Solution
Values That Lean High
Show the problem again
Two risk-neutral bidders have independent private values with cumulative distribution F(v) = v² on [0, 1]. They compete in a sealed-bid first-price auction (the high bid wins and pays its bid) and play the symmetric equilibrium. What is the seller's expected revenue? Give a reduced fraction.
Worked solution
By the revenue equivalence theorem, the revenue equals that of a second-price auction, namely the expected second-highest (here the lower) of two values. Since P(min > t) = (1 − t²)², E[min] = ∫₀¹ (1 − t²)² dt = 1 − 2/3 + 1/5 = 8/15. A direct check: the first-price equilibrium bid is b(v) = v − ∫₀^v F(x) dx / F(v) = v − v/3 = 2v/3, and the maximum M has density 4v³ with mean 4/5, so revenue is (2/3)(4/5) = 8/15.
Source: Original problem written for AxiomIQ; uses the standard first-price equilibrium formula and the revenue equivalence theorem.