Design Multicast Protocols for Non-Cooperative Networks

Transcription

1 Deign Multicat Protocol for Non-Cooperative Networ WeiZhao Wang Xiang-Yang Li Zheng Sun Yu Wang Abtract Conventionally, mot networ protocol aume that the networ entitie who participate in the networ activitie will alway behave a intructed. However, in practice, mot networ entitie will try to maximize their own benefit intead of altruitically contribute to the networ by following the precribed protocol, which i nown a elfih. Thu, new protocol hould be deigned for the non-cooperative networ which i compoed of elfih entitie. In thi paper, we pecifically how how to deign trategyproof multicat protocol for noncooperative networ uch that thee elfih entitie will follow the protocol out of their own interet. By auming that a group of receiver i willing to pay to receive the multicat ervice, we pecifically give a general framewor to decide whether it i poible, and how if poible to tranform an exiting multicat protocol to a trategyproof multicat protocol. We then how how the payment to thoe relay entitie are hared fairly among all receiver o that it encourage collaboration among receiver. A a running example, we how how to deign the trategyproof multicat protocol for the currently ued core-baed multicat tructure. We alo conduct extenive imulation to tudy the relation between payment and cot of the multicat tructure. Index Term Control theory, combinatoric, economic, noncooperative, multicat, payment, haring. I. INTRODUCTION Multicat ha received coniderable attention over the pat few year due to it reource haring capability. In multicat, there i a topology, either a tree or a meh, that connect the ource to a et of receiver, and pacet i only duplicated at the branching node. Numerou multicat protocol have been propoed, and mot of them aumed that the networ entitie, either lin or node, will relay the multicat pacet a precribed by the multicat protocol without any deviation. However, the Internet, which i compoed of different heterogenou and autonomou ytem (AS), raie a doubt about thi common belief. Although multicat benefit the whole ytem by aving bandwidth and reource, it i dubiou that multicat will alo bring benefit to every individual node or lin who relay the pacet. Thu, it i more reaonable to aume that thee AS, probably owned by ome organization and private uer, are elfih: aim to maximize their own benefit intead of faithfully conform the precribed multicat protocol. A networ compoed of elfih AS i generally nown a a non-cooperative networ. Department of Computer Science, Illinoi Intitute of Technology, Chicago, Illinoi, USA. wangwei4@iit.edu, xli@c.iit.edu. The wor of Xiang-Yang Li i partially upported by NSF CCR-374. Department of Computer Science, Hong Kong Baptit Univerity, Hong Kong, China. unz@comp.hbu.edu.h Department of Computer Science, Univerity of North Carolina at Charlotte, North Carolina, USA. ywang3@uncc.edu Nian and Ronen [] tudied the unicat routing problem in non-cooperative networ and introduced the idea of algorithmic mechanim deign: they propoed to give the AS ome proper payment to enure that every AS conform to the precribed protocol regardle of all other AS behavior, which i nown a trategy-proof or truthful. They deigned the payment for unicat by uing the VCG mechanim[], [3], [4], which i conidered a one of the mot poitive reult in algorithm mechanim deign. Unfortunately, VCG mechanim ha it own drawbac. For multicat, if we want to apply VCG mechanim, the multicat tree hould have the leat cot among all tree panning the receiver. However, finding the minimum cot multicat tree i nown to be NP-Hard for both edge weighted networ [4], [5] and node weighted networ [6], [7]. If we init on applying the VCG mechanim to a multicat topology that doe not have the minimal cot, VCG mechanim may fail [8]. Thu, ome payment cheme other than VCG mechanim hould be deigned for multicat. Recently, in [8], the author propoed everal non-vcg trategy-proof payment cheme for everal commonly ued multicat tree. In thi paper, intead of focuing on ome pecific multicat tructure, we tudy whether it i poible to tranform a multicat protocol baed on any given multicat topology to a trategyproof multicat protocol, and if poible, how to deign the trategyproof protocol. Deigning a truthful payment cheme i not the whole tory for many practical application. A natural quetion ha to be anwered i who will afford the payment. A imple olution i that the organization to which the receiver belong pay [8]. However, thi olution i not panacea. In many application uch a video treaming, often the individual receiver have to pay the relay agent to receive the data. How to charge the receiver for multicat tranmiion ha been tudied extenively in literature [9], [], [], [], [3], [4]. In mot of their model, they aumed that ) every receiver ha a valuation for receiving the data and the receiver i elfih, ) all relay agent are cooperative and reveal their true cot, and 3) the multicat tree i formed by the union of the hortet path from the ource to receiver. In the harp contrat, in thi paper, we alo tae the elfih behavior of the relay node (or lin) into account. Thu, we model the networ differently by auming that ) the relay agent are elfih, ) the receiver alway receive the data and pay what they hould pay, and 3) the multicat topology could be any tructure pecified by ome exiting multicat protocol, including tree and mehe. To the bet of our nowledge, thi i the firt paper to conider multicat pricing when the relay agent are non-cooperative. Notice that there i a poible wor

2 left for future exploration: what happen if both the receiver and relay agent are elfih and each receiver ha a valuation and would receive the data if and only if it valuation i greater than what it need to pay according to a trategyproof multicat protocol. One thing we hould point out i that algorithmic mechanim deign i not the only way to achieve trategyproofne. There are lot of literature which ue Nah equilibrium, a tate at which no agent can improve it utility by unilaterally deviating from it current trategy when other agent eep their trategie. Since Nah equilibrium ha a wea requirement for the trategie ued by the agent, it often can achieve a wider variety of outcome. The main contribution of thi paper are two-folded. Firt, we preent a general framewor about whether it i poible, and how if poible, to tranform an exiting multicat protocol to a trategyproof one. We then how how the payment to the relay agent are hared fairly among the receiver. A a running example, we how how to deign a trategyproof multicat protocol, and how the payment are hared among receiver when the leat cot path tree i ued for multicat. We alo conduct extenive imulation to