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3 expand R (u) to R (u) = p u+v xf (x) dx puf (u + v) + pv (1 F (u + v)) u+v h (u + v) F (u + v) + h xf (x) dx ( u+v ) + q E [X] xf (x) dx q (u + v) (1 F (u + v)) + πu L/πu The first-order derivative is easily obtained as R and the second-order derivative is cg (c) dc + L ( ( )) L 1 G πu = pf (u + v) hf (u + v) q (1 F (u + v)) + π 2 R 2 ( ) L2 L = (q p h) f (u + v) πu 3 g πu From the assumption p > q, it is easy to see 2 R 2 for all u >, hence we have the following results. Lemma 1 The revenue function R ( ) is a concave function. L/πu cg (c) dc (2) Lemma 2 The marginal revenue function R( ) is a non-increasing function. Letting R =, we obtain the condition for ū to be optimal as following: (p + h q) F (ū + v) = π L/πū cg (c) dc q (3) Note that (3) is only well-defined for u >. Suppose the marginal revenue of the PPC advertisement is zero at u is no greater than zero, i.e., R (u) lim. u Then the marginal revenue of the PPC advertisement is never positive for all u > by Lemma 2. Hence it is optimal to display no PPC advertisements. On the other hands, if there exists a solution ū >. R (u) lim >, u Theorem 3 The optimal number of impressions for the PPC advertisements is u =, u = ū, R (u) if lim u otherwise where R (u) lim = pf (v) hf (v) q (1 F (v)) + πe [C] (4) u 3

5 Lemma 6 When L and < F (v) < 1, we have πe [C] > p + h = u πe [C] = p + h = u = F 1 (1) v Proof. From (5) and the conditions L and πe [C] p + h, we have R () = (p + h q) F (v) q + πe [C] (p + h q) F (v) q + p + h = (p + h q) (1 F (v)) > Therefore we know u >. Similarly from (2), we have R = (p + h q) F (u + v) q + πe [C] (p + h q) (1 F (u + v)) (7) for all u. If πe [C] > p + h, the inequality (7) strictly holds, hence the marginal revenue is always strictly positive, i.e., R >. Therefore, it is optimal that u. If πe [C] = p + h, the optimality condition gives us F (u + v) = 1 or u = F 1 (1) v For many continuous distribution functions, we also have F 1 (1). Therefore, we can say the following: if the expected marginal revenue of displaying the PPC advertisements exceeds the sum of the marginal revenue and the penalty of the PPV advertisements, it is optimal to display only the PPC advertisements. The following property is natural. Lemma 7 For all L >, we have u u Proof. The inverse cdf F 1 ( ) is a monotonically non-decreasing function. Therefore, from (3) we have since ( L/πū ) π ū = F 1 cg (c) dc q v p + h q ( ) πe [C] q F 1 v p + h q = ū L/πū cg (c) dc for all ū >. Hence u u. From Lemmas 5 and 7, we have the following. cg (c) dc E [C] 5

9 4 35 u * E[min(X u *,v)] 3 IMPRESSIONS Only PPC Mix PPV and PPC 1 5 Only PPV Mix PPV and PPC p (CPM) Figure 3: Optimal strategies vs. CPM, when there is a finite budget limit (L=5,) 4 35 u * E[min(X u *,v)] 3 A IMPRESSIONS Only PPV 5 Mix PPV and PPC π (CPC) Figure 4: Optimal strategies vs. CPC, when there is a finite budget limit (L=5,) 9

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