Selling Multiple Items via Social Networks

Transcription

1 ellng Multple tems va ocal Networks engj Zhao, Bn L, Junpng Xu, ong Hao, and Ncholas R. Jennngs hanghatech Unversty, hangha, hna Unversty of Electronc cence and Technology of hna, hengdu, hna mperal ollege London, London, Unted ngdom ABTRAT We consder a market where a seller sells multple unts of a commodty n a socal network. Each node/buyer n the socal network can only drectly communcate wth her neghbours,.e. the seller can only sell the commodty to her neghbours f she could not fnd a way to nform other buyers. n ths paper, we desgn a novel promoton mechansm that ncentvzes all buyers, who are aware of the sale, to nvte all ther neghbours to jon the sale, even though there s no guarantee that ther efforts wll be pad. Whle tradtonal sale promotons such as sponsored search auctons cannot guarantee a postve return for the advertser (the seller), our mechansm guarantees that the seller s revenue s better than not usng the advertsng. More mportantly, the seller does not need to pay f the advertsng s not benefcal to her. EYWOR Mechansm desgn; nformaton dffuson; revenue maxmsaton AM Reference Format: engj Zhao, Bn L, Junpng Xu, ong Hao, and Ncholas R. Jennngs. 08. ellng Multple tems va ocal Networks. n Proc. of the 7th nternatonal onference on Autonomous Agents and Multagent ystems (AAMA 08), tockholm, weden, July 0 5, 08, FAAMA, 9 pages. NTROUTON Marketng s one of the key operatons for a servce or product to survve. To do that, companes often use newspapers, tv, socal meda, search engnes to do advertsements. ndeed, most of the revenue of socal meda and search engnes comes from pad advertsements. Accordng to tatsta, oogle s ad revenue amounted to almost 79.4 bllon U dollars n 0. However, whether all the advertsers actually beneft from ther advertsements s not clear and s dffcult to montor. Although most search engnes use market mechanms lke generalsed second prce auctons to allocate advertsements and only charge the advertsers when users clck ther ads, not all clcks lead to a purchase [, 4]. That sad, the advertsers may pay user clcks that have no value to them. Therefore, n ths paper, we propose a novel advertsng mechansm for a seller (to sell servces or products) that does not charge the seller unless the advertsng brngs an ncrease n revenue. We model all (potental) buyers of a servce/product as a large socal Proc. of the 7th nternatonal onference on Autonomous Agents and Multagent ystems (AAMA 08), M. astan,. ukthankar, E. André,. oeng (eds.), July 0 5, 08, tockholm, weden. 08 nternatonal Foundaton for Autonomous Agents and Multagent ystems ( All rghts reserved. network where each buyer s lnked wth some other buyers (known as neghbours). The seller s also located somewhere n the socal network. Before the seller fnds a way to nform more buyers about her sale, she can only sell her products to her neghbours. n order to attract more buyers to ncrease her revenue, the seller may pay to advertse the sale va newspapers, socal meda, search engnes etc. to reach/nform more potental buyers n the socal network. However, f the advertsements do not brng any valuable buyers, the seller loses the nvestment on the advertsements. Our advertsng mechansm does not rely on any thrd party such as newspapers or search engnes to do the advertsements. The mechansm s owned by the seller. The seller just needs to nvte all her neghbours to jon the sale, then her neghbours wll further nvte ther neghbours and so on. n the end, all buyers n the socal network wll be nvted to partcpate n the sale. Moreover, all buyers are not pad n advance for ther nvtatons and they may not get pad f ther nvtatons are not benefcal to the seller. Although some buyers may never get pad for ther efforts n the advertsng, they are stll ncentvzed to do so, whch s one of the key features of our advertsng mechansm. Ths sgnfcantly dffers from exstng advertsng mechansms used on the nternet. More mportantly, our advertsng mechansm not only ncentvzes all buyers to do the advertsng, but also guarantees that the seller s revenue ncreases. That s, her revenue s never worse than the revenue she can get f she only sells the tems to her neghbours. A specal case of ths problem was nvestgated by L et al. [8]. They have consdered the settng when the seller sells only one tem and proposed a mechansm, called the nformaton dffuson mechansm, that guarantees that all buyers wll truthfully report ther wllng payments (.e. valuatons) and also nvte all ther neghbours to jon the sale. They have shown that the mechansm gves a revenue whch s at least the revenue the seller can receve wth a second prce aucton among only the seller s neghbours. Ths paper generalses the mechansm proposed by L et al. [8] to settngs where the seller sells multple tems. Ths generalsaton stll guarantees that reportng ther true valuatons and nvtng all ther neghbours s a domnant strategy for all buyers who are aware of the sale. Moreover, the revenue of the seller s also mproved compared wth the revenue she can acheve wth tradtonal market mechansms such as V [,, 5]. Maxmsng the seller s revenue has been well studed n the lterature, but the exstng models assumed that the buyers are all known to the seller and the am s to maxmze the revenue among the fxed number of buyers. ven the number of buyers

