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eBay lets sellers set a reserve, a secret minimum below which the item does not sell. Set it too high and you scare off bidders; too low and you leave money on the table. In the uniform-value benchmark, the right reserve does not depend on how many bidders show up.

Optimal auctions

Revenue equivalence said the format does not matter for revenue. But it took the rules as given. Myerson asked the harder question: across all mechanisms, which raises the most? With independent, risk-neutral bidders whose values come from a known regular distribution, allocate to the highest positive virtual value. For iid uniform values, that is a second-price auction with a reserve of $1/2$ .

From earlier lessons

Two quick ones before we start.

Revenue equivalence says first- and second-price auctions raise...

A reserve price is a threshold below which the item...

Predict, then play

Bidders’ values are uniform on $[0,1]$ . Set the reserve and the number of bidders, and watch the seller’s expected revenue. Then change the number of bidders and see where the peak moves.

Why did that happen?

Take a guess, then reveal the reason

In this iid-uniform model, the reserve is really a price you post to the winning bidder, and the right price to post to a single buyer is $1/2$ , maximizing $p(1-p)$ . Adding more bidders makes it more likely someone clears that bar and pushes up the price they actually pay through competition, so revenue rises. But the threshold itself is a property of one bidder’s value distribution, not of how many bidders there are. So the reserve stays put at $1/2$ . Change the distribution or the model, and the conclusion need not hold.

The math

Definition, virtual value and the optimal auction

For a risk-neutral bidder with value $v$ and distribution $F$ , the virtual value is $\psi(v) = v - \frac{1 - F(v)}{f(v)}$ . In the independent-private-values model, Myerson’s theorem allocates the item to the bidder with the highest nonnegative virtual value. For iid regular distributions, this is a second-price auction with a reserve $r$ solving $\psi(r) = 0$ .

For the uniform distribution, $\psi(v) = 2v - 1$ , so the reserve solves $2r - 1 = 0$ , giving $r^* = \tfrac12$ , independent of $n$ . Intuitively, revenue is expected virtual surplus, and you only want to sell to a bidder whose virtual value is positive, that is, whose value clears $\tfrac12$ .

With a single buyer, this is just posting a price $p$ to maximize $p(1-p)$ , which peaks at $p = \tfrac12$ with revenue $\tfrac14$ . Note the optimal auction sometimes does not sell (when every value is below the reserve), so revenue maximization is not the same as efficiency: the seller accepts occasional no-sales to keep prices up.

Work it out

Find the iid-uniform reserve

The uniform virtual value is $\psi(v) = 2v - 1$ . Set it to zero: $2r - 1 = 0$ , so $r^* = \tfrac12$ . This uses only one bidder’s distribution, so it holds for any number of bidders.

Check the single-buyer case directly: posting price $p$ sells with probability $1-p$ for revenue $p(1-p)$ , maximized at $p = \tfrac12$ , revenue $\tfrac14$ . Now add bidders. Competition raises the price the winner pays (toward the second-highest value), so expected revenue climbs with $n$ , but the reserve that maximizes it stays at $\tfrac12$ in this iid-uniform model. A different value distribution can imply a different reserve.

Key idea

For iid, risk-neutral bidders with regular values, the revenue-maximizing auction allocates to the highest positive virtual value. For uniform values, this is a second-price auction with a reserve of 1/2, whatever the number of bidders, and it deliberately trades some efficiency for revenue.

Check yourself

Check yourselfapply

Bidders' values are iid uniform on [0,1]. What is the optimal reserve, and how does it change with more bidders?

Check yourselfanalyse

Why is the revenue-maximizing auction not the same as the efficient one?

Check yourselfbrings back: revelation principle

How does the revelation principle underpin Myerson's result?

Sources

Myerson, R. B. (1981). Optimal auction design. Mathematics of Operations Research, 6(1), 58-73.