An Extension of Downs Model of Political Competition using Fuzzy Logic (Social Choice under Fuzzy Policy Perception)

OutlineIntroduction

ConceptsFuzzy Concepts of Sets, Number and Operations.

Fuzzy numbers in MVT and DMPCSome conclusions

An Extension of Downs Model of PoliticalCompetition using Fuzzy Logic (Social Choice

under Fuzzy Policy Perception)

Camilo Jose Pecha Garzon

Universitat Autonoma de Barcelona

July 11, 2013

Camilo Jose Pecha Garzon An Extension of Downs Model of Political Competition using Fuzzy Logic (Social Choice under Fuzzy Policy Perception)

OutlineIntroduction



Introduction

ConceptsPreference Relations.Single-Peaked PreferencesMedian Voter TheoremDownsian Partisan Competition and Political Convergence

Fuzzy Concepts of Sets, Number and Operations.

Fuzzy numbers in MVT and DMPCMVT with fuzzy representation-Examples.DMPC with fuzzy representation-Examples.

Some conclusions


OutlineIntroduction



Motivation

“So far as laws of mathematics refer to reality, they are notcertain; and so far as they are certain, they do not refer to reality”.Albert Einstein, Geometry and Experience, cited in [Klir and Yuan,1995].

I Many authors have been demonstrated that MVT’sequilibrium is not stable if there are assumptions like marketimperfections, or asymmetric information, transaction costs,among others. This document intends to show that MVT’sequilibrium is not stable if agents are assumed behave underFuzzy Logic.

I The principal idea here is to include a new tool set thatincludes ways to measure perception and also the implicationof political ideology in that perception, this tool set is calledFuzzy Sets.


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Introduction

I Duncan Black [Black, 1948] proposed a mechanism that isincorporated as a preferences’ aggregation mechanism in avoting process with agenda setting, reaching to a socialchoice. This mechanism was called the median voter theorem(MVT).

I This paper seeks to introduce the fuzzy analysis as a tool tounderstand individual decision making in a society. Inparticular, to show the implications of assuming that agentshas fuzzy choose behavior within the MVT, as well as changesin the results of DMPC that it might generate


OutlineIntroduction



Preference Relations.

A Preference Relation (R) is a subset of Cartesian product ofconsumption set X with X :

R ⊂ X × X .

it satisfies:

I Reflexivity: ∀x ∈ X , (x , x) ∈ R.

I Transitivity: ∀x ∈ X , ∀y ∈ X , ∀z ∈ X ,(x , y) ∈ R ∧ (y , z) ∈ R =⇒ (x , z) ∈ R

I Anti-simetric: ∀x ∈ X , ∀y ∈ X ,(x , y) ∈ R ∧ (y , x) ∈ R =⇒ x = y .


OutlineIntroduction



Single-Peaked Preferences

DefinitionVoter i ’s Policy Preferences are Single-Peaked if and only if:

I q′′ < q′ < qi , or

I q′′ > q′ > qi ,

then V i (q′′) << V i (qi ).Strict Concavity of V i (q) with respect to policy vector is sufficientto ensure that preferences are single-piked. [Acemoglu andRobinson, 2006, pp. 92-98].1

1If 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 choosebetween policies.


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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-PeakedPreferences over Q, then:

1. qM always defeat any other alternative q′ ∈ Q were q′ 6= qM

on a voting over pair of policies.

2. qM is the winner in direct democracy and open agenda.


OutlineIntroduction



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 theelections and can commit with policy proposals. M is the medianvoter and its ideal value qM . If all voters have single-peakedpreferences over Q, then both candidates will chose their proposalssuch that q∗A = q∗B = qM , that constitutes the game’s uniqueequilibrium. [Acemoglu and Robinson, 2006, pp. 92-98].


OutlineIntroduction



Fuzzy Sets

In classical sets, elements belong to the set or not. In fuzzy settheory, elements in the universe belong to the set with a certaindegree. 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]:

µA(x) : x → [0, 1].


OutlineIntroduction



Figure: Young and very young people sets.


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Fuzzy number

Definition (Fuzzy Number)

A fuzzy convex set of real numbers with normalized andcontinuous by parts membership function is called “FuzzyNumber” [Lee, 2005, p. 18].


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Membership function for a given triangular shaped fuzzy number is:

µA(x) =

µL

A(x), if a1 ≤ x ≤ a2,

1, if x = a2,

µRA

(x), if a2 ≤ x ≤ a3,

0, otherwise.

For trapezoidal shaped numbers, assumptions remain but intervalschange.

µA(x) =

µL

A(x), if d ≤ x ≤ e,

1, if e ≤ x ≤ f ,

µRA

(x), si f ≤ x ≤ g ,

0, otherwise.