2 s fxed, f we have some pror nformaton about ther valuatons, Myerson [0] proposed a mechansm by addng a reserve prce to the orgnal V mechansm. Myerson s mechansm maxmses the seller s revenue, but requres the dstrbutons of buyers valuatons to compute the reserve prce. Wthout any pror nformaton about the buyers valuatons, we cannot desgn a mechansm that can maxmse the revenue n all settngs (see hapter of [] for a detaled survey). oldberg et al. [4, 5] have consdered how to optmze the revenue for sellng multple homogeneous tems such as dgtal goods lke software (unlmted supply). Especally, the seller can choose to sell less wth a hgher prce to gan more. n terms of ncentvzng people to share nformaton (lke buyers nvtng ther neghbours), there also exsts a growng body of work [, 7,, ]. Ther settngs are essentally dfferent from ours however. They consdered ether how nformaton s propagated n a socal network or how to desgn reward mechansms to ncentvze people to nvte more people to accomplsh a challenge together. The soluton offered by the MT team under the ARPA Network hallenge s a nce example, where they desgned a novel reward mechansm to share the award f they wn the challenge to attract many people va socal network to jon the team, whch eventually helped them to wn []. The remander of the paper s organzed as follows. ecton descrbes the model of the advertsng problem. ecton brefly revews the mechansm proposed by L et al. [8]. ecton 4 gves our generalsaton and ts key propertes are analysed n ecton 5. Fnally, we conclude n ecton. THE MOEL We consder a seller s sells tems n a socal network. n addton to the seller, the socal network conssts of n nodes denoted by N = {,,n}, and each node N {s} has a set of neghbours denoted by r N {s}. Each N s a buyer of the tems. For smplcty, we assume that the tems are homogeneous and each buyer N requres at most one unt of the tem and has a valuaton v 0 for one or more unts. Wthout any advertsng, seller s can only sell to her neghbours r s as she s not aware of the rest of the network and the other buyers also do not know the seller s. n order to maxmze s s proft, t would be better f all buyers n the network could jon the sale. Tradtonally, the seller may pay some of her neghbours to advertse the sale to ther neghbours, but the neghbours may not brng any valuable buyers and cost the seller money for the advertsement. Therefore, our goal here s to desgn a a knd of costfree advertsng mechansm such that all buyers, who are aware of the sale, are ncentvzed to nvte all ther neghbours to jon the sale wth no guarantee that ther efforts wll be pad. L et al. [8] have shown that ths s achevable when =. n ths paper, we generalze ther approach to. Let us frst formally descrbe the model. Let θ = (v,r ) be the type of buyer N, θ = (θ,,θ n ) be the type profle of all buyers and θ be the type profle of all buyers except. θ can also be represented by (θ,θ ). Let Θ be the type space of buyer and Θ be the type profle space of all buyers. The advertsng mechansm conssts of an allocaton polcy π and a payment polcy x. The mechansm requres each buyer, who s aware of the sale, to report her valuaton to the mechansm and nvte/nform all her neghbours about the sale. Let v be the valuaton report of buyer and r r be the neghbours has nvted. Let θ = (v,r ) and θ = (θ,,θ n ), where θ j = nl f j has never been nvted by any of her neghbours r j or j does not want to partcpate. ven the acton profle θ of all buyers, π (θ ) {0,}, means that receves one tem, whle 0 means does not receve any tem. x (θ ) R s the payment that pays to the mechansm, x (θ ) < 0 means that receves x (θ ) from the mechansm. efnton.. ven an acton profle θ of all buyers, an nvtaton chan from the seller s to a buyer s a buyer sequence of (s,j,,j l,j l+,,j m,) such that j r s, for all < l m j l r j l, r j m and no buyer appears twce n the sequence,.e. t s acyclc. A buyer cannot nvte buyers who are not her neghbours and a buyer who has never been nvted by her neghbours cannot jon the sale, therefore not all acton profles are feasble. efnton.. ven the buyers type profle θ, an acton profle θ s feasble f for all N, θ nl f and only f there exsts an nvtaton chan from the seller s to followng the acton profle of θ. f θ nl, then r r. Let F (θ ) be the set of all feasble acton profles of all buyers under type profle θ. The advertsng mechansm (π,x) s defned only on feasble acton profles. n the followng, we defne the related propertes of the mechansm. efnton.. An allocaton π s feasble f for all θ Θ, for all θ F (θ ), for all N, f θ = nl, then π (θ ) = 0. N π (θ ). A feasble allocaton does not allocate tems to buyers who have never partcpated and t does not allocate more than tems. n the rest of ths paper, we only consder feasble allocatons. efnton.4. An allocaton π s effcent f for all θ Θ, for all θ F (θ ), π arg max π Π π (θ )v N,θ nl where Π s the set of all feasble allocatons. ven a buyer of type θ = (v,r ) and a feasble acton profle θ, the utlty of under a mechansm (π,x) s quaslnear and defned as: u (θ,θ, (π,x)) = π (θ )v x (θ ). For smplcty, we wll use u (θ,θ ) to represent u (θ,θ, (π,x)) as (π,x) s clear and does not change. We say a mechansm s ndvdually ratonal f for each buyer, her utlty s non-negatve when she truthfully reports her valuaton, no matter whch neghbours she nvtes and what the others do. That s a buyer should not lose as long as she reports her valuaton truthfully,.e. she s not forced to nvte her neghbours.

