Spatial Model in Political Science: Dimensions, Stability, and Theorems

1. PART II SPATIAL (EUCLIDEAN) MODEL

2. Definition of Spatial Model • Voter i has ideal (bliss) point xi2<k • Each alternative is represented by a point in <k • A1¸i A2 iff ||xi-A1|| · || xi – A2|| • Can use norms other than Euclidean e.g. ellipsoidal indifference curves

4. Spatial Model • Largely descriptive role rather than normative • The workhorse of empirical studies in political science • k=1,2 are the most popular # of dimensions • In U.S. k=2 gives high accuracy (~90%) , k=1 also very accurate since 1980s, and 1850s to early 20th century.

6. What do the dimensions mean? Different schools of thought • Use expert domain knowledge or contextual information to define dimensions and/or place alternatives • Fit data (e.g. roll call) to achieve best fit • Maximize data fit in 1st dimension, then 2nd • Impute meaning to fitted model

7. 2D is qualitatively richer than 1D x1 A1 A2 x3 x2 A3 A1>A2>A3>A1

8. Condorcet’s voting paradox in Euclidean model x1 A1 A2 x3 x2 A3 Hyperplane normal to and bisecting line segment A1A2

9. Even if all points in <2 are permitted alternatives, no Condorcet winner exists x1 A1 A2 x3 x2

10. Major Question: Conditions for Existence of Stable Point (Undominated, Condorcet Winner) • Plott (67) For case all xi distinct • Slutsky(79) General case, not finite • Davis, DeGroot, Hinich (72) Every hyperplane through x is median, i.e. each closed halfspace contains at least half the voter ideal points. • McKelvey, Schofield (87) More general, finite, but exponential. Are there better conditions?

11. Recognizing a Stable (Undominated) Point is co-NP-complete Theorem: Given x1…xn and x0 in <k, determining whether x0 is dominated is NP-complete. Proof: Johnson & Preparata 1978. Algorithm: In O(kn) given x_1…x_n can find x_0 which is undominated if any point is. Corollary: Majority-rule stability is co-NP-complete.

12. Implications • Puts to rest efforts to find simpler necessary and sufficient conditions • Computing the radius of the yolk is NP-hard • Computing any other solution concept that coincides with Condorcet winner when it exists, is NP-hard