Competitive Diffusion in Social Networks: Quality or Seeding?

Transcription

1 Competitive Diffusion in Socia Networks: Quaity or Seeding? Arastoo Fazei Amir Ajorou Ai Jadbabaie arxiv: v1 [cs.gt] 4 Mar 2015 Abstract In this paper, we study a strategic mode of marketing and product consumption in socia networks. We consider two firms in a market competing to maximize the consumption of their products. Firms have a imited budget which can be either invested on the quaity of the product or spent on initia seeding in the network in order to better faciitate spread of the product. After the decision of firms, agents choose their consumptions foowing a myopic best response dynamics which resuts in a oca, inear update for their consumption decision. We characterize the unique Nash equiibrium of the game between firms and study the effect of the budgets as we as the network structure on the optima aocation. We show that at the equiibrium, firms invest more budget on quaity when their budgets are cose to each other. However, as the gap between budgets widens, competition in quaities becomes ess effective and firms spend more of their budget on seeding. We aso show that given equa budget of firms, if seeding budget is nonzero for a baanced graph, it wi aso be nonzero for any other graph, and if seeding budget is zero for a star graph it wi be zero for any other graph as we. As a practica extension, we then consider a case where products have some preset quaities that can be ony improved marginay. At some point in time, firms earn about the network structure and decide to utiize a imited budget to mount their market share by either improving the quaity or new seeding some agents to incine consumers towards their products. We show that the optima budget aocation in this case simpifies to a threshod strategy. Interestingy, we derive simiar resuts to that of the origina probem, in which preset quaities simuate the roe that budgets had in the origina setup. Department of Eectrica and Systems Engineering and GRASP Laboratory at University of Pennsyvania. arastoo@seas.upenn.edu, ajorou@seas.upenn.edu and jadbabai@seas.upenn.edu. This research was supported by ARO MURI W911NF and AFOSR Compex Networks Program. A preiminary version of this paper has appeared in [1].

2 I. INTRODUCTION Many recent studies have documented the roe of socia networks in individua purchasing decisions [2] [4]. More data from onine socia networks and advances in information technoogies have drawn the attention of firms to expoit this information for their marketing goas. As a resut, firms have become more interested in modes of infuence spread in socia networks in order to improve their marketing strategies. In particuar, considering the reationship between peope in socia networks and their rationa choices, many retaiers are interested to know how to use the information about the dynamics of the spread in order to maximize their product consumption and achieve the most profit in a competitive market. A main feature of product consumption in these settings is what is often caed the network effect or positive externaity. For such products, consumption of each agent incentivizes the neighboring agents to consume more as we, as the consumption decisions between agents and their neighbors are strategic compements of each other. There are diverse sets of exampes for such products or services. New technoogies and innovations, mobie appications (e.g., Viber, WhatsApp), onine games (e.g., Warcraft), socia network web sites (e.g., Facebook, Twitter) and onine dating services (e.g., Zoosk, Match.com, OkCupid) are among many exampes in which consuming from a common product or service is more preferabe for peope. Aso, a main property of many products is substitution. A substitute product is a product or service that satisfies the need of a consumer that another product or service fufis (e.g. Viber and WhatsApp or Gmai and Yahoo emai accounts). In a these exampes, firms might be interested to utiize the socia network among consumers and the positive externaity of their products and services to incentivize a arger consumption of their products compared to riva substitute products. Therefore, it is important for firms to know how to shape their strategies in designing their products and offering them to a set of peope in order to promote their products inteigenty, and eventuay achieving a arger share in the market. In this paper, we study strategic competition between two firms trying to maximize their product consumption. Firms simutaneousy aocate their fixed budgets between seeding a set of costumers embedded in a socia network and designing the quaity of their products. The

3 consumption of each agent is the resut of its myopic best response to the previous actions of its peers in the network. Therefore, a firm shoud provide enough incentives for spread of its product through the payoff that agents receive by consuming it. For this purpose and considering their budgets, firms shoud strategicay design their products and know how to initiay seed the network. We mode the above probem as a fixed-sum game between firms, where each firm tries to maximize discounted sum of its product consumption over time, considering its fixed budget. We describe the unique Nash equiibrium of the game between firms which depends on the budgets of the firms and the network structure. We show that at the Nash equiibrium, firms spend more budget on quaity when their budgets are cose. However, as the difference between budgets increases, firms spend more budget on seeding. We aso show that given equa budget of firms, if seeding budget is nonzero for a baanced graph it wi aso be nonzero for any other graph, and if seeding budget is zero for a star graph it wi be zero for any other graph too. Next, we study a different scenario in which firms produce products with some preset quaities. At some point in time, firms earn about the network structure and dedicate some budget to increase their product consumption. The budget can be spent on new seeding of agents in the socia network and marginay improving the quaity of the products. We derive a simpe rue for optima aocation of the budget between improving the quaity and new seeding which in particuar depends on the network structure and preset quaities of the products. We show that the optima aocation of the budget depends on the entire centraity distribution of the graph. Speciay, we show that maximum seeding occurs in a graph with maximum number of agents with centraities above a certain threshod. Aso, the difference in quaities of firms pays an important roe in the optima aocation of the budget. In particuar, we show that as the gap between the quaities of the products widens, the firms aocate more budget to seeding. We see that the budgets in the first scenario and preset quaities of the second scenario pay simiar roes in the optima aocation. It is worthwhie to note that the probem of infuence and spread in networks has been extensivey studied in the past few years [5] [11]. Aso, diffusion of new behaviors and strategies

