The Price of Anarchy (POA) of network coding and routing based on average pricing mechanism

The Price of Anarchy (POA) of network coding and routing based on average pricing mechanism Gang Wang School of Electronics and Information Engineeri ng Beihang Universit y Beijing, China gwang@buaa.edu.cn Xia Dai School of Electronics and Information Engineeri ng Beihang Universit y Beijing, China sunnydora01@hotmail.co m Abstract  The congestion pricing is an efficien t allocation approach to mediate demand and supply of netw ork resources . Different from the previous pricing using Affine M arginal Cost (AMC), we focus o n studying the game betw een network c oding and routing flow s sharing a single link w hen users are price anticipating based on an Average Cost Sharing (ACS) p ricing mechanism. We characterize the w orst-case efficiency bounds of the ga me c ompared w ith the optimal, i.e., the price-of a narchy (POA), w hich can be l ow bound 50% w ith routing o nly. When both network coding and routing are applied, the POA can be as low as 4 /9. Therefore, netw ork coding cannot improve the POA significantly under the ACS . Moreover, for more efficient use o f limited resources, it indicates the sharing use rs have a higher tendency to choose network coding. Keywords-congestion prici ng ; Average Cost Sharing (ACS); network coding ; POA I. I NTRODUCTI ON T he current Internet is fre quent th at a g roup of diff erent users share the same link transform ing information in a wide variety of real netw ork situa tion s. The congestion pricing , as an effective approach in te rms of improving the effic iency of netw ork resou rce allocation, is s et t o m ediat e dem and and supply of netw ork resources . Since the semin al p aper by Ah lswede et a l. [1] , n etwork coding has show n great potential for improvin g netw ork throughput in communication networks. However, gam e theoretic analysis on th e performance o f n etw ork coding has received more and more attention recently, e.g. , in [2]-[7]. All results in [2]-[4] focus on the case of intra-sessi on network coding p erf ormed by jointly encoding multiple packets from the same user, whereas inter-s ession network coding encoding packets from differ ent users is co nside red in [5-7] . Moreove r, a game theoretic analys is for inter-sess ion network coding of combining netw ork coding and routing flow s on a sing le bottleneck link is investigate d in [5- 7]. In additi on, a k ey perform ance metric named the price of anarchy (PO A) for analyzin g the system , is the ratio between the total payoffs at the w orst Nash equilibrium of th is gam e and the efficie nt payoffs, i .e., the maximum feasibl e total pay offs. The tot al payoffs are denoted by aggreg ate surplus , i.e., aggregate uti lity less aggregate cost. Obviousl y, it indicates that a higher POA denotes a smaller efficiency loss . All the resul ts in [7] are based on the Affine Marginal Cost (AMC) pricing mechan ism similar t o [8] . An Average Cost Sharing (ACS) pricing mechanism noted by Vasilis Ntran os in [9] is more robust that users cannot benefit f rom m erging o r s plittin g th eir rates th an the A MC mechan ism generally, which is more reasonable and fairness to each sharin g user. More specif ically , both the num ber of sharing users and the total rates through the sharing link have been taken into c onside ration . In this pa per, we f ocus o n studying the g ame between netw ork coding and routing flows sharing a single link when users a re p rice anti cipat ing based on the AC S prici ng mechan ism. The innovativ e design of ACS pricing function is         of users sharing the same technique (network coding or routing) . E    cost share charg ed by network is proporti onal to its action ( transm ission rate). The netw ork aggregate surplus is formulated by the total utility of all users minus the total netw ork cost of them. T he key contri butions of this paper is as follows ： 1 ) A complete analysis of a sharing single link system w ith both network coding an d routing flo ws is establish ed b ased on ACS pricing mechanis m. 2 ) POA is calculat ed under th e tw o pricing circu mstan ces, i. e., non-discrim inato ry pricing with routing-only users and discrim inatory pricing with both netw ork coding and routin g . The foll owin g is the o ut line of the remainder of this paper. In Section II, we achieve the results of the average cost sha ring pricing mechanism for a single link o n resource allocation games with routing-only . Then the case wh en some users can jointly perform inter-session netw ork coding will be conside red with extend ed results in Section III . Finally, conclusions are discussed in Secti on IV. II. ACS P RICING M ECHANI SM ----R OUTING G AME A. P roblem formulation In this section, we consider a framework with a single shared lin k 󰇛    󰇜 to allo cate n etw ork capacity eff iciently among a collection of  routin g- only users, i.e., all data that node  receiv e ar e simply for warded to node  , as shown in F ig . 1 . The set of users is denote d b y   󰇝     󰇞 . Each user    is endow ed with a utility function   󰇛   󰇜 depending on their desired data transm ission rate   from its sender   to its receive r   . In addit ion, l et         denote the total rate allo cated at the link , and th e link is overloa ded wh en the sum of rates  through it exceeds the fixed capacity  of the link. A cost sh are for the averag e queue exper ienced by us er  is denoted by   󰇛   󰇜 . Theref ore,   󰇛   󰇜 is the reduction in user    due to the network congestion . Generally ,   󰇛   󰇜 and   󰇛   󰇜 are measured in the same monetary units. Moreove r,   󰇛   󰇜 is the monetary value to each user  of a rate allocation   , while   󰇛   󰇜 is a monetary cost for congestio n at the link to each user  of a rate all ocation   charged by netw ork, respect ively . Figure 1.  