be such that x = kPi. Then mP'i = x and, for all 
1 ≤  r < m, rP'i = rPi if r < k and rP'i = r−1 Pi if r > k. 
 
Given N and A with n ≥  2 < m, a social welfare function is a mapping f : Ln →  L. The 
interpretation is that, for (P1 , …  , Pn) ∈  Ln, f(P1 , …  , Pn) is the social preference that f 
associates with individual preferences P1 , …  , Pn. Social welfare function f is dictatorial 
if there is an i ∈  N such that, for all (P1 , …  , Pn) ∈  Ln, f(P1 , …  , Pn) = Pi; that is, 
individual i (called “dictator”) is such that the social preference always coincides with 
i’s preference. In this context, Arrow’s (1963, p. 97) theorem asserts that, for n ≥  2 < m, 
every social welfare function that satisfies IIA (independence of irrelevant alternatives) 
and A1 (the weak Pareto principle) below is dictatorial. 
 
IIA. For all P  ∈   Ln,  Q  ∈   Ln\{P},  x  ∈   A and y ∈  A\{x}, if P {x,y} = Q {x,y} then 
f(P) {x,y} = f(Q) {x,y}. 
 
A1. For all P ∈  Ln, x ∈  A and y ∈  A\{x}, if xPiy for all i ∈  N then xf(P)y. 
 
Suppose that, for all i ∈  N, x is ranked above y in Pi if, and only if, x is ranked above y 
in Qi. By IIA, x is ranked above y in f(P) if, and only if, x is ranked above y in f(Q). 
Hence, the ranking between x and y at the social level depends only on the ranking 
between these two alternatives at the individual level. By A1, if x is above y in every 
individual ranking in P then x is above y in the social ranking f(P). 
 
Given N and A with n ≥  2 < m, a social choice function is a mapping g : Ln →  A. The 
interpretation is that, for (P1 , …  , Pn) ∈  Ln, g(P1 , …  , Pn) is the social choice that g 
derives from individual preferences P1 , …  , Pn. Social choice function g is dictatorial if 
there is an i ∈  N such that, for all (P1 , …  , Pn) ∈  Ln, g(P1 , …  , Pn) = 1 Pi. A2 next is the 
property that a social welfare function can be reconstructed from a social choice 
function as suggested in Section 1 when the social choice function is merely assumed to 
satisfy A3 (recall footnote 2). By A3, the social choice does not change when a defeated 
alternative is ranked last in all the individual preferences: if Q ∈  Ln is obtained from P 
by ranking x last in every Pi then g(Q) = g(P). 
 
A2. There is a social choice function g : Ln →  A satisfying A3 such that, for all P ∈  Ln 
and 1 ≤  k ≤  m, kf(P) = g(kP), where 1 P = P and, for 2 ≤  t ≤  m, tP = t−1 P↓ g(t−1 P) (tP is 
obtained from t−1 P by ranking g(t−1 P) last in each individual ordering). − 4−  
A3. For all P ∈  Ln and x ∈  A\{g(P)}, g(P↓ x) = g(P). 
 
Lemma 2.1. If n ≥  2 < m and f : Ln →  L satisfies A2 then it satisfies IIA. 
 
Proof. Suppose f does not satisfy IIA: there are P ∈  Ln, Q ∈  Ln\{P}, x ∈  A and y ∈  
A\{x} such that P {x,y} = Q {x,y}, x is ranked above y in f(P) but y is ranked above x in 
f(Q). By A2, let g be the social choice function from which f is defined. Let f(P) = (x1 , 
… , xs, x, y1 , … , yt, y, z1 , … , zu). Hence, by A2, 1 f(P↓ x1 ) = g(P↓ x1 ) = x2 . Similarly, 
with P' = P↓ x1 , 1 f(P'↓ x2 ) = g(P'↓ x2 ) = x3 . Thus, if R ∈  Ln is obtained from P by ranking 
last in each Pi first x1 , then x2 , … , and finally xs then, by A2, 1 f(R) = g(R) = x. 
 
Since f satisfies A2, g satisfies A3 and g(R) = x, 1 f(R↓ y1 ) = g(R↓ y1 ) = g(R) = x. In the 
same vein, with R' = R↓ y1 , 1 f(R'↓ y2 ) = g(R'↓ y2 ) = g(R↓ y1 ) = x. Accordingly, if S ∈  Ln is 
obtained from R by ranking last in each Ri first y1 , then y2 , … , and finally yt and next 
first z1 , then z2 , … , and finally zu then, by A2, 1 f(S) = g(S) and, by A3, g(S) = g(R) = x. 
As a result, S {x,y} = P {x,y} and, for all i ∈  N, the ranking of the last m −  2 alternatives 
in Si is (x1 , … , xs, y1 , … , yt, z1 , … , zu). 
 
Let f(Q) = (a1 , … , ap, y, b1 , … , bq, x, c1 , … , cr). An analogous argument proves that 
1 f(T) = y, where T ∈  Ln is such that T {x,y} = Q {x,y} and, for all i ∈  N, the ranking of 
the last m −  2 alternatives in Ti is (a1 , … , ap, b1 , … , bq, c1 , … , cr). Given that {a1 , … , 
ap, b1 , … , bq, c1 , … , cr} = {x1 , … , xs, y1 , … , yt, z1 , … , zu}, the aim is now to rank the 
last m −  2 alternatives in each Ti in the same way as in each Si without altering the 
socially most preferred alternative. To begin with, identify x1  in {a1 , … , ap, b1 , … , bq, 
c1 , … , cr}. By A3, g(T↓ x1 ) = g(T) so, by A2, 1 f(T↓ x1 ) = 1 f(T) = y. This procedure can be 
applied to, in this order, x2 , … , xs, y1 , … , yt, z1 , … , zu to define V ∈  Ln from T such 
that: (i) by A2, 1 f(V) = y; (ii) V {x,y} = T {x,y}; and (iii) for all i ∈  N, the ranking of the 
last m −  2 alternatives in Vi is (x1 , … , xs, y1 , … , yt, z1 , … , zu). Given this and the fact 
that P {x,y} = Q {x,y}, S {x,y} = P {x,y}, T {x,y} = Q {x,y} and V {x,y} = T {x,y} imply 
S {x,y} = V {x,y}, it follows that S = V. Yet, 1 f(S) = x ≠  y = 1 f(V): contradiction.■ 
 
