1. SPATIAL (EUCLIDEAN) MODEL
PART II
SPATIAL (EUCLIDEAN) MODEL

2. Definition of Spatial Model
Voter i has ideal (bliss) point xi 2 <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 1Dx1 A1 A2 x3 x2 A3
A1 >A2 > A3 > A1

8. Condorcet’s voting paradox in Euclidean model
Hyperplane normal to and bisecting line segment A1A2

9. Even if all points in <2 are permitted alternatives, no Condorcet winner exists

10. Are there better conditions?
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