# Pricing Ad Slots with Consecutive Multi-unit Demand

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3 Pricing Ad Slots with Consecutive Multi-unit Demand 3 Items. Our model considers the geometric organization of ad slots, which commonly has the slots arranged in some sequence (typically, from top to bottom in the right-hand side of a web page). The slots are of variable quality. In the study of sponsored search auctions, a standard assumption is that the quality (corresponding to click-through rate) is highest at the beginning of the sequence and then monotonically decreases. Here we consider a generalization where the quality may go down and up, subject to a limit on the total number of local maxima (which we call peaks), corresponding to focal points on the web page. As we will show later, without this limit the revenue maximization problem is NP-hard. Buyers. A buyer (advertiser) may want to purchase multiple slots, so as to display a larger ad. Note that such slots should be consecutive in the sequence. Thus, each buyer i has a fixed demand d i, which is the number of slots she needs for her ad. Two important aspects of this are sharp multi-unit demand, referring to the fact that buyer i should be allocated d i items, or none at all; there is no point in allocating any fewer consecutiveness of the allocated items, in the pre-existing sequence of items. These constraints give rise to a new and interesting combinatorial pricing problem. Valuations. We assume that each buyer i has a parameter v i representing the value she assigns to a slot of unit quality. Valuations for multiple slots are additive, so that a buyer with demand d i would value a block of d i slots to be their total quality, multiplied by v i. This valuation model has been considered by Edelman et al. [9] and Varian [18] in their seminal work for keywords advertising. Pricing mechanisms. Given the valuations and demands from the buyers, the market maker decides on a price vector for all slots and an allocation of slots to buyers, as an output of the market. The question is one of which output the market maker should choose to achieve certain objectives. We consider two approaches: Truthful mechanism whereby the buyers report their demands and values to the market maker; then prices are set in such a way as to ensure that the buyers have the incentive to report their true valuations. We give a revenue-maximizing approach (i.e., maximizing the total price paid), within this framework. Competitive equilibrium whereby we prescribe certain constraints on the prices so as to guarantee certain well-known notions of fairness and envyfreeness. Envy-free solution whereby we prescribe certain constraints on the prices and allocations so as to achieve envy-freeness, as explained below. The mechanisms we exhibit are computationally efficient. We also performed experiments to compare the revenues obtained from these three mechanisms (See Section 10 in the Appix).

4 4 Xiaotie Deng et al. 1.2 Related Works The theoretical study of position auctions (of a single slot) under the generalized second price auction was initiated in [9, 18]. There has been a series of studies of position auctions in deterministic settings [14]. Our consideration of position auctions in the Bayesian setting fits in the general one dimensional auction design framework. Our study considers continuous distributions on buyers values. For discrete distributions, [4] presents an optimal mechanism for budget constrained buyers without demand constraints in multi-parameter settings and very recently they also give a general reduction from revenue to welfare maximization in [5]; for buyers with both budget constraints and demand constraints, 2-approximate mechanisms [1] and 4-approximate mechanisms [3] exist in the literature. There are extensive studies on multi-unit demand in economics, see for example [2, 6, 10]. In an earlier paper [7] we considered sharp multi-unit demand, where a buyer with demand d should be allocated d items or none at all, but with no further combinatorial constraint, such as the consecutiveness constraint that we consider here. The sharp demand setting is in contrast with a relaxed multi-unit demand (i.e., one can buy a subset of at most d items), where it is well known that the set of competitive equilibrium prices is non-empty and forms a distributive lattice [13, 17]. This immediately implies the existence of an equilibrium with maximum possible prices; hence, revenue is maximized. Demange, Gale, and Sotomayor [8] proposed a combinatorial dynamics which always converges to a revenue maximizing (or minimizing) equilibrium for unit demand; their algorithm can be easily generalized to relaxed multi-unit demand. A strongly related work to our consecutive settings is the work of Michael H. Rothkopf et al. [16], where the authors presented a dynamic programming approach to compute the maximum social welfare of consecutive settings when all the qualities are the same. Hence, our dynamic programming approach for general qualities in Bayesian settings is a non-trivial generalization of their settings. 1.3 Organization This paper is organized as follows. In Section 2 we describe the details of our rich media ads model and the related solution concepts. In Section 3, we study the problem in the Bayesian model and provide a Bayesian Incentive Compatible auction with optimal expected revenue for the special case of the single peak in quality values of advertisement positions. Then in Section 4, we ext the optimal auction to the case with limited peaks/valleys and show that it is NPhard to maximize revenue without this limit. Next, in Section 5, we turn to the full information setting and propose an algorithm to compute the competitive equilibrium with maximum revenue. In Section 6, NP-hardness of envy-freeness for consecutive multi-unit demand buyers is shown. The simulation is presented in Section 10.