3 efnton.5. A mechansm (π,x) s ndvdually ratonal (R) f u (θ,θ ) 0 for all θ Θ, for all N, for all θ F (θ ) such that θ = (v,r ). fferent from the tradtonal mechansm desgn settngs, n ths model, we want to ncentvze buyers to not only just report ther valuatons truthfully, but also nvte all ther neghbours to jon the sale/aucton (the advertsng part). Therefore, we extend the defnton of ncentve compatblty to cover the nvtaton of ther neghbours. pecfcally, a mechansm s ncentve compatble (or truthful) f for all buyers who are nvted by at least one of ther neghbours, reportng ther valuatons truthfully to the mechansm and further nvtng all ther neghbours to jon the sale s a domnant strategy. efnton.. A mechansm (π,x) s ncentve compatble () f u (θ,θ ) u (θ,θ ) for all θ Θ, for all N, for all θ,θ F (θ ) such that θ = θ. ven a feasble acton profle θ and a mechansm (π,x), the seller s revenue generated by (π,x) s defned by the sum of all buyers payments, denoted by R (π,x ) (θ ) = N x (θ ). efnton.7. A mechansm (π,x) s weakly budget balanced f for all θ Θ, for all θ F (θ ), R (π,x ) (θ ) 0. n ths paper, we desgn a mechansm that s and R for the buyers to help the seller propagate the sale nformaton wthout beng pad n advance. THE NFORMATON FFUON MEHANM FOR = n ths secton, we revew the mechansm proposed by L et al. [8] for the case of =. L et al. consdered an advertsng mechansm desgn for a seller to sell a sngle tem n a socal network. The essence of ther approach s that a buyer s only rewarded for advertsng f her nvtatons ncrease socal welfare and the reward guarantees that nvtng all neghbours s a domnant strategy for all buyers. Ther nformaton dffuson mechansm s outlned below: nformaton ffuson Mechansm (M) () ven a feasble acton profle θ, dentfy the buyer wth the hghest valuaton, denoted by. () Fnd all dffuson crtcal buyers of, denoted by. j f and only f wthout j s acton θ j, there s no nvtaton chan from the seller s to followng θ j,.e. s not able to jon the sale wthout j. () For any two buyers,j { }, defne an order such that j f and only f all nvtaton chans from s to j contan. (4) For each { }, f receves the tem, the payment of s the hghest valuaton report wthout s partcpaton. Formally, let N be the set of buyers each of whom has an nvtaton chan from s followng θ, s payment to receve the tem s p = max j N θ j nl v j. 7 4 A B 4 E F H J L M 5 0 O P Q Fgure : A runnng example of the nformaton dffuson mechansm, where the seller s s located at the top of the graph and s sellng one tem, the value n each node s the node s prvate valuaton for recevng the tem, and the lnes between nodes represent neghbourhood relatonshp. Node Y s the node wth the hghest valuaton and, are Y s dffuson crtcal buyers. (5) The seller ntally gves the tem to the buyer ranked frst n { }, let l = and repeat the followng untl the tem s allocated. f s the last ranked buyer n { }, then receves the tem and her payment s x (θ ) = p ; else f v = p j, where j s the (l + )-th ranked buyer n { }, then receves the tem and her payment s x (θ ) = p ; otherwse, passes the tem to buyer j and s payment s x (θ ) = p p j, where j s the (l + )-th ranked buyer n { }. et = j and l = l +. () The payments of all the rest buyers are zero. Fgure shows a socal network example. Wthout any advertsng, the seller can only sell the tem among nodes A, B and, and her revenue cannot be more than 7. f A, B and nvte ther neghbours, these neghbours further nvte ther neghbours and so on, then all nodes n the socal network wll be able to jon the sale and the seller may receve a revenue as hgh as the hghest valuaton of the socal network whch s 0. Let us run M on the socal network gven n Fgure. Assume that all buyers report ther valuatons truthfully and nvte all ther neghbours, M runs as follows: tep () dentfes that the buyer wth the hghest valuaton s Y,.e. = Y. tep () computes = {,}. tep () gves the order of { } as. 8 Y 9 7

