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eBay quietly bids on your behalf up to your maximum, and charges you just enough to win. Sealed procurement makes you commit to one number, blind. Two different rules, and a surprising fact: they raise the seller the same money.

Auctions: first and second price

You value an item, your rivals value it too, and nobody sees anyone else’s number. How you should bid depends on the rules. In a second-price auction, honesty is a dominant strategy. In a first-price auction, you must shave your bid. Yet on average the seller earns the same either way.

From earlier lessons

Two quick ones before we start.

Under incomplete information you best-respond to...

A dominant strategy is one that is best...

Predict, then play

Each round you get a private value and bid against two rivals with unknown values. Toggle between second-price (you pay the runner-up’s bid) and first-price (you pay your own bid), and watch the seller’s average revenue in each.

Why did that happen?

Take a guess, then reveal the reason

In a second-price auction your bid decides only whether you win, not what you pay, so raising it above your value risks winning at a loss and lowering it only forfeits good wins. Bidding your true value is exactly right, whatever others do. In a first-price auction your bid also sets your price, so you shade below your value. Under the revenue-equivalence assumptions, both formats use the same efficient allocation rule and give the lowest type zero expected surplus. The payment identity then makes their expected payments match.

The math

Definition, second-price (Vickrey) auction

The highest bidder wins and pays the second-highest bid. Bidding your true value is a weakly dominant strategy: your bid affects only whether you win, never the price, so you want to win exactly when your value exceeds the price and to lose otherwise, which truthful bidding achieves.

With three bidders valuing 40, 70, and 90, the 90-bidder wins and pays 70, keeping a surplus of 20. No one gains by misreporting.

For the first-price auction with $n$ bidders and values drawn uniformly, the symmetric equilibrium shades each bid to

b(v) = \frac{n-1}{n}\,v.

With $n = 3$ , you bid $\tfrac23 v$ . You pay less when you win, but you win less often. Computing expected seller revenue for values uniform on $[0,1]$ gives $\tfrac{n-1}{n+1}$ in both formats: for $n = 3$ , that is $\tfrac12$ .

Definition, revenue equivalence

Under independent private values, any auction that (in equilibrium) always gives the item to the highest-value bidder and gives a bidder with the lowest possible value zero expected surplus raises the same expected revenue. First-price and second-price auctions both qualify, so they tie.

Work it out

Bid truthfully, then shade

Second-price. Your value is 60. If you bid 60 and the top rival bid is 45, you win and pay 45, surplus 15. Bidding 80 instead changes nothing when you would have won anyway, and risks winning at a price above 60 (a loss). Bidding 50 only makes you lose auctions you would have profited from. So 60, your value, is optimal no matter what others do.

First-price. Now your bid is your price, so bidding 60 on a value of 60 leaves you zero surplus even when you win. With two rivals the equilibrium shades to $b = \tfrac23 \times 60 = 40$ . You win less often but earn a margin when you do. Averaged out, the seller collects the same $\tfrac{n-1}{n+1}$ of the value scale as in the second-price auction.

Key idea

Second-price auctions make honesty dominant; first-price auctions make you shade to (n-1)/n of your value. Despite the different behavior, the seller’s expected revenue is identical.

Check yourself

Check yourselfapply

In a second-price auction, how should you bid relative to your value?

Check yourselfanalyse

First-price bidders shade below their values, yet the seller earns the same as under second-price. How can that be?

Check yourselfbrings back: bayes-nash

Why is the first-price shading rule b(v) = (n-1)/n times v a Bayes-Nash equilibrium?

Sources

Vickrey, W. (1961). Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16(1), 8-37.