Limiting Price Discrimination when Selling Products with Positive Network Externalities

Transcription

1 Limiting Price Discrimination when Selling Products with Positive Network Externalities Luděk Cigler, Wolfgang Dvořák, Monika Henzinger, Martin Starnberger University of Vienna, Faculty of Comuter Science, Austria Abstract. Assume a seller wants to sell a digital roduct in a social network where a buyer s valuation of the item has ositive network externalities from her neighbors that already have the item. The goal of the seller is to maximize his revenue. Previous work on this roblem [9] studies the case where clients are offered the item in sequence and have to ay ersonalized rices. This is highly infeasible in large scale networks such as the Facebook grah: () Offering items to the clients one after the other consumes a large amount of time, and (2) rice-discrimination of clients could aear unfair to them and result in negative client reaction or could conflict with legal requirements. We study a setting dealing with these issues. Secifically, the item is offered in arallel to multile clients at the same time and at the same rice. This is called a round. We show that with O(log n) rounds, where n is the number of clients, a constant factor of the revenue with rice discrimination can be achieved and that this is not ossible with o(log n) rounds. Moreover we show that it is APX-hard to maximize the revenue and we give constant factor aroximation algorithms for various further settings of limited rice discrimination. Introduction With the aearance of online social networks the issue of monetizing network information arises. Many digital roducts such as music, movies, as, e-books, and comuter games are sold via latforms with social network functionality. Often these roducts have so called ositive network externalities: the valuation of a client for a roduct increases (otentially marginal) when a related client (e.g., a friend) buys the roduct, i.e., a roduct aears more valuable for a client if a friend already owns the same roduct. In the resence of ositive network externalities, motivating a client to buy a roduct by lowering the rice he has to ay could incentivize his friends to also buy the roduct. Consequently, it could increase future revenue. Thus, when trying to maximize revenue there is an interesting trade off between the current and the future revenue. We follow the work of Hartline et al. [9] for modeling network externalities and (marketing) strategies in social networks with the goal of maximizing the The research leading to these results has received funding from the Euroean Unions Seventh Framework Programme (FP7/ ) under grant agreement no

2 seller s revenue. We model the seller s information about the clients valuations by using a directed weighted grah or, more generally, by using submodular set functions. For our ositive results we assume that the seller has only incomlete information about the valuations, i.e., we are in a Bayesian setting where the seller is given a distribution of each client s valuation but not the valuation itself. The seller s strategy decides (a) when to offer the roduct to a client and (b) at which rice. Thus each strategy assigns each client a (time, rice)-air. In [9] a setting with full rice discrimination is studied where a seller offers a roduct sequentially for a different rice to different clients. (See [2, 6, 7] for further work on settings with full rice discrimination.) While it gives a good baseline to comare with, this aroach has multile drawbacks. First, rocessing one client after the other and waiting for the earlier client s decision requires too much time if the number of clients is large. Second, rice-discrimination could aear unfair to the clients and could result in a negative reaction of some clients [2]; moreover, it might be in conflict with legal requirements. On the other side offering all clients the same rice (i.e., a uniform rice) reduces the revenue significantly, namely, by a factor of log n, as we show below. Hence, we introduce rounds such that in a round the roduct is offered to a set of clients at the same time and we consider strategies with a limited number of rounds and/or limited rice discrimination. Secifically we study k-round strategies where the roduct is offered to the clients in k rounds with k n such that only clients that have urchased the roduct in a revious round can influence the valuation functions of the clients to whom the roduct is offered in the current round. Following [9] the strategies offer the roduct only once to each client to avoid that clients behave strategically, i.e., wait for rice decreases in the future. We study two tyes of k-round strategies: () k-pd strategies where each client has a ersonalized rice (limited number of rounds), and (2) k-pr strategies where all clients in the same round are offered the same rice (limited number of rounds and limited rice discrimination). Thus, k-pr strategies generalize the simle uniform rice setting, where one might distribute free coies at the beginning and then charge everyone the same rice. The setting in [9] corresonds to the n-pr setting. Throughout the aer we use R to denote the otimal revenue achievable by an n-pr strategy. To summarize we study the natural question of how many different rices/rounds are necessary and sufficient if we want to achieve a constant factor aroximation algorithm. Our main results are: () There is a log n-pr strategy that achieves a constant factor aroximation of R for very general valuation functions, namely robabilistic submodular valuation functions. Thus only log n different rices are necessary. We show that this result is tight (u to constant factors) in two regards: (a) It cannot be achieved with o(log n) rounds, even for very limited valuations with deterministic and additive externalities. (b) There exists a constant c such that it is NP-hard to comute a c-aroximation of R, no matter how many rounds, i.e., maximizing revenue in k-pr strategies (as well as k-pd strategies) is APX-hard. (2) There is a 2-PR strategy that achieves an O(log n)- aroximation of R. (3) We give (nearly) /6-aroximation algorithms for 2

3 the maximum revenue achievable by any k-pr strategy when comared to the otimal k-pr strategy (i.e., not comared with R). All algorithms we resent are olynomial in the number of clients. Interestingly, all of them make very limited but also very natural use of the network structure: They only exloit information about the neighbors of a node and not any global roerties. Discussion of Related work. In the resence of network externalities two main tyes of revenue maximizing strategies have been used in the literature: strategies with rice discrimination (each client ays a different rice), and strategies with uniform rice (every aying client ays the same rice). We have already mentioned the work of Hartline et al. [9] on rice discriminating strategies with n rounds (n-pd). They study so-called influence-and-exloit strategies consisting of two stes: () the influence ste, in which the roduct is given for free to a set of influence nodes; and (2) the exloit ste, in which clients are aroached sequentially and each client is offered the roduct at a ersonalized rice. Hartline et al. give a randomized influence-and-exloit strategy that gives an e 4e 2 - aroximation of the otimal revenue R. Note that the revenue of our generalization to k-pd in Section 3 is quite close to this as it gives an e 4e 2+2/(k ) - aroximation. In articular for k 0 our strategy k-pd(q) achieves more than 95% of the revenue of their strategy. Moreover with the imrovements of Theorem 3 and k 0 we get even better constants than [9]. Additionally, Hartline et al. resent an algorithm that, together with a novel result on submodular function maximization [4], 0.5-aroximates the otimal influence set. Later, Fotakis and Siminelakis [7] studied a restricted model of client valuations and imroved the aroximation algorithms of Hartline et al. [9]. Furthermore, they study the ratio between the otimal strategies and the otimal influence-and-exloit strategies. Babaei et al. [2] exerimentally evaluated several marketing strategies without an influence ste, instead giving the most influential clients discounts. They conclude that discounts increase the revenue in the considered artificial and real networks. Influence-and-exloit strategies with uniform rices have been studied by Mirrokni et al. []. They use generic algorithms for submodular function maximization to obtain an influence set with at least /2 of the revenue of the otimal uniform rice influence-and-exloit strategy, not of R. In both, [9] and [], similar grah models with concave influences (CG) are introduced as model for network externalities. Akhlaghour et al. [] study a different scenario with uniform rices, without any rice discrimination: The roduct is offered on k consecutive rounds to all clients for the same rice, and a client buys the roduct when its value exceeds the rice for the day. They resent an FPTAS for the Basic scenario, where a client buying the roduct immediately influences the valuation of the other clients who then may also buy the roduct in the same round. A round ends (the rice changes) when no client is willing to buy the roduct for the current rice. This model does not fit well to our assumtions of limited time and a large networks. In the Raid scenario, where buyers on the same day do not affect 3

