Generalized Second-Price Auctions

    In an auction, a bidder in position i will never want to pay more than one bid increment above the bid of the advertiser in position (i + 1), and Google adopted this principle in its newly-designed generalized second price (GSP) auction mechanism. In the simplest GSP auction, an advertiser in position i pays a price per click equal to the bid of an advertiser in position (i + 1) plus a minimum increment (typically $0.01). This second-price structure makes the market more user friendly and less susceptible to gaming. 

Note: 

1. Usually in practice, what an advertiser in position i pays is not what the theory states. To maximize revenue, the rank is often not based on bids, but on the product of each advertiser's bid and CTR. 

Then the  advertiser in position i should pay

                          bid(i+1)*CTR(i+1)/CTR(i) for per click.

2. Truth-telling is a dominant strategy under VCG, while truth-telling is not a dominant strategy under GSP. For instance, consider a slight modification of the example from Section 2. There are still three bidders, with values per click of $10, $4, and $2, and two positions. However, the click- through rates of these positions are now almost the same: the first position receives 200 clicks per hour, and the second one gets 199. If all players bid truthfully, then bidder 1’s payoff is equal to ($10 − $4) ∗ 200 = $1200. If, instead, he shades his bid and bids only $3 per click, he will get the second position, and his payoff will be equal to ($10 − $2) ∗ 199 = $1592 > $1200. 

3. If all advertisers were to bid the same amounts under GSP and VCG, then each advertiser’s payment would be at least as large under GSP as under VCG. 

4. GSP can achieve locally envy-free equilibrium B*. In this equilibrium, each bidder’s position and payment is equal to those in the dominant-strategy equilibrium of the game induced by VCG. In any other locally envy-free equilibrium of that GSP can achieve, the total revenue of the seller is at least as high as in B∗.

Reference:
[1]  Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz. Internet Advertising and the Generalized Second-price Auction: Selling Billions of Dollars Worth of Key- words. American Economic Review, 97(1):242–259, 2007.