Generalized Proportional Allocation Mechanism Design for Multi-rate Multicast Service on the Internet

Transcription

1 Generaized Proportiona Aocation Mechanism Design for Muti-rate Muticast Service on the Internet Abhinav Sinha and Achieas Anastasopouos Abstract. In this paper we construct two mechanisms that fuy impement socia wefare maximising aocation in Nash equiibria for the case of singe infinitey divisibe good being demanded by a separate groups of agents, whist being subject to mutipe inequaity constraints. The nature of the good demanded is such that it can be dupicated ocay at no cost. The first mechanism achieves weak budget baance, whie the second is an extension of the first, and achieves strong budget baance at equiibrium. One important appication of these mechanisms is the muti-rate muticast service on the Internet where a network operator wishes to aocate rates among strategic agents, who are segregated in groups based on the content they demand (whie their demanded rates coud be different), in such a way that maximises overa user satisfaction whie respecting capacity constraints on every ink in the network. The emphasis of this work is on fu impementation, which means that a Nash equiibria of the induced game resut in the optima aocations of the centraized aocation probem.. Introduction Aocation of services on the Internet is an important probem, not ony from the system performance point of view but aso from an economic standpoint. As requirements from Internet services increase, having appropriate markets for various services coud ead to more efficient use of avaiabe resources. Once one introduces the notion of aocation of resources based on requiring specific payments for services provided, the next ogica step woud be to mode the system agents as being strategic i.e. utiity-maximising. In a genera informationay (and physicay) decentraised system, such as the Internet, in order to make protocos work the designer woud usuay require dissemination of (oca) information from agents. It is here that enforcing of protocos becomes harder when strategic users are present. For exampe, if a protoco requires agents to take action based in the interest of whoe network then in the absence of true oca information, the designer cannot check whether an agent adheres to the protoco. For strategic agents, the one thing we can ensure is Key words and phrases. Muticast routing, propotiona aocation, game theory, mechanism design, fu impementation, Nash equiibrium.

2 2 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS that actions taken by agents woud be so as to maximise their own profit. So the decentraised information structure aong with strategic agents requires the designer to venture into the fied of mechanism design. Mechanism design focuses on designing contracts which enforce strategic agents to take actions that revea their private information truthfuy. Voting rues, auctions, private and pubic good exchange economies are a few exampes of fieds where Economists have used mechanism design extensivey. In this paper, we focus on a specific aspect of mechanism design, Nash Impementation. Without going into a forma definition, Nash impementation refers to design of contracts such that ony the designer s most preferred outcome is reaised as a resut of interaction between strategic agents (Nash equiibria), whereas in genera mechanism design, other ess preferred outcomes are aso possibe. Thus impementation is more stringent and is capabe of producing better aocations. For impementation, readers may refers to survey artice [] where the discussion is in microeconomics context or one may refer to [2, 3] for mechanism design review with networks and communications appications. Mechanism design for aocation of a singe divisibe good in Internet framework has been discussed in [4, 5, 6], where impementation in genera unicast and muticast has been discussed in [7, 8, 9, ]. A of the above as we as the work here uses Nash equiibrium as a soution concept. Nash equiibrium is generay used as a soution concept for compete information games and we discuss this aspect of modeing in section 4 where information assumptions are stated and aso in the discussion in section 6. We consider the probem of muti-rate muticast service provisioning. Here, on the same network, different services with varying QoS within the same service are provided. The main difference with unicast service is that here agents requesting the same service need not be serviced by estabishing competey separate connections for each; at common inks ony connection for the agent with the highest QoS wi be estabished thereby preventing dupication at that ink. In fact, muticast formaism subsumes the unicast one for any reasonabe system performance metric. Our aim is to buid a mechanism who s Nash equiibria give ony such aocation that maximises the utiitarian socia wefare i.e. sum of agents utiities. Because of the framework of muticast, there are two different aspects of resource aocation that we come across in this paper - private and pubic goods. Overa due to the capacity constraints of the inks on the network, aocation of rate to one group of users wi mean that such additiona rate can no onger be aocated to another group of users sharing this ink - this is the private good aspect. On the other hand within a group since the aocation via the capacity constraint is dictated ony by the highest QoS user from that group on that ink, others in the group can be aocated additiona rate without having to affect somebody ese s aocation - this is the pubic good aspect. The phiosophy here is to aocate rates based on utiitarian socia wefare maximisation, which is a we-known criterion than encompasses fairy genera requirements ike Pareto optimaity and zero independence. In modes without strategic users, researchers have argued for different types of maximization criteria. One exampe is the max-min fairness for muticast is used in [, 2,

