In Search of the Uncovered Set: Supplementary Figures

FIGURE 1.         Ideal Points, the Pareto Set, the 4r Circle, and the Potential Uncovered Set

FIGURE 1A       An Ideal Point Configuration with n = 9

FIGURE 1B       The Pareto Set (Uncovered Set Lies Within It)

FIGURE 1C       Limiting Median Lines

FIGURE 1D       The Yolk and Yolk Triangle

FIGURE 1E       The Yolk and the 4r Circle

FIGURE 1F       The McKelvey Bound on the Uncovered Set

FIGURE 2A       The Uncovered Set with Three Voters (Equilateral Pareto Triangle)

FIGURE 2B       The Uncovered Set with Three Voters (Acute Pareto Triangle)

FIGURE 2C.      The Uncovered Set with Three Voters (Obtuse Pareto Triangle)

FIGURE 2D       The Uncovered Sets with Three Almost Collinear Ideal Points

FIGURE 3A       BJS Figure 2

FIGURE 3B       BJS Figure 1

FIGURE 3C       BJS Figure 4(a)

FIGURE 3D       BJS Figure 4(a) [Clearer Version]

FIGURE 3E       BJS Figure 5

FIGURE 4A       Indifference Curves and the Win Set of a Point Inside the Pareto Set

FIGURE 4B       Indifference Curves and a Win Set of a Point Outside the Pareto Set

FIGURE 4C       Indifference Curves and a Win Set of a Point Outside the Pareto Set with Five Ideal Points

FIGURE 5          Indifference Curves and Win Sets When All Ideal Points are Collinear

FIGURE 6A       Induced Ideal Points on Line L through Point x with Three Voters

FIGURE 6B       Induced Ideal Points on L Through x such that Distinct Voters Jointly Occupy the Median Induced Ideal Point

FIGURE 6C       Induced Ideal Points on L such that L is a Dividing Line Through x and Showing W(x)

FIGURE 6D       Induced Ideal Points on Parallel Lines Lʹ, Lʺ, and Lʹʺ

FIGURE 7          A Non-Limiting Median Line M₅, a Limiting Median Line M₄₅, and a Line Through x₅ that is Not a Median Line

FIGURE 8A       Full Plott Symmetry with Diverse Ideal Points (Showing Limiting Median Lines)

FIGURE 8B       Sufficient Plott Symmetry with Non-Diverse Ideal Points (Showing Limiting Median Lines)

FIGURE 9A       A Non-limiting Median Line M_j

FIGURE 9B       A Limiting Median Line M_ij

FIGURE 9C       A Pair of Limiting Median Lines M_ij and M_jk Showing Angle α_ijk

FIGURE 9D       A “Stand-Alone” Limiting Median Line M_ijk with Collinear Ideal Points

FIGURE 9E       Two Pairs of Limiting Median Lines Through One Ideal Point

FIGURE 10A     A Yolk with a Zero Radius (Plott Symmetry)

FIGURE 10B     A Yolk with a Small Radius

FIGURE 10C     A Yolk with a Large Radius (Showing c, r, and Yolk Triangle)

FIGURE 10D     A Tovey Anomaly

FIGURE 11        A Win Set with a Single Voter

FIGURE 12A     A Circular Win Set W(x) (Yolk with Zero Radius)

FIGURE 12B     Circular Bounds on a Win Set W(x) (Yolk with Small Radius)

FIGURE 12C     Circular Bounds on a Win Set W(x) (Yolk with Large Radius, Point x less than 2r from c)

FIGURE 12D     Cardioid Bounds on a Win Set W(x) (Yolk with Small Radius)

FIGURE 12E      Cardioid Bounds on a Win Set W(x) (Yolk with Large Radius)

FIGURE 13A     W(x) on Line L through Point x

FIGURE 13B     The Win Set of an Ideal Point

FIGURE 14A     A Win Set W(x) with Non-Intersecting Petals

FIGURE 14B     A Win Set W(x) with One Leaf and One Subpetal

FIGURE 14C     Dividing Lines Through Point x

FIGURE 14D     Non-Adjacent Leaves of W(x) (Showing Dividing Lines)

FIGURE 14E     Adjacent but Non-intersecting Leaves of W(x)

FIGURE 15A     An Orderly Win Set W(x) (Point x Outside the Pareto Set)

FIGURE 15B     An Orderly Win Set W(x) (Point x Inside the Pareto Set)

FIGURE 15C     A Disorderly Win Set W(x) with Leaves that Fail to Intersect the Yolk (Point x Outside the Yolk)

FIGURE 15D     A “Highly Disorderly” Win Set W(x) (Point x Inside the Yolk)

FIGURE 16A     A Configuration of 51 Ideal Points From a Bivariate Normal Distribution, Showing All Limiting Median Lines

FIGURE 16B     A Configuration of 51 Ideal Points From a Bivariate Normal Distribution, Showing the Yolk, Yolk Triangle, and a Typical Win Set with 4r Circle and Cardioid Bounds

