# On the interest of introducing randomness in ad-word auctions

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5 new searches λ λ tot keyword search λ tot (1 θ retry ) no retry retry λ tot θ retry Fig. 1: Type- users search behavior. rcs are labeled with the average number of occurrences per time unit. We clearly see in Figure 1 that the total mean number of requests by type- users per time unit verifies which gives, since θ retry < 1, λ tot = λ + θ retry λtot, λ tot = λ λ 1 θ retry = 1 R (1 π 1 c,1 h,1 ). (1) For type-b users, we define the probabilities c B,1, c B,2, h B,2 > 0, and R B in the same way as for type- users. With the same reasoning, the mean number of requests by type-b users per time unit is λ tot B = λ B. B(1 π 2c B,2h B,2) 3 Search engine revenue Let us consider fixed values of the advertiser bids b 1 and b 2, and assume that the corresponding display probabilities are π 1 and π 2 = 1 π 1, with prices per click p 1 and p 2, respectively. The mean revenue (utility) per time unit for the search engine (for the considered keyword auction) is U = p 1 π 1 (λ tot c,1 + λ tot B c B,1 ) +p 2 π 2 (λ tot B c B,2 + λ tot c,2 ) } {{ } } {{ } nb of clicks on ad 1 nb of clicks on ad 2 = λ π 1 (p 1 c,1 p 2 c,2 ) + p 2 c,2 1 R (1 π 1 c,1 h,1 ) + λ B π 1 (p 1 c B,1 p 2 c B,2 ) + p 2 c B,2 B (1 c B,2 h B,2 +π 1 c B,2 h B,2 ),(2) where we used the fact that π 1 + π 2 = 1. To simplify the analysis, we will first assume that users types are symmetric, and that users never click on an ad that does not correspond to their type.

9 Individual rationality: every bidder pays less than his declared willingnessto-pay, so there is always an interest in participating in the auction. Non-negativeness of prices. The VCG mechanism works as follows: Each bidder i is asked to declare its entire valuation (willingness-to-pay) function V i ( ). Then, (allocation rule) the auctioneer computes an allocation vector (π i ) i I that maximizes social welfare, i.e., the sum of bidders declared valuations; (pricing rule) the price that each bidder has to pay equals the loss of social welfare that it imposes the others through its presence. 4.3 pplying VCG Based on (4), declaring one s willingness-to-pay function consists in revealing the four parameters of V i, namely for provider 1: λ, R, v 1, and the product ch. However, as pointed out in Remark 1, some parameters like the retry rate R are more likely to be measured by the search engine itself: We can therefore consider that provider 1 only has to declare the parameters λ, v 1, and ch. Since being truthful is a dominant strategy for advertisers, we consider that they reveal their real valuation parameters. llocation rule. The auctioneer computes the display probabilities π i solving the strictly convex optimization problem max V 1 (π 1 ) + V 2 (π 2 ), (6) π 1,π 2 s.t. π 1+π 2 1 where V i is the declared willingness-to-pay function of advertiser i ( V i = V i if i bids truthfully). Remark that since the functions V i are strictly increasing, the solution of (6) lies in the set where π 1 + π 2 = 1. Therefore, the declared-welfare maximizing allocation ( π 1, π 2 ) is such that π 1 = arg max π 1 [0,1] π 2 = 1 π 1. π 1 λ ch 1 R(1 π 1 ch) v 1 + (1 π 1 )λ B ch 1 R(1 (1 π 1 )ch) v 2 (7) The optimization problem (7) is exactly the same as in (3), with v i playing the role of p i for each advertiser i. Therefore, from Proposition 1, π 1 is given by π 1 = 0 if π 1 = 1 if π 1 = 1 R ch λ v 1 λ Bv 2 λ Bv 2 λ v 1 (1 ch), (1 ch), +R ch ()q λb v 2 λ v 1 q λb v 2 λ 1 v 1 +1 if (1 ch) < λ v 1 λ Bv 2 < (1 ch).

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