Akashi, Koichiro

English

Research Papers in Economics

Hitotsubashi University Repository
Title
A simplified derivation of arrow's impossibility
theorem
Author(s) Akashi, Koichiro
Citation Hitotsubashi Journal of Economics, 46(2): 177-181
Issue Date 2005-12
Type Departmental Bulletin Paper
Text Version publisher
URL http://hdl.handle.net/10086/7654
RightA SIMPLIFIED DERIVATION OF ARROW’S
IMPOSSIBILITY THEOREM

KD>8=>GD A@6H=>
Policy Research Institute, Ministry of Agriculture, Forestry and Fisheries
Kita-ku, Tokyo 1140024, Japan
akashi@a#rc.go.jp
Received August 2005; Accepted September 2005
Abstract
I prove Arrow’s Impossibility Theorem by using elementary Set Theory. The result of the
proof will be available for pedagogical purpose. In the process of the proof, the condition of
Independence of Irrelevant Alternatives plays an essential role.
Keywords: Arrow’s Impossibility Theorem, Set Theory, Condition of Independence of Irrele-
vant Alternatives
JEL Classiﬁcation: D71
I. Introduction
Many studies have done to prove Impossibility Theorem of Arrow (1963). Fishburn
(1970) deﬁned the conditions of the theorem precisely, and proved that all of them are satisﬁed
only when the members of the society is inﬁnite. Sen (1979) proved the theorem by reducing
“the decisive group” into one person. He also analyzed each of the conditions of the theorem
in the light of welfare economics. Suzumura (1988) gave a smart proof by using contraposition
of mathematically inductive method. Takekuma (1997) proved the theorem by the program of
Mathematica. A problem of above articles is that they are too di$cult for undergraduate
students and researchers not being specialized in social choice to understand the proof
completely because they require a certain level of knowledge of social choice and mathematical
ability. On the other hand, Denicolo (1996) gave an elementary proof. The only problem of his
proof is that he assumed the individual preferences to be linear.
This paper intends to prove Arrow’s theorem entirely by using only elemental knowledge
of the Set Theory. First I deﬁne the sets which generate social preferences using the condition
of “Independence of Irrelevant Alternatives”. At the next step I reduce them. Even under-
graduate students can easily understand the whole process of the proof. This paper will be
useful from the pedagogical point of view.
 I wish to thank Professor Kotaro Suzumura for his invaluable comments.
Hitotsubashi Journal of Economics 46 (2005), pp.177-181.  Hitotsubashi UniversityII. Abstract of Arrow’s Impossibility Theorem
Consider an n-person society (n2). Let N{1, 2,…, n} be a set of individuals. X{x,
y,…} denotes the set of social alternatives. There are at least 3 alternatives and the set of
individuals is ﬁnite. Each individual preference (IP) on X is rational (complete, reﬂexive and
transitive). A proﬁle is a list of IPs. For example, in the case of n2a n dm3, the number
of proﬁles is 132169. The social welfare function (SWF) is a rule that assigns a social
preference (SP) to all logically possible proﬁles of IPs. The SP is also assumed to be rational.
Arrow required that the SWF be consistent with the following four rules. Rule 1: The SWF is
deﬁned for all logically possible proﬁles of IPs, i.e., the domain is unrestricted. Rule 2: The
SWF satisﬁes the Pareto Principle; for any x, yX, if all members of the society prefer x to y,
then the society also prefers x to y. Rule 3: The SWF satisﬁes the condition of Independence