Proposition 2.2. If n ≥  2 < m then f : Ln →  L is dictatorial if, and only if, it satisfies 
A1 and A2. 
 
Proof. “⇒ ” If i is a dictator in f then: (i) A1 automatically holds; and (ii) A2 holds with 
g : Ln →  A such that, for all P ∈  Ln, g(P) = Pi. “⇐ ” If f satisfies A2 then, by Lemma 
2.1, it satisfies IIA. As f satisfies IIA and A1, it is dictatorial by Arrow’s theorem.■ 
 − 5−  
A4. For all P ∈  Ln and x ∈  A, if g(P) = x then there is no y ∈  A\{x} such that, for all i ∈  
N, yPix. 
 
The credit of the following result should be attributed to Referee 2. 
 
Proposition 2.3. If n ≥  2 < m then g : Ln →  A is dictatorial if, and only if, it satisfies 
A3 and A4. 
 
Proof. “⇒ ” It is easy to see that if g is dictatorial then A3 and A4 hold. “⇐ ” Suppose g 
satisfies A3 and A4 but it is not dictatorial. As Referee 2 observes, if g satisfies A4 then 
the social welfare function f defined from g as suggested in Section 1 is well-defined 
(because winning alternatives that have been moved to the last position will never be 
winning again) and satisfies A1. Hence, being A3 true for g, f satisfies A2 and, by 
Proposition 2.2, f has some dictator i ∈  N. If i is not a dictator in g then there is P ∈  Ln 
with g(P) ≠  1 Pi. By A2, 1 f(P) ≠  1 Pi, contradicting the fact that i is a dictator in f.■ 
 
Referee 2 suggests connecting Proposition 2.3 with the fact (Corollary 2.5) that social 
choice functions satisfying the weak Pareto principle (see A4) and Maskin (1999, p. 28) 
monotonicity (see A5) are dictatorial (see Muller and Satterthwaite (1977)). 
 
A5. For all P ∈  Ln, Q ∈  Ln\{P} and x ∈  A, if g(P) = x and, for all i ∈  N and y ∈  A\{x}, 
xPiy implies xQiy then g(Q) = x. 
 
Lemma 2.4. A5 is equivalent to A3. 
 
Proof. “⇒ ” Let g(P) = x and z ∈  A\{x}. Then Q = P↓ z is such that, for all i ∈  N and y ∈  
A\{x}, xPiy implies xQiy. By A5, g(Q) = x; that is, g(P↓ z) = g(P). “⇐ ” Let g(P) = x. If, 
for all i ∈  N and y ∈  A\{x}, xPiy implies xQiy then Q must come from P by exchanging 
contiguous alternatives (different from x) in the individual rankings. It therefore suffices 
to show that g(Q) = g(P) when Q differs from P only in that, for some i ∈  N, v ∈  A\{x} 
and z ∈  A\{v, x}, vPiz and zQiv, where P and Q are such that, for no t ∈  A, vPitPiz and, 
for no t ∈  A, zQitQiv. Suppose g(Q) ≠  x. Without loss of generality, assume v ≠  g(Q). By 
A3, g(P↓ v) = g(P) = x and g(Q↓ v) = g(Q) ≠  x. But P↓ v = Q↓ v: contradiction.■ 
 
Corollary 2.5. If n ≥  2 < m then g : Ln →  A is dictatorial if, and only if, it satisfies A4 
and A5. 
 
Proof. Lemma 2.4 and Proposition 2.3.■ − 6−  
The preceding results provide additional links between Arrow’s and the Gibbard 
(1973)-Satterthwaite (1975) theorem: the condition A3 is related to the condition IIA 
for social welfare functions through Lemma 2.1 and also to strategy-proofness of social 
choice functions, through Lemma 2.4 and the equivalence of A5 to strategy-proofness. 
 
Referee 2 wonders whether Proposition 2.2 can be straightforwardly adapted to the case 
in which social indifference is allowed. In this case, since positions in the social 
preference can be filled by sets of alternatives and not just by singletons, social choice 
functions have to be replaced in A2 by social choice correspondences, which map 
preference profiles into sets of alternatives. Nevertheless, Lemma 2.1 does not hold for 
social choice correspondences. To see this, let g be the weak Pareto correspondence, 
where g(P) = {x ∈  A: there is no y ∈  A\{x} such that, for all i ∈  N, yPix}. With N = {1, 
2} and A = {x, y, z}, the profile P with P1  = (z, x, y) and P2  = (y, z, x) results in a social 
preference f(P) with y indifferent to z and both preferred to x, whereas the profile Q with 
preferences Q1  = P1  and Q2  = (z, y, x) results in a social preference f(Q) with z preferred 
to x and x indifferent to y. Thus, P {x,y} = Q {x,y} but f(P) {x,y} ≠  f(Q) {x,y}. 
 
 
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From social choice functions to dictatorial social welfare functions

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From social choice functions to dictatorial social welfare functions

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