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

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Fuzzy logic applied to the Median Voter theorem and the Downs Political competition model

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

1. 1. 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 Jos´e Pecha Garz´on Universitat Aut´onoma de Barcelona July 11, 2013 Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
2. 2. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Introduction Concepts Preference Relations. Single-Peaked Preferences Median Voter Theorem Downsian Partisan Competition and Political Convergence Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC MVT with fuzzy representation-Examples. DMPC with fuzzy representation-Examples. Some conclusions Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
3. 3. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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
4. 4. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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 Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
5. 5. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
6. 6. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
7. 7. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
8. 8. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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]. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
9. 9. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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]. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
10. 10. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Figure: Young and very young people sets. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
11. 11. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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]. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
12. 12. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
13. 13. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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 Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
14. 14. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Figure: ¯K = (a/b/c) Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
15. 15. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Figure: ¯F = (d/e/f /g). Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
16. 16. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
17. 17. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
18. 18. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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 Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
19. 19. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
20. 20. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
21. 21. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
22. 22. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Median Voter Theorem with fuzzy representation. Figure: Policy as a Real Number. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
23. 23. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Figure: Fuzzy and not fuzzy number over policy. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
24. 24. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions Figure: Fuzzy numbers ¯qi1, ¯qi0 and ¯q . Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
25. 25. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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 Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
26. 26. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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). Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
27. 27. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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 . Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
28. 28. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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 . Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
29. 29. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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). Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
30. 30. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
31. 31. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC Some conclusions DMPC with fuzzy representation-Example 1. Figure: qA and qB diﬀerent proposals as fuzzy numbers. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
32. 32. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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). Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
33. 33. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
34. 34. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
35. 35. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
36. 36. Outline Introduction Concepts Fuzzy Concepts of Sets, Number and Operations. Fuzzy numbers in MVT and DMPC 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. Camilo Jos´e Pecha Garz´on An Extension of Downs Model of Political Competition using F
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