tudy the relation between payment and cot of the multicat tructure. Our imulation how that by only overpaying a mall amount to the relay node (or lin), each relay node (or lin) will declare it true cot to maximize it profit. The ret of the paper i organized a follow. We introduce ome preliminarie, related wor, our communication model, and the problem to be olved in Section II. In Section III, we dicu the exitence of truthful payment and how to find it if a given multicat tructure i ued. We how how to deign a truthful multicat protocol baed on a pecific routing topology in Section IV. Several other important iue are dicued in Section V. The performance tudy of our propoed truthful core-baed multicat protocol i preented in Section VI. We conclude our paper in Section VII. II. TECHNICAL PRELIMINARIES A. Algorithmic Mechanim Deign In a tandard model of algorithm mechanim deign, there are n agent {,,,n}. Each agent i {,,n} ha ome private information t i, called it type, e.g. it cot to forward a pacet in a networ environment. All agent type define a profile t (t,t,,t n ). Each agent i declare a valid type τ i which may be different from it actual type t i and all agent trategy define a declared type vector τ (τ,,τ n ). A mechanim M (O, P) i compoed of two part: an output function O that map a declared type vector τ to an output o and a payment function P that decide the monetary payment p i P i (τ) for every agent i. Each agent i ha a valuation function w i (t i,o) that expreed it preference over different outcome. Agent i utility or called profit i u i (t i,o)w i (t i,o)+p i. An agent i i aid to be rational if it alway chooe it trategy τ i to maximize it utility u i. Let τ i (τ,,τ i,τ i+,,τ n ), i.e., the trategie of all other agent except i and τ i t i (τ,τ,,τ i,t i,τ i+,,τ n ). A mechanim i trategyproof if for every agent i, revealing it true type t i will maximize it utility regardle of what other agent do. In thi paper, we are only intereted in mechanim M (O, P) that atify the following three condition: ) Incentive Compatibility (IC): For every agent i, w i (t i, O(τ i t i ))+p i (τ i t i ) w i (t i, O(τ))+p i (τ) τ. ) Individual Rationality (IR): It i alo called Voluntary Participation. Every participating agent mut have a nonnegative utility, i.e., w i (t i, O(τ i t i )) + p i (τ i t i ). 3) Polynomial Time Computability (PC): O and P are computed in polynomial time. VCG MECHANISM: Arguably the mot important poitive reult in mechanim deign i what i uually called the generalized Vicrey-Clare-Grove (VCG) mechanim [], [3], [4]. A direct revelation mechanim M (O(t), P(t)) belong to the VCG family if () the output O(t) computed baed on the type vector t maximize the objective function g(o, t) i w i(t i,o), and () the payment to agent i i P i (t) j i w j(t j, O(t))+h i (t i ). Here h i () i an arbitrary function of t i. It i proved in [4] that a VCG mechanim i truthful, i.e., atifying the IC property. Under mild aumption, VCG mechanim are the only truthful implementation [5] for utilitarian problem, i.e., g(o, t) i w i(t i,o). B. Networ Model and Problem Statement Conider any communication networ G (V,E,c), where V {v,,v n } i the et of communication terminal, E {e,e,,e m } are the et of lin. Every agent i in the networ ha a private cot c i to tranmit a unit ize of data. Here agent could be either terminal or lin whoever could behave elfihly. If agent are terminal then G i node weighted; if agent are lin then G i lin weighted. Given a et of terminal Q {q,q,,q r } V who are willing to receive the data, we will deign a multicat protocol that ) contruct a topology (a tree, a meh, a ring, etc) that pan thee receiver; ) calculate a payment for each relay agent according to a payment cheme that i trategy-proof; 3) charge each receiver according to a pricing cheme that i reaonable. We will formally define what i reaonable in ubection III-C. For the convenience of our analyi, we aume that q i the ource node in one pecific multicat and the ize of the data i normalized to. We alo aume throughout thi paper that agent in the networ will not collude to improve their profit together. In order to prevent the monopoly, we aume the networ i bi-connected. One thing we hould highlight here i that intead of reinventing the wheel by deigning ome new multicat tructure, we focu on how we can deign a truthful payment cheme for the exiting multicat protocol to enure that they wor correctly even in non-cooperative networ. Baed on the truthful payment cheme we deigned, we further tudy how we charge the receiver in a reaonable way. Given a tructure H G, weueω(h) to denote the um of the cot of all agent in thi networ. If we change the cot of any agent i (lin e i or node v i )toc i, we denote the new networ a G (V,E,c i c i ), or imply c i c i. If we remove

3 one agent i from the networ, we denote it a c i. Denote G\e i a the networ without lin e i, and denote G\v i a the networ without node v i and all it incident lin. For the implicity of notation, we will ue only the cot vector c to denote the networ G (V,E,c) if no confuion i caued. C. Related Wor Routing ha been part of the algorithmic mechanim-deign from the very beginning. Nian and Ronen [6] provided a polynomial-time trategyproof mechanim for unicat routing in a centralized computational model. Each lin e of the networ i an agent and ha a private cot t e of ending a meage. Their mechanim i eentially a VCG mechanim. The reult in [6] i extended in [5] to deal with unicat problem for all pair of terminal. They aume there i a traffic demand T i,j from a node i to a node j. They alo gave a ditributed method to compute the payment. Anderegg and Eidenbenz [6] recently propoed a imilar routing protocol for wirele ad hoc networ baed on VCG mechanim again. In [9], Wang and Li propoed an aymptotically optimum centralized method to compute the payment for unicat and howed that there i truthful mechanim that can prevent colluion. For multicat, Feigenbaum et. al [3] aumed that there i a univeral tree panning all receiver and for every ubet R of receiver, the panning tree T (R) i merely a part of the univeral tree induced by receiver et R. They alo aumed that there i a publicly nown lin cot aociated with each communication lin and receiver q i will report a number w i, which i the amount of money he/he i willing to pay to receive the data, which may be different from it true privately nown valuation w i. The ource node then elect a ubet R Q of receiver according to ome criteria. They tudied how to elect receiver and propoed to ue Shapely value and marginal cot to hare the lin cot. Maximizing profit in trategy-proof multicat wa tudied in [7], [8] ([8] i baed on cancellable auction [9]). Sharing the cot of the multicat tructure among