of Irrelevant Alternatives. Hence, for any x, yX, the SP on x and y are determined from the
IPs only on x and y. Rule 4: Nondictatorship; there is no dictator in the society. The dictator
is deﬁned as person d; for any x, yX, whenever person d prefers x to y, the society prefers
x to y. “Arrow’s Impossibility Theorem” is expressed as follows: There is no SWF that satisﬁes
Rule 1, Rule 2, Rule 3 and Rule 4.
III. Proof
1. Formal preparation
A. Process of the proof
I treat only the SWF that satisﬁes Rule 1, Rule 2 and Rule 3. I consider sets of individuals
that generate SP under such rules, and verify that no SWF satisﬁes Rule 4 (Nondictatorship).
First, I verify the above results in the case of 3 alternatives and n (n2) individuals. Next, I
expand the results into the case of m alternatives.
B. Deﬁnition of symbols
In this paper, I use the following notations for simplicity:
“x() y” implies that x is preferred (preferred or indi#erent, indi#erent) to y,a n d
“x() y” implies that y is preferred (preferred or indi#erent) to x.
Especially, I use the following notation “”.
“A*xy” implies that all members of Set A* prefer x to y.
“Sxy” implies that the society prefers x to y.
C. Deﬁnition of sets that generate social preferences
Using the rules of “Independence of Irrelevant Alternatives”, I would like to deﬁne
following non-empty sets A*,…, F* that generate social preferences regarding only two
alternatives.
1
1 Non-emptiness is guaranteed by the Pareto Principle.
[December =>IDIHJ76H=> ?DJGC6A D; :8DCDB>8H +12A*xy  Sxy (1.a)
B*yz  Syz (1.b)
C*zx  Szx (1.c)
D*xy  Sxy (1.d)
E*yz  Syz (1.e)
F*zx  Szx (1.f)
Next, I deﬁne AG as the group of sets that satisfy condition (1.a). The number of elements
of AG is ﬁnite. In the same way, I deﬁne BG,…, FG. I deﬁne A as a set composed of the
minimum number among AG . Likewise, I deﬁne Sets B,…, F.
2. Proof
Step 1. ABC (for any A, B, C).
2
See Appendix.
Step 2. A (BC) consists of only one person (named i) (for any A, B, C).
See Appendix.
Step 3. DEF (for any D, E, F).
The proof is similar to that of Step 1.
Step 4. D (EF) consists of only one person (named j) (for any D, E, F).
The proof is similar to that of Step 2.
Step 5. For i and j mentioned above, ij. (Hence, person i(j) is a dictator.)
(Proof) Assume ij. There exist the following preferences: ixy, jxy. Then, the
SP is determined as follows: Sxy, Sxy. This SP is irrational. Hence, ij.
Step 6. In the case of m alternatives, the SWF that satisﬁes Rule 1, Rule 2 and Rule 3 must
be dictatorial.
(Proof) See Appendix.
AEE:C9>M
(1) Proof of Step 1
A. I prove that for any A, B, C, ABf, BCf, CAf.
(Proof) Assume ABf. Consider the following IPs (in the case of n2, AB AB is
empty): Azxy, Byzx, AB ABzx. Then, the SP is determined as follows:
Sxy, Syz and Szx. This is irrational. Therefore, the assumption is false, and
ABf. The proof of BCf and CAf are similar.
B. I prove that for any A, B, C, ABC.
(Proof) First, I prove AB(AB).
2 I consider the possibility of the existence of multiple sets that satisfy the requirements of Set A in AG.A st o
B,…F, the situations are similar.
2005] 6 H>BEA>;>:9 9:G>K6I>DC D; 6GGDL’H >BEDHH>7>A>IN I=:DG:B +13I divide set A and set Ba s follows:
A(AB){A(AB)}, B(AB){B(AB)}.
where {A(AB)}{B(AB)}f.
1) Suppose {A(AB)}f and {B(AB)}f.
Consider the following IPs: (AB)xyz,{ A(AB)}xy,{ B(AB)}yz.
Then, SP is determined as follows: Sxy (deﬁnition of Set A), Syz (deﬁnition of Set B),
Sxz (transitivity of the SP). Once IPs on x and z of all the members are determined, SP
