SPATIAL (EUCLIDEAN) MODEL

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PART II

• SPATIAL (EUCLIDEAN) MODEL

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Definition of Spatial Model

• Voter i has ideal (bliss) point xi 2 ltk
• Each alternative is represented by a point in ltk
• A1 i A2 iff xi-A1 xi A2
• Can use norms other than Euclidean e.g.
  ellipsoidal indifference curves

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Spatial Model

• Largely descriptive role rather than normative
• The workhorse of empirical studies in political
  science
• k1,2 are the most popular of dimensions
• In U.S. k2 gives high accuracy (90) , k1 also
  very accurate since 1980s, and 1850s to early
  20th century.

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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

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2D is qualitatively richer than 1D
x1
A1
A2
x2
x3
A3
A1 gtA2 gt A3 gt A1
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Condorcets voting paradox in Euclidean model
x1
A1
A2
x2
x3
A3
Hyperplane normal to and bisecting line segment
A1A2
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Even if all points in lt2 are permitted
alternatives, no Condorcet winner exists
x1
A1
A2
x2
x3
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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?

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Recognizing a Stable (Undominated) Point is
co-NP-complete

• Theorem Given x1xn and x0 in ltk, determining
  whether x0 is dominated is NP-complete.
• Proof Johnson Preparata 1978.
• Algorithm In O(kn) given x_1x_n can find x_0
  which is undominated if any point is.
• Corollary Majority-rule stability is
  co-NP-complete.

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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