ON THE SIZE AND LOCATION OF THE YOLK IN SPATIAL VOTING GAMES: RESULTS USING CYBERSENATE

SOFTWARE

Nicholas R. Millerwith

Joseph Godfrey

Harvard/MIT Seminar on Positive Political EconomyMarch 19, 2009

Overview• In two-dimensional spatial voting games, the yolk is the

set of points bounded by the smallest circle that intersects every median line.

• It has been clear on theoretical grounds that important characteristics of such games depend critically on the location and, especially, the size of the yolk, but until recently it has been difficult make useful generalizations about these matters.

• Until a few years ago, discussions of the yolk were based on hand-constructed illustrative voting games with a small number (typically about three to nine) of voters, in which the yolk is typically quite large relative to the distribution of ideal points (e.g., the Pareto set).

• In contrast, this paper uses CyberSenate, a computer program developed by one of us, to examine the location and size of the yolk in larger scale voting games with varying characteristics.

CyberSenate• CyberSenate software provides a flexible tool for the

geometric analysis of two-dimensional spatial voting games.

• Configurations of ideal points can be created and modified by point and click methods, be generated by Monte Carlo routines, or be derived from empirical data (e.g., interest group ratings, NOMINATE SCORES).

• Indifference curves, (limiting) median lines, Pareto sets, win sets, yolks, cardioid bounds, uncovered set approximations, and other constructions can be quickly generated on screen.

• For me CyberSenate is a dream come true. [Live demonstration]

Sheplse & Weingast, “Uncovered Sets and Sophisticated Voting Outcomes,” AJPS,

Feb. 1984

• What is happening in these figures is that the yolk [which is not shown] is shrinking as the number of voter ideal points increases

Figure for n = 9 was drawn incorrectly

The Yolk and Plott Symmetry• The center c of the yolk indicates the generalized center

of the configuration of ideal points in the sense the median. – The importance of this generalized median point in turn is

suggested by the Median Voter Theorem in one-dimensional spatial voting games.

• The size of the yolk is given by its radius r. – If r = 0, there is sufficient “Plott symmetry” that all median lines

intersect in a single point, which is a “total median” and Condorcet winner.

– Otherwise, the magnitude the yolk radius r indicates the extent to which the configuration of ideal points departs from one exhibiting Plott symmetry.

The Yolk and Win Sets• More generally, as r increases, win sets become more

irregular, majority rule becomes more “chaotic,” and voting outcomes can more readily be manipulated by an agenda setter.

The Yolk and Win Sets (cont.)• Every win set W(x) intersects the yolk; and

– if x lies at least 2r from c, c lies in W(x); and– if x lies at least 3r from c, W(x) contains the yolk.

Circular Bounds on Win Sets• More generally,

– any point x lying at distance d from the center of the yolk c beats all points more than d + 2r from the center of the yolk,

– and x is beaten by all points closer than d−2r to the center of the yolk;

– put otherwise, the boundary of W(x) everywhere falls between two circles centered on the yolk with radii of d + 2r and d − 2r respectively (the inner constraint disappears if d < r and the two circles coincide if r = 0).

Cardioid Bounds On Win Sets

• Tighter bounds on W(x), especially in the vicinity of x itself, are provided by the outer and inner cardioids, the eccentricity of which depend on the size of the yolk.

The Yolk and Global Cycling• In the event of Plott symmetry (r = 0), c is the Condorcet

winner and no majority preference cycles exist anywhere in the space.

• But McKelvey’s Global Cycling Theorem tells that, in the event Plott symmetry fails in even the slightest degree, a global cycle engulfs the entire space.– In particular, a path of majority preference can be constructed

between any two points in the space, so that even a point y that lies far beyond the periphery of the ideal point distribution can, in some finite number of steps, indirectly beat a point x that is centrally located within the distribution.

• However, the length and complexity of the required path from y to x depend on the size of the yolk — the smaller yolk, the longer and more convoluted the majority preference path must be,– because any win set W(x) protrudes only slightly beyond the

circle centered on c and passing through x.

