When a government sells radio spectrum or a search engine sells the space beside your query, billions ride on the auction rules. The 2016-17 US incentive auction paid broadcasters about 20 billion. Design is not academic here.
Auctions in the wild
Auction theory earns its keep in two of the biggest markets built by economists: government spectrum sales and online advertising. Both show the theory in tension with practice. Generalized-second-price-style rules are common in search advertising, despite not making truthful bids optimal with multiple slots.
VCG charges each winner the...
The generalized second price auction is truthful with...
Predict, then play
Bid for two ad slots under the generalized second price, then toggle to VCG and compare the charges.
Why did that happen?
Take a guess, then reveal the reason
The generalized second price looks like the truthful second-price auction, and for a single slot it is. But with several slots, the price on the top slot takes no account of what a lower slot would cost, so a bidder can sometimes do better by shading under the bid above it and buying a cheaper slot instead. When that trade pays, truthful bidding is not an equilibrium. In the standard separable-click model, a lowest locally envy-free GSP equilibrium lands on the efficient assignment and VCG prices. Other GSP bid profiles need not. Spectrum auctions face harder problems still: goods that are complements, where bidding on pieces risks the exposure problem.
The real episodes
The FCC has auctioned spectrum since 1994 using simultaneous multiple-round designs; the 2016-17 incentive auction repurposed broadcast spectrum, paying broadcasters to give it up and reselling it: 10.05 billion paid to broadcasters, and more than $7 billion to the Treasury. In the textbook GSP model, advertisers are ranked by bid and each pays the next bid down per click. It is not truthful with multiple slots, but its lowest locally envy-free equilibrium reproduces the efficient VCG outcome under the model’s separable-click assumptions.
Two practical wrinkles. Ad auctions run billions of times a day, so even small design choices (reserves, quality scores) move enormous revenue, and the GSP’s non-truthfulness means bidders and platforms both invest in bid optimization. Spectrum auctions add combinatorial structure: licenses are worth more together, so bidding on a subset exposes a firm to winning a useless fragment (the exposure problem), which package-bidding designs (Ausubel-Milgrom) address.
Work it out
Why GSP is not truthful, yet works
Take two slots, worth 100 and 80 clicks, and advertisers valuing clicks at 10, 6, 4. VCG charges each the externality it imposes: the top advertiser pays 440, the second 320. Under GSP with truthful bids, the top slot’s price is the next bid (6) over its 100 clicks, so 600, well above VCG’s 440. Is that enough to make the top bidder move? Holding slot 1 nets 100(10 - 6) = 400; shading to just under 6 buys slot 2 at the third bid, netting 80(10 - 4) = 480. It is, so truthful bidding is not an equilibrium.
Under the standard model, a lowest locally envy-free equilibrium assigns slots efficiently and charges the VCG amounts. So that equilibrium gives the search engine the efficient, well-priced outcome VCG would give, through a simpler-looking rule, even though bidding your value is not itself optimal. Arbitrary GSP bids do not carry this guarantee. That gap is why advertisers optimize bids and why the theory matters in practice.
Key idea
Spectrum and ad auctions are auction theory at industrial scale. Generalized second price is not truthful with multiple slots. Under standard assumptions, its lowest locally envy-free equilibrium reproduces the efficient VCG outcome, but arbitrary bids do not.
Check yourself
Sources
- Edelman, B., Ostrovsky, M., and Schwarz, M. (2007). Internet advertising and the generalized second-price auction. American Economic Review, 97(1), 242-259.
- Milgrom, P. (2004). Putting Auction Theory to Work. Cambridge University Press.