# Sharing Online Advertising Revenue with Consumers

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11 in the range of p i p i. The expected values of all advertisers decrease, including the second highest expected value. Hence, the search engine is better off by setting α = 0. A.2 Proof of Theorem 2 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 no cashback: z III i = max γ i g i (p i (1 γ i ))(p i (1 γ i ) c i ) g i (p i )(p i c i ) = z 0 i, for every 0 α 1. Since the revenues in both cases are the second highest expected values, we have R 0 = z 0 σ 0{2} ziii σ 0{2} ziii σ III{2} = RIII. A.3 Proof of Theorem 3 Suppose the advertiser is required to pay a fixed fraction θ of the search engine s revenue as cash back. The dominant strategy for all advertisers is to bid b V i =. Advertiser i s expected value before cash-back, denoted as ziv i is p i c i 1+θ i = g i(p i )(p i c i ). 1 + θ z IV It is easy to see that the ranking of advertisers in this mechanisms is the same as the ranking when the search engine pays a fixed fraction δ of its revenue as cash back, i.e., σ V = σ IV. Thus, advertiser σ IV {1} is the winner and pays p V c = g σ IV{2}(p σiv{2})(p σiv{2} c σiv{2}) (1 + θ)g σiv{1}(p σiv{1}) = piv c 1 + θ (9) The final price to the user is p = p σiv{1} x σ IV{2}g(p σiv{2})(p σiv{2} c σiv{2}) (1 + θ)x σiv{1}g(p σiv{1}) The revenue to the search engine is = p σiv{1} θ 1 + θ piv c. 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.

12 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 R 0 = x σiv{1}(1 kp σiv{1})p IV c Its revenue with cash-back is = 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 x σiv{1}kδ (1 δ) ( p IV ) 2 c δxσiv{2}(1 kp σiv{2})p σiv{2} 0, which gives that ( ) 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.

13 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 )) = 0.25.

16 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, 2.4.

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