4 tep (4) defnes the payments p for all nodes n { }, whch are p =, p = 7 and p Y = 9, the hghest valuaton wthout, and Y s partcpaton respectvely. tep (5) frst gves the tem to node ; s not the last ranked buyer n { } and v p, so passes the tem to and her payment s p p = ; s not the last ranked buyer, but v = p Y, therefore receves the tem and pays p. All the rest of the buyers, ncludng Y, pay nothng. n the above example, M allocates the tem to node and pays 7, but s does not receve all the payment, and she pays an amount of for the advertsng. Therefore, the seller receves a revenue of from M, whch s more than two tmes the revenue she can get wthout any advertsng. Note that only buyer s rewarded for the nformaton propagaton as the other buyers are not crtcal for nvtng. L et al. showed that M has the followng desrable propertes. Theorem. (L et al. [8]). M s ncentve compatble (.e. reportng valuatons truthfully and nvtng all neghbours s a domnant strategy) and ndvdually ratonal. The revenue of the seller gven by M s at least the revenue gven by V wth the seller s neghbours only. 4 ENERALE M FOR ELLN MULTPLE TEM n ths secton, we present our generalsaton of M for a seller to sell multple homogeneous tems n a socal network. We assume that each buyer requres at most one tem. learly, we cannot smply run M multple tmes to solve the problem, as buyers who receve the tem earler would pay more, whch s not ncentve compatble. Before we ntroduce the mechansm, we need some addtonal concepts. efnton 4.. ven a feasble acton profle θ of all buyers, for any two buyers j N such that θ,θ j nl, we say s j s crtcal parent f wthout s partcpaton, there exsts no nvtaton chan from the seller to j. f s j s crtcal parent, then j s s crtcal chld. Let P (θ ) be the set of all crtcal parents of and (θ ) be the set of all crtcal chldren of under acton profle θ. f a buyer j P (θ ) does not nvte any of her neghbours r j, cannot jon the sale, whle f does not nvte any of her neghbours r, (θ ) cannot jon the sale. efnton 4.. ven a feasble acton profle θ F (θ ), we defne a partal order θ on all,j N such that θ j f and only f P j (θ ). t s clear that f θ j and j θ l, then θ l. Take the example gven n Fgure, f all buyers act truthfully,.e. θ = θ, for buyer M, we have P M (θ ) = {, }, M (θ ) = {O} and θ. Furthermore, t s easy to check that P A (θ ) = and A (θ ) =, as A s drectly lnked to s and wthout A s partcpaton, all other buyers can stll receve nvtatons. ven a feasble acton profle θ, we defne an optmal allocaton tree based on the effcent allocaton and ther crtcal parents. efnton 4.. ven θ F (θ ), an optmal allocaton tree of θ, denoted by T opt (θ ), s defned as: 7 4 A B 4 E F H J L M 5 0 O P Q Fgure : The optmal allocaton wth fve tems, where yellow nodes receve tems and red nodes are ther crtcal parents. WH= H M 8 4 W= 5 W= WM= Y Y W= W= WY= 7 W= Fgure : The optmal allocaton tree T opt (θ ) of the allocaton gven n Fgure, where the weght besde each node ndcates the number of tems allocated to and s crtcal chldren (θ ). T opt (θ ) s rooted at s, all buyers n { N π ef f (θ ) = }, denoted by N opt, and all ther crtcal parents P opt = N opt P (θ ) are the nodes of T opt (θ ), where π ef f s an effcent allocaton, the path from s to each node N opt P opt s gven by P (θ ) and the order θ n the form of (s,j,,j l,j l+,,j m,) where all j l,j l+ P (θ ) and j l θ j l+. f P (θ ) =, then the path s (s,). For each node n T opt (θ ), we defne a weght w (T opt (θ )) = {j N opt j = P j (θ )}, whch s the total number of tems allocated to and (θ ) under the effcent allocaton π ef f (θ ). Let hldren() be the set of all drect chldren of n T opt (θ ). Take the example gven n Fgure agan, f = 5 and all buyers act truthfully, we have the effcent allocaton as gven n Fgure and ts optmal allocaton tree T opt (θ ) gven n Fgure. 7

5 Now we are ready to defne our generalzaton of M. eneralzed nformaton ffuson Mechansm (M) ven the buyers type profle θ and ther acton θ F (θ ), compute the optmal allocaton tree T opt (θ ). Let W be the set of buyers who receve an tem n M (the wnners), ntally W =. For each W, we defne etfrom() N opt to be the buyer from whom the tem was taken by from the effcent allocaton. etfrom() = ndcates that takes the tem from herself. Allocaton: The allocaton s done wth a F-lke procedure. Let Q be a last n frst out (LFO) stack, ntally Q s empty. The seller s gves w (T opt (θ )) tems to each hldren(s) and adds all hldren(s) nto Q. Repeat the allocaton process defned n the followng untl Q s empty. Payment: For all N, her payment s: W (W v ) f W, W W f P j (θ ) \ W, j W 0 otherwse. where W s defned n the allocaton secton and W s defned by a feasble allocaton π as: Maxmse: W = π j (θ )v j j N ubject to: N = N \ = {} (θ ) j N receved,π j (θ ) = N receved = W P (θ ) j N out,π j (θ ) = 0 N out = {j N receved j = etfrom(l), l N r eceved } The ntuton behnd the allocaton of M s that f a buyer does not receve an tem n the effcent allocaton but her crtcal chldren receve tems, then the buyer may take an tem from one of her crtcal chldren, but not from any other buyer who s not her crtcal chld. Furthermore, the buyer only takes an tem f her valuaton s bg enough, otherwse passng t to her chldren gves her a hgher utlty. n the defnton of M, N receved s the set of s crtcal parents who have already receved an tem before tems are passed to. N out s the set of buyers who receve an tem n the effcent/optmal allocaton, but receve no tems under M as ther tems have been taken by ther crtcal parents n N receved. Buyer can take an tem n M f receves an tem n the socal-welfare-maxmsng allocaton when buyers from do not partcpate, s crtcal parents who have receved tems,.e. N receved, stll receve tems, and all buyers n N out except for do not receve tems. The Allocaton of M () Remove a node from Q, add to W f receves an tem n the followng feasble allocaton π: Maxmse: W = π j (θ )v j j N ubject to: N = N \ = (θ ) P( (θ ) ) (P( (θ ) )) P( (θ ) ) = {l l P j (θ ) θ l} j (θ ) (P( (θ ) )) = j (θ ) j P( (θ ) ) j N receved,π j (θ ) = N receved = W P (θ ) j N out,π j (θ ) = 0 N out = {j N receved j = etfrom(l), l N r eceved } where (θ ) s the set of top ranked crtcal chldren of accordng to ther reported valuaton (from hgh to low). f (θ ) <, then = (θ ) = (θ ). () f W : f j hldren() w j (T opt (θ )) = w (T opt (θ )), set etfrom() =, otherwse, let k = w (T opt (θ )), and out be the buyer wth the k -th largest valuaton report n the subtree (of T opt (θ )) rooted at and w out (T opt (θ )) 0, for all j P out (θ ) {out} f θ j, set w j (T opt (θ )) = w j (T opt (θ )), and set etfrom() = out. () For each chld j of, f w j (T opt (θ )) > 0, gve w j (T opt (θ )) tems to j and add j nto Q. Let (π M,x M ) be the allocaton polcy and payment polcy of the M. t s easy to verfy that M s the same as M when =. onsder the socal network gven n Fgure wth = 5, the M runs as follows. Frstly t computes the optmal allocaton treet opt (θ ) whch s depcted n Fgure, and ntalzes the weght w of each buyer n the tree accordng to w (T opt (θ )). Ths weght may be updated n the followng process. Then compute the allocaton of M whch s partally shown n Fgure 4: () Frstly, the seller gves tems to and tems to, and the stack Q contans, and the wnner set W s empty (shown n Fgure 4(a)). () Then buyer s popped out of Q and s dentfed as a wnner and added n W (shown n Fgure 4(b)). Here, s crtcal chldren (θ ) = {H,, J,M,O}, and = (θ ) = {H,, J,M,O}. Note that (θ ) and are

6 Q=[,] W={} 4 Q=[H,] W={} W= W= W= 9 W= 7 H M 5 W= 4 W= W=0 9 W= Y 0 WY= WM= Y 0 W= H Q=[] W={,H} WY= M 5 4 (b) W= W= Q=[,] W={,H} 4 W=0 9 W= 7 WH= M 5 Y 0 W= W= 5 W= H 7 WH= (a) W= that s crtcal chldren (θ ) = {E, F,,,L,Y,P,Q }, (θ ) = {,,L,Y,P }, P ( (θ ) ) = {F,,Y }, and (P ( (θ ) )) = {L,Y,P,Q }. Nr eceved = Nout =. Thus, = {F,,,L,Y,P,Q }. f we remove from N, W wll allocate tems to {H,M,,A,E}, and therefore cannot wn. (7) eep checkng for the rest of the buyers of,,y, we wll end up wth the allocaton gven n Fgure 5(). Lastly, compute the payments for all buyers accordng to the allocaton. The correspondng payments of the traders n T opt (θ ) are gven n Fgure 5(). All the other buyers who are not n W receve no tems and pay zero. WH= H W=0 W= 9 WY= 7 W= WH= M 5 (c) WY= Y 0 (d) Theorem 5.. The generalzed nformaton dffuson mechansm s ndvdually ratonal. Fgure 4: A runnng example of M Q=[] W={,H,,Y,} 4 W= W=0 Q=[] W={,H,,Y,} H W= 9 W= 4 W= X=0 W=0 X=0 W= X=- WH= M 5 Y 0 7 WY= W= H 9 W= X=0 WH= XH=5 M 5 () X=0 Y 0 7 W= X=4 WY= XY=4 () Fgure 5: The outcomes of M normally not the same f the graph becomes complex. r eceved = N out =. f we remove from N, then N wll receve an tem under W, so add to W. PROPERTE OF M n ths secton, we prove that our generalzed nformaton dffuson mechansm s ndvdually ratonal, ncentve compatble and mproves the seller s revenue, compared wth the revenue the seller can get wthout any advertsng. Therefore, the seller s ncentvzed to apply our mechansm. () n step () of the allocaton process, as dd not receve an tem n the optmal allocaton, we have to remove one tem ntally allocated to s crtcal chldren by the optmal allocaton. We choose the chld wth the lowest valuaton among all s crtcal chldren who stll have