Expected revenue of a Vickrey auction

You run a sealed-bid second-price (Vickrey) auction for one item among 4 bidders whose values are i.i.d. Uniform(0,1)(0,1). Since truthful bidding is dominant, the price you collect is the second-highest value. What revenue do you expect (to two decimals)?

Show hints (2)+
  1. Revenue is the second-highest of 4 uniforms, not the maximum.
  2. The kk-th smallest of nn uniforms has mean kn+1\tfrac{k}{n+1}; the 2nd-highest of 4 is the 3rd smallest.

Answer

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0.6 (± 0.01)

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