4 through oca coordination games has been an active fied of research [12] [19]. Goya and Kearns proposed a game theoretic mode of product adoption in [20]. They computed upper bounds of the price of anarchy and showed how network structure may ampify the initia budget differences. Simiary, in [21] Bimpikis, Ozdagar and Yidiz proposed a game theoretic mode of competition between firms which can target their marketing budgets to individuas embedded in a socia network. They provided conditions under which it is optima for the firms to asymmetricay target a subset of the individuas. Aso, Chasparis and Shamma assumed a dynamica mode of preferences in [10] and computed optima poicies for finite and infinite horizon where endogenous network infuences, competition between two firms and uncertainties in the network mode were studied. The main contribution of our work is to expicity study the tradeoff between investing on quaity of a product and initia seeding in a socia network. Our mode is simiar to the mode proposed in [22], however, instead of pricing strategy in [22], the notion of quaity is introduced and the tradeoff between quaity and seeding is studied. Aso, our mode is tractabe and aows us to characterize the exact product consumption at each time, instead of ower and upper bounds provided in [23], [24]. II. THE SPREAD DYNAMICS There are n agents V = {1,..., n} in a socia network. The reationship among agents is represented by a directed graph G = (V, E) in which agents i, j V are neighbors if (i, j) E. The weighted adjacency matrix of the graph G is denoted by a row stochastic matrix G where the ij-th entry of G, denoted by g ij, represents the strength of the infuence of agent j on i. For diagona eements of matrix G, we have g ii = 0 for a agents i V. We assume that there are two competing firms a and b producing product a and b. Each agent has one unit demand which can be suppied by either of the firms. We define the variabe 0 x i (t) 1 and 0 1 x i (t) 1 as the consumption of the product a and b by agent i at time t. Denote by q a, q b ɛ > 0 the quaity of product a and b respectivey, where ɛ has an infinitesima vaue. The vaues of q a and q b can be interpreted as the payoff that any two agents i and j woud achieve if they both consume the same product. In other words, we can assume

5 q a and q b are payoffs of the foowing game x j 1 x j x i q a x i x j 0 1 x i 0 q b (1 x i )(1 x j ) Since agents benefit from the same action of their neighbors, this game coud be thought of as a oca coordination game. From the above tabe it foows easiy that the payoff of agents i and j from their interaction is u ij (x i, x j ) = q a x i x j + q b (1 x i )(1 x j ). We aso assume that each agent benefits from taking action x i irrespective of actions taken by its neighbors. We assume the isoation payoff of consuming x i and 1 x i from product a and b is represented by the foowing quadratic form functions u a ii = q a (αx i βx 2 i ), u b ii = q b [α(1 x i ) β(1 x i ) 2 ], where α and β are parameters of the isoation payoff. This forms of payoff indicates that a product with higher quaity has a higher isoation payoff. The tota isoation payoff of agent i can be written as u ii (x i ) = {q a (αx i βx 2 i )} + {q b ( α(1 xi ) β(1 x i ) 2) }. In order to have nonnegative isoation payoff for x i = 0 and x i = 1, we assume β α. Assuming quadratic form function for the isoation payoff not ony makes the anaysis more tractabe, but aso is a good second order approximation for the genera cass of concave payoff functions. By changing the variabes x i = 1 + y 2 i after simpification we get u ij (y i, y j ) = q a ( y i)( y j) + q b ( 1 2 y i)( 1 2 y j), u ii (y i ) = (q a + q b )( α 2 β 4 βy2 i ) + (q a q b )(α β)y i.

6 Therefore, the tota utiity of agent i from taking action y i is given by U i (y i, y i ) = (q a + q b )( α 2 β 4 βy2 i ) + (q a q b )(α β)y i n + q a g ij ( y i)( 1 n 2 + y j) + q b g ij ( 1 2 y i)( 1 2 y j). j=1 j=1 (1) In the above equation y i denotes an action vector of a agents other than agent i. From equation (1) we can see that product a and b have a positive externaity effect in the network, meaning that the usage eve of an agent has a positive impact on the usage eve of its neighbors. Therefore, it foows that q a and q b in addition to the payoff of a oca coordination game, can be interpreted as coefficients of network externaity of product a and b respectivey. We assume agents repeatedy appy myopic best response to the actions of their neighbors. This means that each agent, considering its neighbors consumptions at the current period, chooses the amount of the product that maximizes its current payoff, as its consumption for the next period. In other words, consumption of agent i at time t + 1 is updated as foows y i (t + 1) = arg max y i U i (y i (t), y i (t)). The above equation resuts in the foowing update dynamics y i (t + 1) = ( 1 n 2β ) q a q b n g ij y j (t) + ( 4β(q a + q b ) ) g ij + ( (α β)(q a q b ) ). 2β(q a + q b ) j=1 Therefore, the consumption of the product a can be written as the foowing inear update j=1 dynamics form y(t + 1) = ( 1 2β )G y(t) + ((1 + 2(α β))(q a q b ) ) 1. (2) 4β(q a + q b ) Simiary, for the consumption of the product b we have 1 x i (t) = 1 2 y i(t). Assumption 1: We assume 1 + α 2β. This assumption guaranties that 0 x i (t) 1 for a i and a t under the update rue (2).

7 Using the above assumption and defining W ( 1 2β )G, equations (2) can be written as u a ( 1 + 2(α β) ( )( q ) a q b ) 1, (3) 4β q a + q b y(t + 1) = W y(t) + u a. The above equation can be expanded as t 1 y(t) = W t y(0) + W k u a. (4) Therefore, the consumption of agents depends on the initia preferences, i.e. y(0), the quaity of product a and b, i.e. q a and q b, and the structure of the network, i.e. the matrix G. In the next section we discuss how firms can expoit this information in order to maximize their product consumption and aso characterize the unique Nash equiibrium of the game payed between two firms. k=0 III. OPTIMAL BUDGET ALLOCATION In this section we describe the game between firms where each firm aims to maximize the consumption of its product over an infinite time horizon given a fixed budget. Each firm has an initia budget that it can either invest on quaity or spend it on promoting its product by seeding some of the agents, or both. This initia seeding can be viewed as free offers to promote the products in the network. We define the utiity of each firm as the discounted sum of its product consumption over time U a = U b = δ t 1 T ((0.5) 1 + y(t)), t=0 δ t 1 T ((0.5) 1 y(t)). t=0 Each firm has a imited budget K a, K b that can spend on either initia seeding, i.e. S a and S b, or designing the quaity of its product, i.e. q a and q b, or both. Seeding S a and S b wi change