A McKelvey trajectory leading from c to “any-where.”

"Intransitivities in Multidimensional Voting Models and Some Implications for Agenda Control," Journal of Economic Theory, 1976

The Yolk and Covering• A point x lying at

distance d from the center of the yolk c covers all points more than d + 4r from the center of the yolk and is covered by all points closer than d−4r.

• As a corollary, given a status quo x located at distance d from c, an amendment agenda, and sophisticated voting, an agenda setter can design an agenda that produces an outcome at most d + 4r from the center of the yolk.

The Yolk and the Uncovered Set• The uncovered set

(which may constitute a plausible bound on the likely out-comes of many types of spatial voting games) lies within a circle centered on c and with a radius of 4r (McKelvey, 1986).

• Research using Cyber-Senate indicates that the uncovered set typically is– “compactly”

shaped,– approximately

centered on the yolk, and

– has a “radius” of about 2r to 2.5r.

The Yolk and the Un-covered Set (cont.)

Miller, “In Search of the Uncovered Set,” Political Analysis, 2007

The Yolk and Ideal Point Configurations

• Intuitively, we probably expect that the yolk – is centrally located (relative to the configuration of ideal points),

and – tend to shrink in size as the number of voters increases.

• However, has been difficult to confirm these intuitions or even to state them in a theoretically precise fashion. – Feld et al. (1988) took a few very modest first steps.– Koehler (1990) took more substantial steps discussed below.

Feld, Grofman, and Miller, “Centripetal Forces in Spatial Voting Games: On the Size of the Yolk,” Public Choice, 1988

David H. Koehler, “The Size of the Yolk: Computations of Odd and Even-Numbered Committees.” Social Choice and Welfare, 1990

The “Almost Surely Shrinking Yolk”• Tovey (1990) took a major theoretical step by showing

that, – if (and only if) ideal point configurations are random samples

drawn from any “centered” probability distribution, – the expected yolk radius approaches zero as the number of ideal

points increases without limit.

A probability distribution is centered around a point c if every line through c has half the probability mass on either side. Bivariate uniform and normal distributions are centered, as are nonstandard (e.g., correlated) normal distributions (among many others).

• But Tovey’s theoretical result left two important questions open.

Craig A. Tovey, “The Almost Surely Shrinking Yolk.” Naval Postgraduate School, Monterey, California, 1990.

How Fast Does the Yolk Shrink?

• First, at what rate does the yolk shrink as the number of voters increases?

– For example, does a yolk with n = 101 or n = 435 look more like a yolk

• in a committee-sized configuration of n = 9 to n = 25 ideal points, or

• in an electorate-sized configuration of n = 100,000 to n = 100,000,000 ideal points?

The Effects of Clustering?• Second, what is the

impact on the size (and location) of the yolk of various patterns of “non-random” clustering within configurations of ideal points?– And does yolk size

still decrease as the number of voters increases?

• Such clustering is typically seen in empirical ideal point data — for example, the Congressional ideal point configurations generated by the NOMINATE procedure of Poole and Rosenthal.

Yolk Sizes in Ideal Points Configurations Drawn From a Uniform Distribution

• Some years ago, Koehler and Binder (1990) developed a computer program (an ancient ancestor of Cyber-Senate) to compute yolks in two dimensions. – Koehler (1990) calculated yolk locations and sizes in

ideal point configurations randomly drawn from an underlying uniform distribution over a 10 × 10 square.

• He drew twenty five configurations for each of n = 25, n = 51, and n = 75, plus one configuration for every n from 3 through 101.

– This produced two main findings. • First, the average yolk size was fairly small, relative to the 10

x 10 square, compared with yolk sizes found in small hand-constructed configurations three or five voters.

• Second, within the range of configurations studied, the average yolk radius clearly declined as n increased.