receiver wa tudied in [], [7], [], [], [4], [3] o ome fairne i accomplihed. In [8], Wang et al. tudied how to deign trategyproof multicat protocol for variou multicat tree when the relay terminal or lin are elfih and the receiver will relay the data for peer receiver for free. III. CHARACTERIZATION OF TRUTHFUL MULTICAST ROUTING Several multicat topologie have been propoed and ued in practice and it i expected that more topologie will be propoed in the near future. It will be difficult if not impoible to deign a trategyproof multicat mechanim for each of thee topologie individually. Thu, intead of tudying ome pecific multicat topologie, we preent a general framewor to decide whether there i, and how to deign if it exit, a trategyproof mechanim for any given multicat topology. We alo conider how to charge the receiver to cover the total payment to the relay agent. Intuitively, we may till want to ue the VCG payment cheme for thee multicat topologie. Notice that an output function of a VCG mechanim i required to maximize the total valuation of agent. Thi mae the mechanim computationally intractable in many cae, e.g., multicat. Notice that replacing the optimal algorithm with non-optimal approximation uually lead to untruthful mechanim [8]. Thu a mechanim other than VCG i needed when we cannot find the optimal olution or the objective i not to maximize the total valuation of all agent. Thi paper preent the firt general framewor to deign trategyproof mechanim for multicat in which we cannot find the tructure with minimum total cot. A. Exitence of the Truthful Payment Mechanim Before we deign ome truthful payment cheme for a given multicat topology, we hould decide whether uch payment cheme exit or not. Following definition and theorem will preent a ufficient and neceary condition for the exitence of the truthful payment cheme. Definition : A method O computing a multicat topology atifie the monotone property (MP) if for every agent i and fixed c i, following condition i atified: If agent i i elected a a relay agent with cot c i, then it i alo elected with a maller cot c i. Obviouly, the above condition i equivalent to the following condition: There exit a threhold value κ i (O,c i ) uch that if i i elected a a relay agent, then it cot i at mot κ i (O,c i ). For the convenience of our preentation, we ue O i (c) (repectively ) to denote that agent i i elected (repectively not elected) to the multicat topology when the cot vector i c. Theorem : Given a method O computing a multicat topology, there exit a payment P uch that M (O, P) i trategyproof iff O atifie monotone property. Proof: We firt prove if there exit a truthful payment baed on O then O atifie the monotone property. We prove it by contradiction by auming there i a truthful payment cheme P baed on O that doe not atify MP. From the definition of MP, there exit an agent i and two cot vector c i c i and c i c i, where c i <c i uch that O i (c i c i )and O i (c i c i ). Let p i (c i c i )p i and p i(c i c i )p i. Conider a networ with a cot vector c i c i, the utility for agent i when it reveal it true cot i u i (c i )p i. When agent i lie it cot to c i, it utility become p i c i. Since payment cheme P i truthful, we have p i >p i c i. Similarly we conider another networ with a cot vector c i c i. Agent i utility i p i c i when it reveal it true cot. Similarly, if it lie it cot to c i, it utility i p i. Since payment cheme P i truthful, p i <p i c i. Thu, we have p i c i >p i >p i c i. Thi inequality implie that c i >c i, which i a contradiction. We then prove that if O atifie the monotone property then there exit a truthful payment baed on O. We prove it by contructing the following payment cheme P for a given a networ G (V,E,c).

4 Algorithm Payment Scheme P : For any agent i not elected to relay, it payment i. : For any agent i elected to relay, it payment i κ i (O,c i ). From the definition of MP, the IR property i obviou. Thu we only need to prove that the payment cheme P atifie IR. We prove it by cae. Cae : Agent i lie it cot upward to c i or downward to c i, but it doe not change the output whether agent i i elected or not. Notice for fixed c i, when the output of agent i doe not change, it payment i the ame. Thu, agent i utility eep the ame which in turn implie that agent i doe not have incentive to lie in thi cae. Cae : Agent i i elected when it reveal it actual cot c i, and it lie it cot upward to c i uch that it i not elected. From the property of MP, we now c i κ i (O,c i ). Thi enure that agent i get non-negative utility when it reveal it actual cot c i. When i lie it cot to c i, it get zero payment and zero utility. Therefore, agent i won t lie in thi cae. Cae 3: Agent i i not elected when it reveal it actual cot c i, and it lie it cot downward to c i uch that it i elected. Similarly, we have c i κ i (O,c i ), which implie that agent i get a non-poitive utility. Comparing with the zero utility when agent i reveal it true cot, agent i alo ha no incentive to lie in thi cae. Thi finihe our proof. Actually, if we require that relay agent who are not elected hould receive zero payment, our payment cheme illutrated by Algorithm i the only trategy-proof payment cheme. The proof i omitted here due to pace limit. B. Rule to Find the Truthful Payment Scheme Given a multicat tructure atifying MP, it eem quite imple to find a truthful payment cheme by applying Algorithm. However, ometime the proce to find the threhold value in Algorithm i far more complicated. A to our nowledge, our approach preented later i the firt ever effort to find the threhold value efficiently. Intead of trying to give a unified approach that can find the threhold value for all multicat topologie atifying MP, we preent ome ueful technique to find threhold value under certain circumtance. Our general approach wor a follow. Firt, given an output method O that compute a multicat tructure, we decompoe it into everal impler output method. We then find the threhold value for each of the decompoed method. Finally, we calculate the original threhold value by combining the threhold value for thoe decompoed method. ) Simple Combination: Given a multicat method O, let κ(o,c) denote a n-tuple vector (κ (O,c ),κ (O,c ),,κ n (O,c n )). Here, κ i (O,c i ) i the threhold value for agent i when the multicat topology i computed by O and the cot c i of all other agent are fixed. We then preent a imple but ueful technique to find the threhold value. Theorem : Given n multicat method O,, O n atifying monotone property, and κ(o i,c) i the threhold value for O i where i n. Then the output method O(c) O (c) O (c) O n (c) alo atifie monotone property. Moreover, the threhold value for O i κ(o,c) max i n {κ(oi,c)}. The proof of thi theorem i quite imple and i omitted here. We will how how to ue thi imple combination technique in Section IV. ) Round-baed Method: Many multicat topologie are contructed in a round-baed manner: for each round they elect ome unelected agent, update the problem and the cot profile if neceary. Following i a general characterization of a round-baed method that contruct a multicat topology. Algorithm A Round-Baed Multicat Method : Set r and c () c and Q () Q initially. : repeat 3: Let O r be a determinitic method that decide in round r whether agent i i elected or not. 4: Update the networ cot vector and receiver et, i.e., we obtain a new networ cot vector c (r+) and receiver et Q (r+) according to a update rule U r : U r : O r [c r,q (r) ] [c (r+),q (r+) ]. 