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Statistics Help

**Question: **A seller with two items x and y considers running an auction. There are two potential bidders A and B with valuations:

The spectrum auction may be an example. The government is selling the rights to transmit cell-phone signals over speciﬁc bands in NYC () and Boston (). A large nation-wide cellphone service provider (A) believes that it is only worthwhile if they get the rights to both cities. A small provider (B) wants the right for only one city, as it is not capable of serving both cities.

(a) Suppose the seller runs a modiﬁed ﬁrst-price auction. Each bidder submits bids (i.e., three valuations, one for each item or the package), then the seller chooses the revenue-maximizing assignment, and the bidder pays her bid for an item, or the package, given by the seller. Would bidders submit their true valuations? Find any equilibrium bids in this example Edit

**Answer: **Many popular search engines run an auction to determine the placement of advertisements next to search results. Current auctions at Google and Yahoo! let advertisers specify a single amount as their bid in the auction. This bid is interpreted as the maximum amount the advertiser is willing to pay per click on its ad. When search queries arrive, the bids are used to rank the ads linearly on the search result page. Advertisers seek to be high on the list, as this attracts more attention and more clicks. The advertisers pay for each user who clicks on their ad, and the amount charged depends on the bids of all the advertisers participating in the auction. We study the problem of ranking ads and associated pricing mechanisms when the advertisers not only specify a bid, but additionallyexpress their preference for positions in the list of ads. In particular, we study prefix position auctions where advertiser i can specify that she is interested only in the top κi positions. We present a simple allocation and pricing mechanism that generalizes the desirable properties of current auctions that do not have position constraints. In addition, we show that our auction has an envy-free [1] or symmetric [2] Nash equilibrium with the same outcome in allocation and pricing as the well-known truthful Vickrey-Clarke-Groves (VCG) auction. Furthermore, we show that this equilibrium is the best such equilibrium for the advertisers in terms of the profit made by each advertiser. We also discuss other position-based auctions.

In the sponsored search market on the web, advertisers bid on keywords that their target audience might be using in search queries. When a search query is made, an online (near-real time!) auction is conducted among those advertisers with matching keywords, and the outcome determines where the ads are placed and how much the advertisers pay.

Current Auctions. Consider a specific query consisting of one or more keywords. When a user issues that search query, the search engine not only displays the results of the web search, but also a set of “sponsored links.” In the case of Google, Yahoo, and MSN, these ads appear on a portion of the page near the right border, and are linearly ordered in a series of slots from top to bottom.

Formally, for each search query, we have a set of n advertisers interested in advertising. This set is usually derived by taking a union over the sets of advertisers interested in the individual keywords that form the query. Advertiser i bids bi , which is the maximum amount the advertiser is willing to pay for a click. There are k < n positions available for advertisements. When a query for that keyword occurs, an online auction determines the set of advertisements, their placement in the positions, and the price per click each has to pay. 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. Edit

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