8 the initia consumption of products a and b by S a S b and S b S a respectivey. Therefore, the amount that agents initiay consume from product a and b wi be x(0) = (0.5) 1 + S a S b and 1 x(0) = (0.5) 1 + S b S a. This means that if both firms seed an agent equay then the agent has no preference for one product over the other, i.e. y(0) = 0. This assumption can be justified since agents shoud be initiay indifferent between products before their consumption and reaizing the quaity of products if initia seedings by firms are equa. In order to have 0 x i (0) 1 and 0 1 x i (0) 1 for a agents i, we impose the constraints S a 0.5 and S b 0.5. This means that firms can initiay seed agents up to their demand capacity which is 0.5 for a agents. Using equations (3) and (4) and defining the centraity vector v by v = (I δw T ) 1 1 where agents are ordered from the highest to the owest centraity, i.e. v 1 v 2 v n, and noting that v i = 2βn, the utiities of firms can be written as 2β δ n U a = ( 2(1 δ) ) + vt Sa v T Sb + λ( q a q b ), q a + q b n U b = ( 2(1 δ) ) + vt Sb v T Sa + λ( q b q a ), q a + q b (5) where λ = δ(1 + 2(α β))n 2(1 δ)(2β δ). (6) We assume the cost of each unit of quaity is given by c q and the cost of each unit of initia seeding is given by c s. Therefore, the game between the firms can be written as max ( S a,q a s.t. c s S a 1 + c q q a = K a, for firm a, and max S b,q b ( n 2(1 δ) ) + vt Sa v T Sb + λ( q a q b ), q a + q b n 2(1 δ) ) + vt Sb v T Sa + λ( q b q a ), q a + q b s.t. c s S b 1 + c q q b = K b,

9 for firm b. Since the effect of the action of S b is decouped from S a in U a, therefore, the optimization probem of firm a is equivaent to max S a,q a v T Sa + λ( q a q b q a + q b ), s.t. c s S a 1 + c q q a = K a. Simiary, for firm b we have max S b,q b v T Sb + λ( q b q a q a + q b ), s.t. c s S b 1 + c q q b = K b. Remark 1: It can be easiy shown that for a seeding budget S a 1, the optima seeding strategy is to seed the agents in the order of their centraities (from highest to owest) unti we either ran out of budget or a the agents are seeded. Therefore, an optima action ( S a, q a ) is fuy determined from ( S a 1, q a ), thus reducing the action space of firm a to ony q a, given its budget constraint. Simiar argument hods for firm b. Therefore, we may ook at the utiities U a and U b as functions of (q a, q b ) under the optima seeding and fixed budgets. In order to study the existence and uniqueness of the Nash equiibrium for the above game, we use a variation of the we-known Sion s minimax theorem (see [25] for the origina Sion s theorem) as beow. Lemma 1: Consider a two person zero-sum game, on cosed, bounded, and convex finitedimensiona action sets Ω 1 Ω 2, defined by the continuous function L(x 1, x 2 ). Let L(x 1, x 2 ) be stricty convex in x 1 for each x 2 Ω 2 and stricty concave in x 2 for each x 1 Ω 1. Then the game admits a unique pure strategy Nash equiibrium. Proof: See Theorem A.4 on page 286 in [26]. In the foowing theorem we characterize the Nash equiibrium of the game payed between firms. Theorem 1: Consider firms a and b with utiity functions U a and U b as described in (5). The

10 game between firms admits a unique Nash equiibrium of form ṽ qa = (2λ)( c s )( c q (ṽ k + ṽ ) ), 2 q b = (2λ)( c s )( c q (ṽ k + ṽ ) ), i < k, i <, Sa i = K a c s k 1 2λṽ 2 (ṽ k +ṽ i = k, ) 2 0 i > k, ṽ k Sb i = K b c s 1 2λṽ k 2 (ṽ k +ṽ i =, ) 2 0 i >, (7) for some v k ṽ k v k 1 and v ṽ v 1 that satisfy 0 Sa k = K a k 1 2λṽ c s 2 (ṽ k + ṽ ) < 1 2 2, 0 Sb = K b 1 2λṽ k c s 2 (ṽ k + ṽ ) < 1 2 2, (8) where ṽ k = v k if S a k > 0 and ṽ = v if S a > 0. 1 Proof: Given the optima seeding of each firm, i.e. seeding agents from the highest to the owest centraity, as discussed in Remark 1, the tradeoff between seeding amount and quaity can be soved by optimizing U a and U b with respect to q a and q b respectivey. The action space of firms, i.e. ɛ q a Ka c q set. Aso, U a + U b = and ɛ q b K b c q, is a cosed, bounded, and convex finite-dimensiona n, hence, the game is a fixed-sum game and can be transformed to (1 δ) a zero sum game by subtracting the constant vaue of n 2(1 δ) from U a and U b. The term v T Sa in U a is piecewise inear in S a 1 and thus in q a, under optima seeding. Using this, it is easy to see that U a (q a, q b ) is stricty concave in q a for each q b, and stricty convex in q b for each q a via a simiar argument. Therefore, based on Lemma 1, the game admits a unique Nash equiibrium. Assume that the first (k 1) and ( 1) agents are fuy seeded by firms a and b respectivey at equiibrium. Then, from the budget constraints we have S a k = Ka c s and S b = K b c s k 1 2 ( cq c s )q a, 1 2 ( cq c s )q b, therefore, by pugging in the vector of optima seeding S a and S b 1 We define v 0. If Sa n = 1 or 2 S b n = 1, then ṽn vn. 2

11 as described earier, the optimization probem of firms is given by max ɛ q a Ka cq max ɛ q b K b cq ( 1 k 1 2 ) v i + ( K a k 1 ( c q )q a )v k + λ( q a q b ), c s 2 c s q a + q b i=1 ( ) v i + ( K b 1 ( c q )q b )v + λ( q b q a ). c s 2 c s q a + q b i=1 If 0 < Sa k < 1 and 0 < 2 S b < 1, the first order optimaity condition requires taking the derivative 2 of the two equations above with respect to q a and q b and setting them to zero Soving equations above we get ( c q 2λq b )v k + ( c s (q a + q b ) ) = 0, 2 ( c q 2λq a )v + ( c s (q a + q b ) ) = 0. 2 v qa = (2λ)( c s )( c q (v k + v ) ), 2 v k qb = (2λ)( c s )( c q (v k + v ) ), 2 where 1 k, n are integers that must satisfy conditions in (8) for ṽ k = v k and ṽ = v. If S a k = 0 and S b = 0, the first order optimaity condition is as foows v k ṽ k v k 1, v ṽ v 1, (9) where q b ṽ k = (2λ)( c s )( c q (q a + q b ) ), ṽ 2 = (2λ)( c s )( ), (10) c q (q a + q b ) 2 q a and if S a n = 1 2 or S b n = 1 2 then ṽ n v n. We can sove q a and q b in terms of ṽ k and ṽ as described in (7). Coroary 1: If firms have equa budgets K a = K b = K, then in the unique symmetric Nash