FIGURE 16C     A Configuration of 101 Ideal Points From a Bivariate Normal Distribution, Showing All Limiting Median Lines and the Yolk

FIGURE 16D     A Configuration of 251 Ideal Points From a Bivariate Normal Distribution, Showing All Limiting Median Lines and the Yolk

FIGURE 16E      A Configuration of 435 Ideal Points From a Bivariate Normal Distribution, Showing All Limiting Median Lines and the Yolk

FIGURE 17A     Expected Yolk Radius (Lowess Smoothing) By the Number of Ideal Points, with n = 3 through n = 435 Drawn Randomly from a Bivariate Normal Distribution

FIGURE 17B     Expected Yolk Radius (Lowess Smoothing) By the Number of Ideal Points, with n = 3 through n = 435 (Log Scale) Drawn Randomly from a Bivariate Normal Distribution

FIGURE 17C     Expected Yolk Radius (Lowess Smoothing) By the Number of Ideal Points, with n = 3 through n = 435 Drawn Randomly from a Bivariate Uniform Distribution

FIGURE 17D     Expected Yolk Radius (Lowess Smoothing) By the Number of Ideal Points, with n = 3 through n = 435 (Log Scale) Drawn Randomly from a Bivariate Uniform Distribution

FIGURE 17E      Expected Yolk Radius (Plot of Means) By the Number of Ideal Points (n = 3 through n = 101) Drawn Randomly from a Bivariate Uniform Distribution [Bräuninger, Figure 5]

FIGURE 17F      FIGURE 17A Rescaled to Match FIGURE 17E

FIGURE 18A     A Highly Circular Win Set with n = 101 Drawn from a Bivariate Normal Distribution

FIGURE 18B     Figure 18A Showing Subpetals

FIGURE 18C     A Highly Circular Win Set with n = 251 Drawn from a Bivariate Normal Distribution

FIGURE 18D     A Highly Circular Win Set with n = 435 Drawn from a Bivariate Normal Distribution

FIGURE 19A     The Location and Size of the Yolk with Two Closely Balanced Clusters of Ideal Points

FIGURE 19B     FIGURE 19A after One Ideal Point “Turns Over”

FIGURE 20A     Line L through Point x and Median Line M_i Perpendicular to L

FIGURE 20B     Line L_ji through Point x and Median Line M_ji Perpendicular to L_ji (Showing Locus of Reflection Points)

FIGURE 20C     Line L_jk through Point x and Median Line M_jk Perpendicular to L_jk (Showing Locus of Reflection Points)

FIGURE 21        Angle Between M_ij and M_jk and L_ij and L_jk

FIGURE 22        Phantom Voters i and k on the “Stand-Alone” Limiting Median Line M_ijk

FIGURE 23A     BJS Figure 2 with Panels and Voters Labeled and Showing Effective Pareto Sets

FIGURE 23B     Added Panel (g) for BJS Figure 2 Showing Limiting Median Lines and the Effective Pareto Set

FIGURE 23C     Added Panel (h) for BJS Figure 2 Showing Limiting Median Lines and the Effective Pareto Set

FIGURE 24A     A Win Set W(x) in Panel (a) of BJS Figure 2 with x in Pareto Set (Showing Limiting Median Lines and Subpetals)

FIGURE 24B     A Win Set W(x) in Panels (b)-(e) of BJS Figure 2 with x in Pareto Set but Outside Effective Pareto Set (Showing Limiting Median Lines and Subpetals)

FIGURE 24C     A Win Set W(x) in Panel (f) of BJS Figure 2 with x in Pareto Set but Outside Effective Pareto Set (Showing Limiting Median Lines and Subpetals)

FIGURE 24D     A Win Set W(x) in Panel (f) of BJS Figure 2 with Ideal Points 2 and 3 Diverging

FIGURE 24E      A Win Set W(x) in Added Panel (g) of BJS Figure 2 with x in Effective Pareto Set (Showing Limiting Median Lines and Subpetals)

FIGURE 24F      A Win Set W(x) in Added Panel (h) of BJS Figure 2 with x in Effective Pareto Set (Showing Limiting Median Lines and Subpetals)

FIGURE 25A     Figure 3 from Miller (2002) Showing Possibility of Local Covering of Point x Outside of Pareto Set

FIGURE 25B     Figure 4 from Miller (2002) Showing Possibility of Local Covering of any Point x with Plott Symmetry

FIGURE 26A     Local Covering of Point y Outside the Pareto Set by Point x

FIGURE 26B     Local Covering of Point y Inside the Pareto Set but Outside the Effective Pareto Set by Point x (with Plott Symmetry) [BJS Figure 2(a)]

FIGURE 26C     Local Covering of a Point y Inside the Pareto Set but Outside the Effective Pareto Set by Point x [BJS Figures 2(b)-(e)]