4 each other (like in our setting) they show that no constant factor aroximation is ossible and give an O(log k n) aroximation. The main difference to our setting is that in [] the roduct has to be offered to every client (not having the roduct) in each round and thus there is no influence round where clients get the roduct for free. For non-digital goods Ehsani et al. [6] study revenue maximizing strategies with both rice discrimination and uniform rices with full information about the clients valuations and with roduction costs er unit in a setting where clients arrive randomly. They give an FPTAS for the otimal uniform rice. Recall that we are studying digital goods without full information about the valuation functions. Haghanah et al. [8] study submodular network externalities for bidders in auctions and rovide auctions that give a 0.25-aroximation of the otimal revenue. In our models the strategies have to offer items in rounds since the clients are only influenced by clients, who bought the roduct in a revious round; that is an imortant difference to the auctions in [8]. Structure of the aer. In Section 2, we resent our model for networks externalities as well as the different kinds of marketing strategies we consider in this aer. In Section 3, we study the effect of restricting the number of rounds, i.e., the effect of offering the roduct to several clients in arallel, but allowing full rice discrimination. In Section 4 we comare the otimal revenue achievable with individual rices against the otimal revenue achievable with uniform rices. Efficiently comutable k-pr strategies are studied in Section 5. In Section 6 we discuss several extensions of our model. Finally, in the Aendix we rovide all omitted roofs. 2 Preliminaries We are given a network G = (V, E) of n clients V and edges E V V that reresent their relationshis. Suose that we want to offer a digital roduct (i.e., the unit costs of the roduct are zero and we can roduce an arbitrary number of coies) to each client i V for some rice i R 0 and maximize our revenue. We call a client that has bought our roduct active client or buyer; otherwise we call him inactive client. We define the valuation of client i V by v i : V \ {i} R 0, such that the valuation of i only deends on his neighbors, i.e., for A being the set of active clients and N i being the set of neighbors of i in G holds v i (A) = v i (A N i ). We try to exloit this deendency of the valuation on the status of the neighbors (called externality) by offering the roduct to clients in a certain order, i.e., we want to comute an order on the clients. Furthermore, we restrict ourselves to a single offer to each client. We assume that clients are individually rational and have quasi-linear utilities. Thus client i buys the roduct if and only if the rice i is not larger than the valuation v i (A). 4

5 Valuation functions. We describe the different models for externalities for a client i V and his active neighbors B. Our main focus is on submodular valuation functions, based on the intuition that the ositive influence of a fixed neighbor does not increase when the set of active neighbors grows. Next we define the models we consider in the aer: Simle Additive Model (SA). There are non-negative weights w i,i and w i,j for (i, j) E. The valuation of i is given by v i (B) = j B {i} w i,j. Deterministic Submodular Model (DS). The valuation of i is given by v i (B) = g i (B), where g i : V \ {i} R 0 is a monotone, submodular set function. Probabilistic Submodular Model (SM) [8]. The valuation of i is given by v i (B) = ṽ i g i (B) where g i : V \ {i} [0, ] is a (ublicly known) monotone, submodular function with g i (V \ {i}) = and the rivate value ṽ i 0 is drawn from a (ublicly known) distribution with the CDF F i. In the first two models the seller has full information about the valuation while in the SM-model she only knows the distribution. We have that the DS-model generalizes the SA model and the SM-model generalizes both the SA and the DS-model. To simlify the resentation in this aer we state ositive results for the SM-model and hardness results (whenever ossible) for the SA-model. For all the models we call v i ( ) the intrinsic valuation of client i V. Note that v i (V \ {i}) is the maximum valuation of client i V in each model. We will use this fact for uer bounds on the revenue any strategy can extract from a client. Seller information. By the revious definitions the valuation functions model the information of the seller about the real valuation of the clients. If the seller has full information, the seller maximizes her revenue by setting i = v i (B), where B are the active neighbors of i V. For the case of incomlete information rice setting is more challenging. In articular, multile rices could maximize the exected ayment, the so-called myoic rices, and we do not know in general how likely it is that the client accets one of those myoic rices. Definition. Given a client i V, and the set of active clients B V \ {i}, the myoic rices of client i are defined as argmax R 0 P [v i (B) ]. If P [v i (B) 0] = 0 we define zero as the unique myoic rice. The frequently used monotone hazard rate condition imlies that there is a f unique myoic rice. The hazard rate of a PDF f i,b is defined to be i,b (y) F i,b (y) for y 0. If the hazard rate of f i,b is a monotone non-decreasing function of y 0 where y satisfies F i,b (y) < we say that F i,b has a monotone hazard rate. In the SM-model with monotone hazard rates we assume that for each i V the CDF F i has monotone hazard rate. Note that for full information models, F i,b (y) = 0 and f i,b (y) = 0 for all y < v i (B) and F i,b (y) = for all y v i (B). Lemma ([9]). In the SM-model with monotone hazard rates each client has a unique myoic rice which is acceted with robability at least e. For a set of active clients B we denote the unique myoic rice of client i as ˆ i (B). A set function f : 2 S R is called submodular if for all X Y S and each x S \ Y it holds that f(x {x}) f(x) f(y {x}) f(y ). 5