6 6 Xiaotie Deng et al. To put it in words, in a BIC mechanism, no player can improve her expected utility (expectation taken over other players bids) by misreporting her value. An IC mechanism satisfies the stronger requirement that no matter what the other players declare, no player has incentives to deviate. 2.2 Competitive Equilibrium and Envy-free Solution In Section 5, we study the revenue maximizing competitive equilibrium and envyfree solution in the full information setting instead of the Bayesian setting. An outcome of the market is a pair (X, p), where X specifies an allocation of items to buyers and p specifies prices paid. Given an outcome (X, p), recall v ij = v i q j, let u i (X, p) denote the utility of i. Definition 2. A tuple (X, p) is a consecutive envy-free pricing solution if every buyer is consecutive envy-free, where a buyer i is consecutive envy-free if the following conditions are satisfied: if X i, then (i) X i is d i consecutive items. u i (X, p) = (v ij p j ) 0, j X i and (ii) for any other subset of consecutive items T with T = d i, u i (X, p) = (v ij p j ) (v ij p j ); j X i j T if X i = (i.e., i wins nothing), then, for any subset of consecutive items T with T = d i, (v ij p j ) 0. j T Definition 3. (Competitive Equilibrium) We say an outcome of the market (X, p) is a competitive equilibrium if it satisfies two conditions. (X, p) must be consecutive demand envy-free. The unsold items must be priced at zero. We are interested in the revenue maximizing competitive equilibrium and envy-free solutions. 3 Optimal Auction for the Single Peak Case The goal of this section is to present our optimal auction for the single peak case that serves as an elementary component in the general case later. En route, several principal techniques are examined exhaustively to the extent that they can be applied directly in the next section. By employing these techniques, we show that the optimal Bayesian Incentive Compatible auction can be represented by a simple Incentive Compatible one. Furthermore, this optimal auction can be implemented efficiently. Let T i (v i ) = E v i [t i (v)], P i (v i ) = E v i [p i (v)] and ϕ i (v i ) = v i 1 F i(v i ) f i (v i ). From Myerson work [15], we obtain the following three lemmas.

7 Pricing Ad Slots with Consecutive Multi-unit Demand 7 Lemma 1 (From [15]). A mechanism M = (x, p) is Bayesian Incentive Compatible if and only if: a) T i (x) is monotone non-decreasing for any agent i. b) P i (v i ) = v i T i (v i ) v i T v i (z)dz i Lemma 2 (From [15]). For any BIC mechanism M = (x, p), the expected revenue E v [ i P i(v i )] is equal to the virtual surplus E v [ i ϕ i(v i )t i (v)]. Lemma 3. Suppose that x is the allocation function that maximizes E v [ϕ i (v i )t i (v)] subject to the constraints that T i (v i ) is monotone non-decreasing for any bidders profile v, any agent i is assigned either d i consecutive slots or nothing. Suppose also that vi p i (v) = v i t i (v) t i (v i, s i )ds i (2) v i Then (x, p) represents an optimal mechanism for the rich media advertisement problem in single-peak case. We will use dynamic programming to maximize the virtual surplus in Lemma 2. Since the optimal solution always assigns to [s] consecutively, we can boil the allocations to [s] down to an interval denoted by [l, r]. Let g[s, l, r] denote the maximized value of our objective function i ϕ i(v i )t i (v) when we only consider first s buyers and the allocation of s is exactly the interval [l, r]. Then we have the following transition function. g[s 1, l, r] g[s, l, r] = max g[s 1, l, r d s ] + ϕ s (v s ) r j=r d s +1 q j Our summary statement is as follows. g[s 1, l + d s, r] + ϕ s (v s ) l+d s 1 j=l q j Theorem 1. The mechanism that applies the allocation rule according to Dynamic Programming (3) and payment rule according to Equation (2) is an optimal mechanism for the banner advertisement problem with single peak qualities. (3) 4 Multiple Peaks Case Suppose now that there are only h peaks (local maxima) in the qualities. Thus, there are at most h 1 valleys (local minima). Since h is a constant, we can enumerate all the buyers occupying the valleys. After this enumeration, we can divide the qualities into at most h consecutive pieces and each of them forms a single-peak. Then using similar properties as those in Lemma 5 and 6 (see Appix), we can obtain a larger size dynamic programming (still runs in polynomial time) similar to dynamic programming (3) to solve the problem.

