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1 Bidding Costs and Broad Match in Sponsored Search Advertising Wilfred Amaldoss Kinshuk Jerath Amin Sayedi Duke University Columbia University UNC Chapel Hill November 2013 (Manuscript in progress; please do not circulate.)

14 with valuation v is ( EU B,B (v, E) = 1 v+e R v+e (v R)p(y)dydx + 2E max(r,v E) E max(r,v E) x R (v y)p(y)dydx Therefore, the expected utility of Advertiser 1 using broad match, conditional on Advertiser 2 using broad match as well, is ). (6) EU B,B (E) = 1 0 U B,B (v, E)dv = 1 ( ( 5 + E)E 2 + 5( 1 + R) 2 (1 + 2R) ), (7) 30 where the expression on the right hand side assumes that R [E, 1 E]. From the search engine s point of view, this case is equivalent to a second-price auction in which advertisers valuations come from the probability distribution function p(x). It is easy to see that Myerson s virtual value is 0 when x = 1/2. Therefore, the optimum reserve price in this case is R = 1/2, confirming our earlier assumption of R [E, 1 E]. Case 2: Advertiser 1 uses exact match but Advertiser 2 uses broad match. In this case, the expected utility derived by Advertiser 1 with valuation v is EU X,B (v, E, c) = R 0 (v R)p(y)dy + v R (v y)p(y)dy c. (8) Let τ B be the value of v at which this expected utility is zero. Upon assuming τ B 1 E, the critical value of τ B simplifies to τ B = 2c + R 2. (9) Therefore, Advertiser 1 participates in the auction if and only if v τ B, and its expected utility is 1 EU X,B (E, c) = U X,B (v, E, c)dv (10) τ B = 1 ( E c ( ) 2c + R (1 ( + R ))) 2c + R 2. 13

21 Using the above equations to simultaneously solve for τ L and τ H, we obtain ( /2 τ H = R + 2c L + R 2 (2c L + R 2 ) 2 4cH) 2 (20) ( /2. τ L = 2c L + R 2 + (2c L + R 2 ) 2 4cH) 2 (21) Notice that if τ L > 1, an L-type advertiser does not participate in the auction for any valuation. In other words, if c L > 1+c2 H R2 2, the L-type advertiser stays out of the auction. In this case, we abuse the notation to define τ L = 1; when τ L = 1, we have τ H = R + c H. The search engine s revenue is given by (1 τ L )τ H R + (1 τ H )τ L R + (τ L τ H )(1 τ L ) τ L + τ H 2 + (1 τ L ) 2 ( 2τ L + 1 ). (22) 3 We are unable to analytically calculate the optimum value of R due to tractability issues. Therefore, we resort to numerical analysis. Figure 2 presents the optimum value of R for different values of c L and c H. We note that the optimum reserve price is a decreasing function of c H. However, it is a non-monotone function of c L. Furthermore, there is a discontinuity in the optimum reserve price when the L-type advertiser drops out of the auction. If c L is sufficiently high and c H is sufficiently low, the L-type advertiser does not participate in the auction any more. In this case, the search engine sets the reserve price independent of c L, and the optimum reserve price, at the point of discontinuity, jumps to a higher value (given by R = 1 c H 2 ). Note that after the discontinuity, the optimum reserve price is constant in c L, while before the discontinuity, the optimum reserve price could be increasing or decreasing in c L (depending on value of c H ). The above result pertaining to the optimum reserve price decreasing in c H is not new, and is simply an extension of the result in Proposition 1. Interestingly, however, the opposite result could prevail for c L : the search engine may increase the reserve price as the bidding cost of the L-type advertiser increases. The intuition is as follows. Existence of the L-type advertiser forces the optimum reserve price to be lower than what it would be for the H-type advertiser only. However, as the L-type advertiser s bidding cost increases, the search engine may benefit from sacrificing the L-type advertiser in order to extract higher revenue from the H-type advertiser. In other words, 20

22 c H c L Figure 2: Optimum value of R (represented by the contours) for different values of c L and c H, where c H < c L from the search engine s perspective, as the bidding cost of the L-type advertiser increases, the advertiser becomes less valuable for improving the search engine s profits. Therefore, for large enough c L, the search engine increases the reserve price, which induces the L-type advertiser to exit, but extracts more revenue from the H-type advertiser. It is interesting that as c H increases, the L-type advertiser becomes more likely to be sacrificed. For example, when c H = 0, a change in c L from 0.30 to 0.31 decreases the optimum reserve price. However, if c H = 0.25, the same change in c L increases the optimum reserve price. The reason is as follows: When c H is small (and the reserve price is large), the marginal effect of reserve price on the search engine s revenue from the H-type advertiser is low. Therefore, the search engine could more easily lower the reserve price to accommodate the L-type advertiser s change of bidding cost. However, as c H increases, the marginal effect of the reserve price on the search engine s revenue from the H-type advertiser increases. Therefore, for large enough c H, an increase in c L leads to an increase in the optimum reserve price (sacrificing the L-type advertiser). We state the following proposition. 21

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