12 equiibrium of the game between firms we have q a = q b = ( λ 2 )(c s c q )( 1 ṽ ), S a i = S b i = for some v ṽ v 1 that satisfy 1 1 i <, 2 K c s 1 λ 2 2ṽ i =, 0 i >, (11) where ṽ = v if S a = S b > S a = S b = K c s 1 2 λ 2ṽ < 1 2, (12) Equation (7) indicates that the Nash equiibrium depends on both the budgets of the firms, i.e. K a and K b, centraity distribution of agents in the network, i.e. v. We wi discuss the effect of each of these factors on the Nash equiibrium in the foowing subsections. A of our anaysis here is for firm a and simiar resuts can be shown for firm b as we. For simpicity, we ony discuss seeding budget; quaity budget can be found easiy using the budget constraint. A. Effect of Budget of Firms on Firms Decisions: In this subsection we study how the budget of each firm, i.e. K a and K b, can infuence the Nash equiibrium. As it can be seen from (7), the Nash equiibrium depends on both ṽ k and ṽ, which in turn depend on firm s and its riva s budgets, i.e. both K a and K b. In the first proposition, we compare the seeding budget and quaity of two firms at the Nash equiibrium with respect to their budgets. We first prove the foowing emma. Lemma 2: At the Nash equiibrium, if q a < q b, then S a 1 S b 1. Proof: If q a < q b, then from (7), we have ṽ < ṽ k. If ṽ < ṽ k then either k < or = k. If k < it is obvious to see that S a S b. If = k then we have two cases: If 0 < S a k < 1 2, then based on (9) we have ṽ k = v k = v ṽ which is a contradiction with ṽ < ṽ k. If S a k = 0, then obviousy S a k S b and therefore, S a S b. If S a n = 1 2, then ṽ < ṽ k = ṽ n v n, hence, Sb n = 1. This finishes the proof. 2 2 We define v 0. If Sa n = Sb n = 1, then ṽn vn. 2

13 The next proposition states that the firm with higher budget surpasses the riva in both quaity and seeding. Proposition 1: The firm with higher budget has higher seeding budget and quaity, i.e. if K b K a, then S b 1 S a 1 and qb q a. Proof: Suppose that qa < qb. From Lemma 2 we have Sa 1 S b 1, which contradicts with K b K a. Aso, suppose S a 1 < S b 1, then from Lemma 2 we have qa qb, which contradicts with K b K a. This competes the proof. In the next proposition we expain how the seeding budget and quaity at the Nash equiibrium vary with K a and K b. Proposition 2: Given a fixed graph, the optima seeding Sa 1 and quaity qa at the Nash equiibrium are increasing functions of K a. Furthermore, Sa 1 is a decreasing function of K b if K b K a and an increasing function of K b if K a K b. Proof: First note that Sa 1, Sb 1, qa, qb (and as a resut ṽ k and ṽ ) are continuous functions of K a and K b. To see this, et B(q a, q b, K a, K b ) denote the best response of the firms to quaities (q a, q b ) when the budgets are (K a, K b ). It foows from the continuity of the best response and compactness of action spaces that the set {(qa, qb ) B(q a, qb, K a, K b ) = (qa, qb )}, that is the equiibrium space, is cosed. This impies that the graphs of the functions qa(k a, K b ) and qb (K a, K b ) are cosed and thus are continuous. Now, if 0 < S a k < 1 2, then ṽ k = v k. If K a marginay increases, then, using the continuity of the equiibrium, the eve k and as a resut ṽ k does not change. Thus, given fixed K b, the constraint for firm b in (8) and hence ṽ does not change. Therefore, if K a marginay increases, from the Nash equiibrium in (7), qa does not change and Sa k marginay increases. If Sa k = 0 and v k < ṽ k < v k 1, and K a marginay increases, from the continuity of Nash we sti have v k < ṽ k < v k 1 and as a resut S a k = 0 does not change and hence, q a marginay increases. If Sa k = 0 and ṽ k = v k or ṽ k = v k 1, and K a marginay increases, either we have v k < ṽ k < v k 1 which means Sa k = 0 does not change and qa marginay increases, or ṽ k does not change. In this atter case, given the fixed budget K b, the constraint for firm b in (8) wi remain unchanged and hence ṽ wi not change. Therefore, from the Nash equiibrium in (7), qa does not change and

14 as a resut Sa k marginay increases. It is to be noted here that the cases where ṽ k moves above v k 1 or beow v k are not feasibe as they wi cause a jump in the seeding budget, contradicting the continuity of equiibrium. The anaysis for the case when Sa n = 1 and ṽ 2 n v n is quite simiar. Therefore, S a 1 K a 0 and q a K a 0. For the second part of the proposition, if S a k = 0 and v k < ṽ k < v k 1 and K b marginay increases, from continuity of ṽ k we sti have v k < ṽ k < v k 1, and therefore, S a k = 0 and given the fixed K a, q a does not change. Hence, we ony need to consider the case where 0 < S a k < 1 2 and ṽ k = v k, or S a k = 0 and ṽ k = v k or ṽ k = v k 1. In these cases, it is easy to see that either ṽ k or S a k remains unchanged. In the atter case, (given the fixed K a ) q a does not change. Simiar argument hods for when S a n = 1 2 and ṽ n v n. Therefore, we ony need to consider the case where ṽ k does not change. From the first part of the proposition, q b is an increasing function of K b. Aso, from Proposition 1, if K b K a (K a K b ), then q b q a (q a q b ). Therefore, if K b K a (K a K b ) and K b marginay increases, equations (10) impies that q a must marginay increase (decrease) or does not change so that ṽ k remains fixed. Hence, given constant K a, S a k marginay decreases (increases) or does not change. Therefore, for K b K a and S a 1 K b 0, for K a K b. S a 1 K b 0, Proposition 2 impies that when K b K a, the higher the budget of the riva firm, the ower the seeding budget of firm a, i.e., if K b K b K a then, S a(k b ) 1 S a(k b ) 1. On the other hand, when competing with a firm which has a higher budget, i.e. K a K b, the higher the budget of the riva firm, the higher firm a shoud spend on seeding. In other words, if K a K b K b then, Sa(K b ) 1 S a(k b ) 1. Combining these two resuts, we can see that given a fixed vaue of K a, the seeding budget of firm a is increasing with the difference K a K b. The seeding budget attains its minimum when K b = K a, impying that the firm shoud aocate more budget to quaity to distance itsef from the riva firm. However, as the gap between budget widens, competition in quaities becomes ess effective and firms spend more budget on seeding.