OutlineIntroduction



α-Cuts

“slices” through a fuzzy set that produce crisp sets2. Being A afuzzy set and 0 < α ≤ 1, A’s α-cuts are given by:

µA[α] = {x ∈ X |µA(x) ≥ α}

I supp(A) = {x ∈ X |µA(x) > 0}I core(A) = {x ∈ X |µA(x) = 1}.I Convex if and only if:

µA(λx1 + (1− λ)x2) ≥ min{µA(x1), µA(x2)}I Normal3 if and only if

∃x ∈ A such that µA(x) = 1.2crisp sets are non fuzzy sets, they are classical sets3Normality does not apply to all fuzzy sets, there are cases in which the

maximum value of membership function is less than 1.Camilo Jose Pecha Garzon An Extension of Downs Model of Political Competition using Fuzzy Logic (Social Choice under Fuzzy Policy Perception)

OutlineIntroduction



Figure: K = (a/b/c)


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Figure: F = (d/e/f /g).


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Inequalities

To compare a fuzzy number and a real number it is used “d≤”

ordination. The following rule is one that can used to comparefuzzy numbers:If K = (a/b/c) is a fuzzy number and θ a real number:

I θd≤ K if θ ≤ a.

I θd< K if θ < a.

I θd≥ K if θ ≥ c .

I θd> K if θ > c .


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Comparison Indexes

Other comparison ways uses indexes (Onwards CI ) with (β)parameter which is a Decision Maker’s (DM) optimism, pessimismor neutrality measure and in this analysis represents the policyobserver’s (voter) left-right political ideology. If RA,B(β) is CI

between A and B then:

1. if RA,B(β) > 0, then Ad> B.

2. if RA,B(β) = 0, then Ad= B.

3. if RA,B(β) < 0, then Ad< B.


OutlineIntroduction



Here it is used Index constructed by [Chen and Lu, 2002]4 becauseits interpretation is applicable to the case of political attitudes andits α-cut structure can measure political voters’ attributes.

4Othe CI are those proposed by Liu and Han [Liu and Han, 2005] and Liouand Wang [liou and Wang, 1992] who develop indexes from membershipfunction


OutlineIntroduction



I If qi is the policy preferred by voter i (i = 1, 2, . . . ,W ) andαk = k/n with k ∈ {0, 1, 2, ..., n}, n ∈ N, αk -cut is µqi [αk ]and represents voter i ’s “position” with respect to k-th policycomponent. For example, for “tax level”, each α-cut belongsto voter’s position over infrastructure (or investment, orincome redistribution) components that will affect the voter’spreference over tax level.

I li ,k = min{x |x ∈ µqi [αk ]}, ri ,k = max{x |x ∈ µqi [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 thei-th voter. Third and fourth equalities are the average and thedispersion of k-th policy component for the i-th voter,respectively.


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I ∆i ,k(β) = βri ,k + (1− β)li ,k , is the valuation of the i-thvoter’s political ideology. This equation weight perceptionover 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), leftideology, if β = 1/2 ideology is moderated or center.

I ηai ,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 bycandidate is perceived by the voters and how much thisinformation is distorted5.

5As ηi,k tends to infinity when δi,k tends to zero, ηai,k lies between 0 and 1.


OutlineIntroduction



I 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)]× ηas,k/η

at,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 everykind of voter (left, centre, right),

2. if Rs,t(β) = 0 (qsd= 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.


OutlineIntroduction



Median Voter Theorem with fuzzy representation.

Figure: Policy as a Real Number.


OutlineIntroduction



Figure: Fuzzy and not fuzzy number over policy.


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Figure: Fuzzy numbers qi1, qi0 and q′.


OutlineIntroduction



DefinitionVoter i ’s Preferences are Single-Peaked with fuzzy numbers if andonly if:6

I q′′d< q′

d< qi , or

I q′′d> q′

d> qi ,

then V i (q′′) < V i (qi ). To ensure the policies ordination, it isnecessary that CI satisfies the following:

1. q′′d< q′ if and only if Rq′′,q′(β) < 0 and q′

d< qi if and only if

Rq′,qi(β) < 0, or

2. q′′d> q′ if and only if Rq′′,q′(β) > 0 and q′

d> qi if and only if

Rq′,qi(β) > 0.

6Given that for thr ordination relation is necessary to make somecomparisons, it is used the comparison index between fuzzy numbers (CI ) inthe following definition [Liu and Han, 2005] and [Chen and Lu, 2002].Camilo Jose Pecha Garzon An Extension of Downs Model of Political Competition using Fuzzy Logic (Social Choice under Fuzzy Policy Perception)

OutlineIntroduction



MVT with fuzzy representation-Example 1.