8 8 Xiaotie Deng et al. Theorem 2. There is a polynomial algorithm to compute revenue maximization problem in Bayesian settings where the qualities of slots have a constant number of peaks. Now we consider the case without the constant peak assumption and prove the following hardness result (see the proof in Appix). Theorem 3. (NP-Hardness) The revenue maximization problem for rich media ads with arbitrary qualities is NP-hard. 5 Competitive Equilibrium In this section, we study the revenue maximizing competitive equilibrium in the full information setting. To simplify the following discussions, we sort all buyers and items in non-increasing order of their values, i.e., v 1 v 2 v n. We say an allocation Y = (Y 1, Y 2,, Y n ) is efficient if Y maximizes the total j Y i v ij is maximized over all the possible allocations. social welfare e.g. i We call p = (p 1, p 2,, p m ) an equilibrium price if there exists an allocation X such that (X, p) is a competitive equilibrium. The following lemma is implicitly stated in [13], for completeness, we give a proof below. Lemma 4. Let allocation Y be efficient, then for any equilibrium price p, (Y, p) is a competitive equilibrium. By Lemma 4, to find a revenue maximizing competitive equilibrium, we can first find an efficient allocation and then use linear programming to settle the prices. We develop the following dynamic programming to find an efficient allocation. We first only consider there is one peak in the quality order of items. The case with constant peaks is similar to the above approaches, for general peak case, as shown in above Theorem 3, finding one competitive equilibrium is NP-hard if the competitive equilibrium exists, and determining existence of competitive equilibrium is also NP-hard. This is because that considering the instance in the proof of Theorem 3, it is not difficult to see the constructed instance has an equilibrium if and only if 3 partition has a solution. Recall that all the values are sorted in non-increasing order e.g. v 1 v 2 v n. g[s, l, r] denotes the maximized value of social welfare when we only consider first s buyers and the allocation of s is exactly the interval [l, r]. Then we have the following transition function. g[s 1, l, r] r g[s, l, r] = max g[s 1, l, r d s ] + v s j=r d s +1 q j (4) l+ds 1 g[s 1, l + d s, r] + v s j=l q j By tracking procedure 4, an efficient allocation denoted by X = (X 1, X 2,, X n) can be found. The price p such that (X, p ) is a revenue maximization competitive equilibrium can be determined from the following linear programming.

9 Pricing Ad Slots with Consecutive Multi-unit Demand 9 Let T i be any consecutive number of d i slots, for all i [n]. max i [n] j Xi p j s.t. p j 0 j [m] p j = 0 j / i [n] Xi (v i q j p j ) (v i q j p j ) i [n] j Xi j T i (v i q j p j ) 0 i [n] j X i Clearly there is only a polynomial number of constraints. The constraints in the first line represent that all the prices are non negative (no positive transfers). The constraint in the second line means unallocated items must be priced at zero (market clearance condition). And the constraint in the third line contains two aspects of information. First for all the losers e.g. loser k with X k =, the utility that k gets from any consecutive number of d k is no more than zero, which makes all the losers envy-free. The second aspect is that the winners e.g. winner i with X i must receive a bundle with d i consecutive slots maximizing its utility over all d i consecutive slots, which together with the constraint in the fourth line (winner s utilities are non negative) guarantees that all winners are envy-free. Theorem 4. Under the condition of a constant number of peaks in the qualities of slots, there is a polynomial time algorithm to decide whether there exists a competitive equilibrium or not and to compute a revenue maximizing revenue market equilibrium if one does exist. If the number of peaks in the qualities of the slots is unbounded, both the problems are NP-complete. Proof. Clearly the above linear programming and procedure (4) run in polynomial time. If the linear programming output a price p, then by its constraint conditions, (X, p ) must be a competitive equilibrium. On the other hand, if there exist a competitive equilibrium (X, p) then by Lemma 4, (X, p) is a competitive equilibrium, providing a feasible solution of above linear programming. By the objective of the linear programming, we know it must be a revenue maximizing one. 6 Consecutive Envy-freeness We first prove a negative result on computing the revenue maximization problem in general demand case. We show it is NP-hard even if all the qualities are the same. Theorem 5. The revenue maximization problem of consecutive envy-free buyers is NP-hard even if all the qualities are the same.

10 10 Xiaotie Deng et al. Although the hardness in Theorem 5 indicates that finding the optimal revenue for general demand in polynomial time is impossible, however, it doesn t rule out the very important case where the demand is uniform, e.g. d i = d. We assume slots are in a decreasing order from top to bottom, that is, q 1 q 2 q m. The result is summarized as follows. Theorem 6. There is a polynomial time algorithm to compute the consecutive envy-free solution when all the buyers have the same demand and slots are ordered from top to bottom. The proof of Theorem 6 is based on bundle envy-free solutions, in fact we will prove the bundle envy-free solution is also a consecutive envy-free solution by defining price of items properly. Thus, we need first give the result on bundle envy-free solutions. Suppose d is the uniform demand for all the buyers. Let T i be the slot set allocated to buyer i, i = 1, 2,, n. Let P i be the total payment of buyer i and p j be the price of slot j. Let t i denote the total qualities obtained by buyer i, e.g. t i = j T i q j and α i = iv i (i 1)v i 1, i [n]. Theorem 7. The revenue maximization problem of bundle envy-freeness is equivalent to solving the following LP. Maximize: s.t. n α i t i i=1 t 1 t 2 t n T i [m], T i T k = i, k [n] (5) Through optimal bundle envy-free solution, we will modify such a solution to consecutive envy-free solution and then prove the Theorem 6. See the appix. 7 Conclusion and Discussion The rich media pricing models for consecutive demand buyers in the context of Bayesian truthfulness, competitive equilibrium and envy-free solution paradigm are investigated in this paper. As a result, an optimal Bayesian incentive compatible mechanism is proposed for various settings such as single peak and multiple peaks. In addition, to incorporate fairness e.g. envy-freeness, we also present a polynomial-time algorithm to decide whether or not there exists a competitive equilibrium or and to compute a revenue maximized market equilibrium if one does exist. For envy-free settings, though the revenue maximization of general demand case is shown to be NP-hard, we still provide optimal solution of common demand case. Besides, our simulation shows a reasonable relationship of revenues among these schemes plus a generalized GSP for rich media ads. Even though our main motivation arises from the rich media advert pricing problem, our models have other potential applications. For example TV ads can also be modeled under our consecutive demand adverts where inventories of a commercial break are usually divided into slots of fixed sizes, and slots have