5: until the deired property of the multicat topology i met 6: Return the union of the relay agent in all round a the final output. Here, every agent can be elected at mot once. To help the undertanding of general round-baed method, we preent a multicat topology that i contructed in uch way. The example we ued i the polynomial time method in [5] that find a multicat topology whoe cot i no more than time of the minimum cot Steiner tree (MCST) in a lin weight networ. For the completene of our preentation, we review their method here. Algorithm 3 Lin Weighted Multicat Structure [5] : repeat : Find one receiver in the receiver et Q, ay q i, that i cloet to the ource, i.e., the LCP(, q i,d) ha the leat cot among the hortet path from to all receiver. 3: Connect q i to the ource uing the leat cot path between them. Update the cot of all edge on thi path a. Remove q i from the receiver et Q. 4: until no receiver remain Here no receiver remain correpond to the deired propertie of general round-baed method; LCP (, q i,d) in round r correpond to O r ; updating cot of edge on LCP (, q i,d) to and removing q i from Q i the update rule U r. Figure how how to apply Algorithm 3. Initially, the receiver et i Q Q {q,q } and the lin cot are hown in Figure. The firt round elect the nearet receiver q from

5 , and it correponding path q i elected. Remove q from Q and et cot of lin and q to. The networ in the end of firt round i hown in Figure (b). In the econd round, the receiver et i Q {q }, and the leat cot path from to q i q intead of the leat cot path q in original networ. The final multicat tree, hown a olid line in Figure, i the union of the two path..5 v v q q q q q q (a) The original (b) Networ after (c) Networ after networ firt round econd round Fig...5 Illutration of Algorithm (3). Here, i the ource node. Definition : An updating rule U r i aid to be croingindependent if for any unelected agent i: c (r+) i and Q (r+ ) do not depend on c (r) i. Fixed c (r) i,ifd(r) i c (r) i then d (r+) i c (r+) i. Theorem 3: A round-baed multicat method O atifie MP if, for every round r, method O r atifie MP and the updating function U r i croing-independent. Proof: For an agent i, fix the original cot c i of all other agent. We prove that if i i elected when the original cot vector i a {c i,c i }, then it i alo elected when the original cot vector i b {c i,c i } uch that c i <c i. Without lo of generality aume that i i elected in round r under cot vector a. Then under cot vector b, if agent i i elected before round r, our claim hold. Otherwie, in round r, a (r) i b (r) i and a (r) i >b (r) i ince agent i i not elected in the previou round. Notice i i elected in round r under cot vector a (r) i, thu i i alo elected in round r under cot vector b (r) i from the monotone property of the method O r. Thi finihe the proof. Theorem 3 preent a ufficient condition for the exitence of truthful payment cheme for a round-baed multicat method. Following, we how how to find the threhold value for any elected agent. The proof of the correctne of thi algorithm i omitted here due to the pace limit, refer to the full verion for detail. q.5 v q (a) The multicat tree q (.4).5.4(.9) v (.5) 5.6(.).5 (.5) q (b) Payment for elected lin Fig.. Payment calculation baed on LST found by Algorithm (3). We ue the ame networ in Figure to illutrate how to find the threhold value for edge baed on the multicat.5 Algorithm 4 Computing payment for elected agent baed on round-baed multicat method O : Initially et the cot of to and r. : repeat 3: Find the threhold value for agent baed on O r under cot vector c (r) and receiver et Q(r). Let l r κ (O r,c r ) be the threhold value found. Here we et l r if agent cannot be elected in thi round for any cot. 4: Update cot vector and receiver et to obtain the new cot vector c (r+) and Q (r+). Set r r +. 5: until a valid output i found 6: Fix c and aume x i the payment for agent. Let x r be the cot for agent in round r if the original cot i c x. Then x the minimum value that atifie the following inequation: x i l i for i r. tree found by Algorithm 3. In the firt round, can not be elected, thu l. In econd round, it i eay to oberve that when cot i maller than.9, the path q i elected and when cot i greater than.9, path q i elected. Thu, the threhold value for in thi round i l.9. Notice the updating by Algorithm 3 doe not change the cot of an unelected agent, thu the final threhold value i imply the maximum of l and l, which i.9. In other word, we have to pay lin.9. Similarly, we can find all elected edge threhold value a hown in Figure (b): the number in the parenthei are the threhold value. C. Reaonable Charging Scheme For a given et of receiver, after we calculate the payment p for every relay agent baed on a declared cot vector d, it i natural to a who will pay thee payment. Two poible payment model have been propoed in the literature. ) Outide ban or Group payment model: an outide ban or an organization to which the receiver belong will pay all thee relay agent. ) Payment haring model: each receiver i hould pay a reaonable haring S i of the total payment. We will addre what reaonable mean later. For outide ban model, the only thing we hould care i how to find the truthful payment cheme for the given multicat topology, which ha been addreed in the previou ubection. In practice, it i often the cae that the receiver have to hare the payment among themelve. Thu, we will tudy how to hare the payment fairly. Notice that the payment haring i different from the traditional cot haring. How to hare the multicat cot among the receiver ha been tudied previouly in [7], [], [3], [9], in which the cot of relay agent are public and the multicat topology i a fixed tree. Mot of the literature ued the Equal Lin Split Downtream (ELSD) pricing cheme to charge receiver: the cot of a lin i hared equally among all it downtream receiver. A we will how later, if we imply ue the ELSD a our charging cheme to hare the payment, it uually i not reaonable in common ene.

