Fuzzy Logic Approach to Downsian Model of Political Competition

Outline
Introduction
Concepts
Fuzzy Concepts of Sets, Number and Operations.
Fuzzy numbers in MVT and DMPC
Some conclusions
An Extension of Downs Model of Political
Competition using Fuzzy Logic (Social Choice
under Fuzzy Policy Perception)
Camilo José Pecha Garzón
Universitat Autónoma de Barcelona
July 11, 2013
Camilo José Pecha Garzón An Extension of Downs Model of Political Competition using F

Outline
Introduction
Concepts
Some conclusions
Introduction
Concepts
Preference Relations.
Single-Peaked Preferences
Median Voter Theorem
Downsian Partisan Competition and Political Convergence
MVT with fuzzy representation-Examples.
DMPC with fuzzy representation-Examples.
Some conclusions

Outline
Introduction
Concepts
Some conclusions
Motivation
“So far as laws of mathematics refer to reality, they are not
certain; and so far as they are certain, they do not refer to reality”.
Albert Einstein, Geometry and Experience, cited in [Klir and Yuan,
1995].
Many authors have been demonstrated that MVT’s
equilibrium is not stable if there are assumptions like market
imperfections, or asymmetric information, transaction costs,
among others. This document intends to show that MVT’s
equilibrium is not stable if agents are assumed behave under
Fuzzy Logic.
The principal idea here is to include a new tool set that
includes ways to measure perception and also the implication
of political ideology in that perception, this tool set is called
Fuzzy Sets.Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F

Outline
Introduction
Concepts
Some conclusions
Introduction
Duncan Black [Black, 1948] proposed a mechanism that is
incorporated as a preferences’ aggregation mechanism in a
voting process with agenda setting, reaching to a social
choice. This mechanism was called the median voter theorem
(MVT).
This paper seeks to introduce the fuzzy analysis as a tool to
understand individual decision making in a society. In
particular, to show the implications of assuming that agents
has fuzzy choose behavior within the MVT, as well as changes
in the results of DMPC that it might generate

Outline
Introduction
Concepts
Some conclusions
Preference Relations.
A Preference Relation (R) is a subset of Cartesian product of
consumption set X with X:
R ⊂ X × X.
it satisﬁes:
Reﬂexivity: ∀x ∈ X, (x, x) ∈ R.
Transitivity: ∀x ∈ X, ∀y ∈ X, ∀z ∈ X,
(x, y) ∈ R ∧ (y, z) ∈ R =⇒ (x, z) ∈ R
Anti-simetric: ∀x ∈ X, ∀y ∈ X,
(x, y) ∈ R ∧ (y, x) ∈ R =⇒ x = y.

Outline
Introduction
Concepts
Some conclusions
Single-Peaked Preferences
Definition
Voter i’s Policy Preferences are Single-Peaked if and only if:
q < q < qi , or
q > q > qi ,
then V i (q ) << V i (qi ).
Strict Concavity of V i (q) with respect to policy vector is sufficient
to ensure that preferences are single-piked. [Acemoglu and
Robinson, 2006, pp. 92-98].1
1
If q ≤ q ≤ qi
or q ≥ q ≥ qi
, and V i
(q ) ≤ V i
(qi
), and function V i
(q)
is not strictly concave, a potential result is that voter is indifferent to choose
between policies.

Outline
Introduction
Concepts
Some conclusions
Median Voter Theorem
Proposition (Median Voter Theorem)
Consider a set of policies Q ⊂ R; q ∈ Q a policy and median voter
(M) with ideal value qM. If all individuals have Single-Peaked
Preferences over Q, then:
1. qM always defeat any other alternative q ∈ Q were q = qM
on a voting over pair of policies.
2. qM is the winner in direct democracy and open agenda.

Outline
Introduction
Concepts
Some conclusions
Downsian Partisan Competition and Political Convergence
Proposition (Down’s Political Convergency Theorem)
Consider a vector of Proposals (q∗
A, q∗
B) ∈ Q × Q were Q ⊂ R, and
two candidates, A and B, that only care about winning the
elections and can commit with policy proposals. M is the median
voter and its ideal value qM. If all voters have single-peaked
preferences over Q, then both candidates will chose their proposals
such that q∗
A = q∗
B = qM, that constitutes the game’s unique
equilibrium. [Acemoglu and Robinson, 2006, pp. 92-98].