15 B. Effect of Network Structure on Firms Decisions In this subsection we study the effect of network structure on the Nash equiibrium. Since we aready studied the effect of the budget on the Nash equiibrium, for the rest of this subsection we assume K a = K b = K so that we can observe ony the effect of the network structure. We first focus on two we studied graphs, i.e. star and baanced graphs, and highight how they can refect important properties of the seeding budget. Before continuing further, we first formay define these two graphs and find their network centraities in the next emma. Definition 1: A star graph is a directed graph in which there is an edge from any noncentra agent i V {j} to the centra agent j with the weight g ij = 1 and there are edges from the centra agent j to a noncentra agents i V {j} such that i g ji = 1. Definition 2: A baanced graph is a directed graph in which the in-degree of each agent is equa to its out-degree, i.e. j g ji = j g ij = 1. Lemma 3: The centraity of the agents in a baanced graph is given by v = graph, the centraity of the centra agent is and the centraity of non centra agents is vh s = 1 + δ(n 1) 2β 1 (, δ 2β )2 v s = ( δ Moreover, for any arbitrary graph G, v v 1 v s h. δ 2β(n 1). 2β )2 2β. In a star 2β δ Proof: First part simpy foows from the fact that v = (I δw T ) 1 1, where W is given by (3), and that for any agent i in a baanced graph g ji = g ij = 1. For the star graph, noting that v = 1 + δw T v, we can obtain vh s = 1 + δ(n 1)vs, 2β v s δvh s = 1 + 2β(n 1), soving which we can find v s h and vs as given in the emma.

16 Aso, for any arbitrary graph G, v 1 v i n = v. To show v 1 v s h, using v = 1 + δw T v for a j 1 we can obtain This yieds n j=1 v j 1 + ( δ 2β )g 1jv 1. v j (n 1) + (1 + δ 2β )v 1. Appying simpe agebra aong with the fact that v j = 2βn 2β δ eads to v 1 v s h. The next proposition provides a condition for seeding profitabiity of any genera graph. Aso, the seeding budget of star and baanced graphs are compared and it is shown that the graph with higher seeding budget can be any of the two, depending on the budget. Proposition 3: If seeding budget is nonzero for a baanced graph, it wi be nonzero for any other graph too. On the other hand, if seeding budget is zero for a star graph, it wi aso be zero for any other graph. Moreover, if λ 2 v < K c s seeding budget than a star graph, and if λ 2v s h < n + λ, a baanced graph has a arger 2 2v s < K c s < 1 + λ, a star graph has a arger seeding 2 2 v budget than a baanced graph. For n + λ < K 2 2v s c s < n + λ they have the same seeding budget. 2 2 Proof: If seeding budget is nonzero for a baanced graph, then according to (12) we have λ < K 2 v c s. As a resut, for any other graph we wi have λ 2v 1 < K c s, since according to Lemma 3 v v 1. This means that there exists at east one agent that must be seeded. On the other hand, if seeding budget is zero for a star graph, then we must have K c s for any agent i of any arbitrary graph, therefore, K c s other graph. λ 2v i λ. Since we know v s 2vh s h v i and no agent can be seeded in any For the second part of the proposition, denote quaity and seeding budget of baanced and star graphs by q r, S r 1 and q s, S s 1 respectivey. If 1 + λ < K 2 2 v c s < n + λ, then seeding budget is 2 2v s nonzero for baanced graph and hence, q r = ( cs c q ) λ. This impies S 2 v r 1 = K c s λ > 1. For star 2 v 2 graph λ 2 v < K c s < n 2 + λ 2v s impies λ 2v s h < K c s. Therefore, the centra agent in star graph must be seeded, i.e. S a1 = 1 2. If S a 2 = 0, then S s 1 = 1 2 and ceary S s 1 < S r 1. If S a2 > 0, then a noncentra agent must be seeded and we must have q s = ( cs c q ) λ. This impies q 2v s r < q s

17 and as a resut S s 1 < S r 1. If λ 2v s h S s 1 > 0 since < K c s < λ 2 v, then we have two cases. If K c s λ 2 v then S r 1 = 0. On the other hand λ 2v s h < K c s. Therefore, ceary S r 1 < S s 1. So et s assume λ 2 v < K c s < λ 2 v. This impies seeding budget is nonzero for baanced graph and hence, q r = ( cs c q ) λ. As a resut, 2 v S r 1 = K c s λ < 1. Now again consider two cases. If K 2 v 2 c s < 1 + λ, then q 2 2vh s s = ( cs c q ) λ, and 2vh s hence q s < q r. This impies S r 1 < S s 1. Otherwise, if λ 2v s h resut, again we have S r 1 < 1 2 S s 1. K c s then S s As a If n 2 + λ 2v s < K c s < n + λ, then a agents in star graph are seeded up to agents maximum 2 2 demand capacities which is 0.5 for each agent. Aso, since v s < v, we have n 2 + λ 2 v < K c s. Hence, a agents in baanced graph are aso seeded up to agents maximum demand capacities. Therefore, both graphs have the same seeding budget. This competes the proof. The next proposition provides us with a ower and an upper bound for minimum and maximum seeding budget. In order to characterize the graphs with maximum and minimum seedings for a given budget K, we need to introduce a few notations first. Definition 3: Define v max = max v, i.e. the maximum of the -th centraity v among a possibe graphs subject to v i = 2βn. We can see that vmax 2β δ 1 = vh s and v max = nδ + 1, (13) (2β δ) for 2. Simiary, define v min = min v, i.e. the minimum of the -th centraity v among a possibe graphs subject to v i = 2βn. It is easy to see that vmin 2β δ 1 = v, v2 min = v s, and vmin = 1 for 3. Proposition 4: Let (, ṽ max ) be the unique pair satisfying condition (12) where v max ṽ max v max 1 and if 0 < S a in (12), then ṽ max = v max. The maximum seeding budget occurs in any graph for which ṽ = ṽ max. An exampe for such a graph is an -star graph if ṽ max and an ( 1)-star graph if ṽ max > v max = v max. 3 Simiary, et (, ṽ min ) be the unique pair satisfying condition (12) where v min ṽ min v 1 min and if 0 < Sa in (12), then ṽ min = v min. The 3ṽmax n vn max if Sa n = 1. 2

18 minimum seeding budget occurs in any graph for which ṽ = ṽ min. An exampe for such graphs is the baanced graph for = 1, the star graph for = 2, and any graph with n 2 agents with centraity of one for 3. 4 Proof: Let G be a graph attaining the maximum seeding (thus the minimum quaity) and denote its corresponding equiibrium with (, ṽ ). Note that, since in a graph with ṽ = ṽ max the first ( 1) agents are fuy seeded. Now, if >, then from ṽ v max 1 vmax v max ṽ max it foows that ṽ ṽ max and. But, then both pairs (, ṽ ) and (, ṽ max ) cannot satisfy (12). Therefore, in a graph with maximum seeding we shoud have =. Now, if ṽ max < ṽ, then v max < ṽ v max 1 proof, we aso need to show that ṽ = ṽ max that for ṽ max = v max ( 1)-star graph with v 1 =... = v 1 = v max 1 the minimum seeding budget is simiar., which contradicts the uniqueness of the pair (, ṽmax ). To compete the is achievabe. It is quite straightforward to show an -star graph with v 1 =... = v = v max, and for ṽ max > v max admit (, ṽmax) as the equiibrium. The proof for Exampe 1: As a numerica exampe for the minimum and maximum seeding budgets, we consider a network with n = 15 agents with budget K = 2, quaity and seeding costs of c s = c q = 1 and parameters of α = β = 1 and δ = 0.5. For this exampe from equations (6) we have λ = 5. As a resut, we can see that for = 3 and v max 3 = < S a 3 = 1 16 < star with the seeding budget of v = 4 3, condition 0 < S a 1 an from (13), condition in (12) is satisfied. Therefore, a graph with the maximum seeding budget is a = 1 8 < 1 2 as iustrated in Fig. 1. Aso, we can see that for = 1 and in (12) is satisfied. Thus, baanced graph has the minimum seeding budget of 1 8. Given vs h = 4.8 and vs = 1.08, it can be seen that in star graph ṽ 2 = 5 3 and star graph has a seeding budget of 0.5 which is neither a minimum nor a maximum. As we saw, the structure of the graphs with minimum and maximum seeding budget depends on the budget. However, for certain vaues of budget K the seeding budget wi be independent of the structure of the graph, as described in the next proposition. Proposition 5: If K c s < λ 2v s h no graph can be seeded. On the other hand, if K c s > n 2 + λ 2 a 4ṽmin n vn min if Sa n = 1. 2

19 Fig. 1. A graph with maximum seeding budget graphs can be seeded up to agents maximum demand capacities. Proof: The maximum possibe centraity happens for the centra agent of the star graph as shown in Lemma 3. As a resut, if K c s < λ, then we have K 2vh s c s < λ 2v i for a i in any graph and from condition (12) no agent can be seeded. Aso, since from definition 1 v i for a i, if K c s > λ 2 then we have K c s > n 2 + λ 2v i for a agents in any graph, and any graph can be seeded up to agents maximum demand capacities which is 0.5 for each agent. IV. SEEDING VERSUS QUALITY IMPROVEMENT In this section we describe a scenario in which firms have aready produced their products with some preset quaity. We assume at some point in time, say t = 0, firms earn about the network structure and utiize a fixed budget to maximize their margina utiity by either marginay improving the quaity of their products or new seeding some agents to change their consumption towards their products or both. Since the products have been in the market for a whie, we assume agents have aready decided on their consumption from products a and b which are denoted by x(0) and 1 x(0) respectivey. Each firm has a imited budget, i.e. K a and K b, that can spend on either new seeding, i.e. S a and S b, or enhancing the quaity of its product, i.e. q a and q b, or both. New seeding S a and S b wi change the initia consumption of products a and b by S a S b and S b S a respectivey. In order to have 0 x i (0) 1 and 0 1 x i (0) 1 for a agents i, we impose the constraints S a + y(0) 0.5 and S b y(0) 0.5. This means that firms can initiay seed agents up to their demand capacity. From equation (5) the margina change in the utiity of firm a and b resuted from the new

20 budget K a and K b are given by U a = v T Sa v T Sb + 2λq b q a (q a + q b ) 2 2λq a q b (q a + q b ) 2, U b = v T Sb v T Sa + 2λq a q b (q a + q b ) 2 2λq b q a (q a + q b ) 2. We assume the cost of improving the quaity by q is given by c q q and c q is a arge number, and aso the cost of each unit of new seeding is given by c s. Each firm maximizes its margina utiity given its fixed budget. Since the effect of the action of firm b, i.e. S b and q b, is decouped from that of the action of firm a in U a, thus firm a shoud sove the foowing optimization probem max S a, q a v T Sa + 2λq b q a (q a + q b ) 2, s.t. c s S a 1 + c q q a = K a. (14) Simiary, for the firm b we have max S b, q b v T Sb + 2λq a q b (q a + q b ) 2, s.t. c s S b 1 + c q q b = K b. (15) From equations (14) and (15) it can be seen that the optima strategy of each firm is independent of the action of the other firm. It is to be noted that despite the independence of the actions, the optima strategy of each firm depends on the state (i.e., quaity) of the riva firm. This resuts in a Nash equiibrium to be simpy the pair of the optima actions of the firms. In the next Theorem we describe a simpe rue for the optima aocation of the budget for each firm and discuss the resuting Nash equiibrium. Theorem 2: For firm a, it is more profitabe to seed agent j rather than enhancing the quaity of its product if v j > vc a where q b vc a (2λ)( c s )( ). (16) c q (q a + q b ) 2 Simiary, for firm b, it is more profitabe to seed agent j rather than enhancing the quaity of

21 its product if v j > v b c where q a vc b (2λ)( c s )( ). (17) c q (q a + q b ) 2 Moreover, any pair of the optima strategies of the firms described by the above threshod rues describes a Nash equiibrium. Proof: From equation (14) and (15) the reative margina utiity to cost for spending budget to seed agent j is v j c s. Therefore, it is aways more profitabe to seed an agent with higher centraity. Aso, the reative margina utiity to cost for spending budget on enhancing quaity of product a is 2λq b c q(q a+q b ) 2 according to (14). Therefore, for firm a it is more profitabe to seed agent j rather than enhancing the quaity of its product iff v j c s > 2λq b c q (q a + q b ) 2. This competes the proof. Simiar story hods for firm b. Moreover, since the best response of each firm resuting from equations (14) and (15) is independent of the action of the other firm, any Nash equiibrium of the game between firms is simpy a pair of firms best responses. Coroary 2: If firms have equa quaities q a = q b = q, for both firms a and b, it is more profitabe to seed agent j rather than enhancing the quaity of their products if v j > v c where v c ( λ 2 )(c s )( 1 ). (18) c q q Moreover, any pair of the optima strategies of the firms described by the above threshod rues describes a Nash equiibrium. Remark 2: If we compare the threshods with quaities in Section III q b vc a (2λ)( c s )( c q (q a + q b ) ), 2 vb c (2λ)( c s )( ), (19) c q (q a + q b ) 2 q a ṽ qa = (2λ)( c s )( c q (ṽ k + ṽ ) ), 2 q b = (2λ)( c s )( ), (20) c q (ṽ k + ṽ ) 2 we can see a simiarity as foows: In equation (19), q a and q b determine v a c and v b c which in turn ṽ k

22 determine the trade off between S and q according to Theorem 2. In Section III, K a and K b determine ṽ k and ṽ based on the inequaities in (8) and ṽ k and ṽ determine q a and q a according to (20), which in turn determine the trade of between S and q based on the budget constraint. Therefore, as it wi be discussed ater, we can achieve simiar resuts for the effect of q a and q b on the optima budget aocation, as we did for the effect of K a and K b on the Nash equiibrium. Foowing the above theorem, the optima aocation of the budget for each firm is to foow a so caed water-fiing strategy, that is, to start seeding in the order of agents centraities unti the centraity fas beow the threshod given by (16) for firm a or (17) for firm b (in which case the firm spends the rest of the budget on improving the quaity), or the firm runs out of budget. Aso, the amount that agents can be seeded is up to their demand capacity, i.e. S max a = (0.5) 1 y(0) > 0 and S max b = (0.5) 1 + y(0) > 0. Aso, note that if the centraity of any agent is equa to the threshod defined in (16) or (17), then firms are indifferent between seeding that agent and quaity improvement. Equations (16) and (17) indicate that the optima aocation depends on quaity of products, i.e. q a and q b and centraity distribution of agents in the network, i.e. v. In what foows, we wi study the effect of each of these factors on the optima aocation of the firms in more detais. A of our anaysis here is for firm a and simiar resuts can be shown for firm b as we. For simpicity, we ony discuss optima seeding budget; optima quaity improvement budget can be found easiy using the budget constraint. A. Effect of Quaity of Products on Firms Decisions: In this subsection we study how the quaity of each product, i.e. q a and q b, can infuence the optima aocation of seeding and quaity improvement budgets. As it can be seen from equation (16), the threshod vc a depends on both firm s and its riva s quaities, i.e. both q a and q b. In the next proposition, we compare the seeding budget of two firms in the optima aocation with respect to their quaities. Proposition 6: Given an equa budget, the firm with higher quaity aso has higher seeding budget, i.e. if q a q b, then S a 1 S b 1. Proof: From equations (16) and (17) it can be easiy seen that if q a q b, then vc b vc a.

23 As a resut, more agents satisfy the condition (17) for firm b compared to firm a and therefore, S a 1 S b 1. This resut is due to diminishing return of quaity which means if a firm aready has a high quaity it woud profit ess by spending on quaity improvement and it woud be better for the firm to invest on seeding. Aso, note that the resut of Proposition 6 is simiar to the resut of Proposition 1. The ony difference is that instead of budgets K a and K b in Proposition 1, quaities q a and q b in Proposition 6 pay the roe of the budgets whie comparing the seedings of the firms. In the next proposition we expain how the optima seeding budget vary with q a and q b. Proposition 7: Given a fixed graph, the optima seeding budget is an increasing function of q a. Furthermore, it is a decreasing function of q b if q b q a and an increasing function of q b if q a q b. Proof: The optima seeding budget is a decreasing function of the threshod vaue vc a. Aso, it is easy to see that the threshod vaue of v a c is a decreasing function of q a. This impies the first part of proposition. For the second part, it is easy to see that the threshod vaue of vc a is a decreasing function of q a. This impies the first part of proposition. For the second part, it is easy to see that the threshod vaue of vc a is an increasing function of the quaity of product b, when q b q a and a decreasing function of the quaity of product b, when q b q a. This competes the proof. Proposition 7 impies that a higher quaity in a firm s product resuts in a higher seeding budget in the optima aocation. This can be due to the diminishing return of quaity: when quaity is higher there is ess need for quaity improvement and it woud be more profitabe to spend on seeding. Furthermore, when q b q a, the higher the quaity of the riva firm s product, the ower the seeding budget of firm a, i.e., if q a q b q b then, S a(q b ) 1 S a(q b ) 1. On the other hand, when competing with a firm whose product has a higher quaity, i.e. q b q a, the higher the quaity of the riva firm s product, the higher firm a shoud spend on seeding. In other words, if q a q b q b then, Sa(q b ) 1 S a(q b ) 1. Combining these two resuts, we can see that given a fixed vaue of q a, the seeding budget of

24 firm a is increasing with the difference q a q b. The seeding budget attains its minimum when q b = q a, impying that the firm shoud aocate more budget to quaity improvement to distance itsef from the riva firm. However, as the gap between quaities widens, competition in quaities becomes ess effective and firms spend more budget on seeding. Aso, note that the resut of Proposition 7 is simiar to the resut of Proposition 2. The ony difference is that seeding budgets vary with q a and q b in Proposition 7, whereas they vary with K a and K b in Proposition 2. B. Effect of Network Structure on Firms Decisions: In this subsection we study the effect of network structure on the optima aocation of the budget for seeding and quaity improvement. First we define seeding capacity of a graph. Definition 4: The seeding capacity of a graph is the amount that it can be seeded in the optima aocation when there is no budget constraint. We first focus on two we studied graphs, i.e. star and baanced graphs, and highight how they can refect important properties of the seeding budget. The next proposition provides a condition for seeding profitabiity of any genera graph. Aso, the seeding capacity of star and baanced graphs are compared and it is shown that the graph with higher seeding capacity can be any of the two, depending on the threshod vaue of vc a in (16). Proposition 8: If seeding capacity is nonzero for a baanced graph, it wi be nonzero for any other graph too. On the other hand, if seeding capacity is zero for a star graph, it wi aso be zero for any other graph. Moreover, if v s < v a c < v, a baanced graph has a arger seeding capacity than a star graph, and if v < vc a < vh s, a star graph has a arger seeding capacity than a baanced graph. For 1 < v a c < v s they have the same seeding capacity. Proof: If seeding capacity is nonzero for a baanced graph, then we have v a c < v. As a resut, for any other graph we wi have v a c < v max, where v max = max v i, since according to Lemma 3 v v max. This means that there exists at east one agent that must be seeded. On the other hand, if seeding capacity is zero for a star graph, then we must have vc a > vh s. Since we know v s h v i for any agent i of any arbitrary graph, therefore, v a c > v i and no agent can be seeded in any other graph. For the second part of the proposition, if v s < v a c < v, then seeding

25 capacity for the star graph wi be Sa max i, where Sa max 1 Sa max 2 Sa max n are eements of the demand capacity vector S max a and agent i is the centra agent. However, for the baanced graph a agents can be seeded up to their maximum demand capacities and the seeding capacity wi be S a max 1. On the other hand, if v < vc a < vh s, sti seeding for the star graph wi be Smax a i, however, no agent can be seeded in the baanced graph. For 1 < vc a < v s, agents in both graphs can be seeded up to S max a 1. It is easy to see that Proposition 8 presents very simiar resuts as Proposition 3. The next proposition provides us with a ower and an upper bound for minimum and maximum seeding capacities. Proposition 9: If 1 < vc a < vh s, the maximum seeding capacity is given by S a max 1 = k i=1 S max a i, where nδ k = min{, n}. (21) (vc a 1)(2β δ) On the other hand, the minimum seeding capacity is S max a n v s < v a c < v, and is zero if v < v a c < v s h. + S max a n 1 if 1 < vc a < v s, is Smax a n Proof: From condition (16) the more agents with centraities above the threshod v a c, the more seeding budget can be aocated. Therefore, the maximum number of k agents with if centraities above the threshod v a c must be found. Since v i 1 for a agents, first a centraity of one is given to each agent and then the remainder of the centraity sum is distributed among maximum number of agents so that each agent receives at east v a c 1, making its overa centraity greater than v a c. It is easy to see that the number of such agents is upper bounded by 2βn 2β δ n vc a 1. This aong with the fact that 1 k n resuts in (21). Note that, in order to compete the proof, we shoud aso provide an exampe achieving this maximum capacity. For k = 1, the maximum seeding capacity is ceary achieved by the star graph with the seeding capacity of Sa max 1. For k 2, a graph with argest seeding capacity is the one with k centra agents having

26 the argest demand capacities and with equa centraities of ṽ s h = nδ k(2β δ) + 1, where k is given in (21), and the remainder n k agents with the minimum centraity of ṽ s = 1. For the graph with minimum seeding capacity, simiar to the proof of Proposition 8, we have minimum seeding capacity of S max a n in star graph if v s < v a c < v, and zero in baanced graph if v < vc a < vh s. For the case where 1 < va c < v s, et i be the agent with the highest centraity. Ceary, v s < v i vh s. Now, considering the fact that sum of the centraities is fixed, there is an agent j V {i} for which v s v j. This impies that there exist at east two agents whose centraities are above v a c. An exampe of a graph with exacty two centraities above v a c is a directed star graph where a edges are directed towards the center except one edge which goes both ways. It can be seen from both Proposition 9 and Proposition 4 that graphs with simiar structures attain maximum and minimum seeding in both scenarios. Exampe 2: As a numerica exampe for the minimum and maximum seeding capacities, we consider a network with n = 15 agents with demand capacity vectors of S max a = S max b = (0.5) 1, quaities of q a = q b = 1, quaity and seeding costs of c s = c q = 1 and parameters of α = β = 1 and δ = 0.5. For this exampe from equations (16) and (17) we have v a c = v b c = 2.5 and as a resut, from equation (21) we get k = 3. Therefore, a graph with the maximum seeding capacity is a 3-star with seeding capacity of 1.5 as iustrated in Fig. 1. Aso, since v = 4 3 < va c, v b c < v s h = 4.8, a baanced graph has the minimum seeding capacity of zero. A star graph has a seeding capacity of 0.5 which is neither a minimum nor a maximum. Note that in both Exampe 1 and Exampe 2 a graph with the maximum seeding budget and capacity is a 3-star graph and a graph with the minimum seeding budget and capacity is a baanced graph. A star graph has neither a minimum nor a maximum seeding budget and capacity in both exampes. As we saw, the structure of the graphs with minimum and maximum seeding capacity depends on the threshod vaue of v a c. However, for certain vaues of v a c the seeding capacity wi be