6 Given a et of receiver R, let P(R, d) p (R, d) denote the total payment to all relay agent. For a charging cheme S, let S i (R, d) denote the charge (or called haring) to receiver i. We call a charging cheme S reaonable or fair if it atifie the following criteria. ) Nonnegative Sharing (NNS): Any receiver q i haring hould not be negative. In other word, we don t pay the receiver to receive. ) Cro-Monotone (CM): For any two receiver et R R containing q i : S i (R,d) S i (R,d). In other word, for a given networ, receiver i haring doe not increae when more receiver require ervice. 3) No-Free-Rider (NFR): The haring S i (R, d) of a receiver q i R i never le than R of it unicat haring S i (q i,d). Thi guarantee that the haring of any receiver will not be too mall. 4) Budget Balance (BB): The payment to all relay agent hould be hared by all receiver, i.e., P(R, d) q S i R i(r, d). Notice the definition of reaonable can be changed due to different requirement. For example, a common criterion for multicat charging cheme i to maximize networ welfare: elect a ubet of receiver uch that the networ welfare i maximized. Here, the networ welfare i defined a the total valuation of all elected receiver minu the cot of the networ providing ervice. Since in our model we do not conider receiver valuation, we will only focu on budget balance intead of maximizing the networ welfare. In literature, the Shapely value [8] i one of the mot commonly ued charging cheme to achieve BB and CM. By auming a univeral multicat tree and the publicly nown lin cot, Feigenbaum et al. [3] proved that ELSD charging cheme i a Shapely Value. Unfortunately, the ELSD charging cheme i not alway fair if we want to hare the payment. Lemma 4: For tree LST, ELSD haring i not fair. Proof: A a running example, we will ue the multicat tree, denoted by LST, found by Algorithm 3 to how that ELSD i not fair. We till ue the ame networ hown in Figure (a). Let Q q,q be receiver. The multicat tree LST (Q) i hown in Figure (c). Tree LST (q ) and LST (q ) are hown in Figure 3 (a) and (b) repectively. Fig. 3. v.5(.6) q q q (.4).5 (.5) q (a) LST (q ) (b) LST (q ) LST (q ) and LST (q ) and their correponding payment(3). We now how that ELSD i not fair in thi ituation. Figure 3 (a) and (b) illutrate the payment P (q ).5 and P (q ).9. If we ue ELSD a our charging cheme, the haring by q i S (q q,c) which i larger than it haring S (q,c).6 when q i the only receiver. Thu, it violate the property CM. It implie that ELSD i not a fair charging cheme for multicat topology LST. Furthermore, uing the ame example, we how by contradiction that there i no charging cheme atifying both CM and BB. Lemma 5: For multicat topology LST, there i no charging cheme that atifie both CM and BB for a truthful payment cheme. Proof: For the ae of contradiction, we aume that a charging cheme S atifie both CM and BB. From the property of BB, we have S (q,c).6, S (q,c).9 and S (q q,c)+s (q q,c)6.4. From CM, we have S (q q,c) S (q,c).6 and S (q q,c) S (q,c).9. Combining thee two inequalitie, we obtain 6.4 S (q q,c)+s (q q,c) , which i a contradiction. Thu, given an arbitrary multicat topology and it correponding truthful payment cheme, a fair charging cheme may not exit at all. It i attractive and important to find the neceary and ufficient condition for the exitence of a fair charging cheme for a given multicat topology. IV. CASE STUDY: CORE-BASED MULTICAST In thi ection, we illutrate how to deign a truthful multicat protocol for the currently ued core-baed multicat which ue the leat cot path tree (LCPT) a it topology. Here, we aume that the networ i modelled a a lin weighted graph. All our reult preented in thi ection alo apply to the cae when the networ i modelled a a node weighted graph. Given a et of receiver R, we firt compute the leat cot path, denoted by LCP(, q i,d), between the ource and every receiver q i Q under the reported cot profile d. The union of all leat cot path between the ource and the receiver i called leat cot path tree, denoted by LCP T (R, d). A. Payment Scheme Intuitively, we may ue the VCG payment cheme in conjunction with the LCPT tree tructure a follow. The payment p to each lin e that i not in LCPT i and the payment to each lin e on LCPT i p ω(lcp T (R, d )) ω(lcp T (R, d)) + d. In other word, the payment i it declared cot plu the difference between the cot of the leat cot path tree without uing e and the cot of the leat cot path tree. We how by example that the above payment cheme i not trategyproof. In other word, if we imply apply VCG cheme on LCPT, a lin may have incentive to lie about it cot. Figure 4 illutrate uch an example where lin can lie it cot to improve it utility. The payment to lin e 4 i and it utility i alo if it report it cot truthfully. The total payment to lin e 4 when e 4 lie it cot down to 4 i ω(lcp T (R, c 4 ))

7 q q q 3 Fig q q q q q q 3 (a) Networ (b) LCPT (c) LCPT after lie VCG mechanim i not truthful for LCPT ω(lcp T (R, c 4 d 4 ))+d and the utility of lin become u 4 (c 4 d 4 )6 8 8, which i larger than u 4 (c). Thu lin e 4 ha incentive to lie, which implie that VCG mechanim i not truthful. With the failure of the VCG mechanim, we may doubt whether there exit a truthful payment cheme baed on LCPT. Remember LCPT i formed by union of leat cot path. By applying Theorem, we conclude that LCPT atifie MP. Thu, there exit a truthful payment cheme and the truthful payment can be found according to Theorem a following. For each receiver q i R, we find the leat cot path from the ource to q i, and compute an intermediate payment p i to lin e on LCP(, q i,d) uing the VCG payment cheme for unicat p i d + LCP(, q i,d ) LCP(, q i,d). Here LCP(, q i,d) denote the total cot of the leat cot path LCP(, q i,d). The final payment to lin e LCP T i p max q pi i Q () The payment to a lin i zero if it i not on LCPT. Let u illutrate the above payment cheme for LCPT by a running example in Figure 4. If lin report it cot 8 truthfully, then it get payment ince it i not in the LCPT. If lin report a cot 4, it i now in the LCPT, a hown in Figure 3 (c). It payment then become max(p,p,p 3 ), where p LCP(, q,d v4 ) LCP(, q,d) Similarly, p 