PART II SPATIAL EUCLIDEAN MODEL Definition of Spatial
- Slides: 12
PART II SPATIAL (EUCLIDEAN) MODEL
Definition of Spatial Model • Voter i has ideal (bliss) point xi 2 <k • Each alternative is represented by a point in <k • A 1 ¸i A 2 iff ||xi-A 1|| · || xi – A 2|| • Can use norms other than Euclidean e. g. ellipsoidal indifference curves
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 1980 s, and 1850 s to early 20 th century.
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 1 st dimension, then 2 nd Impute meaning to fitted model
2 D is qualitatively richer than 1 D x 1 A 2 x 3 A 3 x 2 A 1 >A 2 > A 3 > A 1
Condorcet’s voting paradox in Euclidean model x 1 A 2 x 3 A 3 x 2 Hyperplane normal to and bisecting line segment A 1 A 2
Even if all points in <2 are permitted alternatives, no Condorcet winner exists A 1 x 1 A 2 x 3 x 2
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, De. Groot, Hinich (72) Every hyperplane through x is median, i. e. each closed halfspace contains at least half the voter ideal points. • Mc. Kelvey, Schofield (87) More general, finite, but exponential. Are there better conditions?
Recognizing a Stable (Undominated) Point is co-NP-complete Theorem: Given x 1…xn and x 0 in <k, determining whether x 0 is dominated is NPcomplete. 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-NPcomplete.
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