Uniform Distributions (cont.)• Hug (1999) subsequently used Koehler’s program to extend his estimates to

larger-scale ideal point configurations. – Hug drew samples of five configurations for each of a variety of n’s

ranging up to n = 1001 plus a single configuration of n = 2001.

• Bräuninger (2007, Figure 5) used his own computer program to randomly select 4000 voter configurations from an underlying uniform distribution over a 10 × 10 square, for odd n’s running from 3 to 101.

• Bianco, Jeliazkov, and Sened (2004) have developed a program to calculate uncovered sets (and, along the way, yolks) for relatively large configurations,– but they have not used it to address the questions posed here.

Simon Hug, “Nonunitary Actors in Spatial Models.” Journal of Conflict Resolution, 1990.Thomas Bräuninger, “Stability with Restricted Preference Maximizing,” Journal

of Theoretical Politics, 2007 Bianco, Jeliazkov, and Sened, “The Uncovered Set and the Limits of Legislative Action,” Political Analysis, 2004

Uniform Distribution (cont.)

• CyberSenate can generate configurations of ideal points drawn randomly from – either bivariate uniform and bivariate normal distributions with

varying means and standard deviations, – compute and display all limiting median lines and the yolk, and – compute the location and size of the yolk.

• The following chart is CyberSenate output that shows all limiting median lines and the yolk for a configuration of 51 ideal points randomly drawn from a bivariate normal distribution centered on (50,50) and with a standard deviation of 15.

CyberSenate Results: Uniform Distribution

• Using CyberSenate, we have computed yolk sizes for 670 ideal point configurations, half drawn from each type of distribution, with various n’s (all odd) ranging from 3 to 2001.

• The results for the 335 configurations drawn from a bivariate uniform distribution over a 10 × 10 square are displayed in Figure 2, which plots the yolk radius for each individual configuration and shows the mean yolk radius for each n. – The figure also shows that yolk sizes are quite stable from

sample to sample in large configurations but highly variable in small configurations.

– It is clear that, once a low threshold of about n = 7 is crossed, the expected yolk radius shrinks as the number of voters increases.

Pooled Results: Uniform Distribution

• The following chart pools data from Koehler, Hug (for n >101 only), Bräuninger, and the previous chart into a single chart. – Koehler’s data for all n from 3 to 101 is shown

individually. Otherwise only the mean for each n is shown).

• It is evident that there is good agreement among all these sources of results within the range of electorate sizes examined.

Projecting to Electorate-Sized Uniform Configurations

• Examination of the preceding charts indicates that, once n is greater than about 15, the mean yolk radius is cut by about 60% for each tenfold increase in the number of voters. – More specifically, given a 10×10 uniform distribution,

the expected yolk radius appears to be approximately

2.5 × .4**log(n).

• Projecting this relationship to larger electorate sizes (far beyond the practical capabilities of CyberSenate) suggests the expected yolk sizes shown in the following chart and table.

Projectionsfor Uniform

Configurations

Normal vs. Uniform Ideal Point Configurations

• Configurations of ideal points uniformly distributed over a square look very artificial, while configurations drawn from a bivariate normal distribution (as in the previous CyberSenate chart) look considerably more natural.

Bivariate Normal Distributions

• The following chart displays individual and mean yolk sizes for 335 ideal point configurations drawn from a bivariate normal distribution – centered on (50,50) – with a standard deviation of 15 each in dimension.

• The results are broadly similar to those for the uniform distribution.

Projecting to Electorate-Sized Normal Configurations

• Examination of the preceding chart suggests that, once n is greater than about 15, the mean yolk radius is cut by a bit over 60% for each tenfold increase in the number of voters. – More specifically, given a normal distribution with an

SD of 15 in each dimension, the expected yolk radius appears to be approximately

11 × .375**log(n).

• Projecting this relationship to larger electorate sizes (far beyond the practical capabilities of CyberSenate) suggests the expected yolk sizes shown in the following chart and table.

