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Statistics Help

**Question: **Suppose the search engine runs a generalized second price (GSP) auction. Find any envy-free equilibrium bids that lead to the prices paid in part (a). Edit

**Answer: **The most common auction mechanism in use today is the generalized secondprice (GSP) auction (sometimes also referred to as the next-price auction). Here the ads are ranked in decreasing order of bid, and priced according to the bid of the next advertiser in the ranking. In other words, suppose wlog that b1 ≥ b2 ≥ . . . ≥ bn; then the first k ads are placed in the k positions, and for all i ∈ [1, k], bidder i gets placed in position i and pays bi+1 per click.1 We note two properties ensured by this mechanism: 1. (Ordering Property) The ads that appear on the page are ranked in decreasing order of bi . 2. (Minimum Pay Property) If a user clicks on the ad at position i, the advertiser pays the minimum amount she would have needed to bid in order to be assigned the position she occupies. Search engine companies have made a highly successful business out of these auctions. In part, the properties above have dissuaded advertisers from trying to game the auction. In particular, the minimum-pay property ensures that an advertiser has no incentive to lower a winning bid by a small amount in order to pay a lower price for the same position. Still, the GSP auction is not truthrevealing, that is, an advertiser may be incentivized to bid differently than her true value under certain conditions [4]. Only recently have we obtained a detailed formal understanding of the properties of this auction. Authors in [1, 2, 4] have analyzed the auction in terms of its equilibria. They show that when the click-through rates are separable, i.e. the click-through rate of an ad at a given position is the product of an ad-specific factor and a position-specific factor, the GSP has a Nash equilibrium whose outcome is equivalent to the famous Vickrey-Clarke-Groves (VCG) mechanism [5–7] which is known to be truthful. [1, 2] go on to show that this equilibrium is envyfree, that is, each advertiser prefers the current outcome (as it applies to her) to being placed in another position and paying the price-per-click being paid. by the current occupant of the position. Further, among all the envy-free equilibria, the VCG equilibrium is bidder-optimal; that is, for each advertiser, her price-per-click is minimum under this equilibrium. We note that when the clickthrough rates are separable, the outcome produced by the VCG mechanism has the ordering property. Authors in [4] also show that even when the click-through rates are arbitrary, there is a pricing method with the ordering property that is truthful. (This pricing method reduces to the VCG pricing method when the click-through rates are separable.) Furthermore, they show that the GSP has a Nash equilibrium that has the same outcome as their mechanism. Together, these results provide some understanding of the current auctions. That in turn provides confidence in the rules of the auction, and helps support the vast market for search keyword advertising. Edit

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