tems, whch s buyer M, and update the weghts of M and M s crtcal parents to (as shown n Fgure 4(b)). ntutvely, takes the tem from M, so we set etfrom() = M. (4) n step () of the allocaton process, gves H one tem and adds H nto Q. (5) Then n the next teraton, buyer H s popped out and s added nto W (as shown n Fgure 4(c)). Here H =, N Hr eceved = {} and N Hout = {M }. () Then s popped out and s not dentfed as a wnner. just gves tems to and tem to and adds, nto Q (as shown n Fgure 4(d )). t s worth mentonng Proof. ven the buyers type profle θ and ther acton profle θ F (θ ), to prove M s ndvdually ratonal, we need to show that for all N, u (θ,θ, (π M,x M )) 0 for all θ = (v,r ). From the defnton of M, for any buyer N, we have ether u (θ,θ ) = 0 or u (θ,θ ) = W W. Accordng to the defntons of W and W, we have W W, because N N and the optmzaton under N cannot be worse than that under N. Therefore, we have u (θ,θ ) 0. Theorem 5.. The generalzed nformaton dffuson mechansm s ncentve compatble,.e. reportng valuaton truthfully and nvtng all neghbours s a domnant strategy for all buyers who are aware of the sale. Proof. ven the buyers type profle θ and ther acton profle θ F (θ ), to prove M s ncentve compatble, we need to show that for each buyer N such that θ, nl: fx s nvtaton to be r, reportng v truthfully maxmse s utlty. fx s valuaton report to be v, nvtng all s neghbours r maxmse s utlty. We wll prove the above for buyers n three dfferent groups: () all buyers who receve one tem,.e. W. () all buyers who are not n W, but are crtcal parents of W,.e. W P (θ ) \ W. () all buyers who are not n the frst two groups. roup (): for each W, her utlty s u (θ,θ ) = v + (W v ) W. Fx s nvtaton to be r, then s fxed.

7 f reports v truthfully,.e. v = v, then u (θ,θ ) = W W. nce W s the optmal socal welfare under the constrants that cannot nfluence, f can msreport v to change the allocaton to ncrease v + (W v ), then t contradcts that W s the optmal socal welfare. Furthermore, W s ndependent of and we have W W as N N. Therefore, s utlty s maxmsed as soon as s stll n W. Now f msreports v such that does not receve an tem and becomes a crtcal parent of W n group (). n ths case, s utlty s u (θ,θ ) = W W. nce v s not consdered any more n W, t means that W s at most the socal welfare when v s consdered. Therefore, the utlty s not better than reportng v. Lastly f msreports v such that s n group (), then u (θ,θ ) = 0, whch s not better than reportng v truthfully. Therefore, fxng s nvtaton to be r, reportngv truthfully maxmzes s utlty. Fx s valuaton report to be v, change s nvtaton to be any subset r of r. For any θ = (v,r ) and θ = (v,r ) such that r r r, then we have (θ,θ ) (θ,θ ), because nvtng more neghbours wll brng more buyers to jon the sale. When we remove the buyers whose valuaton s among the top largest from (θ,θ ) and (θ,θ ) respectvely, let the correspondng be,r,r and. nce (θ,θ ) (θ,θ ), we have more buyers whose acton s not nl n N,r than those n N,r. Therefore, we get W,r.e. W,r s maxmsed when θ = θ. W,r, s maxmsed when r = r. Thus, u (θ,θ ) roup (): for each j W P j (θ ) \ W, her utlty s u (θ,θ ) = W W. Fx s nvtaton to be r, then s fxed. f msreports v, but s stll n group (), then u (θ,θ ) does not change, as both W and W are ndependent of v. f msreports v such that W, then u (θ,θ ) = v + (W v ) W. f v + (W v ) s greater than W when reports v, t contradcts that W s optmal. Thus, s utlty s not better than reportng v. cannot msreports v to become a member of group (). Fx s valuaton report to be v, nvtng all neghbours r maxmses s utlty. The proof s the same as the proof for group (). roup (): for each who s not a member of group () or group (), her utlty s u (θ,θ ) = 0. Fx s nvtaton to be r, may msreport v to become a member of W. However, n ths case, u (θ,θ ) = v + (W v ) W. We know that f reports v truthfully, we have W = W. f msreports to get v + (W v ), then v + (W v ) must be less than or equal to W. Therefore, t s not worth msreportng v. Fx s valuaton report to be v, f nvtng all r does not brng to group () or group (), then nvtng less wll keep n group (). Therefore, nvtng all r s the best that can do to optmse her utlty. Next we show that the seller s revenue s mproved wth M compared wth the revenue she can get wth other truthful mechansms wthout advertsng, especally we compare t wth V. Wthout advertsng, the seller can only