11 Pricing Ad Slots with Consecutive Multi-unit Demand 11 various qualities measuring their expected number of viewers and corresponding attractiveness. With an extra effort to explore the periodicity of TV ads, we can ext our multiple peak model to one involved with cyclic multiple peaks. Besides single consecutive demand where each buyer only have one demand choice, the buyer may have more options to display his ads, for example select a large picture or a small one to display them. Our dynamic programming algorithm (3) can also be applied to this case (the transition function in each step selects maximum value from 2k + 1 possible values, where k is the number of choices of the buyer). Another reasonable extension of our model would be to add budget constraints for buyers, i.e., each buyer cannot afford the payment more than his budget. By relaxing the requirement of Bayesian incentive compatible (BIC) to one of approximate BIC, this extension can be obtained by the recent milestone work of Cai et al. [5]. It remains an open problem how to do it under the exact BIC requirement. It would also be interesting to handle it under the market equilibrium paradigm for our model. References 1. Saeed Alaei. Bayesian combinatorial auctions: Expanding single buyer mechanisms to many buyers. In Proceedings of the 52nd IEEE Symposium on Foundations of Computer Science (FOCS), pages , L. Ausubel and P. Cramton. Demand revelation and inefficiency in multi-unit auctions. In Mimeograph, University of Maryland, Sayan Bhattacharya, Gagan Goel, Sreenivas Gollapudi, and Kamesh Munagala. Budget constrained auctions with heterogeneous items. In Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 10, pages , New York, NY, USA, Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. An algorithmic characterization of multi-dimensional mechanisms. In Proceedings of the 43rd annual ACM Symposium on Theory of Computing, Yang Cai, Constantinos Daskalakis, and S. Matthew Weinberg. Optimal multidimensional mechanism design: Reducing revenue to welfare maximization. In FOCS 2012, Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, Estelle Cantillon and Martin Pesorfer. Combination bidding in multi-unit auctions. C.E.P.R. Discussion Papers, February Ning Chen, Xiaotie Deng, Paul W. Goldberg, and Jinshan Zhang. On revenue maximization with sharp multi-unit demands. In CoRR abs/ , G. Demange, D. Gale, and M. Sotomayor. Multi-item auctions. The Journal of Political Economy, pages , Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz. Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords. American Economic Review, 97(1): , March R. Engelbrecht-Wiggans and C.M. Kahn. Multi-unit auctions with uniform prices. Economic Theory, 12(2): , M. Feldman, A. Fiat, S. Leonardi, and P. Sankowski. Revenue maximizing envyfree multi-unit auctions with budgets. In Proceedings of the 13th ACM Conference on Electronic Commerce, pages ACM, 2012.

12 12 Xiaotie Deng et al. 12. M.R. Garey and D.S. Johnson. Complexity results for multiprocessor scheduling under resource constraints. SIAM Journal on Computing, 4(4): , F. Gul and E. Stacchetti. Walrasian equilibrium with gross substitutes. Journal of Economic Theory, 87(1):95 124, Sébastien Lahaie. An analysis of alternative slot auction designs for sponsored search. In Proceedings of the 7th ACM conference on Electronic Commerce, EC 06, pages , New York, NY, USA, Roger B. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58 73, Michael H. Rothkopf, Aleksandar Pekeč, and Ronald M. Harstad. Computationally manageable combinational auctions. Management science, 44(8): , L.S. Shapley and M. Shubik. The Assignment Game I: The Core. International Journal of Game Theory, 1(1): , Hal R. Varian. Position auctions. International Journal of Industrial Organization, 25(6): , 2007.