6 We will also assume that we can comute this myoic rice in olynomial time. (Marketing) Strategies. A marketing strategy determines in each round to which clients and at what rices the roduct is offered in this round. This choice may deend on the already visited clients, the active clients and the number of rounds remaining. Definition 2. A k-round (marketing) strategy is a robabilistic function s that mas (C, B, j) to (V j, ), where C V are the clients visited so far, B C are the buyers in the revious rounds, j [k] is the current round, V j V \ C are the clients in round j, and : V j R 0 gives the rices for clients in V j. 2 In the following we consider two classes of k-round strategies, (i) one where each client gets an individual rice, i.e., the rounds only influence the valuations and lace no additional restrictions on the rice, and (ii) one where the seller must set uniform rices in each round, and for each of them also the subclass of Influence and Exloit ()-strategies, where the seller offers the roduct for free to the clients of the first round, i.e., the rice in the first round is fixed to 0. In -round strategies the seller has to offer the roduct to all clients at the same time and thus network externalities do not come into lay at all. Price discrimination. [k-pd] This class contains all k-round strategies. If we set k = V we get the class of all ossible strategies. For k 2 we consider the subclass [k-pd ] of k-round influence and exloit strategies, where the seller gives the roduct for free to the clients selected in the first round. Uniform rices er round. [k-pr] This class contains all k-round uniform rice strategies, where all clients visited in the same round are offered the same rice. For k 2 the k-round uniform rice influence and exloit strategies [k-pr ] are the k-pr strategies with the first round having uniform rice 0. Given a strategy s we use R(s) to denote the (exected) revenue obtained by s. By R k-pd, R k-pd, Rk-PR and R k-pr we denote the otimal revenues achievable by the above classes of strategies, where R = R n-pd is the otimal revenue achievable by any strategy. Restricted to k-rounds and uniform rices, our main goals are to achieve constant factor aroximations of R and R k-pr. 3 Strategies with Individual Prices In this section we analyze k-pd strategies, i.e., k-round strategies with full rice discrimination. We first show that maximizing the revenue of such strategies is comutationally hard even in the SA-model; in articular, we show that it is NPhard to aroximate better than within a factor of 34k/( + 34k). The reduction uses the fact that even in the SA-model, the roblem of maximizing the revenue of k-pd generalizes Maximum k-cut, and the latter is APX-hard [0]. Theorem. Maximizing the revenue of k-pd, res. k-pd, is APX-hard in the SA-model; it is NP-hard to 34k +34k -aroximate R k-pd, res. R k-pd, for k 2. 2 In rincile it suffices that s(,, j) is defined on the ossible outcomes of round j. In articular for j = it has only to be defined for s(,, ). 6

7 On the ositive side we generalize the result in [9] to the SM-model with monotone hazard rates and to k rounds. Secifically, we show that the following strategy gives a constant factor aroximation of the otimal k-round revenue as well as of the otimal revenue R and can be comuted in olynomial time. We will use these results in the following sections as they imly that any k-pr strategy that is an α-aroximation of R k-pd is an O(α)-aroximation of R. Algorithm (Strategy k-pd(q)). Let q be in [0, ].. Assign clients in V indeendently with robability q to set V. Give the clients in V the roduct for free. 2. Partition the clients in V \ V into sets V 2,..., V k s.t. each client is in V j indeendently of the other client with robability ( q)/(k ). 3. Offer the clients in V j the roduct in arallel for their myoic rice. To analyze the strategy we first consider the exected ayment π i (S) = ˆ i (S) P [v i (S) ˆ i (S)] we can extract from a client i given the active clients S. By the definition of the myoic rice, we can show that ˆ i (S) = ˆ i (V \ {i}) g i (S). The crucial idea in Lemma 2 is now that we can lower bound the exected revenue π i (S) collected from client i by the maximum revenue that can be collect from i, namely π i (V \ {i}), multilied by the robability β that a client is in S. Note that β is a function of q. Theorem 2 then determines the value β, which in turn sets q. Finally we use a well-known roerty of submodular functions [9] to lower bound the revenue of k-pd(q). Lemma 2. Let S V \ {i} be the random set of clients and let each client j V \ {i} be in S indeendently with a robability of at least β. Then it holds that E S [π i (S)] β π i (V \ {i}). Theorem 2. Consider the SM-model with monotone hazard rates. For q = e (k ) e (k ) 2e(k ) k+2 it follows that R(k-PD(q)) R 4e(k ) 2k+4 and thus also R k-pd e (k ) 4e(k ) 2k+4 R. For k = 2, buyers are only influenced by clients in the influence set and Theorem 2 gives a 2-PD strategy, i.e., 2-PD(0.5), which achieves at least 4 of the otimal revenue (a similar result was given in [9]). However the main challenge in the above algorithm is to exloit also the externalities from the other receding rounds. The 2-round case will be crucial for our k-pr-algorithm in Section 5. Corollary. Given the SM-model with monotone hazard rates, it follows that R k-pd R k-pd 4 R for k 2. Fotakis and Siminelakis [7] show that myoic rices are not necessarily otimal for -strategies. They rovide -strategies, using lower rices, that beat those of [9] if the valuations follow the uniform additive model. In the following we generalize this idea to (a) submodular valuations with monotone hazard rates and (b) to the k-round setting. That is, we consider strategies that use different discount factors α j for different rounds j. To be more recise in each round the seller charges every client only an α j -fraction of his myoic rice. Moreover, we 7

