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Billie has a car for sale, of quality θ ∈ [0, 1]. Louis can make an offer p ∈ [0, 1] but only Billie knows the quality of the car. The distribution of θ is uniform, and this is common knowledge. The value of the car to Billie is θ, while the value of the car to Louis is 1.5θ (thus for any θ, there is the possibility of efficient trade). Show that Louis offering p = 0 and Billie accepting any p ≥ θ is a Bayes-Nash equilibrium. What does this imply about efficient trade?

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