13 Pricing Ad Slots with Consecutive Multi-unit Demand 13 Appix 8 Examples In the literature, there have been two other types of envy-free concepts, namely, sharp item envy-free [7] and bundle envy-free [11]. Sharp item envy-free requires that each buyer would not envy a bundle of items with the number of her demand while bundle envy-free illustrates that no one would envy the bundle bought by any other buyer. From the definition of those three envy-free concepts, we have the following inclusive relations: sharp item envy-free bundle envy-free, consecutive envy-free bundle envy-free Example 1 (Three types of envy-freeness). Suppose there are two buyers i 1 and i 2 with per-unit-quality v i1 = 10, v i2 = 8 and d i1 = 1, d i2 = 2. The item j 1, j 2, j 3 with quality as q j1 = q j3 = 1 and q j2 = 3. The optimal solution of the three types of envy-freeness are as follows: The optimal consecutive envy-free solution, X i1 = {j 3 }, X i2 = {j 1, j 2 } and p j1 = p j3 = 6 and p j2 = 26 with total revenue 38; Optimal sharp item envy-free solution, X i1 = {j 2 }, X i2 = {j 1, j 3 } and p j1 = p j3 = 8 and p j2 = 28 with total revenue 44; Optimal bundle envy-free solution, X i1 = {j 2 }, X i2 = {j 1, j 3 } and p j1 = p j3 = 8 and p j2 = 30 with total revenue 46; It is well known that a competitive equilibrium always exists for unit demand buyers (even for general v ij valuations) [17]. For our consecutive multi-unit demand model, however, a competitive equilibrium may not always exist as the following example shows. Example 2 (Competitive equilibrium may not exist). There are two buyers i 1, i 2 with values v i1 = 10 and v i2 = 9, respectively. Let their demands be d i1 = 1 and d i2 = 2, respectively. Let the seller have two items j 1, j 2, both with the unit quality q j1 = q j2 = 1. If i 1 wins an item, without loss of generality, say j 1, then j 2 is unsold and p j2 = 0; by envy-freeness of i 1, we have p j1 = 0 as well. Thus, i 2 envies the bundle {j 1, j 2 }. On the other hand, if i 2 wins both items, then p j1 + p j2 v i2 j 1 + v i2 j 2 = 18, implying that p j1 9 or p j2 9. Therefore, i 1 is not envy-free. Hence, there is no competitive equilibrium in the given instance. In the unit demand case, it is well-known that the set of equilibrium prices forms a distributive lattice; hence, there exist extremes which correspond to the maximum and the minimum equilibrium price vectors. In our consecutive demand model, however, even if a competitive equilibrium exists, maximum equilibrium prices may not exist. Example 3 (Maximum equilibrium need not exist). There are two buyers i 1, i 2 with values v i1 = 10, v i2 = 1 and demands d i1 = 2, d i2 = 1, and two items j 1, j 2 with unit quality q j1 = q j2 = 1. It can be seen that allocating the two items to i 1 at prices (19, 1) or (1, 19) are both revenue maximizing equilibria; but there is no equilibrium price vector which is at least both (19, 1) and (1, 19).

14 14 Xiaotie Deng et al. Because of the consecutive multi-unit demand, it is possible that some items are over-priced ; this is a significant difference between consecutive multi-unit and unit demand models. Formally, in a solution (X, p), we say an item j is over-priced if there is a buyer i such that j X i and p j > v i q j. That is, the price charged for item j is larger than its contribution to the utility of its winner. Example 4 (Over-priced items). There are two buyers i 1, i 2 with values v i1 = 20, v i2 = 10 and demands d i1 = 1 and d i2 = 2, and three items j 1, j 2, j 3 with qualities q j1 = 3, q j2 = 2, q j3 = 1. We can see that the allocations X i1 = {j 1 }, X i2 = {j 2, j 3 } and prices (45, 25, 5) constitute a revenue maximizing envy-free solution with total revenue 75, where item j 2 is over-priced. If no items are over-priced, the maximum possible prices are (40, 20, 10) with total revenue Missing Proofs Proof (of Theorem 1). To prove the solution of dynamic programming (3) actually maximizes virtual surplus, we need the following two lemmas. Lemma 5. There exists an optimal allocation x that maximizes i ϕ i(v i )t i (v) in the single peak case, and satisfies the following condition. For any unassigned slot j, it must be that either j > j, slot j is unassigned or j < j, slot j is unassigned. Proof. We pick an arbitrary optimal allocation x that maximizes the summation of virtual values. If x satisfies the property, it is the desired allocation and we are done. Otherwise, we do the following modification on x. Let slot j (1 < j < m) be the unassigned slot between buyers allocated slots. Since the quality function are single peaked, we have q j q j+1 or q j q j 1. We only prove the lemma for the case q j q j+1 and the proof for the other case is symmetric. Let slot j > j be the leftmost assigned slot on the right side of j. We modify x by assigning the buyer i who got the slot j the d i consecutive slots from j. It is easy to check the resulting allocation is still feasible and optimal. Moreover, the slot j becomes assigned now. By keep doing this, we can eliminate all unassigned slots between buyers allocations. Thus, the resulting allocation must be consecutive. Next, we prove that this consecutiveness even holds for all set [s] [n]. That is, there exists an optimal allocation that always assigns the first s buyers consecutively for all s [n]. For convenience, we say that a slot is out of a set of buyers if the slot is not assigned to any buyers in that set. Then the consecutiveness can be formalized in the following lemma. Lemma 6. There exists an optimal allocation x in the single peak case, that satisfies the following condition. For any slot j out of [s], it must be either j > j, slot j is out of [s] or j < j, slot j is out of [s]. Proof. The idea is to pick an arbitrary optimal allocation x and modify it to the desired one. Suppose x does not satisfy the property on a subset [s]. By