sell the tems to her neghbours r s. Assume that r s >, then the revenue of applyng V among r s s R V = v +, where v + s the ( + )- th largest valuaton report among r s. Under V, may mprove the revenue by sellng less. No matter how many tems the seller chooses to sell under V, let be the actually number of tems that the seller s sellng under both V and M. Theorem 5. proves that the revenue of M s not less than the revenue of V when the number of neghbours of the seller s more than. Note that when the number of the seller s neghbours s less than or equal to, the revenue of V s zero. Theorem 5.. The revenue of the generalsed nformaton dffuson mechansm s greater than or equal to v +, where v + s the ( + )-th largest valuaton report among r s, assume that r s >. Before provng the theorem, let us frst show some relatonshp between the payment of a buyer and the payments of her drect chldren n the optmal allocaton tree. Frst of all, gven the buyers acton profle θ, each buyers s payment under the generalsed nformaton dffuson mechansm s equal to: (W V N stll ) (W V N stll v ) f W, (W V N stll ) (W V N stll ) f P j (θ ) \ W, j W 0 otherwse where V N stll = v j j N stll N stll = (N opt \ ( N out )) N receved = {} (θ ) N receved = W P (θ ) N out = {j N receved j = etfrom(l), l N receved } n the payment defnton of M, terms W and W both count the valuatons of all buyers n N stll. t s clear that

8 N stll = k. Therefore, we can remove all the valuatons of N stll from both W and W to get above form of the payments. Followng ths, Lemma 5.4 shows the deeper relatonshp. Lemma 5.4. ven all buyers acton profle θ, under the generalsed nformaton dffuson mechansm, for all j W P j (θ ) \W, let k be the number of tems passed to, and there are m chldren of who have receved tems from denoted by {,, m }, and let k l be the number of tems gves to l, we have W V N stll l (W l V N stll l ) Proof. ven the above form of the payments, we have W s the sum of the top k hghest valuatons among buyers V N st ll n N \ ( N stll ). For each l, W l V N stll s the sum of l the top k l hghest valuatons among buyers n N \ ( l N stll l ). nce W,.e. does not receve any tem, we have k = l k l, N stll N stll l and N stll \N stll l we have l l and N stll l we get l N stll l N stll s the sum of the top k hghest valua- N \ ( N stll ). nce W tons n N \ ( V N stll = k k l. nce k k l, \ N stll, therefore and N \ ( l N stll ) N stll ), whle W l V N stll l the top k l hghest valuatons n a larger set N \ ( l N stll conclude that W V N stll l s the sum of ), we l l (W l V N stll ). l Followng Lemma 5.4, we can further prove that for all buyers W, (W V N stll v ) (W l V N stll ) l l where k = l k l because receves an tem. Furthermore, f k =, then does not gve any tem to her chldren and we have W V N st ll v = 0. Proof Theorem 5.. Followng 5.4, we conclude that for any buyers j W P j (θ ) W, the second term of s payment (W V N st ll or W V N stll v ) s ether 0 or offset by the sum of all the frst terms of the payments of s chldren accordng to Lemma 5.4. Therefore, the sum of all buyers payments s equal to (W V N stll ) + hldren(s) where 0. t s the sum of all the frst terms of the payments of the seller s drect chldren n the optmal allocaton tree T opt (θ ) plus the remanng of all the offsets. Assume s has m chldren n T opt (θ ), denoted by {s,,s m }. The number of tems passed to them are k s,,k sm, where s k s =. For all s {s,,s m }, we have W s V N stll s the s sum of the top k s hghest valuatons among all buyers n N \ ( s Ns stll ). Both s and Ns stll may contan some buyers from s s neghbours r s, but these two sets cannot contan all r s,.e. r s s Ns stll as there are at most tems and r s >, r s s Ns stll f and only f or one of s crtcal chld has the top hghest valuaton report among all buyers N. pecfcally, for each s, s contans at most one buyer from r s, and Ns stll contans at most k s buyers from r s. Therefore, the mnmum of the top k s hghest valuatons among all buyers n N \( s Ns stll ) s at least v +, the top ( + )-th largest valuaton among all buyers n r s. Thus, we have (W V N stll ) + v + hldren(s) ONLUON We have proposed an aucton mechansm that gves sale promotons to a seller to sell multple homogeneous tems va a socal network. t generalses the mechansm proposed by L et al. [8] for a sngle tem settng. The mechansm s run by the seller, and she does