Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 44 / 1 / a

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- Solution a

Solution

An affine function preserves mixtures, so

U_{0} (p λ + (1 - p) ν) - U_{0} (p μ + (1 - p) ν) = p [U_{0} (λ) - U_{0} (μ)] > 0.

Therefore

p λ + (1 - p) ν ≻ p μ + (1 - p) ν .

The positive mixture weight preserves the strict sign; the common component cancels. This is the independence axiom for lottery preferences for the affine utility on probability measures.

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