15 Pricing Ad Slots with Consecutive Multi-unit Demand 15 Lemma 5, there are no unassigned slots in the middle of allocations to set [s]. Then there must be a slot assigned to a buyer i out of the set [s] that separates the allocations to [s]. We use W i to denote the allocated slots of buyer i. Let j and j be the leftmost and rightmost slot in W i respectively. We consider two cases q j q j and q j < q j. We only prove for the first case and the proof for the other case is symmetric. If q j q j, we find the leftmost slot j 1 > j assigned to [s] and the rightmost slot j 2 < j 1 not assigned to [s]. In addition, let i 1 [s] be the buyer that j 1 is assigned to and i 2 > s be the buyer that j 2 is assigned to. In the single peak case, it is easy to check q j q j implies that all the slots assigned to i 2 have higher quality than i 1 s. Thus swapping the positions of i 1 and i 2 will always increase the virtual surplus, i ϕ i(v i )t i (v). By keeping on doing this, we can eliminate all slots out of [s] in the middle of allocation to [s] and attain the desired optimal solution. To complete the proof, it suffices to prove that T i (v i ) is monotone nondecreasing. More specifically, we prove a stronger fact, that t i (v i, v i ) is nondecreasing as v i increases. Given other buyers bids v i, the monotonicity of t i is equivalent to t i (v i, v i ) t i (v i, v i) if v i > v i. Assuming that v i > v i, the regularity of ϕ i implies that ϕ i (v i ) ϕ i (v i ). If ϕ i(v i ) = ϕ i (v i ), then t i(v i, v i ) = t i (v i, v i) and we are done. Consider the case that ϕ i (v i ) < ϕ i (v i ). Let Q and Q denote the total quantities obtained by all the other buyers except buyer i in the mechanism when buyer i bids v i and v i respectively. ϕ i (v i)t i (v i, v i ) + Q ϕ i (v i)t i (v i, v i ) + Q ϕ i (v i )t i (v i, v i ) + Q ϕ i (v i )t i (v i, v i ) + Q. Above inequalities are due to the optimality of allocations when i bids v i and v i respectively. It follows that ϕ i (v i)(t i (v i, v i ) t i (v i, v i )) Q Q ϕ i (v i )(t i (v i, v i ) t i (v i, v i )) Q Q By the fact that ϕ i (v i ) < ϕ i (v i ), it must be t i(v i, v i ) t i (v i, v i). Proof (of Theorem 2). Our proof is based on the single peak algorithm. Assume there are h peaks, then there must be h 1 valleys. Suppose these valleys are indexed j 1, j 2,, j h 1. In optimal allocation, for any j k, k = 1, 2,, h 1, j k must be allocated to a buyer or unassigned to any buyer. If j k is assigned to a buyer, say, buyer i, since i would buy d i consecutive slots, j k may appear in lth position of this d i consecutive slots. Hence, by this brute force, each j k will at most have i d i + 1 mn + 1 possible positions to be allocated. In all, all the valleys have (mn + 1) h possible allocated positions. For each of this allocation, the slots is broken into h single peak slots. We can obtain similar properties as those in Lemma 5 and 6. Without loss of generality, suppose the rest buyers are still the set [n], with non-increasing virtual value. Since the optimal solution always assigns to [s] consecutively, we can boil the allocations to [s] down to

16 16 Xiaotie Deng et al. intervals denoted by [l i, r i ], i = 1, 2,, d, where [l i, r i ] lies in the i-th single peak slot. Let g[s, l 1, r 1,, l d, r d ] denote the maximized value of our objective function i ϕ i(v i )t i (v) when we only consider first s buyers and the allocations of [s] are exactly intervals [l i, r i ], i = 1, 2,, d. Then we have the following transition function. g[s 1, l 1, r 1,, l d, r d ] g[s, l 1, r 1,, l d, r d ] = max i [d] g[s 1, l 1, r 1,, l i, r i d s,, l d, r d ] + ϕ s (v s ) r i j=r i d s +1 q j g[s 1, l 1, r 1,, l i + d s, r i,, l d, r d ] + ϕ s (v s ) l i+d s 1 j=l i q j Proof (of Lemma 4). Since p is an equilibrium price, there exists an allocation X such that (X, p) is a competitive equilibrium. As a result, by envy-freeness, u i (X, p) u i (Y, p) for any i [n]. Let T = [m]\ i Y i, then we have m v ij p j m v ij p j = v ij p j i j Y i j=1 i j X i j=1 i j X i i j X i = u i (X, p) u i (Y, p) = v ij p j i i i j Y i i j Y i = m v ij p j + p j (6) i j Y i j=1 j T where the first inequality is due to Y being efficient and first equality due to u i (X, p) being competitive equilibrium (unallocated item priced at 0). Therefore, j T p j = 0 and the above inequalities are all equalities. i : u i (X, p) = u i (Y, p). Further, because the price is the same, i a loser Z consecutive items and Z = d i, we have u i (Z) 0. i a winner Z consecutive items and Z = d i, we have u i (Y i ) = u i (X i ) u i (Z). Therefore, (Y, p) is a competitive equilibrium. Proof (of Theorem 3). We prove the NP-hardness by reducing the 3 partition problem that is to decide whether a given multi-set of integers can be partitioned into triples that all have the same sum. More precisely, given a multi-set S of n = 3m positive integers, can S be partitioned into m subsets S 1,..., S m such that the sum of the numbers in each subset is equal? The 3 partition problem has been proven to be NP-complete in a strong sense in [12], meaning that it remains NP-complete even when the integers in S are bounded above by a polynomial in n. Given a instance of 3 partition (a 1, a 2,..., a 3n ), we construct a instance for advertising problem with 3n advertisers and m = n + i a i slots. It should be mentioned that m is polynomial of n due to the fact that all a i are bounded by a polynomial of n. In the advertising instance, the valuation v i for each advertiser i