FIGURE 26D     Disorderly Win Set of a Point x Inside the Effective Pareto Set (No Local Covering) [BJS Figures 2(b)-(e)]

FIGURE 26E      Absence of Local Covering Inside the Pareto Set (No Phantoms) [BJS Figure 2(g)]

FIGURE 26F      Absence of Local Covering with Phantoms Not at Vertices of Pareto Set [BJS Figure 2(h)]

FIGURE 27A     Typical Covering at a Distance of Point y by Point x (Points x and y Aligned with and on the Same Side of c)

FIGURE 27B     Typical Covering at a Distance of Point y by Point x (Points x and y Aligned with and on Opposite Sides of c)

FIGURE 27C     Typical Covering at a Distance of Point y by Point x (Points x, y, and c not Aligned)

FIGURE 28A     On the Line Through y and c, Point x Is the Closest Point to y that Covers y

FIGURE 28B     On the Line Through y and c, Point xʹ Covers y But Points Slightly Closer to c Do Not Cover y

FIGURE 28C     On the Line Through y and c, Point xʺ Covers y But Points Between xʺ and xʹ Do Not Cover y

FIGURE 28D     On the Line Through y and c, Point xʹʺ Covers y But Points Between xʹʺ and xʺ Do Not Cover y

FIGURE 29A     The Minimum Distance 4r Sufficient for Covering at a Distance

FIGURE 29B     Possible Covering Relationships for Point x (Based On Size of Yolk Only)

FIGURE 29C     Atypical Covering at a (Small) Distance

FIGURE 30A     A Set UC(x) Demarcated Entirely by Voter Indifference Curves

FIGURE 30B     A Set UC(x) Not Demarcated Entirely by Voter Indifference Curves

FIGURE 31A     The Set UC(x) for a Point x on the Pareto Frontier

FIGURE 31B     The Set UC(x) for a Point x outside the Pareto Set

FIGURE 32A     The 2r Circular Bound on W(c) and 4r Circular Bound on UC(c) [Regular Pentagon Configuration]

FIGURE 32B     The 2r Circular Bound on W(c) and 4r Circular Bound on UC(c) [Irregular Pentagon Configuration]

FIGURE 32C     The d + 4r Circular Bound on UC(x) [Irregular Pentagon Configuration]

FIGURE 33A     UC(c) and UC(X) versus the 4r Circle [Equilateral Pareto Triangle]

FIGURE 33B     UC(c) and UC(X) versus the 4r Circle [Highly Acute Pareto Triangle]

FIGURE 33C     UC(c) and UC(X) versus the 4r Circle [Highly Obtuse Pareto Triangle]

FIGURE 34A     Uncovered Set UC(X) in Additional Panel (g) of BJS Figure 2

FIGURE 34B     Uncovered Set UC(X) in Additional Panel (h) of BJS Figure 2

FIGURE 35A     UC(c), UC(X), and 4r Circle [Regular Pentagon Configuration]

FIGURE 35B     BJS Figure for UC(X) in a Regular Pentagon Configuration

FIGURE 36        UC(X), UC(c), and 4r Circle in a Regular 9-Sided Polygon Configuration

FIGURE 37A     W(c) and UC(c) in an Equilateral Triangle Configuration (Hand Drawn)

FIGURE 37B     W(c) and UC(c) in an Equilateral Triangle Configuration (CyberSenate)

FIGURE 37C     W(c) and UC(c) in a Regular Pentagon Configuration (Hand Drawn)

FIGURE 37D     W(c) and UC(c) in a Regular Pentagon Configuration (CyberSenate)

FIGURE 37E     W(c) and UC(c) in a Regular 9-Sided Polygon Configuration (Incorrectly Hand Drawn)

FIGURE 37F     W(c) and UC(c) in a Regular 9-Sided Polygon Configuration (Correctly drawn by CyberSenate)

FIGURE 38A     The Yolk, W(c), UC(c), UC(X), and 4r Circle for a Scaled-Down Version of BJS Figure 1 (n = 25)

FIGURE 38B     The Yolk, W(c), UC(c), UC(X), and 4r Circle for an Unclustered Configuration (n = 25)

FIGURE 38C     The Yolk, UC(c), UC(X), and 4r Circle for a Bivariate Normal Configuration (n = 25)

FIGURE 38D     The Yolk, UC(c), UC(X), and 4r Circle for a Bivariate Normal Configuration (n = 51)

FIGURE 38E      Figure 19A Showing Approximate UC(X)

FIGURE 39A     The Location and Size of the Yolk with Two Clusters of Ideal Points Where the Majority Cluster Has Greater Vertical Dispersion

FIGURE 39B     The Location and Size of the Yolk with Three Minority Clusters

FIGURE 40A     Mapping Between Points in the “Central Nucleus” and the Boundary of the Uncovered Set (Hartley and Kilgour)

FIGURE 40B     Mapping Between Points in the “Central Nucleus” and the Boundary of the Uncovered Set [Regular Pentagon Configuration]