Consider the two player version of the second price auction and the first price auction, where each player's value is drawn uniformly from .
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It is defined as a ratio with values between 0 and 1: the numerator is the area between the Lorenz curve of the distribution and the uniform distribution line; the denominator is the area under the uniform distribution line.
The quantum ergodicity theorem of Shnirelman, Yves Colin de Verdière, and Zelditch states that a compact Riemannian manifold whose unit tangent bundle is ergodic under the geodesic flow is also ergodic in the sense that the probability density associated to the nth eigenfunction of the Laplacian tends weakly to the uniform distribution on the unit cotangent bundle as n → ∞.