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7 budget, both in P and the non-strict algorithm. Therefore, the total revenue of the non-strict algorithm 1 from inactive advertisers is at most a δ times 1 δ fraction of the revenue of P. Given the assumption that bids are small compared to budgets, we expect that δ to be small. Therefore, the theorem above essentially gives a tight bound on the maximum ratio between the revenue of the two throttling models. 4 An online strict algorithm In this section we present an online algorithm for the strict model under the GSP scheme. Note that the greedy algorithm (for the pay-as-you-bid pricing scheme) is the same as the non-throttling algorithm, so by Corollary 1, it fails to get a good competitive ratio in the strict model. In the next section we also show that the algorithm of Mehta et al is not competitive in this model. Our strict algorithm makes greedy choices at each step and achieves competitive ratio 1 3, even when compared to the offline non-strict optimal algorithm. Further, we will show that the analysis of the strict greedy algorithm is tight, even against a strict offline optimal. The strict greedy algorithm: Upon the arrival of a new query for keyword j: Allocate the query to a proper set with maximum revenue, i.e., proper set S which maximizes π(j, S). Lemma 5 Proper set S with maximum revenue can be found using dynamic programming. Proof : Without loss of generality, assume for i 1, b i b i+1. Define r u,l to be the maximum revenue can be obtained from slots 1 to l when u is in the slot l + 1. We can compute r u,l using the dynamic program below: { max r u,l = {v<u: buθ l > remaining budget of v} {r v,l 1 + b u θ l } l < u (6) 0 l u or l =1 The time complexity of this dynamic program is O( A 2 k). Theorem 6( Let δ) be the maximum ratio between the bid and the budget of an advertisers. The strict algorithm is -competitive with respect to the optimal offline solution for the non-strict model which is 1 δ 3 2δ allowed to use (but not charge) advertisers whose budget have been exhausted. 3 Proof : Consider an optimal non-strict algorithm denoted by OPT. Let π G (t) and π OPT (t) denote the total revenue obtained by the strict greedy algorithm and OPT at time t. Also, let π OPT(t) (A) (resp. π OPT(t)) (I) denote the total revenue obtained by OPT from advertisers who are active (resp. inactive) at time t. In these definitions and in the rest of the proof, when we call an advertiser active/inactive at time t, it is always with 3 Note that the optimal revenue in the non-strict model is greater than or equal to the revenue in the strict model. 7

8 reference to the strict greedy algorithm; the same is true for proper sets. We will relate π OPT(t) (I) and π OPT(t) (A) separately to the revenue of the strict greedy algorithm, π G (t). First, observe that any advertiser who contributes to π OPT(t) (I) must have already used up a fraction (1 δ) of her budget in the strict greedy algorithm at time t. Hence, we have: π OPT(t) (I) π G (t)/(1 δ) (7) t t Also, by Lemma 3, and because the greedy algorithm allocated the query to a proper set with maximum revenue, we have: π (A) OPT(t) 2π G (t) (8) The claim immediately follows from inequalities 7 and A tight example We will now prove that our analysis for the strict greedy algorithm is tight, even against a strict offline optimal algorithm. Consider a scenario with k slots, and 3 + 2k advertisers A 1 ; A 2 ; A 3 ; B 1,B 2,..., B k ; C 1,C 2,..., C k. Advertiser A 1 has budget N; A 2,A 3 have budgets N/k each; B 1,B 2,..., B k have budgets N/(k 1) each; C 1,C 2,..., C k have budgets N/k each. The queries arrive in three phases; each phase has N identical queries. We summarize the queries, the bids, and the actions taken/profit obtained by the strict greedy algorithm as well as an offline algorithm in table 1. The total revenue earned by the online algorithm is N + O(N/k) while the total revenue of the offline algorithm is 3N O(N/k), giving a lower bound of 1 3 on the competitive ratio. Phase Bids Greedy Greedy Offline Offline ranking revenue ranking revenue 1 A 2 :2/K; A 3 :1/K A 2,A 3 N/k Passes 0 2 A 1 : 2; A 2 :1 A 1,B 1... B k Nk/(k 1) A 1,A 2,C 1... C k 1 N(1 + k 1 k ) B 1... B k :1/(k 1) C 1... C k :1/k 3 B 1... B k :1/(k 1) B 1... B k N/(k 1) B 1... B k N (if budget left) Table 1: An example showing that the analysis of greedy is tight. Each phase has N queries. When we do not specify the bid of an advertiser for a query, it is assumed to be 0. 5 Online algorithms for the non-strict model In this section, we show that the greedy algorithm is 1 2-competitive in the non-strict model. But first, we present the modified algorithm of Mehta et al. [11] for the GSP scheme. Define function Φ : [0, 1] [0, e 1 e ] to be Φ(x) =1 e x 1. Also, let f i be the fraction of the spent budget of advertiser i. The (modified) algorithm of Mehta et al. for the GSP scheme: Upon the arrival of a new query for keyword j: Allocate the query to a set S argmax S Sj i S Φ(f i)p(i, j, S). 8