8 also consider different robabilities q j for a client being assigned to round j. The next theorem shows that charging less than the myoic rice can imrove the overall revenue. Algorithm 2 (Strategy k-pd( q, ᾱ)). Let q and ᾱ be vectors of length k with entries in [0, ] and let the entries of q sum u to.. Partition V into sets V,..., V k s.t. each client i V is in V j indeendently of the others with robability q j. 2. Offer the clients i V j the roduct in arallel for rice α j ˆ i where ˆ i is the myoic rice of client i. Theorem 3. Given the SM-Model with monotone hazard rates, for each k there exist vectors q, ᾱ such that R(k-PD( q, ᾱ)) C k R, where C 3 = 0.279, C 5 = 0.298, C 8 = 0.308, and C 0 = 0.3. Comuting C k is a multi-arameter otimization roblem (with 2k arameters) that we solved numerically. More details are rovided in the aendix. Finally, note that the Algorithms and 2 use the network structure only in the comutation of the myoic rices, which only requires to know the active neighbors. 4 Comaring Individual Prices to Uniform Prices In this section we study k-pr strategies where there are k rounds and in each round we offer the roduct to a subset of the clients for a uniform rice. We first analyze the imact that restricting the strategies to be uniform rice strategies has on the otimal revenue for a constant number of rounds. We show that in the SM-model the otimal revenue can decrease in the worst case by a factor of Θ(/n). Thus, if we do not make assumtions on the robability distributions we cannot do better than in each round just selecting the most valuable client and offer the roduct to him for his myoic rice. However, if we consider the SM-model with monotone hazard rates the otimal revenue can decrease in the worst case by a factor of Θ(/ log n). As a result we will focus on models with monotone hazard rates in the remainder of the aer. Theorem 4. Assume that the valuations of the clients follow the SM-model.. For every ε > 0 and k there exists a network and valuations v i such that R k-pr k+ε R n k-pd. 2. For any network and valuations v i, R -PR R n -PD. 3. For any network and valuations v i, R 2-PR n R 2-PD 4n R. The next theorem shows three oints: () Even with monotone hazard rates, no k-pr strategy can be better than a k/h n -aroximation of R k-pd and, thus, also of R. Recall, that the SA-model satisfies the monotone hazard rate condition and is a secial case of a DS and an SM-model and thus these negative results also extend to these models. Thus without rice discrimination within a round no constant factor aroximation of R with o(log n) different rounds exists. 8

9 (2) Even with only 2 rounds the otimal 2-PR strategy achieves an O(log n) aroximation of R 2-PD, which, by Corollary, achieves a 4-aroximation of R. Thus, the otimal 2-PR strategy achieves an O(log n) aroximation of R. This is a large imrovement over the negative result from Theorem 4, which holds for valuations that do not have monotone hazard rates. However, we show in the next section that comuting the otimal 2-PR strategy is NP-hard. Theorem 5. Assume that valuations of the clients follow the SM-model with monotone hazard rates.. For each k there exists a network and valuations v i such that R k-pr k H n Rk-PD (even in the SA-model). 2. For any network and valuations v i, R -PR 3. For any network and valuations v i, e H n R-PD. R 2-PR e H n R 2-PD. Recall that we want to achieve a constant aroximation of R using a uniform rice strategy. Thus, in the next section we give a olynomial time comutable k-pr strategy, with k Θ(log n), that achieves a constant factor aroximation. 5 Strategies with Uniform Prices In the analysis of the algorithms in Section 3 we exloited that the exected revenue from a client i was submodular in the set of active neighbors. This is not true in the PR setting, as we can only extract revenue from a client if his valuation is larger than the uniform rice. Still we can show the following: () We give a olynomial-time aroximation scheme (PTAS) for one round strategies in the SM-model with monotone hazard rates, i.e., for finding the otimal uniform rice for one round. (2) We show that finding an otimal 2-PR strategy is not only NP-hard but also APX-hard, even for the DS-model. (3) We give a constant factor aroximation of R with O(log n) rounds for valuations from the SMmodel with monotone hazard rates. (4) From Section 4 we know that k-pr strategies, where k is a constant, lose a factor of log n of the otimum revenue when comared to R. Thus, in this case the best we can hoe for is a constant factor aroximation of Rk-PR, not of R. We have two such results: (4a) We give a (/6 ε) aroximation of Rk-PR for the SM-model with monotone hazard rates. (4b) We show that under certain conditions we can even give a (/4 ε)-aroximation of R 2-PR. Combined with Theorem 5 this gives an O(log(n)/k)-aroximation of R in k rounds. We first give a PTAS for comuting the otimal rice for the one round setting. This will be a useful tool for the 2-round setting. Algorithm 3. Let c = ( ε) and ε > 0.. Comute ˆ max = max i V ˆ i. 2. For all j {0,..., log c (e n) }: Comute the exected revenue R j for the uniform rice j = ˆmax c. j 3. Return j with maximal R j. 9

10 Theorem 6. Given the SM-model with monotone hazard rates, then for each ε > 0 Algorithm 3 gives a -PR strategy s (i.e., a uniform rice), such that R(s) ( ε) R -PR in olynomial time. The basic idea of the roof is that the otimal uniform rice cannot be less than ˆ max /(e n) and if we ick a rice within ( ε) of we get a ( ε)- aroximation of R -PR. Next we show that for two rounds the roblem becomes APX-hard. That is it is NP-hard to aroximate better than within a factor of 259/260. The roof is via a reduction from the dominated set roblem (see Lemma 9 in the aendix). In this reduction a client has valuation iff at least one of its neighbors is in the dominating set, and 0 otherwise. This function is not additive and thus the result requires the DS-model (and not the SA-model). Theorem 7. Maximizing the revenue of 2-PR, res. 2-PR, is APX-hard for the DS-model (and also for the concave grah models of [9, ]), in articular, it is not aroximable within 259/260. Next we resent the constant factor aroximation of R for k Ω(log n). In the following strategy the set A of clients for the first round is given. We will then choose A using the 2-PD -strategy of Theorem 2 to get the final result. Algorithm 4 (Strategy k-pr(c, A)). Let A V be the influence set and c > be a constant.. Give the roduct to all clients in A for free in the first round. 2. Set (ˆ, ˆ 2,..., ˆ t ) to the myoic rices of the clients in V \A for the influence set A in descending order. 3. Set S j = { i ˆ c ˆ j i > ˆ c j } and select the first k non-emty sets. 4. Each of these sets S j becomes a set of clients that is offered the roduct in one round with uniform rice (j) = min i Sj ˆ i. Our analysis of this strategy only collects revenue for clients in V \ A and only exloits externalities induced by clients in A, i.e., from clients in the first round. Thus this algorithm would have the same erformance if all nodes in V \ A are offered the item in the same round but with k different rices. We denote by k-pr (c, q) the strategy where the influence set A is chosen randomly such that each client is in A with robability q, indeendently of the other clients. For the clients in the selected sets S j we extract at least /c of the revenue the otimal 2-PD strategy would extract from them. Additional in each set there is one client that gets his myoic rice (i.e., an additional ( /c) factor of his otimal revenue). We show in the roof below that this second contribution can be used to comensate for the otimal revenue of the clients which are not in a selected set, resulting in the bound of the next theorem. Theorem 8. Let c > be a constant. Given valuations from the SM-model with monotone hazard rates then for every 2-PD strategy s with influence set A the strategy (k + )-PR(c, A) achieves at least min{ c, (c k ) } of the (c k )+e(n k)(c ) revenue R(s) of s. 0