Outline
Introduction
Concepts
Some conclusions
Fuzzy Sets
In classical sets, elements belong to the set or not. In fuzzy set
theory, elements in the universe belong to the set with a certain
degree. This degree is generated by a Membership Function.
Deﬁnition (Fuzzy Set)
Given X the Universe Set, the set ¯A subset of X (¯A ⊂ X) ;
Membership Function takes elements from X an send these to
[0, 1]:
µ¯A(x) : x → [0, 1].

Outline
Introduction
Concepts
Some conclusions
Figure: Young and very young people sets.

Outline
Introduction
Concepts
Some conclusions
Fuzzy number
Deﬁnition (Fuzzy Number)
A fuzzy convex set of real numbers with normalized and
continuous by parts membership function is called “Fuzzy
Number” [Lee, 2005, p. 18].

Outline
Introduction
Concepts
Some conclusions
Membership function for a given triangular shaped fuzzy number is:
µ¯A(x) =



µL
¯A
(x), if a1 ≤ x ≤ a2,
1, if x = a2,
µR
¯A
(x), if a2 ≤ x ≤ a3,
0, otherwise.
For trapezoidal shaped numbers, assumptions remain but intervals
change.
µ¯A(x) =



µL
¯A
(x), if d ≤ x ≤ e,
1, if e ≤ x ≤ f ,
µR
¯A
(x), si f ≤ x ≤ g,
0, otherwise.

Outline
Introduction
Concepts
Some conclusions
α-Cuts
“slices” through a fuzzy set that produce crisp sets2. Being ¯A a
fuzzy set and 0 < α ≤ 1, ¯A’s α-cuts are given by:
µ¯A[α] = {x ∈ X|µ¯A(x) ≥ α}
supp(¯A) = {x ∈ X|µ¯A(x) > 0}
core(¯A) = {x ∈ X|µ¯A(x) = 1}.
Convex if and only if:
µ¯A(λx1 + (1 − λ)x2) ≥ min{µ¯A(x1), µ¯A(x2)}
Normal3 if and only if
∃x ∈ ¯A such that µ¯A(x) = 1.
2
crisp sets are non fuzzy sets, they are classical sets
3
Normality does not apply to all fuzzy sets, there are cases in which the

Outline
Introduction
Concepts
Some conclusions
Figure: ¯K = (a/b/c)

Outline
Introduction
Concepts
Some conclusions
Figure: ¯F = (d/e/f /g).

Outline
Introduction
Concepts
Some conclusions
Inequalities
To compare a fuzzy number and a real number it is used “
d
≤”
ordination. The following rule is one that can used to compare
fuzzy numbers:
If ¯K = (a/b/c) is a fuzzy number and θ a real number:
θ
d
≤ ¯K if θ ≤ a.
θ
d
< ¯K if θ < a.
θ
d
≥ ¯K if θ ≥ c.
θ
d
> ¯K if θ > c.

Outline
Introduction
Concepts
Some conclusions
Comparison Indexes
Other comparison ways uses indexes (Onwards CI) with (β)
parameter which is a Decision Maker’s (DM) optimism, pessimism
or neutrality measure and in this analysis represents the policy
observer’s (voter) left-right political ideology. If R¯A,¯B(β) is CI
between ¯A and ¯B then:
1. if R¯A,¯B(β) > 0, then ¯A
d
> ¯B.
2. if R¯A,¯B(β) = 0, then ¯A
d
= ¯B.
3. if R¯A,¯B(β) < 0, then ¯A
d
< ¯B.

Outline
Introduction
Concepts
Some conclusions
Here it is used Index constructed by [Chen and Lu, 2002]4 because
its interpretation is applicable to the case of political attitudes and
its α-cut structure can measure political voters’ attributes.
4
Othe CI are those proposed by Liu and Han [Liu and Han, 2005] and Liou
and Wang [liou and Wang, 1992] who develop indexes from membership
function

Outline
Introduction
Concepts
Some conclusions
If ¯qi is the policy preferred by voter i (i = 1, 2, . . . , W ) and
αk = k/n with k ∈ {0, 1, 2, ..., n}, n ∈ N, αk-cut is µ¯qi [αk]
and represents voter i’s “position” with respect to k-th policy
component. For example, for “tax level”, each α-cut belongs
to voter’s position over infrastructure (or investment, or
income redistribution) components that will aﬀect the voter’s
preference over tax level.
li,k = min{x|x ∈ µ¯qi [αk]}, ri,k = max{x|x ∈ µ¯qi [αk]},
mi,k =
(ri,k +li,k )
2 , δi,k = (ri,k − li,k), is the value for the left
and right perceived degree over k-th policy component for the
i-th voter. Third and fourth equalities are the average and the
dispersion of k-th policy component for the i-th voter,
respectively.