Projectionsfor Normal

Configurations

The Expected Yolk Radius is Smaller in a Normal Distri-bution vs. a Uniform Distribution with the Same n and SD

Nonstandard Normal Configurations

• Thus far we have considered only ideal point configurations that have equal dispersion in each dimension.

• The data displayed in the following chart suggests that – if the dispersion (SD) of ideal points in one dimension

is fixed,– the expected yolk radius varies with the square root of

the dispersion (SD) in the other dimension.

Nonstandard Normal Configurations (cont.)

Non-Standard Normal Configurations: Correlated Issue Preferences

• All constructions considered (median lines, yolk, etc.) are invariant under rotation of the (orthogonal) issue axes.

• So the previous consideration applies also to configurations which have same dispersion in both dimensions but in which there is a (more or less strong) correlation between dimension-by-dimension ideal points.

• This is a Cyber-Senate chart with SD(H) = .25 x SD(V), which is then rotated 45°.

• Yolk size is inversely related to the correlation between D1 and D2 with respect to ideal points.

The Impact of Clustering

Clustering can greatly increase yolk size.

Location of the Yolk with Bimodal Clustering• The yolk is

deflected toward the majority cluster.

• This is because the yolk is the generalized (two-dimensional) median, not the general-ized mean.

Bimodal Clustering: Majority ≈ Minority • Almost all

(limiting) median lines form a “bow-tie” pattern, passing through a small area about halfway between the two clusters. – This might

suggest that the yolk lies in this small central region.

Bimodal Clustering: Majority ≈ Minority (cont.)

• But even if MAJ = MIN + 1, there must be at least one (limiting) median line that passes through two ideal points that are both in the majority cluster.– One such median line lies along the centrist face of

the majority cluster, with the entire the minority cluster and the space between them on the other side.

• Since it intersects all median lines, the yolk must lie within the majority side of the bow tie (which essentially forms the yolk triangle) and be nestled against the centrist face of the majority cluster.

Bimodal Clustering Often Produces a “Tovey Anomaly”

The Advantages of Ideological Cohesion

• If the cohesion of one cluster increases (i.e., it becomes more tightly clustered) relative to the cohesion of the other cluster, – the more cohesive cluster is advantaged with respect

to the proximity of• the yolk, and• the uncovered set.

• This is especially true with respect to cohesion in the dimension orthogonal to the dimension of polarization between the clusters.

Majority cohesion reduces

the size of the yolk

and pushes it

further from the minoritycluster

Minority cohesion

increases the size of the yolk and

pushes its boundary toward the

minoritycluster

Bimodal Clustering: Majority >> Minority

• As the relative size of the majority increases, the yolk – shrinks

slowly and– penetrates

the majority cluster.

Clusteringand a Shrinking Yolk?

• With (bimodal) clustering, – the yolk hardly shrinks as the number of ideal

points increases,– especially as the relative size of the majority

increases.

Degree of Polarization and Yolk Size• Consider clusters drawn from two standard bivariate

normal distributions with the same SDs and that – are non-polarized in the vertical dimension,

• i.e., m1v = m2v, but– may be polarized in the horizontal dimension,

• i.e., m2h – m1h ≤ 0.

• “Standardize” polarization as (m2h – m1h) / (SD1 + SD2)

– Given majority and minority clusters such that Maj = Min +1,• initially polarization has almost no effect of yolk size, but• when polarization ≈ 2, yolk size begins to increase

substantially.• Likewise for the deflection of (the center of) the yolk in in the

direction of the majority cluster.

Trimodal Clusters

• The clusters form rough vertices of a triangle.

• The yolk is inscribed within this triangle, – regardless of the relative sizes of the clusters,– unless one cluster is of majority size.

Cohesiveness Makes Little Difference

Four-modal Clusters

• Either – all clusters are external, forming the rough

vertices of a tetragon, or– one cluster is internal, with the others forming

the rough vertices of a triangle.• Either

– no clusters are dummies, or– one cluster is a dummy, or– three clusters are dummies.

Centrist Cluster Is Powerless but “Lucky”

Being a Dummy May Not Hurt