11 Proof. Consider the set V \ A = {,..., n} and let ˆ := (ˆ, ˆ 2,..., ˆ n ) be the vector of (myoic) rices induced by A. W.l.o.g., we assume that the rices and the corresonding clients are sorted in a descending order. By the definition R(s) = n i= ˆ i P [v i ˆ i ]. For each client i we denote the rice charged by the strategy (k +)-PR(c, A) by i. The uniform rice in round j is denoted by (j). Then the revenue of (k + )-PR(c, A) is given by: R((k + )-PR(c, A)) = n i= i P [v i i ] Now, by construction of S j, either i ˆ i/c in the case where i is in one of the selected sets (the first k non-emty sets), or i = 0 otherwise. Let J be the set of the indices of the selected sets and l the largest index among them. Let m be the number of clients that are offered the roduct, i.e., m is the client with the lowest myoic rice in S l. Consider suitable chosen α /c. For each client i in a selected set the algorithm collects at least a revenue of α ˆ i P [v i ˆ i ]. Additionally for each j J there exists at least one client i j S j who is charged his myoic rice and thus the algorithm collects the full revenue ˆ ij P [ ] v ij ˆ ij. R((k + )-PR(c, A)) i m αˆ i P [v i ˆ i ] + ( α) j J (j) P [ v ij (j) ] The first term is an α-aroximation for the revenue of the first m clients. We next relate the second term to the revenue of the remaining clients and comute an aroximation factor α such that: ( α) (j) P [ v ij (j) ] α ˆ i P [v i ˆ i ] j J m+ i n and by the monotone hazard rate condi- By the definition of the sets, (j) ˆ c j tion P [ v ij (j) ] /e. Thus (j) P [ v ij j J (j) ] l k+ j l ˆ c j e = ck l ˆ ( c k ) (c )e = ˆ (c k ) (c )c l e. Using (a) m k, (b) P [v i ˆ i ] and (c) ˆ i ˆ c l for all i m + we get n i=m+ ˆ i P [v i ˆ i ] (n k)ˆ c l. When resolving the inequality ( α)ˆ (c k ) (c )c l e α(n k)ˆ we obtain α c l α = min(β, /c) yields the claim. (c k ) (c k )+e (n k)(c ) =: β. Now setting If the number of rounds is Ω(log n) and we are using the influence set from the 2-PD strategy 2-PD(0.5) in Theorem 2 we get a constant factor aroximation. Corollary 2. Assuming valuations from the SM-model with monotone hazard rates, R(((log c n) + )-PR (c, /2)) R/(4c e) for any constant c >. Proof. Consider the 2-PD strategy s from Theorem 2, i.e., 2-PD(0.5), which is a /4-aroximation of R, i.e., R(s) R/4. Using the influence set A from

12 s, i.e., randomly ocking nodes with robability /2, we get that k-pr(c, A) is equal to k-pr (c, /2). If we set k = log c n + then Theorem 8 shows that R(((log c n) + )-PR (c, /2)) R(S)/(c e). Combining the two results we get a 4e c -aroximation of R. To obtain a k-pr strategy that matches the bound of Theorem 5 one can first construct the ((log c n) + )-PR (c, /2) strategy from above. Then for the k-pr strategy one uses the same influence set for the first round and for the remaining rounds one icks the k rounds with the highest exected ayment in the strategy ((log c n) + )-PR (c, /2). 3 Now let us consider 2-PR strategies. Due to the results in Section 4, the best we can hoe for is a constant factor aroximation of R2-PR, not of R or of R 2-PD. We first show that we can restrict ourselves to aroximating the otimal strategy, as a revenue otimal strategy is within half of the revenue otimal 2-PR strategies. The roof idea is to design two strategies, one for the case that at least half of the revenue of the otimal k-pr strategy comes from the first round, and one or the case that it does not. In either case, at most half of the revenue is lost. Lemma 3. Given valuations v i from the SM-model, then R R k-pr 2 R k-pr. Thus it suffices to aroximate k-pr. We give a simle 2-round algorithm for the SM-model that is based on our -round strategy from Algorithm 3. Algorithm 5 (Strategy PR 0.5 (ε)). Let ε be in R >0.. Assign each client in V to an influence set A, s.t. each client is a member of A indeendently of the others with robability /2. Give the roduct to the clients in A for free in the first round. 2. Use Algorithm 3 to comute a ( ε)-aroximation of the otimal revenue for the given influence set A. In the analysis of Algorithm 5 we first bound the robability that the valuation of a client is larger than a fixed uniform rice. This is different from the aroach in Section 3, where it was sufficient to argue about the exected revenue we collect from a client. Then we use a technique similar to [] to show that R(PR 0.5 (ε)) is a (/8 ε)-aroximation of R 2-PR. Theorem 9. Given valuations v i rates, then R(PR 0.5 (ε)) ( 8 ε) from the SM-model with monotone hazard for every ε > 0. R 2-PR Together with Lemma 3 we then obtain that the above algorithm achieves at least /6 of R 2-PR in the SM-model with monotone hazard rates. By Theorem 5, R 2-PR is a eh n -aroximation of R 2-PD the above strategy is thus also a Θ( H n )- aroximation of R. Corollary 3. Given valuations v i from the SM-model with monotone hazard rates, then R(PR 0.5 (ε)) (/8 ε) 4H n R 2-PD (/8 ε) 6H n R for every ε > 0. 3 The authors are grateful to an anonymous reviewer for ointing this out. 2