3 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 3 3, 4]. On the other hand, in [5, 6] authors have used integer and convex programming to get decentraised agorithms that maximise utiitarian socia wefare in the muticast probem. One can argue in favour of maximising sum of utiities as foows; any aocation which is paretodominated by another cannot be predicted as the outcome for a society since mere exchange of resources wi make everybody better off (a side market woud faciitate this and a designer woud want to avoid this situation). For quasi-inear utiities e.g. utiities with a inear money component, it can be shown that aocations that maximise sum of utiities (without money) are the ony Pareto optima aocations. The work here generaises the idea of proportiona aocation that was introduced in mechanism design framework by [5, 6] for the case of infinitey divisibe singe good (unicast) with ony one capacity constraint and for stochastic contro of networks by [7]. We emphasize, that in the context of this paper, proportiona aocation does not refect an attempt to aocate resources in a fair manner. It is used as a way to transate users demands into actua aocations taking into account the capacity constraints. Additionay, readers may be refer to [8] for a fu impementation mechanism that uses proportiona aocation in the unicast framework. The main advantage of proportiona aocation is the off-equiibrium feasibiity of aocation. This means that even if agents do not pay an equiibrium action profie, the aocation of rates woud sti satisfy capacity constraints of a inks in the network. So the communication system wi perform reiaby even off-equiibrium; actuay the aocation here is aways on the boundary of the feasibe region thereby aways utiising the system resources to the fu extent. This is in contrast to [9], where significant effort has been made to ensure budget baance off-equiibrium but feasibiity isn t ensured off-equiibrium. The second contribution of this work is to demonstrate how the budget baance property can be added to a non-budget baanced mechanism without significant difficuty (at east in this setup). In section 5 we go on show that with the proportiona aocation idea it takes the exchange of one more signa to achieve this. Contrary to [5, 6], the work here ensures fu impementation of socia wefare maximising aocation, so the designer can guarantee that ony the most efficient outcome wi be reached and no other (this kind of guarantee is substantiay harder to make in a game-theoretic framework). From a practica point of view, the work here estabishes a tight upper bound on the number of message exchanges required for impementation for the muticast probem. The agents here are ony required to communicate via announcing signas which consist of demands and prices as opposed to generaised VCG mechanisms (refer to [, 2, 9, 2] or section 5.3 in [2]), which are widey used in mechanism design probems and require announcement of pay-off types (entire vauation function in this case). The remainder of this paper is structured as foows - in section 2 we state and characterise the soution of the Centraised probem that we wish to impement in a decentraised manner. In sections 4 and 5 we describe and prove our mechanisms for the weak and strong budget

4 4 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS baance cases respectivey. In section 6 we discuss reevant iterature and saient features of our mechanism in greater detai. 2. Centraised Probem Formay, maximising socia wefare wi be defined via the centraised probem beow. Consider a set N of Internet agents who have been divided into disjoint groups (an agent is considered as a pair of source and destination users). The set of groups is denoted by K = f; 2; : : : ; Kg and within a group k 2 K, the set of agents by G k. Each group k 2 K has G k agents i.e. jg k j = G k. From a this one can write N = f(k; i) j k 2 K; i 2 G k g and that the tota number of distinct agents is N = P K k= G k. The agents communicate over pre-specified routes on the Internet and the agents have been divided into groups based on the content they demand. Whie the content demanded by different groups is distinct, within a group a agents demand the same content but maybe at different rates. An aocation wi be a vector x of rates which has K = jkj eements, each of which are themseves vectors of sizes G k, k 2 K. For this denote by x ki 2 R + (where R + is the set of non-negative rea numbers) the rate aocated to agent i of group k, of the content demanded by group k (from here on we wi refer to such an agent as agent ki). Agent s vauation for an aocation x can be written as ~v ki (x) = v ki (x ki ) 8 ki 2 N where v ki ( ) : R +! R, for a ki 2 N, which indicates that agent ki s satisfaction depends ony on his information rate aocation x ki. Due to capacity constraints on the utiised inks, aocation to agents is constrained by a number of inequaity constraints - both on the network eve as we as the group eve. Separate Routes and Notation. Each agent has a fixed pre-determined route. The route L ki of agent ki is the set of inks that agent ki uses for his communication, and L = [ ki2n L ki is the set of a avaiabe inks. The set of agents utiising a ink 2 L is defined as N = fki 2 N j 2 L ki g. Aso we define G k = N \ G k, the set of agents from group k who use ink and K is the set of groups that have at east one agent that uses ink i.e. K = fk 2 K j G k 6= ;g. The magnitudes of the sets defined above are L = jlj, L ki = jl ki j, G k = jg kj and N = jn j. In addition to group-wise ordering of agents, we aso have ordering of agents within the group for every ink that is used by that agent. Any agent ki, on ink, wi aternativey be aso referred to as ki 7! g k (i) where g k (i) G k; 8 2 L ki : Here the mapping is such that if for i; j 2 G k and i > j then g k (i) > g k (j). This is done to order agents in a group separatey at every ink. Note that given the previous definitions, this

5 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 5 notation is redundant; however we define it because it wi be usefu ater on. Inverse mapping from ordering within a group and ink wi be denoted by (g k ). The network administrator is interested in maximizing the socia wefare under the ink capacity constraints. This centraized probem is formay defined beow. max x k2k i2g k v ki (x ki ) s.t. x ki 8 ki 2 N (C ) and k2k maxf kjx kj g c 8 2 L (C 2 ) j2gk Specificay, constraints C 2 are the inequaity constraints on aocation, which as mentioned above, can be interpreted as capacity constraint for every ink 2 L, in the network. In this interpretation kj woud be representative of the QoS requirement of agent j combined with the specific architecture on ink. As an exampe, kj = for a inks 2 L R kj ( kj, where kj ) kj represents the packet error probabiity for ink for a packet encoded with channe coding rate R kj. 2.. Assumptions. Our anaysis woud be done under the foowing assumptions. (A) For a agents, v ki ( ) 2 V ki, where the sets V ki are arbitrary subsets of V, the set of a stricty increasing, stricty concave, twice differentiabe functions R +! R with continuous second derivative. (A2) vki () is finite 8 ki 2 N. This aso impies that v ki (x) is finite and bounded 8 ki and 8 x since v ki s are concave. (A3) Every ink has at east two groups that use it, i.e. K L. (A4) The optima soution of the centraised probem is such that on every ink there are at east 2 groups such that each has at east one non-zero component, i.e. if S (x) := fk 2 K j 9 i 2 Gk s.t. x ki > g then the assumption says js (x? )j L (where x? is the optima soution of (CP)). In addition, the coefficients are a stricty positive, i.e. ki > we-posedness of the probem we take c > 8 2 L. 8 2 L ki, 8 ki 2 N. Aso, for Assumption (A) is made in order for the centraized probem to have a unique soution and for this soution to be sufficienty characterized by the KKT conditions. (A2) is a mid technica assumption that is required in the proof of Lemma 4.7. Assumption (A3) is made in order to avoid situations where there is a ink constraint invoving ony one agent. Such case requires specia handing in the design of the mechanism (since in such a case there is no contention at the ink), and destructs from the basic idea that we want to communicate. Finay (A4) is reated

6 6 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS to (A3) and is made in order to simpify the exposition of the proposed mechanism, without having to define corner cases that are of minor importance Necessary and Sufficient Optimaity conditions. Foowing are the KKT conditions, which are generay necessary, but in our case (due to a constraints being affine and strict concavity of v ki ) they wi aso be sufficient. For this we first rewrite the centraised probem by restating the capacity constraints differenty (CP) max x;m k2k i2g k v ki (x ki ) s.t. x ki 8 ki 2 N (C ) and k2k m k c 8 2 L (C 2 ) and kix ki m k 8 i 2 G k; k 2 K ; 2 L (C 3 ) Here the capacity constraints have been rewritten with the introduction of new variabes. The virtua variabes m k represent the weighted maximum rate of group k on ink. It s easy to see that the soution of this probem is the same as the soution of the origina centraised probem as far as optima x is concerned. Now we define the Lagrangian for (CP) L(x; ; ; ) = v ki (x ki ) k2k i2g k 2L k2k i2gk 2L ki kix ki m + k2k m k c ki2n ki x ki Here KKT conditions wi be written without expicity referring to ki s and just using the fact that ki? and kix?? ki = 8 ki 2 N. With the assumptions above, it s easy to see that the KKT conditions beow wi give rise to a unique x? (and m? ) as the optimiser for (CP). KKT conditions: a) Prima Feasibiity: x? ki 8 ki 2 N and and kix? ki m? k k2k m k? c 8 2 L 8 i 2 G k; k 2 K ; 2 L A (5) b) Dua Feasibiity:? 8 2 L, ki 8 ki 2 N, 2 L c) Compimentary m? k k2k c A = 8 2 L ki? kix? ki m k? = 8 i 2 G k; k 2 K ; 2 L

7 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 7 d) Stationarity: and v ki (x? ki ) = v ki (x? ki )? ki 2L ki? ki 2L ki (6)? = i2g k? ki ki 8 ki 2 N if x? ki > ki 8 ki 2 N if x? ki = 8 k 2 K ; 2 L Looking at (5), ki? wi be non-zero ony if kix? ki = mk?, so these can be interpreted as the prices for ony those agents who receive maximum weighted aocation from a group at a given ink. Consequenty, from (6),? wi be the sum of ki? over those agents in a group for whom it is non-zero and it is the same for a groups.? can be thought of as the common tota price subject to each group at ink. 3. Different Formuations of the Centraised Probem The designer s task is to ensure that the above optimum aocation is made. This ceary requires the knowedge of v ki s even when constraints C, C 2 and C 3 are competey known. The premise of our probem is that we are deaing with agents who are strategic and for each of whom, the designer doesn t know their private information i.e. their vauation function v ki ( ). One way forward for the designer coud be to simpy ask each agent to report their private information and announce the soution of (CP), with reported functions in pace of v ki, for aocation. Apart from the fact that asking to report a function creates a practica communication probem, the main probem with this is that the agents coud report untruthfuy and end up getting a stricty better aocation. For exampe, reporting a v ki which has higher derivative than origina at every point. In mechanism design terminoogy, as stated, the aocation function arising out of (CP) isn t even partiay impementabe. Restricting ourseves to a certain cass of utiity functions (quasi-inear utiities), provides additiona fexibiity of penaising agents for reporting untruthfuy by imposing taxes/subsidies. In this way, another reated probem is created which is impementabe, and which we wi aso show to be equivaent to (CP) as far as aocation is concerned. This eads us to the foowing additiona assumption about agents utiities (A5) A agents have quasi-inear utiities, i.e. we can write overa utiity functions as u ki (x; t) = v ki (x ki ) t ki 8 ki 2 N where in addition to aocation we have introduced taxes t (a vector ike x). This can be deduced from the reveation principe. Indeed if there was a mechanism that even partiay impements the aocation function arising out of (CP), then there woud exist aso a truthfu impementation. However, as shown with the above exampe, such an impementation wi aways fai.

8 8 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS Note that under assumption (A5), agent ki pays tax if t ki > and receives a subsidy if t ki <. Taxes affect utiities ineary and overa utiity itsef is vauation after adjustment for taxes (tota monetary representation of one s state of happiness). Because we tak about socia wefare as our main objective, the centraised probem (CP) isn t compete unti we fix who owns the good that is being aocated. Then one wi have to further check whether incuding their wefare in the objective function changes the optimum aocation. As it turns out, under the assumption of quasi-inear utiities and cost of providing the good being zero for the owner, optimum doesn t change even if we invove the seer s wefare. In this regard, there are two interesting ways of reformuating (CP), as eaborated beow. 3.. First Reformuation of CP: Weak budget baance. We now introduce agent as the owner of the good (caed the seer). The seer doesn t have any costs for producing and providing the good, i.e. his vauation is the zero function. This coud be interpreted as the good being aready produced and ready to be provided, so those costs don t come into consideration for the seer as we as the designer. His utiity is inear (since vauation is zero) and his revenue is the tota tax paid by the agents, P ki2n t ki. (CP ) We define centraised probem (CP ) as max x;m;t ki2n u ki (x; t) + t ki ki2n s.t. C and C 2 and C 3 where now, instead of just taking agent s vauations into account, we maximise the sum of their overa utiities, with the addition of seer s utiity (which is ony his revenue) - each agent pays a tax t ki, a of which goes to the seer, who has no vauation and therefore has utiity equa to sum of taxes. Anticipating that a rationa seer wi ony se if his revenue is non-negative we can add a weak budget baance (WBB) constraint, which states (WBB) ki2n t ki : 3.2. Second Reformuation of CP: Strong budget baance. In this case, in contrast to (CP ), there is no separate seer. We can aternativey say that the agents are themseves the owners of the good and are ony ooking to distribute the good (which they coectivey own) in a way such that sum of utiities is maximised. Therefore strong budget baance (SBB) constraint is needed. This means that for the system N, no money has been introduced from the outside and the agents wish that no excess money remain on the tabe either.

9 (CP 2 ) MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 9 The new centraised probem (CP 2 ) resuting from the above interpretation can be stated as max x;m;t ki2n u ki (x; t) (SBB) s.t. C and C 2 and C 3 and ki2n t ki = : The two probems defined above wi be shown to be equivaent to (CP) where since our origina probem (CP) did not invove taxes, we wi tak of equivaence ony in terms of optimum aocation, x?. Note that due to different conditions on taxes in the two, two different mechanisms wi be needed to impement them. It is straightforward to see that (CP 2 ) and (CP) are competey equivaent - due to constraint (SBB), the objective for (CP 2 ) is independent of t and is exacty the same as objective for (CP), with same remaining constraints. Now for (CP ) and (CP 2 ), since the constraints on x are the same in (CP ) and (CP 2 ) and the x dependent part of the objective in (CP 2 ) in independent of t and is the same as the objective of (CP ), we can see that (CP ) and (CP 2 ) are equivaent. The two equivaences above automaticay give the third one i.e. (CP) and (CP ). The above equivaences mean that not ony wi x? be the same, but aso that the necessary and sufficient conditions describing it wi be the same i.e. KKT conditions, for x? and? ;?, wi be exacty the same for a three probems (additionay we wi show (WBB) and (SBB) constraints to be satisfied in respective formuations). This fact wi be used in Sections 4, 5 where the KKT conditions from Section 2 wi be treated as if they have been written for (CP ), (CP 2 ), respectivey. In Section 4, we wi present a mechanism that fuy impements (CP ) in Nash Equiibria (NE), whie in Section 5 we wi modify our mechanism to fuy impement (CP 2 ) in NE. 4. A Mechanism with Weak Budget Baance In this section we refer to (CP ) as the centraised probem. So we have a the agents in N pus the seer and socia wefare is in terms of everyone s utiity (incuding seer s). We wi define a mechanism, in a way that doesn t require knowedge of v ki, whose game-form wi have NE in pure strategies such that the aocation which corresponds to the equiibria of the game-form is same across a equiibria and is equa to the unique optimiser of (CP ), x?. In addition, the mechanism wi be such that everyone invoved (incuding the seer) wi be weaky better-off at equiibrium than not participating at a.

10 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS 4.. Information assumptions. Assume that v ki ( ) is a private information of agent ki and nobody ese knows it 2. Let I c be the set of common information between a agents, containing the information about fu rationaity of each agent. Finay, et I d be the knowedge of the designer, containing the information about constraints C, C 2, C 3, the fact that V ki V ; 8 ki 2 N and that the seer has vauation Mechanism. Formay, we have a set of environments V = ki2n V ki. We have seen from KKT, how each eement of V can be mapped to an aocation x? which maximises socia wefare for that set of utiities. The aocation x? achieves the maximum of (CP ), and correspondingy any tax t satisfying (WBB) woud do. In our mechanism, the designer woud define an action space S ki for each agent ki 2 N. We denote S = ki2n S ki the set of action profies for a agents. In addition the designer defines and announces the contract h : S! R N + R N that maps every vector of messages received from the agents into an aocation vector and a tax vector (thinking of x; t as vectors with N = P k2k G k eements). The designer woud then ask every agent ki 2 N to choose a message from the set S ki based on which aocations (and taxes) woud be made. The seer is not asked to take any action, so as far as strategic decision making is concerned, we don t need to consider him any further. It is impicit in our mechanism in this section that when the tax t is imposed, the seer gets revenue (or utiity) of P ki2n t ki. Specificay, the designer woud ask each agent to report s ki = (y ki ; p ki ) where p ki = (p ki; qki ). This incudes their demand for the good and the price for each constraint 2L ki that they are invoved, which they beieve other(s) shoud pay. In this for every agent and ink, there are two quoted prices - p ki and qki; the first one represents the price for kix ki m k constraint and the second one for the constraint kjx kj m k where kj is the agent that can aternativey be identified by gk (i) +. A this gives us S ki = R + R 2L ki +. For received messages s = (s ; : : : ; s N ; : : : ; s K ; : : : ; s KNK ) = (y; P; Q) = (y ; : : : ; y KNK ; p : : : ; p KNK ) the contract h ki (s) = (h x;ki (s); h t;ki (s)) wi be defined for each ki 2 N as foows. If the received demand vector is y = (y ; : : : ; y KNK ) = then the aocation is x = (x ; : : : ; x KNK ) =. Otherwise it is evauated by first generating a scaing factor r through n k := maxf kiy ki g i2gk 8 k 2 K ; 2 L r = min 2L r 2 This assumption is crucia because it raises the question of the vaidity of NE as a soution concept of the resuting game, since that woud require that a agents have compete information about everyone s utiities. We beieve this is a serious probem in this entire ine of research and that a Bayesian formuation woud be more appropriate. However, in this work we accept the justification weak in our opinion given by Reichestein and Reiter in [22] and Groves and Ledyard in [23].

11 (7) with MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET r = 8 >< >: P P c k2k n k c k2k n k ; if js (y)j 2 f (n k ); if S (y) = fkg + if js (y)j = f (n k ) = f (y i ) i2g k = c n k (n k + ) Using a these previousy defined quantities, the aocation and taxes woud be (8) h x;ki (s) = x ki = ry ki m k := rn k 8 k 2 K ; 2 L For tax we wi first define tota prices, wk; w k for any ink and group k 2 K w k := p ki w k := w jk k nfkgj = (9) w K k : i2g k k 2K nfkg where w k is we-defined due to assumption (A3). () h t;ki (s) = t ki = 2L ki t ki k 2K nfkg where if G k 2 then consider agents kj and ke who have aternate representation on ink as g k (i) and g k (i) + (mod G k), respectivey t ki = x ki kiq kj + (q ki p ke )2 + (w k w k )2 + q kj (p ki q kj )(m k kix ki ) + w k (w k w k )(c k 2K m k ); and if G k = then t ki = x ki ki w k + (w k w k )2 + w k (p ki w k )(m k kix ki ) + w k (w k w k )(c k 2K m k ); Here there are two eves of interactions that the mechanism is deaing with, one among groups for aocation of maximums on each ink and second within each group. Here agents are contesting to demand aocation that makes fu use of the fact that ony maximum at each ink wi give rise to a price on that ink. At any ink and group k, tota price w k is the summation of prices quoted by a the agents in the group at ink. The quantity w k is cacuated by averaging the tota prices for ink over a other groups than k. (v ki ), quoting of prices and demand is used as a way of eiciting v ki (x ki) by comparing it appropriatey with prices. In this vein, we do not wish to infuence p k with prices quoted by groups whose agents aren t using the ink at a, since the price then essentiay doesn t contain any information.

12 2 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS The quantity h x;ki (s) creates aocation by first creating proxies n k for weighted maximum at each ink for each group. Then m k s are created by diating/shrinking n k s on to one of the hyperpanes defined by the second set of constraints in C 2, specificay, that hyperpane for which the corresponding m k s are the cosest to origin (this coud aso be at the intersection of mutipe hyperpanes). Finay aocation x ki is cacuated by diating/shrinking y ki by the same factor. Another way to describe this is to say that the contract diates/shrinks n k to the boundary of the feasibe region defined by the capacity constraints and then aocations within a group are made proportionay. Since a the kj s are positive, this means that a constraints in C 2 are satisfied for the aocation automaticay (shown ater). Additionay, the separate definition for r when js (y)j < 2 is to ensure (as it wi be shown ater) that there are no equiibria where js (y)j < 2. This is required since we are ony deaing with achieving soutions 3 to (CP) which satisfy (A4). The mechanism gives rise to a one-shot game G, payed by a the agents in N, where action sets are (S ki ) ki2n and utiities are given by ^u ki (s) = v ki (x ki ) t ki = v ki (h x;ki (s)) h t;ki (s) 8 ki 2 N We wi say that maximising socia wefare for (CP ) has been fuy impemented in NE, if the outcomes (a possibe NE) of this game produce aocation x? and a agents in N pus the seer are better-off participating in the mechanism than opting out (getting aocation and taxes). The second property is known as individua rationaity Resuts. Theorem 4. (Fu Impementation). For game G, there is a unique aocation, x, corresponding to a NE. Moreover, x = x?, the maximiser of (CP). In addition, individua rationaity is satisfied for a agents and for the seer. The theorem wi be proved by a sequence of resuts, in which a candidate NE of G are characterised by necessary conditions unti ony one famiy NE candidates is eft. We wi then show that G has NE in pure strategies, and that a of them resut in aocation x = x?. Finay, individua rationaity wi be checked. Lemma 4.2 (Prima Feasibiity). For any action profie s = (y; P ) of game G, constraints C and C 2 are satisfied at the corresponding aocation. Proof. Constraint C is ceary aways satisfied. For y = we wi have x = and m =, so constraints C 2 and C 3 are aso ceary satisfied. We wi now show C 2 and C 3 for any y 6= as demand. In that case r < + (since there exists at east one ink q with js q (y)j and thus r q < +). Now, for any ink, we have the foowing two cases. If js (y)j = then the 3 Note that for the given aocation function, js (y)j = ; is equivaent to js (x)j = ;.

13 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 3 aocation for a agents on that ink is zero (aong with the corresponding m k s), so C 2 and C 3 for those inks is satisfied. If js (y)j we have m k = r n k r n c k P n k2k k2k k2k k2k n k = c k k2k where the first inequaity hods because r is the minimum of a r s. The second inequaity wi be equaity if js (y)j 2 and wi be strict ony if js (y)j = (see second sub-case in (7)). For C 2, take any agent ki and ink 2 L ki kix ki = r kiy ki rn k = m k where the inequaity hods because n k is the maximum over kiy ki s for a i 2 G k. Feasibiity of aocation for action profies is a direct consequence of using projections of demand y on to the feasibe region. Now we wi prove that a groups, using a ink, quote the same tota price wk for that ink at any equiibrium, this is brought about by the 3 rd tax term P (wk w k )2. This is a way of threatening agents with higher taxes just for quoting a different price than average, at each ink. Lemma 4.3. At any NE s = (y; P ) of G, for any ink 2 L we have w k = w 8 k 2 K : Aso, for any group k and ink such that G k 2 if we take any agents i; e 2 G k where aternate representation for e is g k (i) + then at equiibrium we wi have q ki = p ke. (which wi denote as p k;e) Proof. First we wi show the second part of the emma, so suppose there are agents i; e 2 G k as above, for whom q ki 6= p ke. Here if agent ki deviates with q ki = p ke then we can write the difference in agent ki s utiity after and before deviation by just comparing tax for ink (since aocation and tax for other inks don t change) ^u ki = (q ki p ke )2 + (q ki p ke )2 = (q ki p ke )2 > which means that the deviation was profitabe. This gives us the second part of the emma. (In addition to defining q ki = p ke = p k;e when G k 2 we wi aso denote p ki = w k = p k;i when G k = fig). For the first part, suppose there is a ink for which (w k ) k2k are not a equa, at equiibrium. Ceary then there is a group k 2 K for which w k > w k (this can be seen from (9)). We wi show that some agent i 2 G k can deviate by reducing price p ki and be stricty better off, thereby contradicting the equiibrium condition. First we wi take the case when the group k is such that G k 2 and then G k =.

14 4 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS Since w k > w k we must have w k > and since w k = P i2g k p ki there must be an agent i 2 G k for whom p ki >. Take deviation by this agent ki as p ki = p ki >, for which we can write the difference in utiity, just as before, as ^u ki = (w k w k ) 2 + (w k w k )2 p k;i (p k;i p k;i )(m k kix ki ) + w k (w k w k )(c m k ) + w k (w k w k )(c m k ) k2k k2k = 2 + 2(w k w k ) + p k;i (m k kix ki ) + w k (c m k ) k2k () ^u ki + 2(w k w k ) + p k;i (m k kix ki ) + w k (c m k ) A = ( + a) k2k where a > because of Lemma 4.2 and the fact that w k > w k. So by taking such that minfa; p kig > >, the above deviation wi be a profitabe one for agent ki. This gives the resut for G k 2. For G k =, say G k = fig, we have that p ki = w k > w k. This again means that p ki > and we take the deviation p ki = p ki > and get ^u ki + 2(w k w k ) + w k (m k kix ki ) + w k (c Foowing the same argument as above we wi get our resut here as we. m k ) A : k2k With Lemma 4.3, we can tak in terms of the common tota price vector at equiibrium rather than different tota price vectors for a agents. In particuar, any NE s = (y; P; Q) can be characterized as s = (y; P ) with P = (p k;i ) ki2n and p k;i = p k;i 2L ki. Later it wi become cear how p k;i and w take the pace of dua variabes ki and when we compare equiibrium conditions with KKT conditions, hence we identify the foowing condition as dua feasibiity. Lemma 4.4 (Dua Feasibiity). p k;i, w 8 i 2 G k, 8 k 2 K and 8 2 L. Proof. This is aso by design, since agents are ony aowed to quote non-negative prices and that w is the sum of such prices. Foowing is the property that soidifies the notion of prices as dua variabes, since here we caim that inactive constraints do not contribute to payment at equiibrium. This notion is very simiar to the centraised probem, where if we know certain constraints to be inactive at the optimum then the same probem without these constraints woud be equivaent to the origina. The 4 th and 5 th terms in the tax function faciitate this by charging extra taxes for inactive

15 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 5 constraints where the agent is quoting higher prices than the average of remaining ones, thereby driving prices down. Lemma 4.5 (Compimentary Sackness). At any NE s = (y; P ) of game G with corresponding aocation x, for any agent i 2 Gk, group k 2 K and ink 2 L we have m k c A = ; k2k p k;i kix ki m k = Proof. Suppose there is a ink for which w > and P k2k m k < c. Take any group k 2 K and an agent i 2 G k such that p ki = p k;i > (there is such an agent because w = P i2g k p ki > ). Take the deviation p ki = p k;i > and we get (using same arguments as in () and noting that w k = w k = w ) ^u ki = + p k;i (m k kix ki ) +w (c m k ) k2k {z } by Lemma 4.2 where a > due to Lemma 4.2 and the assumption that w (c that w (c ) = for a 2 L at equiibrium. P k2k m k C A = ( + a): P k2k m k ) >. This gives us Now suppose there is an agent ki for whom p ki = p k;i > and kix ki < m k. Same as before, we wi take the deviation p ki = p ki >, ^u ki = + p k;i (m k kix ki ) = ( + a) where a > by assumption. This gives us that p k;i (m k kix ki ) = for a ki 2 N and 2 L, at equiibrium. Lemma 4.6 (Stationarity). At any NE s = (y; P ) of game G, and corresponding aocation, x, we have v ki (x ki) = 2L ki p k;i ki 8 ki 2 N if x ki > v ki (x ki) 2L ki p k;i ki 8 ki 2 N if x ki = and (2) w = i2g k p k;i 8 k 2 K ; 8 2 L Proof. (2) is true by construction since we defined w k = P i2g k p ki and by Lemma 4.3 we have p k;i = p ki and w k = w. At any NE, agent ki s utiity in the game ^u ki (s ) = v ki (h x;ki (s )) h t;ki (s ) as a function of his message s ki = (y ki; p ki ), with s ki fixed, shoud have a goba maximum at s ki = (y ki ; p ki ). This woud mean that if this function was differentiabe w.r.t. y ki at s, the partia derivatives

16 6 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS w.r.t. yki at s shoud be. However, since our aocation diates/shrinks demand vector y on to the feasibe region, it coud be the case that increasing and decreasing yki gives aocations ying on different hyperpanes, meaning that the transformation from y to x is different on both sides of y ki and therefore we concude that ^u ki coud be non-differentiabe w.r.t yki at s. Important thing here however is to notice that right and eft derivatives exist, it s just that they may not be equa. Hence we can take derivatives on both sides of y i as (noting that derivative of the other terms in utiity invoving x ki or invoving m k wi be zero due Lemma 4.3) ^u ki y ki #y ki ^u ki y ki ki "y v ki (x v ki (x ki) 2L ki p k;i ki 2L ki p k;i ki y ki #y ki y ki "y ki We wi first show that i =@y i term above (for either equation) is aways positive. If y = then ceary this is true, because if any agent ki demands y ki > whie y ki = then ceary x ki > (in fact the aocation is differentiabe at y = ). If y 6= from (8), we can write where r = r q. ki ki = r + ki = r q + y ki From here we divide our arguments into foowing cases: (A) ki =2 N q (, i =2 Gk); q (B) ki 2 N q, i =2 arg max q j2g fq k kjy kj g and (C) ki 2 N q, i 2 arg max q j2g fq k kjy kj g. (A) Here q =@y i = and this makes = r q >. (B) Since vaue of r q depends ony on the vaue of (n q k ) k2k q, and in this case changes in y ki don t affect n q k we can see q =@y ki = and so = r q >. (C) We divide this case into two cases: js q (y)j 2 or js q (y)j =. If js q (y)j 2 then ki = r q + y ki = r q + y ki! P c q k ( k 2K q nq k ) ki : is either q ki or. If it is then = r q >. Otherwise we have = (rq ) 2 c q n q k k 2K q nfkg which is positive because js q (y)j 2, since then there is at east one positive term in the summation. For js q (y)j =, we wi consider S q (y) = fkg; ese in case S q (y) = fk g 6= fkg, taking the derivative woud give the same expression as above. For S q (y) = fkg we wi get r q = cq n q k c q n q k (nq k + ) = cq n q k n q k +! = cq n q k +

17 ki MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 7 = r q + y q q ki is either or q ki. If it is then = r q > and if it is q ki then we have So we have > in a cases. = (rq ) 2 c q Referring to (3), there are two possibiities, the first term on RHS in both equations in (3) is positive or negative. If it s positive, then we can see from the first equation in (3) that by increasing y ki from y ki (and therefore x ki from x ki ) agent ki can increase his pay-off, which contradicts equiibrium. Now simiary consider the first term in (3) to be negative, then from the second equation in (3), agent ki can reduce y ki from y ki to get a better pay-off. But the downward deviation in y ki is ony possibe if y ki > (, x ki > ). So we concude that v ki (x ki) = 2L ki p k;i ki 8 ki 2 N if x ki > v ki (x ki) 2L ki p k;i ki 8 ki 2 N if x ki = Coecting the resuts of the above emmas, we can concude that every NE satisfies the KKT conditions of the (CP). This means we now have necessary conditions on the NE up to the point of having unique aocation. In the next Lemma we verify the existence of the equiibria that we have caimed. Lemma 4.7 (Existence). For the game G, there exists equiibria s = (y; P; Q) = (y; P ), where corresponding aocation (x ki ) ki2n and prices (p k;i ) ki2n ; 2L and (w ) 2L satisfy KKT conditions as (x? ki ) ki2n, ( ki? )ki2n ; 2L and (? ) 2L, respectivey. Proof. The proof is competed in two parts. Firsty we wi check that for every x that can be a possibe soution to (CP) whie satisfying assumption (A4) there is indeed at east one y 2 R N + such that the aocation corresponding to y is x. (However we do not need to check the same for prices and Lagrange mutipiers (; ) since there it is straightforward). Secondy we wi check that for the caimed NE, there are no uniatera deviations that are profitabe. In ieu of (A4), the optima x? is such that js (x? )j 2 for a inks; aso it is cear that x? and m? are on the boundary of the feasibe region defined by C and C 2. So any vector y which is a scaar mutipe of x? woud give aocation x? (and corresponding m wi be equa to m? ). So in particuar, y = x wi aso do the job. Hence our first task is done. (kindy see more detaied comments at the end of this proof for why (A4) was required here) Now we wi check for profitabe deviations. For this we want to show any action profie that satisfies the hypothesis of above statement, is a NE. Due to assumptions on v i and by

18 8 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS construction, where we have taxes and aocations that are continuous, so we can do this by checking the first and second order conditions for any arbitrary agent, say agent ki. Price derivatives ^u = 2(w k w k ) q kj (m k kix ki ) w k (c m ki k k ) 8 2 L ki s.t. G k 2 2K This is ceary zero at equiibrium due to Lemma 4.3 and 4.5. Simiary we can get price derivative equa to when G k =. Now referring to the differentiabiity arguments made in the proof of Lemma 4.6, we symboicay ^u ki v (x ki) kiq ki kj q kj ki 2L (p ki q kj ki ki ki 2L ki w k (w k w k A 2L ki 2K ki where as before we note ki >, aways (note that we have written the above expression as if a inks have G k 2 but the expression and consequent arguments won t change much even if there are inks where G k = ). The second and third terms above are aways zero at equiibrium (due to Lemma 4.3). So if x ki > at equiibrium then the st term is aso zero as we (due to Stationarity) and if x ki = (, y ki = ) the st term is either negative, making the whoe derivative negative (this is fine since downward deviation isn t possibe from y ki = ), or equa to zero. For the remaining of the proof we wi consider the case where the first order derivatives are equa to zero. Second order partia derivatives are u pp ^u ki@p ki = 2 u p2 ^u ki@p 2 ki = u pq ^u ki@q ki = u pq2 ^u ki@q 2 ki = u qq := u yy ^u ki@q ^u ki = u py := = 2 u qk ^u ki@q k v ki (x ^u ki@y ki = q kj 2L ki kiq ki ki A = u qy ^u ki@y ki x ki! + w k + v ki k A 2K ki These derivatives wi give us a Hessian H of size (2L ki + ) (2L ki + ), where st row and coumn represent y ki and subsequent L ki rows and coumns represent p ki s for different s and

19 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 9 then the ast L ki rows represent q ki s. We want H to be negative definite at equiibrium. Now, st term in u yy is zero at equiibrium, and the 2nd term is stricty negative due to strict concavity of v i. This aong with u pp = u qq = 2 tes us that a diagona entries in H are negative. Aso notice that a off-diagona entries, except the first L ki + in the first row and coumn, are zero. Finay, note that due to assumption (A2), a prices are finite at equiibrium and so u py wi be finite. We wi show that roots of the characteristic poynomia of H (i.e. its eigenvaues) a become negative 8 y such that jyj sufficienty arge. For this, we take a generic matrix A, which is simiar in structure to H and has the same dependence on jyj as H. So A wi be of the form 2 A = 4 A D where D = ( 2)I Lki. In this case we know that eigenvaues of A and D together wi give us a the eigenvaues of A. Ceary eigenvaues of D are 2 repeated L ki times, so a that we now need to do is check whether a eigenvaues of A are negative. Entries in A are a a = a ij = a ji = 8 i; j > ; i 6= j jyj a ii = 2 a i = a i = b i jyj i L ki + where a > (and we don t care about the sign of b i s). We can expicity cacuate ja Ij and write the characteristic equation as! P a Lki Q() = ( 2 ) L ki i= + ( )i b 2 i ( 2 ) L ki = jyj jyj 2 So 2 is a repeated eigenvaue, L ki times. The equation for the remaining two roots can be written as! a C ( 2 ) + jyj jyj = 2 Necessary and sufficient conditions for both roots of this quadratic to be negative are 2 + a jyj! > 2a C jyj + jyj > ; 2 first of which is aways true, since a >. The second one gives jyj > C, which can be satisfied 2a (for arge enough jyj) irrespective of the sign of constant C. Hence we have shown the Hessian H to be negative definite for jyj arge enough. Severa comments are in order regarding the seection of the proportiona aocation mechanism and in particuar (7). If we use pure proportiona aocation i.e. same expression for r for js (y)j 2 and, then irrespective of optima soution of (CP), for game G the stationarity

20 2 ABHINAV SINHA AND ACHILLEAS ANASTASOPOULOS property wi not be satisfied for equiibria with js (y)j. Thus the mechanism wi resut in additiona extraneous equiibria. For this reason we tweak the expression for r when js (y)j, so that we can eiminate these extraneous equiibria - irrespective of the soution of (CP). With this tweak in the expression for r, a KKT conditions become necessary for a equiibria regardess of the vaue of js (y)j. This however creates a probem in the proof of existence of equiibria. In particuar, if x? was such that it had inks where js (x? )j = then in our aocation this woud require y at NE such that js (y)j =. In this case the r used woud be ower than what the proportiona aocation requires (see second sub-case in (7)) and we actuay woud have the probem of possiby not having any y that creates x? as aocation. Hence we have used (A4) to eiminate this case. Lemma 4.8 (Individua Rationaity). At any NE s = (y; P ) of G, with corresponding aocation x and taxes t, we have (4) and u ki (x; t) u ki (; ) ki2n t ki (W BB) 8 ki 2 N Proof. Because of Lemma 4.3, the ony non-zero term in t ki (see ()) at equiibrium is x ki P2L ki kip k;i, which is ceary non-negative. Hence P ki2n t ki at equiibrium. This is the seer s individua rationaity condition. Now if x ki = then we know from Lemma 4.3 and () that t ki = and so (4) is evident. Now take x ki > and define the function f(z) = v ki (z) z 2L ki kip k;i: Note that f() = u ki (; ) and f(x ki ) = u ki (x; t), the utiity at equiibrium. Since f (x ki ) = (Lemma 4.6), we see that 8 < y < x ki, f (y) > since f stricty concave (because of v ki ). This ceary tes us f(x ki ) f(). Now that we have Lemmas characterising NE in the same way as KKT conditions (and individua rationaity), we can compare them to prove Theorem 4.. Proof of Theorem 4.. We know that the four KKT conditions produce a unique soution x? (and corresponding? ). For the game G, from Lemmas we can see that at any NE, aocation x and prices p satisfy the same conditions as the four KKT conditions and hence they give a unique x = x?, as ong as (A5) is satisfied. So we have that the aocation is x? across a NE. This combined with individua rationaity Lemma 4.8, gives us Theorem 4..

21 MECHANISM DESIGN FOR MULTI-RATE MULTICAST SERVICE ON THE INTERNET 2 5. A Mechanism with Strong Budget Baance We now turn our attention to probem (CP 2 ). So in this case we have the agents in N, who are the owners and users that wish to aocate the good amongst themseves in a way that maximises P ki2n u ki. In this case one can now think of taxes as a way of faciitating efficient redistribution of the aready avaiabe good. Since a payments are made amongst agents in N and we have quasi-inear utiities, this ceary tes us that P ki2n t ki must be zero. This interpretation is sighty different from Section 4, where taxes were indeed payments made to the seer for provisioning of the good. A of the above is required to be done again under the assumption of strategic agents, which means the designer (who is a third party) sti has the probem of information eicitation and moreover has to make sure that the weath has to be redistributed in a way that we sti get x? aocation at the a equiibria. Here we wi say that the mechanism fuy impements maximising socia wefare aocation if in addition to the previous conditions, we aso have SBB. 5.. Information assumptions. These are the same as Section 4. For creating a mechanism in this formuation, main difference with the previous section, is that we have to find a way of redistributing the tota tax paid by a the agents. In the ast section we saw that the tota payment made at the equiibrium is B = xki 2L ki kip k;i A = r yki 2L ki kip k;i since a other tax terms were zero at equiibrium. We wi redistribute taxes by modifying tax function for each agent ony using messages from other agents. This has the advantage of keeping our equiibrium cacuations in ine with Section 4, since deviations by an agent woudn t affect his utiity through this additiona term. In view of this, we can express B as foows (5) B = r 2L ki N k j2n nfkig k jp k ;jy k j A ; where each term of the outer summation depends ony on demands of agents other than the ki th one. This means that each term in the parenthesis (scaed by the factor r) can now be used as the desired additiona tax for user ki. Observe, however, that in our mechanism, each agent s demand affects the factor r as we. So, if a agents can agree on vaue of r then we can use that signa to create the term that faciitates budget baance. In ieu of this, our mechanism here works by asking for an additiona signa ki from every agent and imposing an additiona tax of ( ki r) 2, thereby essentiay ensuring that a agents agree on the vaue of r (via ki s) at equiibrium. Finay, we use ki (cf. (9)) as a proxy for r in (5) - somewhat simiar to what we did with w k s. A