6and p 3 7. Then the utility of lin become max(p,p,p 3 ) 87 8, which i le than what it get by reporting it truth cot. B. Ditributed Payment Algorithm Remember that LCPT i baed on the union of the leat cot path from the ource to all receiver. For unicat, Feigenbaum et al. [5] gave a ditributed method uch that each node i can compute a number p ij >, which i the payment to node for carrying the tranit traffic from node i to node j if node i on LCP(i, j, d). The algorithm converge to a table tate after d round, where d i the maximum of diameter of graph G removing a node, over all. We then briefly dicu how to compute the payment for multicat uing LCPT. Our ditributed algorithm ue the algorithm in [5] a the firt phae and i hown a follow. Algorithm 5 Ditributed payment computing : Apply the ditributed algorithm in [5] to compute the payment p q i. After thi tep, every receiver q i will compute the payment p i to each uptream edge e on the leat cot path between and q i. : Every receiver q i end the payment information it computed to it parent. 3: Upon receiving a pacet containing the payment from it child which originated from receiver q i, lin e only eep payment p i and end all remaining payment information to it parent if exit. 4: When lin e receive p i from all it downtream receiver q i, it compute the maximum of them a the it own final payment. C. Payment Sharing Intuitively, we may want to ue ELSD a the charging cheme. Unfortunately, we will how by example that ELSD i not fair when coupled with LCPT. Conider the networ hown by Figure 5 (a). There are two receiver q,q. Path 3 4 () () q q q q q q (a) Networ (b) LCP T (q,d) (c) LCP T (q q,d) Fig. 5. ELSD charging cheme doe not wor for LCPT LCP T (q,d) i hown in Figure 5 (b) and the payment to lin i hown beide the lin cot in parenthei. The total payment to lin on LCP T (q,d) i + 4. If we conider LCP T (q q,d), the payment to lin i hown in Figure 5 (c). If we apply ELSD to hare payment, the payment to lin (which i 6) i plit equally between q and q. Thu, the hared payment of receiver q i 3+5 when the receiver et i {q,q }, while it payment i only 4 when q i the only receiver. Thu, ELSD haring method violate the CM property here, i.e., ELSD i not a fair charging cheme for LCPT. Therefore we hould find ome reaonable charging cheme other than ELSD. In thi paper, we give one fair payment haring method. The baic idea behind our method i that a receiver hould only pay a proportion of the payment that i due to it exitence. Roughly peaing, our payment haring cheme wor a follow. Notice that a final payment to an agent j i the maximum of payment p i j by all receiver. Since different receiver may have different value of payment to agent j, the final payment P j hould be hared proportionally to their value, not equally among them a cot-haring. Figure IV-C illutrate the payment haring cheme that follow. Without lo of generality, aume that p j p j pn j, i.e., p j p n j. We then divide the payment p j into n portion: p j, p j p j,, pi j pi j 3 4 () (6) 3 (6),, p n j pn j. Each portion p i j pi 4 j

8 Fig. 6. Σ i t e pt p t j j n i+ S j p j e i p i pi j j e n n pn j j Share the payment to ervice provider among receiver fairly. i then equally hared among the lat n i+ element, which have the larget n i + payment to S j. Algorithm 6 Fair charging cheme for LCPT. : for edge e LCP T (R, d) do : Let R(e ) be the et of downtream receiver of e, i.e., p max qi R(e ) p i max q i R p i. 3: Sort the receiver in R(e ) according to p i in an acending order. If two or more receiver have the ame value, the receiver with maller ID ran firt. Let σ {σ,σ,,σ R(e ) } be the raning. Here, we add a dummy payment p σ to raning σ. 4: For receiver not in R(e ), it haring of the payment p of lin e i. 5: For a receiver q σa R(e ), it haring of the payment p to lin e i: f σ a (R, d) a x p pσ x R(e ) x + In other word, for two receiver q σx, q σx+ who are conecutive in raning σ, the difference + i hared by all receiver who ran after q σ x. 6: end for 7: The total charge for receiver q i in LCPT i S i (R, d) f j i (R, d) (3) e j LCP T (R,d) We firt illutrate how to charge the receiver q uing Algorithm 6 for a networ repreented by Figure 5. For lin, the two intermediate payment are p and p 6. Firt, we obtain a ran of thee receiver baed on the intermediate payment {q,q }. Then p i equally plit between q and q and p p 4i charged to q alone. Thu, receiver q i charged + 3 totally in LCP T (q q,d), which i maller than the price 4 when q i the only receiver. Thi how that charging cheme decribed by Algorithm 6 i reaonable for thi pecific networ. Following theorem how that it i reaonable for LCPT generally. Theorem 6: The charging cheme defined in Algorithm 6 for LCPT atifie NNS, CM, NFR and BB. Proof: A lin i called an uptream lin of a receiver q i if it i on the unique imple path between the ource and the receiver q i in the multicat tree. Obviouly, our charging cheme atifie NNS ince pσx for any () two receiver q σx and q σx. Remember for a receiver q σa R(e ), it haring of the payment to it uptream lin e i: a fσ a (R, d) pσ x a pσ x R(e ) x + R(e ) pσa x pσ R(e ) pσa R(e ) Thu, the total charge to receiver q σa i S σa (R, d) fσ a (R, d) e LCP T (R,d) e LCP (,q σa,d) pσa R(e ) x e LCP (,q σa,d) Sσa (q σa,d). R(e ) It implie that the charging cheme 6 atifie NFR. Summing f σ a (R) for a from to R(e ), we obtain R(e ) a R(e ) a R(e ) a f σ a (R) a x p σa Thu, we obtain S(R, d) R(e ) a a x R(e ) R(e ) x + R(e) q i R a p σa S i(r, d) e j LCP T (R,d) q i R f j i (R, d) pσ x R(e ) x + a a x f σ a (R, d) p σ x R(e ) x + pσ R(e ) p q i R e j LCP T (R,d) e j LCP T (R,d) f j i (R, d) Thi prove that our charging cheme (6) atifie BB. p j P(R, d) We then how that our cheme doe atify CM. Notice a neceary and ufficient condition for CM i that for any R Q and q j Q R we have S i (R, d) S i (R q j,d) for every q i R. To prove thi, it i ufficient to prove that f i (R) f i (R q j ). Aume q i i raned a in raning σ when the receiver et i R. We prove it by dicuing all poible cae: Cae : p j pi. Let σ be the new raning for receiver et R q j, then q j raned after q i in σ. Thu σ x σ x for x a. In other word, all receiver raned before or at a in raning σ till ha the ame ran in σ. Therefore, f i (R q j) a x a x a x pσ x R(e ) + x + pσ x R(e ) x + pσ x R(e ) x + f i (R) Cae : p j <pi. In thi cae, q j i raned before q i in σ and q i raned a + in σ. Without lo of generality, we aume q j raned b in raning σ. Thu, we have σ x σ x for