on x and z is also determined (Rule3). As for the members not belonging to (AB), IPs on
x and z are not restricted at all, so all logically possible IPs on x and z can be realized (Rule
1). Now (AB)xz and Sxz are realized. Hence, it follows that
(AB)xz  Sxz. (A.1)
Next, consider the following IPs: (AB)xyz,{ A(AB)}yx,{ B(AB)}
zy. Then, there exists the following SP: Syx (deﬁnition of Set A), Szy (deﬁnition
of Set B)a n dSzx (transitivity of the SP). This result contradicts Eq. (A.1).
2) Suppose {A(AB)}f and {B(AB)}f.
Consider the following IPs: (AB)xyz,{ A(AB)}xy. It follows that
(AB)xz  Sxz. (A.2)
Next, consider the following IPs: (AB)xzy,{ A(AB)}zyx. Then, there
exists the following SP: Syx (deﬁnition of Set A), Szy (Pareto Principle) and Szx
(transitivity of the SP). This result contradicts Eq. (A.2).
3) Suppose {A(AB)}f and {B(AB)}f.
The proof is similar to that of 2).
From the results of 1), 2) and 3), {A(AB)}{B(AB)}f. Hence, AB
(AB).
By the same method, ABC can be easily proven.
(2) Proof of Step 2
(Proof) Assume that A (BC) consists of complexed members {1,…r}( r2).
Consider the following IPs: {person 1}xyz, {person 2,…, person r}xyz. Then,
the SP is determined as follows: Sxy and Syz. It follows that
Axz  Sxz. (A.3)
Next, consider the following IPs: {person 1}xzy, {person 2,…, person r}yxz.
Then, there exists the following SP: Syx (deﬁnition of Set A), Szy (deﬁnition of Set
B)a n dSzx (transitivity of the SP). This result contradicts Eq. (A.3).
(3) Proof of Step 6
(Proof) In the case of k (k3) alternatives, consider that Rules 1-3 are satisﬁed and a
dictator exists (named i). In the case of k	1 alternatives, assume that Rules 1-3 are satisﬁed
[December =>IDIHJ76H=> ?DJGC6A D; :8DCDB>8H +2*and person i does not have dictatorial power on the k1 alternative (w). So, without losing
generality, we can assume that there exist the following preferences: {person i}xw,
Swx. Then, consider the following IPs: {person i}xyw, {others}yw. The SP is
determined as follows: Sxy (person i has dictatorial power on x and y), Syw (Pareto
Principle), Swx (person i does not have dictatorial power on w)a n dSyx (transitivity
of the SP). This SP is irrational. Therefore, person i should be a dictator. Hence, in the case
of m alternatives (m is any natural number), a dictator exists.
R:;:G:C8:H
Arrow, K. J. (1963), Social Choice and Individual Values, 2nd edition, New York: Wiley.
Denicolo, V. (1996), “An Elemental Proof of Arrow’s Impossibility Theorem,” Japanese
Economic Review 47, pp. 432-35.
Denicolo, V. (2001), “An Elemental Proof of Arrow’s Impossibility Theorem: Correction,”
Japanese Economic Review 52, pp. 134-35.
Fishburn, P.C. (1970), “Arrow’s Impossibility Theorem: Concise Proof and Inﬁnite Voters,”
Journal of Economic Theory 2, pp. 103-106.
Suzumura, K. (1988), “Reduction of Social Choice Problems: A Simple Proof of Arrow’s
General Possibility Theorem,” Hitotsubashi Journal of Economics 29, pp. 219-21.
Sen, A. K. (1979), “Personal Utilities and Public Judgements: Or What’s Wrong with Welfare
Economics?” Economic Journal 89, pp. 537-558.
Takekuma, S. (1997), “A Proof of Arrow’s Impossibility Theorem by Mathematica,” Hitotsu-
bashi Journal of Economics 38, pp. 139-48.
2005] 6 H>BEA>;>:9 9:G>K6I>DC D; 6GGDL’H >BEDHH>7>A>IN I=:DG:B +2+

A simplified derivation of arrow's impossibility theorem

http://hermes-ir.lib.hit-u.ac.jp/rs/bitstream/10086/7654/1/HJeco0460201770.pdf

A simplified derivation of arrow's impossibility theorem

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics