Information Overload in a Network of Targeted Communication: Supplementary Notes

Transcription

1 Inforation Overload in a Network of Targeted Counication: Suppleentary Notes Tiothy Van Zandt INSEAD 10 June 2003 Abstract These are suppleentary notes for Van Zandt 2003). They include certain extensions. To be in synch, these notes and the ain paper should have the sae date. Author s address: INSEAD Boulevard de Constance Fontainebleau CEDEX France Voice: Fax: Eail: tvz@econ.insead.edu Web: zandtwerk.insead.edu I a indebted to Roy Radner and Peter Linhart for helpful discussions.

2 Contents 1 Model 1 2 Equilibriu 1 3 Strategies that axiize the senders total payoffs 2 4 Type-dependent echaniss for allocating attention 2 5 Allocating attention through unifor surcharges 3 6 Deographic data and types 4 A Proofs 4 References 11

3 Van Zandt : Inforation Overload in a Network of Targeted Counication 1 1 Model Copared to the odel in Van Zandt 2003), our only odification is to allow the counication cost and surcharges to vary aong senders. Let c j be the cost per essage sent by sender j, so that the cost to sender j of targeting types X j T is c j γx j ). The net payoff of sender j given X and c j is π j X; c j ) s j σ j X) c j γx j ). Each n-tuple c c 1,...,c n R n + of counication costs thus defines a gae Γ c in noral for in which the players are the n senders, each sender s strategy set is B, and sender j s payoff function is π j ; c j ). By an equilibriu for Γ c, we ean a pure strategy Nash equilibriu. 2 Equilibriu Let c R n +. ItisstillthecasethatthegaeΓ c can be decoposed into independent single-receiver gaes. For t T,letΓ c t) be the single-receiver gae for type t. This is the gae in noral for in which a) there are n players; b) each player s strategy set is {0, 1}, where 0 eans not send and 1 eans send ; and c) player j s payoff, given the strategy profile x x 1,...,x n {0, 1} n,is { sj t j in{1,/#x}) c j x j =1 u j x; c j,t) 0 x j =0. Proposition 1 Γ c t) has a pure strategy equilibriu for all c R n + and t T. Given a strategy profile X X 1,...,X n, the payoffs in Γ c given X are equal to the averages of the payoffs in the gaes Γ c t) given X 1 t),...,x n t). Thatis: π j X; c j ) = u j X 1 t),...,x n t); c j,t) dγt). Proposition 2 then follows easily. T Proposition 2 Let c R n +. A strategy profile X 1,...,X n is an equilibriu for Γ c if and only if, for γ-a.e.t T, X 1 t),...,x n t) is a pure-strategy equilibriu for Γ c t). Corollary 1 Γ c t) has an equilibriu for all c R n +. The proofs of these two results are the sae as when all senders face the sae cost, and so they are oitted. Assuption 4.1 in Van Zandt 2003) is also aintained in this paper.

4 Van Zandt : Inforation Overload in a Network of Targeted Counication 2 Proposition 3 Let j {1,...,n} and c j R +. Let X j X 1,...,X j 1,X j+1,...,x n B n 1 be a profile of strategies for senders other than j. Then sender j has a unique best response to X j given c j. That is, ax Xj B π j X j,x j ; c j ) has a solution and it is unique up to equivalence. Denote this solution by X j X j ; c j ). 3 Strategies that axiize the senders total payoffs Assuption 4.1 iplies that, for c R n +, {s j t j c j j =1,...,n} contains n distinct nonzero eleents for γ-a.e. t T. Thus, there is a unique strategy profile that axiizes the total net payoffs of the senders in the gae Γ c, and it is given by Y c j { t T s j t j c j > 0&#{i s i t i c i >s j t j c j } < }. Let Y c Y1 c,...,yn c. As when all senders have the sae counication cost, inefficiency fro the point of view of the senders arises fro too uch rather than too little inforation, and tends to be worse when the counication cost is lower. We suarize this in Proposition 4. Part 1 says that the total counication in equilibriu is greater than or equal to the total counication given Y c. Part 2 says that if the senders other than j adopt the strategies Y j, c then sender j wants to target at least the receivers in Yj c and perhaps others. Part 3 says that for counication costs c in a neighborhood of 0, Y c is not an equilibriu for Γ c. Part 4 says that if suppγ) = n 1, then if the senders total payoffs are not axiized in equilibriu, neither are they axiized in equilibriu when the counication cost is lower. Proposition 4 1. Let c R n + and let X 1,...,X n be an equilibriu for Γ c.then n j=1 γx j n j=1 γy c j. 2. Let c R n + and let j 1,...,n.ThenY c j X j Y c j; c j ). 3. Let C be the set of costs c R n + such that Y c is an equilibriu for Γ c. Then C is non-epty and closed and does not contain Suppose suppγ) = n 1. Then C is convex. Furtherore, if c C and c c, then c C. 4 Type-dependent echaniss for allocating attention The use of type-dependent echaniss is essentially the sae as when senders have the sae cost.

5 Van Zandt : Inforation Overload in a Network of Targeted Counication 3 For exaple, suppose the echanis designer uses a price echanis P : T R + so that the surcharge on targeting B T is B P t) dγt). This defines a gae Γc,P ) in which each sender s strategy set is B and sender j s payoff, given X B n,iss j σ j X) X j c j +P t))dγt). If, for each t T, P t) is equal to the +1) st highest value of s 1 t 1 c 1,...,s n t n c n or to 0, whichever is greater, then for each j, Therefore, Y c is an equilibriu for Γ c,p ). zs j t j c j P t) > 0= t Yj c and s j t j c j P t) < 0= t/ Yj c. 5 Allocating attention through unifor surcharges We allow the surcharges to depend on the identify of the sender and denote the by p R n.wesaythatp supports Y c if Y c is an equilibriu of the gae Γ c+p. Proposition 5 confirs, not surprisingly, that we need only consider positive surcharges rather than subsidies. Proposition 5 Let c R n + and let p R n be such that c + p 0 and p supports Y c. Then there is p R n + that supports Y c. Proposition 6 shows that the negative results on using unifor surcharges to support the strategy profile that axiizes the senders payoffs persist even when we allow the surcharges to depend on the identity of the sender. Proposition 6 Let c R n +.Suppose that either a) suppγ) =T,orb) n 1 suppγ) and either >1 or s j c j 0 for soe j. If Y c is not an equilibriu for Γ c then there is no surcharge p R n that supports Y c. Our next result concerns the case where suppγ) = n 1 and = 1. For this result, we assue that the counication cost is the sae for all firs and we restrict attention to identical surcharges for all firs. There ay not be any arginal receivers; so, if sall enough, a surcharge does not induce senders to drop receivers who they should target in the efficient profile. The difficulty is that, in order to support Y c, a surcharge ust be large enough to eliinate inforation overload. The paraeter values for which these two requireents can be reconciled is liited. Proposition 7 Assue suppγ) = n 1 and =1. Let c 0 and suppose that Y c is not an equilibriu for Γ c. Then there is a surcharge that supports Y c if and only if either 1) n =2;

6 Van Zandt : Inforation Overload in a Network of Targeted Counication 4 2) n =3and s 1 i + s 1 j s 1 k for all distinct i, j, k; or 3) n =4and s 1 = s 2 = s 3 = s 4. In this case, the surcharge n ) 1 k=1 s 1 k c supports Y c. 6 Deographic data and types This section describes a ore priitive odel of the receivers and the inforation the senders have about the. Because the purpose is to provide intuition rather than a atheatical fraework to be used elsewhere in the paper, we consider a finite odel. The odel with a large dispersed population of receivers and types of receivers is eant to be an approxiation of this finite odel. There is a finite set A of receivers. The senders have a coon ailing list, which gives the nae and address of each receiver. The ailing list also gives deographic inforation such as age, sex, race, place of residence, job title, and agazine subscriptions. Let D be the finite set of possible deographic characteristics. Let Z: A D be a function that specifies the characteristic of each receiver. The senders also have arketing data, which gives the correlation between these characteristics and the interest in the senders essages. Specifically, let A j A be the set of receivers who are interested in j s essage. For d D, let δ j d) #A j Z 1 d)). #Z 1 d) Here, δ j d) is the proportion of those receivers with deographic characteristic d who are interested in j s essage. Sender j can estiate δ j d) by sapling the population of receivers in Z 1 d), such as with arketing surveys. Let δd) = δ 1 d),...,δ n d) and let T [0, 1] n be the set of types. Then δ: D T.Let M: A T be the coposition δ Z. Receiver a s type is Ma). The nuber of receivers of type t T is γt) #M 1 t). The proportion of receivers of type t 1,...,t n interested is sender j s essage is exactly t j. If the deographic characteristics are good indicators of the interests of the receivers, then γ places greater ass near the corners of T. Otherwise, γ places greater ass near #A 1 /#A,...,#A n /#A. If each receiver is interested in one and only one essage, then γ puts positive weight only on points in n 1. Figure 1 contains an illustration of the link between γ and the senders inforation. A Proofs Proof of Proposition 1. The structure of the gae that we use in this proof is that each player has a fixed payoff of 0 fro not sending a essage and each player s payoff fro sending

7 Van Zandt : Inforation Overload in a Network of Targeted Counication 5 Division between basketball and golf players basketball basketball Height Height Height basketball golf golf golf Incoe Incoe Incoe Support of γ t 2 t 2 t t t t 1 Figure 1. Illustration of the senders inforation. There are two senders, a basketball equipent retailer and a golf equipent retailer. Each receiver plays either basketball or golf, depending only on her incoe and height. The receivers incoe and heights are distributed uniforly on [$0, $100K] [150c, 200c]. The division between basketball and golf players is shown in the upper row for three cases. The senders know the receivers incoes but not their heights. The support of the induced distribution γ on [0, 1] 2 is drawn in the lower row. For the left-hand division, incoe is partly inforative and γ is unifor on the siplex 1. For the iddle division, incoe is uninforative and γ is concentrated on 1/2, 1/2. For the right-hand division, incoe is fully inforative and γ is concentrated on 1, 0 and 0, 1.

8 Van Zandt : Inforation Overload in a Network of Targeted Counication 6 a essage is decreasing in the nuber of other players who also send a essage but does not depend on the identities of these players. Let c R n + and t T. For j {1,...,n}, letl j {0, 1,...,n} be such that sending a essage is a best response for sender j in Γ c t) if and only if at ost l j 1 other senders send essages. Specifically, l j ax { l =0, 1,...,n l =0orsj t j ax {1,/l}) c j 0 }. Renuber the senders if necessary so that l 1 l n. Iagine that the senders sequentially choose to enter send a essage), basing this decision only on the nuber of senders who have already entered. Let k + 1 be the first player to choose not to enter. That is, k ax {j =0, 1,...,n j =0orl j j}. Since sender k finds it profitable to enter, l k k. Since l 1 l k, senders 1,...,k 1 still find it profitable to send a essage given that a total of k senders do so. Since player k + 1 chooses not to enter, l k+1 <k+1. Sincel k+1 l n players k +2,...,n also find it unprofitable to enter given that k players have already done so. Hence, it is an equilibriu that players 1,...,k send a essage and players k + 1,...,n do not. Proof of Proposition 3. Let j {1,...,n}, letc j R +,andletx j X 1,...,X j 1, X j+1,...,x n B n 1 be a profile of strategies for senders other than j. LetXt resp., X t + ) be the set of types t for who j has a strict resp., weak) incentive to send a essage given c j and X j t). That is, X j { t T sj t j in {1,/#X j t)+1)} ) } >c j X + j { t T s j t j in {1,/#X j t)+1)} ) c j }. Then X j and X + j are both best responses. Furtherore, for any best response X j B, X j X j X + j where B B eans that γ-a.e. eleent of B is in B ). The syetric a.s. a.s. a.s. different between X j and X + j is X + j \ X j = { t T s j t j in {1,/#X j t)+1)} ) >c j } n {t T t j =l/)c j /s j )}. Assuption 4.1 iplies that γ{t T t j =l/)c j /s j )} = 0 for l =,...,n. Hence X j and X + j, and any other best response X j,areequivalent. l= Proof of Proposition We show that for γ-a.e. t T,#Xt) #Y c t). Let t be such that X 1 t),...,x n t) is an equilibriu for Γ c t); this holds γ-a.e. according to

9 Van Zandt : Inforation Overload in a Network of Targeted Counication 7 Proposition 4.2. If at least essages are sent in this equilibriu, i.e. if #Xt), then #Xt) #Y c t) since#y c t) receivers are never overloaded in the efficient strategy profile). Otherwise, #Xt) <and each sender s essage is processed for sure, as would also be the case if one ore sender sent a essage. Assue that s j t j c j 0 for each j, which holds for γ-a.e. t T. Then sender j sends a essage in this equilibriu if and only if s j t j c j > 0. This is true also for the efficient strategy profile when this inequality holds for at ost senders. Hence, #Xt) =#Y t). 2. Let t Y c j.then#y c jt) 1ands j t j c j > 0. Therefore, and t X j Y c j; c j ). s j t j in { 1,/#Y c j t)+1) }) > c j, 3. Suppose c = 0. For any t T, sending a essage is a strictly doinant strategy for sender j in the single-receiver gae for type t unless t j = 0. Assuption 4.1 iplies that γ{t T t j =0} = 0. Hence, in the gae Γ c the strategy T or a set equivalent to T is a strictly doinant strategy for each player. However, this is not the efficient strategy profile since then γ-a.e. receiver would be overloaded. Thus, 0 / C. Let c R n + \ C. Then by part 2), there is j {1,...,n} such that Up to equivalence, γ X j Y c j; c j ) \ Y c j ) > 0. Xj Y j; c c j ) \ Yj c { } = t T +1 s jt j c j > 0&#{i s i t i c i >s j t j c j } { = t T +1 s jt j c j > 1 { η &# i s it i c i >s j t j c j + 1 } η η=1 } T η. Hence, there is η such that γt η > 0. Let c belong to the nbd. of c such that c i c i < 1/2η) for all i {1,...,n}. Then, for all t T η, +1 s jt j c j > +1 s jt j c j 1/2η) > 0 and # { i s i t i c i >s j t j c j} > #{i si t i c i >s j t j c j +1/η)}, and thus t Xj Y j; c c j) \ Yj c.thenγxj Y j; c c j) \ Y j) c > 0andc / C. Thus,C is closed. Finally, note that if s j = c j for all j, thenyj c = and this is also a doinant strategy for j; therefore, s 1,...,s n C and C. 4. Assue that suppγ) = n 1. Let F be the set of subsets of {1,...,n} with +1 eleents. Given F F and c R n +,lett F c) be the unique eleent of R +1 such that, for all i, j F, j F tf j c) =1ands j t F j c) c j = s i t F i c) c i. The following lea is proved below.

10 Van Zandt : Inforation Overload in a Network of Targeted Counication 8 Lea 1 Y c is an equilibriu for Γ c if and only if for every F F and j F, 1) +1 s jt F j c) c j 0. Let F F and c R n +.Thereisλ R such that, for every j F, s j t F j c) c j = λ, and thus t F j c) = λ+c j s j.then1= i F tf j c) = i F λ = 1 i F s 1 i Therefore, equation 1) is equivalent to 1 i F s 1 i +1 i F s 1 i i F s 1 i c i λ+c i s i,andso c i. + c j ) c j 0. Rearranging, ) 2) 1 s 1 i c j + s 1 i c i 1. i F i F Therefore, because of the clai and because Equations 1 and 2 are equivalent, { C = c R n + F F, j F : ) 1 s 1 i c j + } s 1 i c i 1, i F i F which is convex since it is defined by linear inequalities. The coefficients of the linear inequalities defining C are positive, so that if c C and c c, thenc C. This ends the proof, except for the proof of the lea. Proof of Lea 1. First, we prove the forward iplication by proving its contrapositive. Suppose there are F Fand j F such that +1 jt F j c) c j > 0. Then also s j t F j c) c j > 0. Perturbing t F c) by subtracting a sall aount fro t F j c) and adding the sae to t F i c) for soe i j, we obtain t 1,...,t n n 1 such that +1 jt j c j > 0and {i s i t i c i >s j t j c j } F\,{j}. Thus, U { t n 1 #{i s i t i c i >s j t j c j } & } +1 s jt j c j > 0 is a non-epty and open subset of n 1. Since suppγ) = n 1, γu > 0. Also, U Xj Y j; c c j ) \ Yj c. Therefore, Y c is not an equilibriu for Γ c. Conversely, suppose equation 1) holds for all F F and j F. Let j {1,...,n} and let t Yj c. If s j t j c j 0, then t j / Xj Y j; c c j ). Assue instead that s j t j c j > 0. Since t Yj c,#{i t Yi c } = and F {i t Yi c } {j} F γ-a.s. Assue so. Then, for every i F \{j}, s i t i c i >s j t j c j, which iplies that s j t F j c j >s j t j c j and thus +1 s jt j c j < +1 s jt F j c j < 0. The second inequality follows fro equation 1). Then t Xj Yj c ; c j ). Therefore, for all j, Xj Y j; c c j ) \ Yj c = ; by part 2 of this proposition, Y c is an equilibriu for Γ c.

11 Van Zandt : Inforation Overload in a Network of Targeted Counication 9 Proof of Proposition 5. Let p j ax{0,p j } for every j {1,...,n} and let p p 1,...,p n. SinceY c is an equilibriu for Γ p+c, Yj c = Xj Y j; c c j +p j ) for every j {1,...,n}. Thus, if p j = p j,thenyj c = Xj Y j; c c j + p j). Otherwise, p j =0>p j,and Y c j X j Y c j; c j ) X j Y c j; c j + p j ) = Y c j. The first inclusion holds by Proposition and the second inclusion holds because p j < 0 and Xj is onotone decreasing in the cost of counication. Therefore, Yj c = Xj Y j; c c j +p j) for every j {1,...,n}, andy c is an equilibriu for Γ c+p. Proof of Proposition 6. Let p R n +.ThenY c is an equilibriu for Γ c+p if and only if, for all j {1,...,n}, 3) 4) s j t j c j p j for γ-a.e. t Yj c,and +1 s jt j c j p j for γ-a.e. t T \ Yj c. Suppose that the assuptions hold but that there is p R n + that supports Y c.weshow that Y c is an equilibriu for Γ c. Let j {1,...,n}. If s j c j s j c j > 0. Let ɛ>0andlet 0, then Y c j = X j Y c j; c j )=. Suppose instead that U { t T ɛ>s j t j c j > 0&#{i s i t i c i >s j t j c j } < }, which is open and which is a subset of Y c j.chooset j 0, 1) such that ɛ>s j t j c j > Suppose suppγ) =T.Since 0,...,t j,...,0 U, U is non-epty and thus γu > Suppose n 1 suppγ). If >1, choose any i j. Ifthereisî such that sî cî 0, let i =î. Then let t n 1 be such that t j = t j and t i =1 t j. In either case, t U n 1,andsoU n 1 and γu > 0. Since U Yj c and γu > 0, equation 3) iplies that p j ɛ. Since this holds for every ɛ>0, p j =0. SinceYj c is an equilibriu for Γ c+p, we again obtain Yj c = Xj Y j; c c j ). Therefore, Y c is an equilibriu for Γ c. Proof of Proposition 7. Various steps are stated as leas. The assuptions i) suppγ) = n 1 and ii) = 1 are aintained throughout. Let v 2 ax =ax t n 1 v 2 t) andv 1 in = in t n 1 v 1 t). Lea 2 Suppose Y c is not an equilibriu of Γ c. Then v 1 in 1/2)v 2 ax is necessary and sufficient for there to be soe p>0 such that Y c is an equilibriu of Γ c+p.

12 Van Zandt : Inforation Overload in a Network of Targeted Counication 10 Proof of Lea 2. Suppose Y c is not an equilibriu of Γ c. It follows fro Corollary A.1 that 1/2)v 2 t) >cfor soe t n 1 and hence that 1/2)v 2 ax >c. Suppose that v 1 in 1/2)v 2 ax. Letp =1/2)v 2 ax c, sop>0. Then v 1 in c + p) 0 and hence v 1 t) c + p) 0 for all t n 1. Furtherore, 1/2)v 2 ax c + p) = 0 and hence 1/2)v 2 t) c + p) 0 for all t n 1. It follows fro Lea A.1 that Y c is an equilibriu of Γ c+p. Suppose instead that v 1 in < 1/2)v 2 ax. We consider two cases: 0 <p<1/2)v 2 ax c and p 1/2)v 2 ax c. Consider the first case. Let U {t T 1/2)v 2 t) c + p) > 0}, which is a set of types for which Y c t) is not an equilibriu fro Lea A.1). Since v 2 is continuous, U is open; since 1/2)v 2 ax c + p) > 0, we have U n 1. Hence, γu > 0 and Y c is not an equilibriu of Γ c+p. Let instead p 1/2)vax c,sothatv 2 in c+p) 1 < 0. Let U = {t T p>v 1 t) c>0}, which again is a set of types such that Y c t) is not an equilibriu of Γ c+p t). The condition v 1 t) c>0 eans that a highest valuation sender should send to t according to Y c t), but the condition v 1 t) c + p) < 0 eans that every sender s doinant strategy in Γ c+p t) is to not send a essage.) Then U n 1 because a) v 1 is continuous, b) vin 1 <c+ p, c) vax 2 >chence there is a t n 1 such that v 1 t) >c), and d) n 1 is connected. Continuity of v 1 also iplies that U is open. Hence, γu > 0andY c is not an equilibriu of Γ c+p. Lea 3 Renuber the senders if necessary so that s 1 s n.then n ) 1 5) vin 1 =, 6) v 2 ax k=1 s 1 k = ) s s Proof. The t that iniizes v 1 t) on n 1 is such that t 1 s 1 = = t n s n, which eans that t j =1/s j )/ n ) k=1 s 1 k and hence v 1 t) =s j t j = n ) 1. k=1 s 1 k The t that axiizes v 2 t) on n 1 has positive values only for senders 1 and 2, who have the two highest values of s j ; t is then such that t 1 s 1 = t 2 s 2. Hence, t 1 =1/s 1 )/1/s 1 +1/s 2 ) and v 2 t) = ) 1. s s 1 2 Lea 4 v 1 in 1/2)v 2 ax if and only if condition 1), 2), or 3) of Proposition 7 holds. Proof. Renuber the senders if necessary so that s 1 s n. Then the inequality vin 1 1/2)vax 2 can be written as 7) n ), s k s 1 s 2 k=1 8) n 1 s k s 1 s 2 k=3

13 Van Zandt : Inforation Overload in a Network of Targeted Counication 11 If n = 2, then equation 8) iposes no restriction of all. If n = 3, then equation 8) becoes s 1 3 s s 1 2 which in turn iplies s 1 i s 1 j + s 1 k for other perutations {i, j, k} of {1, 2, 3}). If n = 4, then equation 8) becoes s s 1 4 s s 1 2. Since s 1 s 2 s 3 s 4, this holds if and only if s 1 = s 2 = s 3 = s 4. Suppose n>4. Then the left-hand side of equation 8) is the su of three or ore ters, each of which is larger than the two ters sued on the right-hand side. Hence, the left-hand side is necessarily larger than the right-hand side. This concludes the proof of Proposition 7. References Van Zandt, T. 2003). Inforation overload in a network of targeted counication. INSEAD.