17 Pricing Ad Slots with Consecutive Multi-unit Demand 17 is 1 and his demand d i is defined as a i. Moreover, for any advertiser, his valuation distribution is that v i = 1 with probability 1. Then everyone s virtual value is exactly 1. To maximize revenue is equivalent to maximize the simplified function i j x ijq j. Let B = i a i/n. We define the quality of slot j is 0 if j is times of B + 1, otherwise q j = 1. That can be illustrated as follows } {{ } B } {{ } B 0... } 1 1 {{ 1 } 0 B It is not hard to see that the optimal revenue is i a i if and only if there is a solution to this 3 partition instance. Proof (of Theorem 5). We prove the NP-hardness by reducing the 3 partition problem that is to decide whether a given multi-set of integers can be partitioned into triples that all have the same sum. More precisely, given a multi-set S of n = 3m positive integers, can S be partitioned into m subsets S 1,..., S m such that the sum of the numbers in each subset is equal? The 3 partition problem has been proven to be NP-complete in a strong sense in [12], meaning that it remains NP-complete even when the integers in S are bounded above by a polynomial in n. Given a instance of 3 partition (a 1, a 2,..., a 3n ). Let B = i a i/n. we construct a instance for advertising problem with 3n + 1 advertisers and m = B n + i a i slots. It should be mentioned that m is polynomial of n due to the fact that all a i are bounded by a polynomial of n. In the advertising instance, the valuation v i for each advertiser i is 1 and his demand d i is defined as a i and there is another buyer with valuation 2 for each slot and with demand B + 1. The quality of each slot j is 1. It is not hard to see that the optimal revenue is nb + 2(B + 1) if and only if there is a solution to this 3 partition instance, the optimal solution is illustrated as follows } {{ } B+1 1 }{{} unassigned } {{ } B 1 }{{} unassigned } {{ } B 1 }{{} unassigned } {{ } B Proof (of Theorem 7). Recall P i denote the payment of buyer i, we next prove that the linear programming (5) actually gives optimal solution of bundle envyfree. By the definition of bundle envy-free, where buyer i would not envy buyer i + 1 and versus, we have v i t i P i v i t i+1 P i+1 (7) v i+1 t i+1 P i+1 v i+1 t i P i (8) Plus above two inequalities gives us that (v i v i+1 )(t i t i+1 ) 0. Hence, if v i > v i+1, then t i t i+1. From (7), we could get P i v i (t i t i+1 ) + P i+1. The maximum payment of buyer i is P i = v i (t i t i+1 ) + P i+1, (9)

18 18 Xiaotie Deng et al. together with t i t i+1, implying (7) and (8). Besides the maximum payment of n is P n = t n v n. (9) together with t i t i+1 and P n = t n v n would make everyone bundle envy-free, the arguments are as follows. All the buyers must be bundle envy free. By (9), we have P i P i+1 = v i (t i t i+1 ), hence P i = n 1 k=i v k(t k t k+1 ) + P n. Noticing that if t i = 0, then P i = 0, which means i is loser. For any buyer j < i, we have P j P i = i 1 k=j v k(t k t k+1 ) i 1 k=j v j(t k t k+1 ) = v j (t j t i ). rewrite P j P i v j (t j t i ) as v j t i P i v j t j P j, which means buyer j would not envy buyer i. Similarly, P j P i = i 1 k=j v k(t k t k+1 ) i 1 k=j v i(t k t k+1 ) = v i (t j t i ), rewrite P j P i v i (t j t i ) as v i t i P i v i t j P j, which means i would not envy buyer j. Now let s calculate n i=1 P i based on (9) using notation t n+1 = 0, one has n P i = i=1 = = n i=1 n [ n 1 ] v k (t k t k+1 ) + P n = k=i k=1 i=1 k v k (t k t k+1 ) = n kv k t k k=1 n i=1 k=i n kv k (t k t k+1 ) k=1 n (k 1)v k 1 t k = k=1 n α i t i i=1 n v k (t k t k+1 ) We know the revenue maximizing problem of bundle envy-freeness can be formalized as (5). Since consecutive envy-free solutions are a subset of bundle envy-free solutions, hence the optimal value of optimization (5) gives an upper bound of optimal objective value of consecutive envy-free solutions. Noting optimization LP (5) can be solved by dynamic programming. Let g[s, j] denote the optimal objective value of the following LP with some set in [j] allocated to all the buyers in [s]: s Maximize: α i t i Then s.t. i=1 t 1 t 2 t s T i [j], T i T k = i, k [s] g[s, j 1] g[s, j] = max j g[s 1, j d] + α s u=j d+1 q u (10) Next, we show how to modify the bundle envy-free solution to consecutive envyfree solutions by properly defining the slot price of T i, for all i [n]. Suppose the optimal winner set of bundle envy-free solution is [L]. Assume the optimal allocation and price of bundle envy-free solution are T i = {j i 1, j i 2,, j i d } with j i 1 j i 2 j i d and P i respectively, for all i [L].

