Introduction to spatial modeling. (a mostly geometrical presentation)

Transcription

1 Introduction to spatial modeling (a mostly geometrical presentation)

2 Alternatives X = R n e.g. (x 1,x 2,, x n ) X Alternatives are infinite set of policies in n- dimensional Euclidean space Each dimension is an issue or characteristic of policy: Economic liberalism Civil liberties Taxation Redistribution Defense spending Welfare spending Trade protection Immigration Lecture 4 Spatial Model 2

3 Preferences Preferences are satiable Each agent has an ideal point Utility declines as a distance from ideal point increases k U( x) = α θ j = 1 Linear j x j j k j = 1 ( ) U( x) = α θ Quadratic j x j j 2 j indexes dimensions α j = weight on dimension j θ j = ideal policy on dimension j Lecture 4 Spatial Model 3

4 One dimension Preferences satisfy single-peaked property Black s median voter theorem applies Lecture 4 Spatial Model 4

5 Two dimensions Median voter theorem does not apply Can we guarantee transitivity of MR? Can it be generalized to 2 dimensions? Lecture 4 Spatial Model 5

6 Utility function Lecture 4 Spatial Model 6

7 Lecture 4 Spatial Model 7 Projection onto policy space ( ) ( ) ( ) ( ) ( ) ( ) ( ) θ θ θ θ θ θ + = = = y x u y x u y x y x, U

8 Indifferent between x and y Lecture 4 Spatial Model 8

9 Projection onto policy space w P z P x I y Lecture 4 Spatial Model 9

10 Effect of weights Equal weights: Indifference circle Different weights: Indifference ellipse α 1 = α 2 α 1 < α 2 Lecture 4 Spatial Model 10

11 Cut point Midpoint between two alternatives, divides ideal points x c = x + 2 y y Lecture 4 Spatial Model 11

12 Cut point Midpoint between two alternatives, divides agents with opposing preferences θ x c = x + 2 y y d(θ,x) d(θ,y) xp θ y Lecture 4 Spatial Model 12

13 Cut point Midpoint between two alternatives, divides ideal points x c = x + 2 y y τ d(τ,x) d(τ,y) yp τ x Lecture 4 Spatial Model 13

14 Cutting lines Set of points equidistant between two alternatives Convenient way to determine preferences x y All voters with ideal points on this side of line: xpy All voters with ideal points on this side of line: ypx Lecture 4 Spatial Model 14

15 Useful sets P i (x) = i s preferred-to set of x Set of policies an individual prefers to x (Interior of indifference curve through x) W(x) = Majority rule winset of x Set of all policies that some majority prefers to x Finding winsets Step 1. For each majority coalition, find intersection of preferred-to sets Step 2. Winset is union of sets in Step 1. Lecture 4 Spatial Model 15

16 Finding W(Q) Set of policies coalition {1,2} prefers to Q Lecture 4 Spatial Model 16

17 Finding W(Q) Set of policies coalition {1,3} prefers to Q Lecture 4 Spatial Model 17

18 Finding W(Q) NOTE: This figure is incorrect since P2 s indifference curve should go through Q instead of P3. Set of policies coalition {2,3} prefers to Q Lecture 4 Spatial Model 18

19 Finding W(Q) NOTE: This figure is incorrect since P2 s indifference curve should go through Q instead of P3. Majority rule winset of Q Lecture 4 Spatial Model 19

20 Finding W(Q) NOTE: This figure is incorrect since P2 s indifference curve should go through Q instead of P3. Unanimity rule winset of Q Lecture 4 Spatial Model 20

21 Plott conditions The core is non-empty if and only if ideal points are distributed in a radially symmetric fashion around a policy x* and x* is a voter s ideal point Radial symmetry means that pairs of ideal points form lines that intersect x* with x* between the pair of ideal points Lecture 4 Spatial Model 21

22 Examples Lecture 4 Spatial Model 22

23 Examples: Plott conditions hold P2 has an empty winset Condorcet Winner Lecture 4 Spatial Model 23

24 Examples: Plott conditions hold P2 has an empty winset Condorcet Winner Lecture 4 Spatial Model 24

25 Examples: Plott conditions violated Plott conditions are violated W(P2) nonempty Lecture 4 Spatial Model 25

26 Example: Plott conditions violated Plott conditions are violated W(P2) nonempty Lecture 4 Spatial Model 26

27 Constructing a preference cycle Majority {P1, P4, P5} votes for B over A Lecture 4 Spatial Model 27

28 Constructing a preference cycle Majority {P1, P2, P5} votes for C over B Lecture 4 Spatial Model 28

29 Constructing a preference cycle Majority {P2, P3, P4} votes for A over C Lecture 4 Spatial Model 29

30 Top cycle set Alternatives in the top cycle set Defeat all alternatives outside the set Preference cycles over the alternatives in the set Example: apb, bpc, cpa, apd, bpd, cpd T = {a,b,c} Lecture 4 Spatial Model 30

31 McKelvey s Theorem Given the spatial model, the majority rule core is either non-empty or the top cycle set is T = X. Lecture 4 Spatial Model 31

32 McKelvey s Theorem (corollary) If the Plott conditions are not satisfied, then for any two points x and y, there exists a finite chain of policies {a 1,a 2,,a n } such that xpa 1 Pa 2 Pa n Py Lecture 4 Spatial Model 32

33 Construct a chain from y to x Lecture 4 Spatial Model 33

34 Note that x is majority preferred to y! Lecture 4 Spatial Model 34

35 W(x) Lecture 4 Spatial Model 35

36 z 1 P x Lecture 4 Spatial Model 36

37 W(z 1 ) Lecture 4 Spatial Model 37

38 z 2 P z 1 P x Lecture 4 Spatial Model 38

39 y P z 2 P z 1 P x Lecture 4 Spatial Model 39

40 Although x P y, we have the chain: y P z 2 P z 1 P x Lecture 4 Spatial Model 40

41 Implications Plott conditions are very rarely satisfied In two dimensions, we can cycle over every policy McKelvey s Theorem does not predict chaos All preference aggregation rules are problematic, including majority rule Preference aggregation alone is insufficient to understand political outcomes Lecture 4 Spatial Model 41