13 The above results can be generalized to k-pr strategies. Theorem 0. Given valuations v i from the SM-model with monotone hazard rates, for any k 2 and ε > 0 there exists a olynomial-time comutable k-pr strategy s such that R(s) ( 8 ε) R k-pr and thus R(s) ( 6 ε) R k-pr. The roof of Theorem 0 exloits that (i) we only lose a constant factor when ignoring the influence between the latter rounds and that (ii) when given the influence set and an instance without influence between the other clients we can give an aroximation scheme for the otimal uniform rices and comute the corresonding round assignment. Mirrokni et al. [] study a different less general model, called concave grah model (CG). They give an algorithm that, under certain assumtions, finds an influence set A which achieves at least /2 of the revenue achieved by the otimal influence set. We extend this result to 2-PR strategies in the SM-model. The key ideas of the roof are that () if the seller charges a uniform rice sufficiently close to the otimal rice then she only loses an ε of the revenue and (2) once the seller has fixed the osted rice, under the assumtions of the theorem, the exected revenue is a submodular function of the influence set. Theorem. Let µ > 0, M v 0 and M 0 such that M M v µ. For each client, let their valuations follow the SM-model with the following additional assumtions: g i ( ) µ > 0, and ṽ i is drawn from a robability distribution F i whose robability density function f i is ositive, differentiable, non-decreasing on (0, M v ) and for all x (0, M v ), f i (x) f for some constant f, and F i (0) = 0, F i (M v ) =. Let the rice be in the interval 0 M. Then for every ε = o( V ) there is an algorithm finding a 2-PR strategy s, i.e., an influence set A and a uniform rice for the second round, such that R(s) ( 2 ε) R 2-PR. By Lemma 3, the algorithm of Theorem achieves at least (/4 ε) of R 2-PR. 6 Extensions of the Model Our models and results can be extended in several directions. First, one can consider more general classes of externalities. In the General Monotone Model (GM) the valuation v i (B) of i is drawn from a (known) distribution with the CDF F i,b such that P [v i (B) ] P [v i (B ) ] for all B B and R 0. Notice that the roofs of Theorems 5 and 8 do not exloit the fact that the valuations are from the SM-model and thus extend to the GM-model. Second, the monotone hazard rate condition can be relaxed. For all theorems, excet Theorem 3, the crucial art we use is that there is a myoic rice with a certain accetance robability. The exact aroximation bound then deends on this accetance robability. If one can guarantee a higher accetance robability than /e also the resented aroximation guarantees imrove. For instance, if one considers only uniform distributions for ṽ i then the accetance robability of 3

14 the myoic rice is /2 and the aroximation guarantee of Theorem imroves to (k )/(3k 2). Third, one can consider different classes of marketing strategies. For instance strategies where clients are slit into k grous and l rounds such that (a) each client belongs to exactly one grou and round, (b) all clients in the same grou are offered the same rice indeendent of their round and (c) only clients that have urchased the roduct in a revious round can influence the valuation functions of the clients in the current round, but this influence is indeendent of their grou. For instance, a grou could model all clients that live in the same country, referred clients, or an age grou. Our results extend to this setting as well. That is, one can get a constant factor of R using O(log n) different grous/rices, but not with o(log n) different grous/rices. References. Akhlaghour, H., Ghodsi, M., Haghanah, N., Mirrokni, V.S., Mahini, H., Nikzad, A.: Otimal iterative ricing over social networks. In: 6th WINE. LNCS, vol. 6484, (200) 2. Babaei, M., Mirzasoleiman, B., Jalili, M., Safari, M.A.: Revenue maximization in social networks through discounting. Social Network Analysis and Mining 3(4), (203) 3. Barlow, R.E., Marshall, A.W.: Bounds for distributions with monotone hazard rate, I. The Annals of Mathematical Statistics 35(3), (964) 4. Buchbinder, N., Feldman, M., Naor, J.S., Schwartz, R.: A tight linear time (/2)- aroximation for unconstrained submodular maximization. In: 53rd FOCS (202) 5. Chlebík, M., Chlebíková, J.: Aroximation hardness of edge dominating set roblems. Journal of Combinatorial Otimization (3), (2006) 6. Ehsani, S., Ghodsi, M., Khajenezhad, A., Mahini, H., Nikzad, A.: Otimal online ricing with network externalities. Information Processing Letters 2(4), 8 23 (202) 7. Fotakis, D., Siminelakis, P.: On the efficiency of influence-and-exloit strategies for revenue maximization under ositive externalities. In: 8th WINE. LNCS, vol. 7695, (202) 8. Haghanah, N., Immorlica, N., Mirrokni, V., Munagala, K.: Otimal auctions with ositive network externalities. ACM Transactions on Economics and Comutation (2), 3: 3:24 (203) 9. Hartline, J., Mirrokni, V.S., Sundararajan, M.: Otimal marketing strategies over social networks. In: 7th (2008) 0. Kann, V., Khanna, S., Lagergren, J., Panconesi, A.: On the hardness of aroximating max k-cut and its dual. Chicago Journal of Theoretical Comuter Science 997 (997). Mirrokni, V.S., Roch, S., Sundararajan, M.: On fixed-rice marketing for goods with ositive network externalities. In: 8th WINE. LNCS, vol. 7695, (202) 2. Oliver, R.L., Shor, M.: Digital redemtion of couons: satisfying and dissatisfying effects of romotion codes. Journal of Product & Brand Management 2(2), 2 34 (2003) 4

15 A Proofs of Section 3 We will make use of the following two lemmata. Lemma 4 ([9]). Consider a monotone submodular function f : 2 X R and a subset S X. Consider a random set S by choosing each element of S indeendently with robability at least β. Then E [f(s )] β f(s). Lemma 5. Given the SM-model it holds that ˆ i (S) = ˆ i (V \ {i}) g i (S) for all S V \ {i}. Proof. Note that v i (V \ {i}) = ṽ i g i (V \ {i}) = ṽ i and that ṽ i = v i (S)/g i (S). By the definition of the myoic rice ˆ i (S), we( know that [ setting = ˆ i (S) ]) maximizes the revenue P [v i (S) ] = g i (S) g P i(s) v i (V \ {i}) g i(s). Since g i (S) does not deend on, it follows that setting = ˆi(S) g maximizes i(s) P [v i (V \ {i}) ]. Thus, ˆ i (V \ {i}) = ˆi(S) g and ˆ i(s) i(s) = ˆ i (V \ {i}) g i (S). Proof of Lemma 2. For the random set S it holds that E S [π i (S)] = () E S [ˆ i (S) P [v i (S) ˆ i (S)]] = (2) E S [ˆ i (V \ {i}) g i (S)P [v i (V \ {i}) ˆ i (V \ {i})]] = (3) E S [g i (S)] ˆ i (V \ {i}) P [v i (V \ {i}) ˆ i (V \ {i})] (4) β π i (V \ {i}). (5) The equality of (3) follows from Lemma 5; the equality of (4) follows since only g i (S) deends on S (and not on ). From Lemma 4 we get E S [g i (S)] β g i (V \ {i}) = β, and thus, (5) follows. Proof of Theorem 2. We use B l to denote the clients that become active in round l and l(i) to denote the round in which client i is offered the roduct. The robability that for two given clients i j V it holds that j l<l(i) B l under [ the condition that i V is P [j V ]+P [j V ] P j ] <l<l(i) B l i, j V. [ It holds that P [j V ] = q and P [j V ] = q. P j ] <l<l(i) B l i, j V is given by the robability that l(j) < l(i) times by the robability that client j accets the offer. Furthermore P [l(j) < l(i) i, j V ] = k 2 k 2 because l(i) < l(j) and l(j) < l(i) have equal robability and with robability k 2 k it holds that i and j are not assigned to the same round. Finally, note that we offer the roduct for the myoic rice and thus, by the monotone hazard rate condition, the accetance robability of j is at least /e. Thus, β := q + ( q) k 2 k 2 e is a lower bound of the robability that j l<l(i) B l under the condition that i V. 5

