# Sharing Online Advertising Revenue with Consumers

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9 The revenue to the search engine is Rc V = x σiv{1}g(p σiv{1} θ 1 + θ piv c ) piv c 1 + θ. (10) When θ = δ/(1 δ), both the final prices to the user and the search engine revenue are the same for the two cases. A.4 Proof of Theorem 4 Note that since the ranking does not change, and the pricing scheme is as given, the per-conversion payment does not change with δ. The revenue of the search engine with cashback is R IV = g σiv{1}(p σiv{1} δp IV z σiv{2} c ) g σiv{1}(p σiv{1}) piv c (1 δ). The revenue of the search engine without cashback is obtained by setting δ = 0 in the above expression. Thus, when there exists δ > 0 such that g σiv{1}(p σiv{1} δp c )(1 δ) g σiv{1}(p σiv{1}), cashback can increase revenue of the search engine. When functions g are linear, g i = x i (1 kp i ) where 0 < x i 1 and k > 0, and c i = 0, the search engine s revenue without cash-back is Its revenue with cash-back is R 0 = x σiv{1}(1 kp σiv{1})p IV c = x σiv{2}(1 kp σiv{2})p σiv{2}. R IV = x σiv{1}(1 kp σiv{1} + kδp IV c )(1 δ)p IV c. (11) In order for search engine to be better off giving cash-back, we need that which gives that x σiv{1}kδ (1 δ) ( p IV ) 2 c δxσiv{2}(1 kp σiv{2})p σiv{2} 0, ( ) 2 x σiv{1} 1 kpσiv{1} 1 δ ( kx σiv{2} 1 kpσiv{2}) pσiv{2} δ 1 (1 kp σ IV{1}) kp IV. c = (1 kp σ IV{1}) kp IV c When p σiv{1} + p IV c > 1/k, δ is greater than 0, i.e., revenue sharing with the user actually increases the search engine s expected revenue. A.5 Proof of Theorem 5 For a fixed advertiser i, the expected value is always at least as high when she chooses the discount factor γ i herself as compared to when the search engine chooses α: z III i = max γ i g i (p i (1 γ i ))(p i (1 γ i ) c i ) g i (p i (1 α))(p i (1 α) c i ) = z I i, for every 0 α 1. Since the revenues in both cases are the second highest expected values, we have R I = z I σ I{2} ziii σ I{2} ziii σ III{2} = RIII. 9

10 A.6 Proof of Theorem 6 If the conditions are satisfied, it can be seen that the rankings for the two schemes, which are according to g i (p i (1 β))(p i c i ), and g i (p i (1 β))(p i (1 β) c i ) are the same. Specifically, the top two advertisers are the same. We have R II = g 2 (p 2 (1 β))(p 2 c 2 ) βp 1 g 1 (p 1 (1 β)) g 2 (p 2 (1 β))(p 2 c 2 ) βp 2 g 2 (p 2 (1 β)) = g 2 (p 2 (1 β))(p 2 (1 β) c 2 ) = R I, where we used the condition on the rankings in the second line, and the fact that the rankings are identical in the final step. A.7 Proof of Proposition 5 We use two examples to prove the proposition. Case 1 describes a situation where the optimal revenue obtained scheme II is larger than the maximum possible revenue generated by scheme II. Case 1: Suppose p i = x i (1 0.1p i ). There are 3 advertisers A, B, and D, who have p A = 6, p B = 8, p D = 7, and x A = 0.7, x B = 0.3, x D = 0.9. With advertisers bidding on both revenue sharing fractions and payment to search engine, the second highest expected value is from bidder A, with a value of 1.75, which is the expected revenue to the search engine. When search engine pays a fixed fraction of his profit as cashback, the ranking of the advertisers does not change with the fraction of cash back. The revenue to the search engine is maximized at δ = max(0, 1 2 ( p A 0.1p c )) = At this value of α, the search engine s revenue is R = g D (p D δ p c ) (p c δ p c ) = So the optimal expected revenue of the search engine is higher than that with the former case. For comparison, the expected revenue of the search engine with no cash-back at all is The example in the other direction is not too surprising. Case 2: Suppose p i = x i (1 0.1p i ). There are 3 advertisers A, B, and D, who have p A = 8, p B = 7, p D = 6, and x A = 0.8, x B = 0.7, x D = 0.3. With advertisers bidding on both revenue sharing fractions and payment to search engine, the second highest expected value is from bidder B, with a value of 1.75, which is the search engine s expected revenue. When search engine pays a fixed fraction of his profit as cashback, the ranking of the advertisers does not change with δ, and the revenue to the search engine is maximized at δ = max(0, 1 2 ( p A 0.1p c )) = At this value of δ, the search engine s expected revenue is R = g A (p A δ p c ) (p c δ p c ) = So the optimal expected revenue of the search engine in the later case is less than that in the former case. For comparison, the expected revenue of the search engine revenue with no cash-back at all is B Examples Example 1 Suppose g i (p i ) = 1 0.1p i. There are three advertisers A, B, and C competing for one advertising slot. Their prices are p A = 6, p B = 9, and p C = 10; c = 0 for all advertisers. Figure 1(a) plots 10

12 SE s Expected Revenue C Value SE s Expected Revenue Value B A D A B C D Price Price After Cash Back (a) No Cashback (b) Cashback with γ A = 0, γ B = 0.44, and γ C = 0.5 Fig. 2. Example 2 Values of advertisers at the Nash Equilibrium But in the fourth scheme, p c = , and p A +p c is greater than 10, which is the condition for existence of cashback in the fourth scheme. At this p A, advertiser A is still the winner, since his expected value without cashback is g A (p A )p A = 2.1, which is greater than ) Example 4 Suppose g i (p i ) = x i (1 0.1p i ), and there are two advertisers A and B, with x A = x B = 1. Suppose that the posted prices are p A = 5 and p B = 6. In this case, there is no cashback in the fourth scheme, since p c = 4.8, and p A + p c < 10. However, the revenue maximizing cashback from the third scheme gives a revenue of 2.5, which is greater than the revenue without cashback,

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