Outline
Introduction
Concepts
Some conclusions
∆i,k(β) = βri,k + (1 − β)li,k, is the valuation of the i-th
voter’s political ideology. This equation weight perception
over policies’ li,k and ri,k with respect to β. If β ∈ (0.5, 1],
voter has a right political ideology, and if β ∈ [0, 0.5), left
ideology, if β = 1/2 ideology is moderated or center.
ηa
i,k = 1 − 1
1+ηi,k
, were ηi,k = mi,k/δi,k is the signal-noise ratio
of each policy component, i.e what for that proposed by
candidate is perceived by the voters and how much this
information is distorted5.
5
As ηi,k tends to inﬁnity when δi,k tends to zero, ηa
i,k lies between 0 and 1.

Outline
Introduction
Concepts
Some conclusions
¯qs and ¯qt, two perception over fuzzy policy proposals
(proposals as fuzzy numbers), CI from [Chen and Lu, 2002] is:
Rs,t(β) =
n
k=1
αk × [∆s,k(β) − ∆t,k(β)] × ηa
s,k/ηa
t,k
n
k=1
αk
.
1. if Rs,t(β) > 0 (¯qs
d
> ¯qt), then perception over policies ¯qs and
¯qt is that ¯qs is superior to ¯qt for any value of β, i.e for every
kind of voter (left, centre, right),
2. if Rs,t(β) = 0 (¯qs
d
= ¯qt), both policies are perceived as equal
by voter with any political tendency (∀β), and
3. if Rs,t(β) < 0 (¯qs
d
< ¯qt), ¯qt is perceived as superior than ¯qs by
any type of voters.

Outline
Introduction
Concepts
Some conclusions
Median Voter Theorem with fuzzy representation.
Figure: Policy as a Real Number.

Outline
Introduction
Concepts
Some conclusions
Figure: Fuzzy and not fuzzy number over policy.

Outline
Introduction
Concepts
Some conclusions
Figure: Fuzzy numbers ¯qi1, ¯qi0 and ¯q .

Outline
Introduction
Concepts
Some conclusions
Definition
Voter i’s Preferences are Single-Peaked with fuzzy numbers if and
only if:6
¯q
d
< ¯q
d
< ¯qi , or
¯q
d
> ¯q
d
> ¯qi ,
then V i (¯q ) < V i (¯qi ). To ensure the policies ordination, it is
necessary that CI satisfies the following:
1. ¯q
d
< ¯q if and only if R¯q ,¯q (β) < 0 and ¯q
d
< ¯qi if and only if
R¯q ,¯qi
(β) < 0, or
2. ¯q
d
> ¯q if and only if R¯q ,¯q (β) > 0 and ¯q
d
> ¯qi if and only if
R¯q ,¯qi
(β) > 0.
6
Given that for thr ordination relation is necessary to make some
comparisons, it is used the comparison index between fuzzy numbers (CI) in
the following definition [Liu and Han, 2005] and [Chen and Lu, 2002].Camilo José Pecha Garzón An Extension of Downs Model of Political Competition using F

Outline
Introduction
Concepts
Some conclusions
MVT with fuzzy representation-Example 1.
Figure: Policies q and qM
as real numbers (left) and policies q and qM
as fuzzy numbers (right).

Outline
Introduction
Concepts
Some conclusions
Based on classical analysis, in an electoral race where there are two
options, q and qM such that q < qM (q =46/7 and qM=7) as is
shown in Figure 7 [Chen and Lu, 2002, pp 1462 and 1463], society
will choose option preferred by median voter. This is due to that
voters with ideal policy q such that q > qM will vote for qM
because this option implies the minimum decrease in their utility
function compered to the utility lose generated by option q .

Outline
Introduction
Concepts
Some conclusions
On the other hand, if the analysis is performed on policies ¯q and
¯qM, two fuzzy numbers (Figure 7), conclusions may diﬀer.
According to the example, ¯q = (94
35/46
7 /10) and ¯qM = (2/7/9) are
now fuzzy numbers that represents voters’ perceptions over
policies. Lets say that ¯q is a policy such that ¯q
d
< ¯qM, then, will
remain ¯qM socially preferred to ¯q ? [Chen and Lu, 2002] shown
that R¯q ,¯qM
=0.002 for β=1, which means that voters with ¯q
d
> ¯qM
perceve the inequality between ¯q and ¯qM as ¯q
d
> ¯qM, hence ¯qM is
not the winner. Voters with ¯q
d
< ¯qM and voters with ¯q
d
> ¯qM will
vote for policy ¯q .