16 Let S i be l<l(i) B l for all i V. It follows from Lemma 2, the definition of β, and i V π i(v \ {i}) R that R(k-PD(q)) = i V P [i V ] E [π i (S i ) i V ] i V P [i V ] β π i (V \{i}) = i V ( q) β π i(v \{i}) ( q) β R; e (k ) the maximal value of ( q) β is 4e(k ) 2k+4 and can be obtained by setting q = e (k ) 2e(k ) k+2. A. Proof of Theorem 3 For a roof of Theorem 3 we need the following lemmas. Lemma 6 ([3]). Given a distribution function F satisfying (i) the monotone hazard rate condition and (ii) F (0 ) = 0 then x log( F (x)) and [ F (x)] x are non-increasing in x. The following lemma generalizes Lemma for discounted rices. Lemma 7. Assume the SM-model with the monotone hazard rates, then P [v i (B) α ˆ i (B)] P [v i (B) ˆ i (B)] α ( e ) α. Proof. Consider the distribution F of valuation v i. Then P [v i (B) α ˆ i (B)] = F (α ˆ i (B)). By Lemma 6, ( F (α ˆ i (B))) α ( F ( i (B)). Now as P [v i (B) ˆ i (B)] = ( F ( i (B)) we obtain the first inequality. For the second inequality recall that by the monotone hazard rate condition and Lemma, P [v i (B) ˆ i (B)] /e. To analyze the strategy we again consider the exected ayment π i,α (S) = α ˆ i (S) P [v i (S) α ˆ i (S)] the seller gets from client i when giving a discount factor α. we can extract from a client i given the active clients B. Next we generalize Lemma 2 for k-pd( q, ᾱ) strategies. Lemma 8. For S V \ {i}: π i,α (S) α e α π i (S). Proof. Consider the exected ayment π i,α (S) from client i when getting a discount factor of α. Using Lemma 7 we get π i,α (S) α ˆ i (S) P [v i (S) ˆ i (S)] α and by the definition of π i (S) the latter is equal to α P [v i (S) ˆ i (S)] α π i (S). Again alying Lemma 7 results π i,α (S) α (/e) α π i (S). Proof of Theorem 3. Let k be the size of the vectors, then R(PD( q, ᾱ)) = k i V j= P [i V j] E Sl [π i,α (S j )]. The robability β j that an arbitrary client i is active in round j is given by the robability that i is in one of the revious rounds and buys the roduct. Thus β j j l= q l ( ) αl e. Next we use the resented lemmas and linearity of exectation to relate the revenue of PD( q, ᾱ) 6

17 to R. R(PD( q, ᾱ)) k [ q j E Sl αj e αj π i (S j ) ] i V j= ( k j ( ) ) αl q j q l α j e αj π i (V \ {i}) e i V j= l= ( k j ( ) ) αl q j q l α j e αj π i (V \ {i}) e j= l= i V ( k j ( ) ) αl q j q l α j e αj e R j= l= Hence to maximize the aroximation guarantee we consider ( k j ( ) ) αl q j q l α j e αj. e j= l= Certain assignments to q, ᾱ and the corresonding aroximation ratios can be found in Table. Table. Otimal values q, α for k-pd( q, ᾱ) strategies and the corresonding aroximation ratio. k q ᾱ arox 2 (0.5,0.5) (0,) (0.432,0.267,0.299) (0,0.757,) (0.399,0.87,0.99,0.23) (0,0.644,0.850,) (0.369,0.38,0.5,0.64,0.76) (0,0.5,0.707,0.866,) (0.353,0.3,0.20,0.29,0.37,0.45) (0,0.447,0.632,0.774,0.894,) (0.342,0.096,0.00,0.06, (0,0.408,0.577,0.707, 0.2,0.8,0.24) 0.86,0.92,) (0.333,0.084,0.086,0.090, (0,0.377,0.534,0.654, ,0.0992,0.03,0.08) 0.755,0.845,0.925,) (0.326,0.075,0.075,0.0783,0.08, (0,0.353,0.5,0.62,0.707, 0.085,0.088,0.092,0.095) 0.790,0.866,0.935,) (0.320,0.068,0.067,0.069,0.07, (0,0.333,0.47,0.577,0.666, 0.074, 0.077, 0.080,0.083,0.085) 0.745,0.86,0.88,0.942,) 0.3 B Proofs of Section 4 Proof of Theorem 4. () Consider the following examle of the marketing roblem. The clients are given by V = {,..., n}, and for given arameter λ 7