not need to pay n advance for gettng the promotons. The mechansm ncentvzes all buyers who are aware of the sale to do free promotons to ther neghbours, because ther promotons wll be rewarded f some buyers nvted by them buy the tems n the end. Besdes the free advertsng part, all buyers wll also truthfully report ther valuatons to compete for the sale wth people they have nvted. Eventually, buyers who are closer to the seller wll have a hgher lkelhood to wn tems than ther chldren, because ther chldren cannot partcpate n the sale wthout ther promotons/nvtatons. Ths s the key to guarantee that all buyers are happy to nvte more buyers to compete wth themselves for the lmted resources. nce buyers who are closer to the sellers have the ablty to control whch neghbours they want to promote to, they can also control how the tems are allocated. For example, when a seller sells multple dentcal tems to fxed number of buyers, the seller can choose to sell less wth hgher payments to maxmse her revenue. Ths also apples to the buyers n our settng and they can nvte more or less neghbours to control how many tems are sold to ther chldren, whch may gve them dfferent rewards/utltes. On the other hand, buyers chldren can also manpulate n order to satsfy ther parents needs. Therefore, t s extremely challengng to defne a truthful mechansm n more complex settngs. To prevent buyers manpulatons mentoned above, we have carefully chosen the allocaton and payments. n partcular, the defnton of for each buyer n M plays the essental role to stop ther manpulatons. ven, buyer s payment does not depend on how many tems her chldren get, therefore, she s not ncentvsed to control how many tems her chldren wll get. n ths paper, we assumed that each buyer only requres at most one tem. We wll easly lose the control f they requre more than one tem wth dfferent margnal valuatons. Offerng truthful mechansms for general combnatoral valuaton settngs s hghly demanded. Furthermore, we assumed that nvtng neghbours n the socal network does not ncur a cost, e.g. postng an advertsement va facebook or twtter. However, there mght be a cost to do so, so a new mechansm wll be requred to guarantee that buyers promoton costs wll be covered. A specal socal network wth publc dffuson/transfer costs was studed by L et al. [9], but coverng dffuson costs n general networks s stll open.

9 REFERENE [] Edward H larke. 97. Multpart prcng of publc goods. Publc choce, (97), 7. [] Benjamn Edelman, Mchael Ostrovsky, and Mchael chwarz nternet Advertsng and the eneralzed econd-prce Aucton: ellng Bllons of ollars Worth of eywords. Amercan Economc Revew 97, (007), [] Yuval Emek, Ron ard, Moshe Tennenholtz, and Avv Zohar. 0. Mechansms for mult-level marketng. n Proceedngs of the th AM conference on Electronc commerce. AM, [4] Andrew V oldberg and Jason Hartlne. 00. ompettve auctons for multple dgtal goods. n European ymposum on Algorthms. prnger, [5] Andrew V oldberg, Jason Hartlne, and Andrew Wrght. 00. ompettve auctons and dgtal goods. n Proceedngs of the twelfth annual AM-AM symposum on screte algorthms. ocety for ndustral and Appled Mathematcs, [] Theodore roves. 97. ncentves n teams. Econometrca: Journal of the Econometrc ocety (97), 7. [7] avd empe, Jon lenberg, and Éva Tardos. 00. Maxmzng the spread of nfluence through a socal network. n Proceedngs of the nnth AM nternatonal conference on nowledge dscovery and data mnng. AM, 7 4. [8] Bn L, ong Hao, engj Zhao, and Tao Zhou. 07. Mechansm esgn n ocal Networks. n Proceedngs of the Thrty-Frst AAA onference on Artfcal ntellgence [9] Bn L, ong Hao, engj Zhao, and Tao Zhou. 08. ustomer harng n Economc Networks wth osts. n Proceedngs of the 7th nternatonal Jont onference on Artfcal ntellgence and the rd European onference on Artfcal ntellgence. [0] Roger B Myerson. 98. Optmal aucton desgn. Mathematcs of Operatons Research, (98), [] Noam Nsan, Tm Roughgarden, Eva Tardos, and Vjay V Vazran Algorthmc game theory. Vol.. ambrdge Unversty Press ambrdge. [] alen Pckard, We Pan, yad Rahwan, Manuel ebran, Rley rane, Anmol Madan, and Alex Pentland. 0. Tme-crtcal socal moblzaton. cence 4, 055 (0), [] Everett M Rogers. 00. ffuson of nnovatons. mon and chuster. [4] Hal R Varan Onlne ad auctons. The Amercan Economc Revew 99, (009), [5] Wllam Vckrey. 9. ounterspeculaton, auctons, and compettve sealed tenders. The Journal of Fnance, (9), 8 7.