Outline
Introduction
Concepts
Some conclusions
MVT with fuzzy representation-Example 2.
Figure: Policies qM
and q as real numbers (left) and policies q and qM
as fuzzy numbers (right).

Outline
Introduction
Concepts
Some conclusions
Now, in an electoral race where there are other two options, q and
qM such that q > qM (q =0.7 and qM=0.5) as shown Figure 8,
based in the classical analysis qM again defeats q . Assuming
options as fuzzy numbers ¯q =(0.35/0.5/1.0) and
¯qM=(0.15/0.7/0.8) (Figure 8), results will change. [Chen and Lu,
2002] found that R¯qM ,¯q =-0.077 for β=0, i.e, voters who have a
political left ideology perceive as better option that one that lies in
the right of median voter’s preference. If so, policy ¯q defeat policy
¯qM. So voters with ¯q
d
> ¯qM as voters with ¯q
d
< ¯qM perceives ¯q
over ¯qM.

Outline
Introduction
Concepts
Some conclusions
DMPC with fuzzy representation-Example 1.
Figure: qA
and qB
diﬀerent proposals as fuzzy numbers.

Outline
Introduction
Concepts
Some conclusions
Figure 9 shows a pair of policies proposed by candidates A and B,
qA = 0, 5 and qB = 0, 7, respectively [Chen and Lu, 2002, p. 1462]
and [Liu and Han, 2005, p.1747]. It is assumed that the best
policy for the median voter is such that qM ∈ [qB, 0, 75). If the
analysis is based on a classical way, proposals will be ordered as
qA < qB < qM, which implies that the candidate B is the winner.
Now, if the proposed policies are perceived in a fuzzy way by the
median voter (as in Figure 9), results vary. The index constructed
in [Liu and Han, 2005] says that if voters are neutral such that
β = 0, 5 or near 0.5, proposals are perceived such that ¯qA
d
= ¯qB
and in a very probably manner, equal in a fuzzy way to ¯qM. This
example illustrates how a voter with a center-wing political
ideology (tentatively the median voter) and fuzzy logic behavior,
perceive proposed options as equal (both core and supp α-Cuts of
each policy are diﬀerent).

Outline
Introduction
Concepts
Some conclusions
DMPC with fuzzy representation-Example 2.
Figure: qA
and qB
equal proposals (left), ¯qA and ¯qB diﬀerent proposals
(right).
Figure 10 shows an example where both candidates having the
same proposal it is not maintained the classical convergence
equilibrium: although the two candidates who know their proposals
are the same, voter’s behavior under fuzzy logic makes him/her
perceived these as two diﬀerent proposals.

Outline
Introduction
Concepts
Some conclusions
Two identical proposals qA and qB as shown in Figure 10 and qM,
the best policy for the median voter such that
qA = qB = qM = 0, 5, if the analysis is done from the classical
point of view, this represents the model equilibrium. If proposals
are represented in terms of median voter’s fuzzy behavior, the
result changes. According to [Chen and Lu, 2002] and [Liu and
Han, 2005] the voter perceives inequality between these proposals,
giving greater importance to the platform which he perceived
nearest to ¯qM. In Figure 10 [Liu and Han, 2005, pp.1747-1748]
shows proposals with the same core but with diﬀerent supp, this
generates platform ¯qB
d
> ¯qA, which implies that the candidate B
wins since the voter is in a place such that ¯qA
d
> ¯qM, or candidate
A will be the winner if voter is somewhere such that ¯qA
d
> ¯qM.

Outline
Introduction
Concepts
Some conclusions
Result shown are:
1. by defining relations between fuzzy sets, preferences properties
are held, in particular fuzzy transitivity solves Arrow’s
aggregation problem without affecting negatively any society
actor with the final election;
2. it was tested that MVT does not held by using fuzzy concepts;
3. also, it was proved that classic DMPC equilibrium is not
unique and does not always apply in fuzzy extension.

Outline
Introduction
Concepts
Some conclusions
Results allow to interpret fuzzy logic as a tool to understanding the
decision-making process in the real and subjective world. It is
possible that decisions made in a real environment are not taken
with complete certainty and are not explained by traditional MVT
and that the outcome of the classic DMPC does not explain social
political decisions (such as the choice of ultra-right candidates in
the northern European or left in some areas of Latin America).
Particularly, the fuzzy extension shows that an equal platform
between candidates could be perceived by voters as diﬀerent and
hence a winner will rise, without changes in the candidates’
political platform.

Fuzzy Logic Approach to Downsian Model of Political Competition

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