Q#1: Second price (Vickrey) auction

Suppose you’re running a second-price auction. In this auction, the highest bidder will win, but will pay the auctioneer (you) the value of the second-highest bid. Assuming there are two bidders bidding on one item, and the bidder knows his own valuation but sees the valuation of the rival as uncertain and distributed uniformly in the unit interval, calculate the expected revenue when the reserve price is 1/2.

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TRY yourself before proceeding

This is a Statistics based question and relies on your knowledge of the Vickrey (or Second price) auction. I know what you are thinking, “OH NO, Math!”, but lets walk through the process.

First what is a Vickrey auction? In this auction, everyone bids without knowing what others bid and the highest bidder pays what the second highest bid for the item was. Still confused? Maybe the diagram below can help clarify.

Now onto how to solve this question. We will utilize Bayesian-Nash Equilibrium theory which in game theory makes the assumption that players will play the strategy that gives themselves the highest payoffs. Therefore the expected return for us the Auctioneer is dependent on all the possible outcomes of plays by the two bidders. The three possible plays are:

1. No one bids
2. One bidder bids and the other stays out
3. Both bid at their optimal reserve price (in this case 1/2)

The solution requires some integration over the given statement that the bidder’s valuation is a uniform valuation. The details can be found here: https://math.stackexchange.com/questions/845869/expected-revenue-obtained-by-the-vickery-auction-with-reserve-price-1-2. In short, it is suffice to know that for a Vickrey auction under unit normal distribution the expected revenue R is equal to one-third times the bidders’ reserve price squared minus four-thirds times the bidders’ reserve price cubed:

R = 1/3 + r² - 4/3r³

Plugging in r = 1/2

R = 1/3 + 1/4 -(2/12) = 5/12