19 Pricing Ad Slots with Consecutive Multi-unit Demand 19 Proof ( of Theorem 6). Define the price of T i iteratively as follows: p j L k = v L q j L k, for all k [d]; p j i k = v i (q j i k q j i+1) + p k j i+1 for k [d] and i [n] k Now we could see that the price defined by above procedure is still a bundle envy-free solution. This is because by induction, we have P i = d k=1 p jk i. Hence, we need only to check the prices defined as above and allocations T i constitute a consecutive envy-free solution. In fact, we prove a strong version, suppose T i s are consecutive from top to down in a line S, we will show each buyer i would not envy any consecutive sub line of S comprising d slots. For any i, Case 1, buyer i would not envy the slots below his slots. for any consecutive line T bellow i with size d, suppose T comprises of slots won by buyer k(denoted such slot set by U k ) and k + 1(denoted such slot set by U k+1 and let l = U k+1 ) where k i. Recall that t i = j T i q j, then p j p j = v i (t i t i+1 ) + P i+1 j T i j T = v i (t i t i+1 ) + v i+1 (t i+1 t i+2 ) + + P k j U k U k+1 p j j U k U k+1 p j = v i (t i t i+1 ) + v i+1 (t i+1 t i+2 ) + + p j j T k \U k l = v i (t i t i+1 ) + v i+1 (t i+1 t i+2 ) + + v k (q j k u q j k+1) u v i (t i t i+1 ) + v i (t i+1 t i+2 ) + + u=1 l u=1 v i (q j k u q j k+1) u j U k+1 p j = v i t i v i q j. (11) j T Rewrite j T i p j j T p j v i t i v i j T q j as v i t i j T i p j v i j T q j j T p j we get the desired result. Case 2, buyer i would not envy the slots above his slots. for any consecutive line T above i with size d, suppose T comprises of slots won by buyer k(denoted such slot set by U k ) and k 1(denoted such slot set by U k 1

20 20 Xiaotie Deng et al. and let l = U k 1 ) where k i. Recall that t i = j T i q j, then = = p j p j = j T i j T d u=d l+1 d u=d l+1 d u=d l+1 v k 1 (q j k 1 u v k 1 (q j k 1 u v i (q j k 1 u p j j U k 1 U k j T i p j q j k u ) + p j j T k j T i p j q j k u ) + v k (t k t k+1 ) + + v i 1 (t i 1 t i ) q j k u ) + v i (t k t k+1 ) + + v i (t i 1 t i ) = v i q j v i t i. (12) j T Rewrite j T p j j T i p j v i j T q j v i t i as v i t i j T i p j v i j T q j j T p j we get the desired result. 10 Simulation We did a stimulation to compare the expected revenue among those pricing schemes. The sampling method is applied to the competitive equilibrium, envyfree solution, Bayesian truthful mechanism, as well as the generalized GSP, which is the widely used pricing scheme for text ads in most advertisement platform nowadays. The value samples v come from the same uniform distribution U[20, 80], and each group V k contains n samples, V k = {vk 1, v2 k,, vn k }. With a random number generator, we produced 200 group samples {V 1, V 2,, V 200 }, they will be used as the input of our simulation. For the parameters of slots, we assume there are 8 slots to be sold, and fix their position qualities: Q = {q 1, q 2, q 3, q 4, q 5, q 6 } = {0.8, 0.7, 0.6, 0.5, 0.4, 0.3} The actually ads auction is complicated, but we simplified it in our simulation, we do not consider richer conditions, such as set all bidders budgets unlimited, and there is no reserve prices in all mechanisms. we vary the group size n from 5 to 12, and observe their expected revenue variation. From j = 1 to j = 200, at each j, invoke the function EF (V j, D, Q), GSP (V j, D, Q), CE (V j, D, Q) and Bayesian (V j, D, Q) respectively. Thus, those mechanisms use the same data from the same distribution as inputs and compare their expected revenue fairly. Finally, average those results from different mechanisms respectively, and compare their expected revenue at sample size n.

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