Figure: Policies q′ and qM as real numbers (left) and policies q′ and qM

as fuzzy numbers (right).


OutlineIntroduction



Based on classical analysis, in an electoral race where there are twooptions, q′ and qM such that q′ < qM (q′=46/7 and qM=7) as isshown in Figure 7 [Chen and Lu, 2002, pp 1462 and 1463], societywill choose option preferred by median voter. This is due to thatvoters with ideal policy q such that q > qM will vote for qM

because this option implies the minimum decrease in their utilityfunction compered to the utility lose generated by option q′.


OutlineIntroduction



On the other hand, if the analysis is performed on policies q′ andqM , two fuzzy numbers (Figure 7), conclusions may differ.According to the example, q′ = ( 94

35/467 /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 qd< qM , then, will

remain q′M socially preferred to q′? [Chen and Lu, 2002] shown

that Rq′,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 qd< qM and voters with q

d> qM will

vote for policy q′.


OutlineIntroduction



MVT with fuzzy representation-Example 2.

Figure: Policies qM and q′ as real numbers (left) and policies q′ and qM

as fuzzy numbers (right).


OutlineIntroduction



Now, in an electoral race where there are other two options, q′ andqM such that q′ > qM (q′=0.7 and qM=0.5) as shown Figure 8,based in the classical analysis qM again defeats q′. Assumingoptions as fuzzy numbers q′=(0.35/0.5/1.0) andqM=(0.15/0.7/0.8) (Figure 8), results will change. [Chen and Lu,2002] found that RqM ,q′=-0.077 for β=0, i.e, voters who have apolitical left ideology perceive as better option that one that lies inthe right of median voter’s preference. If so, policy q′ defeat policy

qM . So voters with qd> qM as voters with q

d< qM perceives q′

over qM .


OutlineIntroduction



DMPC with fuzzy representation-Example 1.

Figure: qA and qB different proposals as fuzzy numbers.


OutlineIntroduction



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 bestpolicy for the median voter is such that qM ∈ [qB , 0, 75). If theanalysis is based on a classical way, proposals will be ordered asqA < qB < qM , which implies that the candidate B is the winner.Now, if the proposed policies are perceived in a fuzzy way by themedian voter (as in Figure 9), results vary. The index constructedin [Liu and Han, 2005] says that if voters are neutral such that

β = 0, 5 or near 0.5, proposals are perceived such that qAd= qB

and in a very probably manner, equal in a fuzzy way to qM . Thisexample illustrates how a voter with a center-wing politicalideology (tentatively the median voter) and fuzzy logic behavior,perceive proposed options as equal (both core and supp α-Cuts ofeach policy are different).


OutlineIntroduction



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 thesame proposal it is not maintained the classical convergenceequilibrium: although the two candidates who know their proposalsare the same, voter’s behavior under fuzzy logic makes him/herperceived these as two different proposals.


OutlineIntroduction



Two identical proposals qA and qB as shown in Figure 10 and qM ,the best policy for the median voter such thatqA = qB = qM = 0, 5, if the analysis is done from the classicalpoint of view, this represents the model equilibrium. If proposalsare represented in terms of median voter’s fuzzy behavior, theresult changes. According to [Chen and Lu, 2002] and [Liu andHan, 2005] the voter perceives inequality between these proposals,giving greater importance to the platform which he perceivednearest 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 qBd> qA, which implies that the candidate B

wins since the voter is in a place such that qAd> qM , or candidate

A will be the winner if voter is somewhere such that qAd> qM .


OutlineIntroduction



Result shown are:

1. by defining relations between fuzzy sets, preferences propertiesare held, in particular fuzzy transitivity solves Arrow’saggregation problem without affecting negatively any societyactor 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 notunique and does not always apply in fuzzy extension.


OutlineIntroduction



Results allow to interpret fuzzy logic as a tool to understanding thedecision-making process in the real and subjective world. It ispossible that decisions made in a real environment are not takenwith complete certainty and are not explained by traditional MVTand that the outcome of the classic DMPC does not explain socialpolitical decisions (such as the choice of ultra-right candidates inthe northern European or left in some areas of Latin America).Particularly, the fuzzy extension shows that an equal platformbetween candidates could be perceived by voters as different andhence a winner will rise, without changes in the candidates’political platform.


Education

￼An Extension of Downs Model of Political Competition using Fuzzy Logic (Social Choice under Fuzzy Policy Perception)

An Extension of Downs Model of Political Competition using Fuzzy Logic (Social Choice under Fuzzy Policy Perception)