18 with 0 < λ <, and for each B V the valuation functions v i are given by for 0, P [v i (B) ] = λ i for 0 < ˆ i, 0 for > ˆ i, with ˆ i := ( n j=i λj ). Thus, the valuations of the clients are not influenced by the other clients, and R k-pd does not deend on k ( R k-pd = R -PD ). Note that for each client i, ˆ i is the otimal myoic rice, since for any rice with 0 < < ˆ i it holds that P [v i ] = P [v i ˆ i ], and hence P [v i ] ˆ i P [v i ˆ i ]. The revenue with rice discrimination and arameter λ is then R λ PD := n ˆ i P [v i ˆ i ] = i= n i= λ i n j=i λj. As there is no influence between clients we have that R k-pr k R -PR. Hence, we first consider R -PR. The otimal uniform rices have to be in the set {ˆ, ˆ 2,..., ˆ n } for the same reason as these are otimal myoic rices: using a rice such that ˆ i < < ˆ i does not increase the robability P [v j ] for any client j, but it does decrease the revenue since the rice is lower. The revenue for any uniform rice ˆ i {ˆ, ˆ 2,..., ˆ n } is Rˆi -PR := ˆ i n P [v j ˆ i ] = n j=i λj j= n λ j =. Thus, all the rices in the set {ˆ, ˆ 2,..., ˆ n } are otimal uniform rices, and R -PR = Rˆi -PR. The limit of the ratio R -PR for λ 0 RPD λ + is j=i lim λ 0 + R-PR R λ PD = n = λ lim i λ 0+ i= n n. j=i λj Hence, for any ε > 0 we can find a λ (0, ) and a value n 0 such that for n > n 0 it holds that k R k-pr R -PR + ε n Rλ PD = + ε n R k-pd. (2) For each i let ˆ i denote the myoic rice ˆ i ( ). The revenue for the otimal -PD strategy is given by R -PD = i V ˆ i P [v i ˆ i ]. As uniform rice we set = ˆ j such that j argmax ˆ i P [v i ˆ i ]. Hence the revenue R -PR is i V given by R -PR = i V P [v i ] max i V ˆ i P [v i ˆ i ] V R -PD. R 2-PD, consider the otimal 2-PD strategy with influ- (3) For R 2-PR n ence set A and a 2-PR strategy with the same influence set. By definition there is no revenue in the first round and in the second round we are faced with 8

19 the roblem of aroximating -PD with -PR, according to (2) we can get a /n aroximation. Finally for n R 2-PD 4n R recall that, by Corollary, we have R 2-PD R. 4 Proof of Theorem 5. () Consider the following examle of the marketing roblem with valuations from the SA-model. The clients are given by V = {,..., n} and, for each B V the valuation functions v i are given by v i (B) = /i. The total revenue with rice discrimination is then R -PD = n i= i = H n. The otimal uniform rice is going to be of the form /i: using a rice such that i < < i does not increase the number of buyers, but it does decrease the revenue since the rice is lower. The revenue R -PR (/j) for a uniform rice /j is then (/j) {i V /i /j} =, and thus, R -PR =. Moreover, as there is no influence between the clients we have that R k-pr k R -PR k H n R-PD = k H n Rk-PD. (2) Let ˆ := (ˆ, ˆ 2,..., ˆ n ) be the vector of otimal myoic rices for the clients. W.l.o.g., we assume that the rices are sorted in a descending order. Consider the -PR strategy s that selects the myoic rice ˆ i with robability i H n. Note that n i= i H n = H n n i= i =. The exected revenue from using these uniform rices is then R(s) = n i= n i= n i= i H n i H n i H n n P[v j ˆ i ] ˆ i j= i P[v j ˆ i ] ˆ i j= i j= e ˆ i = e H n n ˆ i. At the same time, the revenue with rice discrimination R -PD is n i= ˆ i P[v i ˆ i ] n i= ˆ i. Therefore, we get R -PR R(s) e H R-PD n. (3) Consider the otimal 2-PD strategy with influence set A and a 2-PR strategy with the same influence set. By definition there is no revenue in the first round and in the second round we are faced with the roblem of aroximating -PD with -PR, according to (2) we can get a e H n aroximation. i= C Proofs of Section 5 Proof of Theorem 6. Consider the otimal uniform rice. Given the monotone hazard rate condition it is easy to show that ˆ max. We next show that also ˆmax e n. Towards a contradiction assume that not. Then the revenue is 9

20 bounded by n < ˆmax e. But using ˆ max as unique rice gives at least a revenue ˆ max e, a contradiction to the otimality of. Given the above bounds we know that Algorithm 3 considers a rice j such that j ( ε). Let R( ) denote the revenue achieved by the uniform rice then: R( j ) = i V j P [v i ( ) j ] i V ( ε) P [v i ( ) ] = ( ε) R() Hence, Algorithm 3 gives a ( ε) of R -PR. Lemma 9 (Maximum dominated set). The Maximum dominated set roblem, i.e., comuting a maximum set S V s.t. for each b S there is an a V \ S with (a, b) E, is APX-hard (not aroximable within 259/260). Proof. It is known that the Minimum dominating set roblem for grahs with bounded degree is APX-hard [5]. Let us assume that for each ε > 0 there is an ( ε)-aroximation algorithm for the Maximum dominated set roblem. Consider an arbitrary grah of degree at most with Ot M being a minimum dominating set and Ot S = V \ Ot M being the corresonding maximum dominated set. Further let A S be the dominated set returned by the aroximation algorithm and A M = V \ A S then A M = n A S n ( ε) Ot S = n Ot S + ε Ot S = Ot M + ε Ot S. Now as the grah has degree clearly Ot S Ot M and thus A M ( + ε) Ot M. With ε = δ/, we would obtain ( + δ)-aroximation algorithms for Minimum Dominating Set, contradicting the APX-hardness. Using a result from [5], showing that Minimum dominating set on grahs with degree 5 can not be aroximated within 53/52, we obtain the hardness of the Maximum dominated set roblem for constants < 259/260. Proof of Theorem 7. This is by a reduction from the Maximum dominated set roblem. For an instance G = (V, E) of Maximum dominated set, consider a social network with the DS-model such that for all clients the valuations v i (B) are if B N i and 0 otherwise, where N i denotes the neighbors of i in G. Now as we can extract at most revenue from each client the otimal uniform rice in this setting is always =. Thus maximizing the total revenue is equivalent to comuting a maximum dominated set which is APX-hard. Finally observe that the valuation functions v i can also be modeled in the convex grah models by a convex function f(x) = x for x < and f(x) = for x and deterministic edge weights w i,j = for (i, j) E. Proof of Lemma 3. Consider a k-pr strategy s maximizing the sellers revenue, and let V be the set of clients visited in the first round. Let R be the exected revenue from the clients in V and R 2 the exected revenue of the clients in V \ V. If R R 2 then consider the k-round strategy s which coincides with s excet that the uniform rice in the first round is zero. Clearly 20