9 x<band σ x σ x+ for x>b. Therefore, f i (R q j) b x a+ x pσ x R(e ) + x + pσ x a+ R(e ) x + + xb+ For the firt part of the equality we have b x b x b x b x pσ x R(e ) x + pσ x R(e ) x + pσ x R(e ) x + pσ x R(e ) x + + pσ b + pσ b + pσ b For the econd part of the equality we have a+ xb+ a+ xb+ a xb pσ x R(e ) x + p σ x p σ x R(e ) x + pσ x R(e ) x + pσ b Combining the above two, we obtain fi (R q j) b pσ x a+ R(e ) x + + x b x a xb a x pσ x R(e ) x + pσ x R(e ) x + pσ x R(e ) x + pσ x R(e ) x + pσ b R(e ) b + pσ b R(e ) b + pσ b R(e ) b + pσ b pσ b R(e ) b + pσ b R(e ) b + xb+ + pσ b pσ b pσ x R(e ) x + pσ b R(e ) b + + pσ b R(e ) b + f i (R) Thi immediately implie that our charging cheme atifie CM. Thi finihe the proof of Theorem 6. D. Ditributed Charge Calculation Notice, if we implement the payment haring cheme in a centralized way, for every lin, it need to tore up to Q r intermediate payment. Thu the total pace needed i O(nr). In practice, it may be more deirable to implement a ditributed payment haring cheme. In the following, we preent a ditributed algorithm that implement our payment haring cheme that require at mot O(r) pace for each lin and total meage at mot O(r h), where h i the height of the tree. In our ditributed algorithm, for any lin e in LCP T, we not only need it final payment p, but alo need the intermediate payment p j for every downtream receiver q j. We aume that thi i already available through our ditributed payment computing method. In our ditributed charge cheme, at every lin e we ue MD [i] to tore the payment it and all it uptream agent will receive from the receiver q i. Our ditributed charging cheme i implemented in a top-down fahion from the ource to all receiver. Algorithm 7 Ditributed charging cheme : Initially, the ource node end all it children in the multicat tree a vector MD for all receiver. : Every lin e in LCP T, upon receiving a charging vector MD from it parent, update the charge for each of it downtream receiver q i a MD [i] MD[i] + f i (R(e )). Here, f i (R(e )) i calculated according to Algorithm 6. 3: If lin e ha more than one downtream receiver, it contruct a new charge vector MD j {MD[i ],MD[i ],,MD[i R(ej) ]} for every downtream adjacent lin e j. Here, the charge MD[i t ] ( t R(e j ) ) i for receiver q it who i a downtream receiver of lin e j. Then end vector MD j to lin e j. If lin e ha only one downtream receiver q i then e imply end the modified charge MD to it downtream lin. 4: Every receiver q i will finally receive a charge which i equal to equation (3). V. OTHER ISSUES AND OPEN QUESTIONS A we mentioned early, thi paper i the firt tep to explore the general networ protocol deign when relay agent are non-cooperative. There are many intereting and important iue that have been untouched and left for further tudy. We jut lit a few here. Colluion: Throughout thi paper, we aume all agent will not collude together to manipulate the protocol. It i intereting to tudy what will happen when agent will collude and how to find truthful mechanim that are reitent to colluion. Our conjecture i that no truthful multicat protocol that can prevent the colluion from an initial wor proved in [9] for unicat. Ditributed Computing: One thing we hould notice i that thee agent running the ditributed algorithm are indeed noncooperative. How to enure they implement the correct ditributed algorithm we deigned alo i an important quetion we have to conider. Receiver Valuation: So far, we aume that the receiver will pay the fair amount of haring of payment to receive data uing multicat. In practice, each receiver often ha a valuation to indicate how much it i willing to pay to receive the information. Receiver will chooe to receive the information if and only if the charge i at mot it valuation. Furthermore, receiver could alo be non-cooperative and elfih: each receiver will alway maximize it profit by manipulating it

10 reported valuation. Thi mae the multicat deign even harder and it i a very promiing and intereting future reearch direction. It i well-nown that a cro-monotone cot haring cheme implie a group-trategyproof mechanim [7]. Unfortunately, we can how that the imple application of a cro-monotone payment-haring mechanim doe not imply a group-trategyproof mechanim at all. The elfih relay agent could lie up or downward it cot to improve it utility. VI. PERFORMANCE STUDY We conduct extenive imulation to tudy the performance of trategy-proof multicat routing baed on LCPT. Remember that the payment of LCPT i at leat the actual cot of LCPT. For a LCPT T, let c(t ) be it cot and P(T ) be the total payment to all relay agent. We define the overpayment ratio (OR) of T a OR(T ) P(T ) c(t ). (4) In the wort cae, the ratio OR(T ) could be a large a O(n) for a networ of n node [3], even for the unicat pecial cae. Notice there are ome other definition about overpayment ratio in the literature. In [3], the author propoed to compare the total payment P(T ) with the cot of the new LCPT obtained from the graph G\T, i.e., removing T from the original graph G. In addition to the overpayment ratio, we propoe another metric to meaure the performance of the trategy-proof multicat baed on LCPT. Remember that the payment to relay agent are hared among receiver. Thu, for each receiver, it i more intereted in how much extra it hould pay to guarantee the truthfulne of the lin. Given the LCPT T for a et of receiver R, let m i (R, T ) be the price that receiver q i i charged to receive the information if the lin are cooperative. Notice that S i (R, T ) i the amount that receiver q i i charged to receive the data if the lin are non-cooperative. We define the Price-Cot-Ratio (PCR) a PCR(q i,t) S i(r, T ) m i (R, T ). (5) In our experiment, we generate random networ with n node, where n i a parameter. In order