Using CyberSenate Software in Theoretical Research

1. Using CyberSenate Softwarein Theoretical Research Nicholas R. Miller Based on “In Search of the Uncovered Set,” Political Analysis, forthcoming APSA

2. CYBERSENATE • CyberSenate was developed by Joseph Godfrey of the WinSet Group, LLC. • It can create configurations of ideal points • by point and click methods, • generate them by Monte Carlo methods, or • derive them from empirical data. • Indifference curves, median lines, Pareto sets, win sets, yolks, cardioid bounds on win sets, uncovered set approximations, and other constructions can be generated on screen. • CyberSenate produced all but four of the figures that follow. APSA

3. CyberSenate for me is a dream come true -- compare the figures below with those that follows. APSA

4. Review: The Covering Relation Alternative x covers alternative y iff (a) x beats y, and (b) x beats every alternative that y beats, so (c) W(x) is a proper subset of W(y). The covering relation is transitive so (d) maximal (uncovered) alternatives exist under relevant circumstances. Strategic Property: an uncovered point beats every other alternative in no more than two steps. APSA

5. Given finite alternatives and a majority preference tournament, UC(X) • coincides with the Condorcet winner (if it exists); • is a subset of the top cycle set; and • is a subset of the Pareto set. The size of UC(X) depends on the degree of intransitivity in the tournament. APSA

6. In a spatial context (with Euclidean preferences) • The same three properties hold. • However, (b) loses its punch since (in the absence of “Plott symmetry” and a Condorcet winner) the top cycle encompasses the entire space. • However, in a spatial context one additional bound on the uncovered set is known. APSA

7. In a (two-dimensional Euclidean) spatial context • In the unlikely event that a Condorcet winner exists, the uncovered set coincides with it (as in non-spatial context). • The uncovered set lies within the Pareto set (as in non-spatial context). • The uncovered set lies within a circle centered on the yolk with a radius four times that of the yolk (McKelvey, 1986). APSA

8. But one big problem has remained: • Beyond these three points, we have had only incomplete and rough knowledge concerning the location, size, and shape of the uncovered set in a spatial context. • Motivation for work by Bianco, Jeliazkov, and Sened, “The Uncovered Set and the Limits of Legislative Action,” Political Analysis, Summer 2004. • BJS use a grid-search algorithm to generate pictures of uncovered sets. APSA

9. My follow-up work using CyberSenate • The second (Pareto) bound operates through proximatecovering and is overgenerous in principle, though probably not in practice. [Role of invisible voters] • The third (4r) bound operates through distant covering and is overgenerous in both principle and practice, as the uncovered set is typically contained within a circle centered on the yolk with a radius only a bit larger than twice that of the yolk. • The first point (UC(X) = {CW}) has long been recog-nized as a corollary of (3), since a yolk of zero radius implies a Condorcet winner. It is here show that (1) is also a corollary of (2) [refined to refer to the visible Pareto set]. • Connection is also made with Schofield’s local covering and heart. APSA

10. Refining the Pareto bound on UC(X) An orderly win set W(x) • lies entirely to one side of some (dividing) line; • in the vicinity of x, W(x) is a subset of some individual preference set; • precludes local cycling; and • produces “localcovering” (Schofield) APSA

11. In contrast, A disorderly win set W(x) • does not lie to one side of any line through x; • even in the vicinity of x, W(x) is not a subset of any individual prefer-ence set; • produces local cycling; and • precludes (any kind of) covering. APSA

12. A win set is disorderly iff • It is surrounded by [limiting] median lines. • It lies inside the heart [Schofield] (or locally uncoveredset) APSA

13. The Covering Relation Recall that point x covers point y iff • x beats y, and • x beats every point y beats, • so W(x) is a proper subset of W(y). Thus in spatial context, x covers y iff the boundary of W(y) literally encloses the boundary of W(y). APSA

14. In proximate mode, x covers y iff • x also covers every point between x and y (so covering operates between neighboring points) so • W(x) is simply a slightly shrunked replica of W(y), which means that • x must be closer than y to ideal point of every voter whose indifference curve through y dematcates part of the boundary of W(y). APSA

15. Point y is most obviously proximately covered by point x if • y lies outside the Pareto set, x Pareto dominates y; therefore • i.e., x is closer to every ideal point than y is, so • the Pareto bound on the uncovered set results from proximate covering. APSA

16. In contrast, local covering (Schofield) • can operate within the Pareto Set (with orderly win sets), but • does not imply (“global”) covering (proximate or otherwise). APSA

17. But proximate covering can occur within the Pareto set if some voters are invisible APSA

18. The Pareto bound on UC(X) is refined to the visible Pareto bound • If an ideal point configuration exhibits Plott symmetry, all voters except one are invisible, so proximate covering pares the uncovered set down to the ideal point of the one visible voter. • Otherwise, in so far as the demarcation of the uncovered set results from proximate covering, it has straight-line boundaries, since the (visible) Pareto set has straight-line boundaries [see BJS, Figure2]. APSA

19. Distant covering and refining the 4r bound on UC(X) If point x covers point yat a distance: • x does not cover points on between x and y, and • W(x) is not simply a shrunken replica of W(y). Rather W(x) is somewhat (or totally) differently shaped from W(y) but, at the same time, W(x) is sufficiently smaller than W(y) (because x is sufficiently closer to the center of the yolk than y is) that its differently shaped (and perhaps very disorderly) boundary is nevertheless enclosed within the boundary of W(y). APSA

20. Typical Covering at a Distance APSA

21. Atypical Covering at a Distance While every point x is covered (at a distance) by any point 4r closer to the center of the yolk, we cannot specify a minimum distance necessary for such covering. APSA

22. The set of points UC(c) [not covered by the center of the yolk] is off special interest The 4r bound on UC(X) is actually a bound on UC(c), UC(X) being a subset of all sets UC(x), including UC(c). Typically UC(c) does not extend to the 4r circle and approaches it only at particular points. APSA

23. UC(c), UC(X), and the 4r Circle APSA

24. Generalizations concerning UC(X)based on distant covering • For points x close to c, UC(x) sets are irregularly shaped with points emanating from a central core. • The central cores of all such UC(x) sets substantially coincide but their points emanate in offsetting directions. • Therefore, in forming intersections these points are sniped off, leaving UC(X) as essentially the common core of all sets UC(x). • UC(X) is more compact than the individual UC(x) sets, is approximately centered on the yolk, with a boundary generally lying about 2r to 2.5r from c. • Moreover, points x beyond about 0.5r from c do not cover any otherwise uncovered points closer to c such sets UC(x) are irrelevant to demarcating the boundary of the uncovered set. APSA

25. Importance of the Size and Location of the Yolk • both for size and location of UC(X), and • as important parameter in its own right. Tovey (1990): if ideal point configurations are random samples drawn from any “centered” distribution, the expected size of the yolk approaches zero as the number of ideal points increases without limit. Two follow-up questions: • At what rate does the yolk shrink? • What are the effects of non-random clustering? APSA

26. CyberSenate Simulations of Yolk Sizes APSA

27. Pooled Simulation Data APSA

28. A NORMAL IDEAL POINT CONFIGURATION APSA

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32. But clustering can greatly increase yolk size APSA

33. The yolk is a generalized median, not a generalized mean APSA

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