9 We develop a primal-dual scheme to analyze this algorithm which is similar to the approach proposed by Buchbinder et al. [3]. But, before the analysis, we first observe that this algorithm cannot be used under the strict model. The reason is that set S which maximize i S Φ(f i)p(i, j, S) may contain advertisers with no remaining budget. For instance, consider the first example in Section 3. In the strict model, the algorithm presented above allocates each query to the same set of advertisers as the non-throttling algorithm. Hence, by Corollary 1, it fails to be competitive. However, we prove that the algorithm is (1 1 e )-competitive in the non-strict model. Theorem 7 The non-strict algorithm is (1 1 e )-competitive with respect to the optimal offline solution of the non-strict model. Proof : In the online setting, without loss of generality, we assume that there is one query from each keyword, i.e., n j = 1 for every keyword j. We construct a feasible solution for the primal and the dual. We describe how to update the primal and dual variables, after arrival of each new query, such that it maintains the feasibility; also, the ratio of the values of the primal and the dual be greater than or equal to 1 1 e. Therefore, by the end of the algorithm, we have a solution which is within a ratio of 1 1 e of the optimal solution of the primal linear program. We initialize all of the variables to zero, and update them once a query shows up. In particular we let β i = ef i 1 e 1, which implies 1 β i = e e 1 Φ(f i). Consider query j, and assume that the algorithm allocates it to set S which maximizes: (1 β i )p(i, j, S) Let α j equal to this maximum. Therefore, the dual remains feasible. i S Recall that β i = ef i 1 e 1 ; hence, β i is updated multiplicatively as long as β i < 1. By the assumption that bids are small compared to the budgets, the increase in β is equal to: c i (β i + 1 B i e 1 ) where c i = min{p(i, j, S), (1 f i )B i } is the amount i is charged. Also, we increase y i by p(i, j, S) c i. Therefore, the primal remain feasible; and its value is increased by i S π(j, S) y i = i S c i. The increase in the value of the dual is equal to: α j + i S (β i + 1 e 1 ) c i B i = (1 β i )c i + (β i + 1 B i e 1 )c i i S i S e = c i e 1 i S Therefore, the value of the primal remains within a e 1 e the proof. ratio of the value of the dual, which completes Note that the above proof works for any pricing scheme, and is not specific to GSP. Therefore, as long as we can find a set S argmax S Sj { i S Φ(f i)p(i, j, S)} in polynomial time, the above theorem guarantees a competitive ratio of 1 1/e. For GSP, such a set can be find in polynomial time using dynamic programming similar to Lemma 5. The proof of the following lemma is omitted. Lemma 8 Set S argmax S Sj i S Φ(f i)p(i, j, S) can be found in polynomial time using dynamic programming. Moreover, the same dynamic program can be used as the separation oracle to solve LP 2. 9

10 Also observer that if one can find a set S such that Φ(f i )p(i, j, S) λ max{ Φ(f i )p(i, j, S )}, S i S i S for a constant λ 1, then the approximation ratio of the algorithm is at least λ(1 1 e ). Now we show that the greedy algorithm is essentially 1 2-competitive in the non-strict model. Proposition 9 The non-strict ( greedy ) algorithm that upon the arrival of a new query allocates it to a set with the maximum revenue is -competitive with respect to the optimal non-strict offline solution. 1 δ 2 δ Proof : It is easy to see that t π(i) OPT(t) t π G(t)/(1 δ), where all the variables are defined with respect to the non-strict greedy algorithm. Because the greedy non-strict algorithm can choose the same set that is chosen by the optimal offline algorithm at this step, we have π OPT(t) (A) π G (t). The claim follows immediately by combining these two inequalities. 6 Conclusion In this paper we studied the problem of online competitive ad space allocation when the pricing scheme is GSP. We observe a difference in the performance of the allocation algorithm between pay-as-you-bid and the GSP pricing schemes. For example, under the GSP scheme, if the search engine rapidly exhausts the budget of an advertiser with high value, the total revenue may significantly decrease in the future. This is the reason that the algorithm proposed for the pay-as-you-bid model fail to be competitive in the online strict model. We studied different models of throttling and their effects on the revenue. From algorithmic point of view, the main open problem is to close the gap for the competitive ratio in the strict model. We provided a lower bound of 1 3, and an upper bound of 1 2 against the optimal non-strict algorithm. With respect to the optimal strict algorithm, by a reduction from the single slot model, we have an upper bound of (1 1 e ) [11]. Another big open question in this area is to combine the algorithmic problem of online ad allocation with a game-theoretic analysis of the auction mechanism. Neither our work nor any of the predecessors of this work [11, 3, 9, 6, 7] consider the strategic behavior of the bidders. This is mainly due to difficulties surrounding repeated auctions and the lack of a satisfactory game theoretic analysis of such auctions. In particular, the folk theorem proves that repeated games have a large set of equilibria, showing that the usual equilibrium concepts do not have good predictive power for such games. The challenge is to come up with a reasonable game theoretic model which limits the range of strategic options of the bidders to avoid running into the folk theorem, while at the same time does not ignore the strategic behavior of the agents altogether. Recently, Feldman et al. [5] made an interesting attempt to get around these difficulties. They proposed a truthful mechanism for the offline setting, assuming the utilities of agents is the number of clicks (not click times value) they receive. References [1] Zoë Abrams, Ofer Mendelevitch, and John Tomlin. Optimal delivery of sponsored search advertisements subject to budget constraints. In Proceedings of the 8th ACM Conference on Electronic Commerce (EC), pages , [2] Ian Ayres and Peter Cramton. Deficit reduction through diversity: How affirmative action at the fcc increased auction competition. Stanford Law Review, 48: ,

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