to enure the networ i bi-connected, the average node degree hould be greater than log n with high probability. Firt, for every node u, we randomly draw a number from [α log n, 5α log n] a it degree d u, where α i a parameter. A random graph atifying thee degree requirement i then generated. The length of each edge i then uniformly drawn from ditribution [, ]. By chooing different parameter, we tudy what apect of the networ affect the OR and PCR. To compute the probability ditribution, we generate 4 different networ and compute the number of intance that fall in ome pecific interval. For other imulation, given all fixed parameter, we generate 3 different networ intance and compute the performance accordingly. A. Effect of Networ Size In thi imulation, we fix the parameter α to 3logn, which mean that node degree are drawn from a uniform ditribution [ 3, 5 3 ] with average. We alo fix the ize of receiver et R to 5. We meaure the performance of our trategyproof multicat protocol baed on the following four metric: Average Overpayment Ratio (AOR), Maximum Overpayment Ratio (MOR), Average Price-Cot-Ratio (APCR) and Maximum Price-Cot-Ratio (MPCR). Figure 7 (a) and (b) plot the ditribution of the average overpayment ratio and the average PCR when the number of node are and 5. Oberve that the probability ditribution of AOR (alo APCR) for different networ ize are imilar. Figure 7 (c) how that the AOR, MOR and APCR do not change when the number of networ node grow from to 5. On the other hand, MPCR fluctuate and i much larger than the other three metric. Thu, we conclude that the number of node do not affect the overpayment ratio and price-cot-ratio in random networ. B. Effect of Networ Denity Since the difference in the networ ize do not affect the performance of our trategy-proof protocol, we then tudy other effect by fixing the networ ize ( in the reult reported here). We pecifically tudy the effect of the networ denity by changing the node degree parameter α. Figure 8 (a) and (b) how the ditribution of AOR and APCR repectively when the node degree are drawn from two uniform ditribution [log, 5 log ] and [ log, log ]. Figure 8 (c) how that the AOR, MOR and APCR change when the networ denity change. It i intereting to oberve that both AOR and APCR firt decreae when the networ denity (i.e., the average node degree) increae from to 3, and then increae lightly when the networ denity increae from 3 to 4. They both become teady when the networ denity i greater than 4. It i intereting to analyze thi phenomenon theoretically. C. Performance Comparion with Unicat In thi imulation, we compare the average cot and payment per receiver in multicat baed on LCPT with thoe of unicat. We randomly generate n terminal where n varie from to 5. The degree of each node i randomly drawn from the uniform ditribution [log n, 5 log n]. For a pecific networ, we average the cot and payment for all receiver. Figure 9 (a) plot the cot and payment for multicat and unicat per receiver when the number of receiver i 5, while Figure 9 (b) how the reult when % of node are receiver. Oberve that the average cot and payment per receiver for multicat baed on LCPT i maller than the average cot and payment per receiver for unicat repectively. Furthermore, under mot of the cae, the payment per receiver for LCPT payment i even maller than the cot per receiver for unicat. Thi enure u that multicat not only ave the total reource, but alo benefit the individual receiver even in elfih networ. We then vary the networ ize among,, 3, 4, 5 and the number of receiver from to

11 Node Number Node Number 5.3 Node Number Node Number Probability..5. Probability..5. Overpayment Ratio/ Price Cot Ratio AOR MOR APCR MPCR.5.5 Average Overpayment Ratio Average Price Cot Ratio Node Number (a) Probability ditribution of AOR (b) Probability ditribution of APCR (c) Average and Maximum OR/PCR Fig. 7. The average overpayment ratio and price cot ratio do not depend on the networ denity α α.4.35 α α.65.6 AOR MOR APCR Probability Probability Over Payment Ratio/Price Cot Ratio Average Overpayment Ratio Average Price Cot Ratio Average Node Degree (a) Probability ditribution of AOR (b) Probability ditribution of APCR (c) Average and Maximum OR/PCR Fig. 8. The overpayment ratio and price cot ratio depend on the networ denity. 6 5 Unicat Cot Unicat Payment LCPT Cot LCPT Payment 6 4 Unicat Cot Unicat Payment LCPT Cot LCPT Payment Reveiver cot and payment 3 Reveiver cot and payment 8 Cot number of node Fig number of node Number of Node 5 5 Number of Receiver (a) 5 receiver (b) % node are receiver (c) Varying receiver number The cot and payment per receiver for unicat and multicat baed on LCPT Figure 9 (c) how the unicat cot (the red urface) and the LCPT baed multicat payment (the blue urface). From the reult of previou three imulation, we oberve that AOR and APCR are both quite mall for a random networ, and even the MOR i maller than.7 generally. Thu, we conclude that the theoretical wort cae almot urely will not happen in a random networ. VII. CONCLUSION AND FUTURE WORKS In thi paper we give a trategyproof payment and charging mechanim that timulate cooperation for multicat in a elfih networ. We aumed that a group of receiver i willing to pay to receive the data. Each poible relay agent ha a privately nown cot of providing the relay ervice. In a multicat cheme, each elfih relay agent firt i aed to declare a cot for relaying data for other node. In return, it will get a payment baed on the reported cot of all relay agent that can provide the ervice. The objective of every individual relay agent i then to maximize it profit. A multicat protocol i aid to be trategyproof if no peculation and counter peculation happen, i.e., every relay agent will maximize it profit when it truthfully report it cot. It i well-nown that the traditional protocol deigned for conforming agent cannot prevent the elfih agent from manipulating it cot to it benefit. Intead of redeigning the wheel, it i preferred to enhance an exiting multicat protocol to deal with elfih agent. In thi paper, we pecifically gave a general rule to decide whether it i poible, and how to if