routing-only users sharing a singl e link Next, we m ake the follo wing assumptions regarding   󰇛   󰇜 and   󰇛   󰇜 . Assumption 1 : There is a differentiab le, convex, an d non-decreasin g price function 󰇛  󰇜 over    , with 󰇛  󰇜   and  󰇛 󰇜   as    , and w e suppose the price fun ction is linea r for sim plicity . Assumption 2 : F or each transm ission rate     ,     , the cost f uncti on fo r the   user is m odeled as   󰇛   󰇜        󰇛 󰇜 subjecte d to    . He re   󰇛   󰇜 is convex and non-decreasin g. Moreove r, let    󰇛   󰇜  denote the cost incurre d at th e lin k w hen the total al locate d rat e is    . Similarly ,    󰇛   󰇜  is als o convex and non-decre asing. Assumption 3 : For each    , utility function   󰇛   󰇜 is concave, non-negative, strictly increasing, and differenti able . Moreover, for simplicity and generality , suppose the utility function is also linear for all users. That is   󰇛   󰇜      ,    , w here utility param eter     f or all users. In gener al, since the utility functions are local to the users and are not know n at each link, the eff icient res ource allo cation needs to be done via pricing . T heref ore, the pricing mechanism applied is very signif icant f or the sy stem . Propositio n 1 : Suppose ass um ption 1 holds. There exist two techniques for data transm ission on the shared single link: netw ork coding and routing. We defin e the foll owin g ACS pricing m echanism referrin g to [ 9]:  󰇡       󰇢 , (1) where   denotes the number of users sharing one of the two independent single links (ro uting o r netw ork coding ), and     is the sum of r ates for each user  through the sing le link w ith the sam e techn ique ,        . Given the rate v ecto r    󰇛         󰇜 from the use rs, the capacity of single communication link 󰇛    󰇜 is su ppose d as    , then             . For sim plicity and generality , we assume that                  in the r est of this p aper. In this scenar io, w e only take th e routing into account . Thus we set a single scalabl e price for the share d link to all the routing u sers:  󰇛  󰇜   󰇡       󰇢 , (2) where           is the sum of rates fr om all the routing users,  is the number of r outin g users sharing the same link 󰇛    󰇜 . Obvious ly, the price is affected by the sum of all routing  rates and the total num ber of routing users sharing the sam e link. Furtherm ore, the cost of each user    for its t ransm ission ra te   is   󰇛   󰇜      󰇛  󰇜 . Efficiency is defined in terms of the aggregate value of a netw ork allocat ion in [8 ]. Similarly, w e also analy ze the perform ance of the pricing m echanism from the po int of efficient all ocation, w hich ca n be characterized one via maxim izing aggreg ate su rplus as an optim al solu tion: maxim ize    󰇛   󰇜      󰇛   󰇜  , (3) subject to     ,        (4) The above objective functi on (3) is referre d as the aggregate su rplus . Then we should co nsi der the surplus of each user    for a price  󰇛  󰇜   , where they will play a game to acquire a share of link because users are anticipatin g the effect of their rates o n the resulting price . T h e notation   is denoted the vector o f all rates chosen by users o the r than  . Then given   , each u ser  will ch oose     to m axim ize its sur plus:  󰇛      󰇜      󰇟   󰇛   󰇜    󰇛   󰇜 󰇠     ,       . (5) In fact , the decision made by user  depends on the r ates selected by o ther users, leading to a resource allocat ion game among all routing users. A normal form of the game   is given as follow s:       󰇝   󰇞  󰇝   󰇛       󰇜 󰇞  , (6) where   󰇝     󰇞 is the set of players (sharing users ) , the r ate   is the st rategy o f each play er    , an d 󰇝   󰇛       󰇜 󰇞  󰇝   󰇛       󰇜      󰇛       󰇜 󰇞 is the set o f surplus functions that each play er    wishes to maxim ize. Furtherm ore,   󰇛       󰇜    󰇛   󰇜     󰇡       󰇢 . (7) Without loss of g enerali ty, f or each use r    ,   󰇛       󰇜       . Then w e can inf er th at     󰇛  󰇜     . In game   , each user    strate gically selects its rate     to max imize its payoff functi on   󰇛       󰇜 . From this model we ca n conclude that there may                 Cost:           .   .    Price:  exist such stabl e strategies, which are identif ied as Nash Equili bria. A system is said to be in a Nash Equili brium when no individual user    can increase its own p ay off by means of changing its strateg y   ultim ately. A Nash equilibrium of g ame   can be defined as a rate vecto r     , such tha t for all users    ,     , we have   󰇛         󰇜    󰇛        󰇜 . B. Nash Equilibrium and POA We first show that a Nash equilibrium exists fo r this gam e under the new average c ongesti on pricing mechanism . The intereste d readers can find the proof with reference to [8]. Therefore , we a chieve th at (i) Gam e   al w ays has a unique N ash equil ibrium . ( ii ) Given   is an optimal solu tion for function (3) and   is a feasible Nash Equilibriu m for game   , we want to get the ch aracte r min imize                                        , (8) where   and   shoul d satisfy       and       , respectiv ely. From the given conditions   󰇛   󰇜      and          , th e foll owin g inequali ty hold s:                                                                     󰇛     󰇜            . (9) Propositio n 2 : Suppose that Assumptions 1 , 2 and 3 hold , and that 󰇛 󰇜   , for some    . Also,   󰇛  󰇜   and       are assum ed. We prove that     . Proof : First, we show without loss of gener ality that   󰇛       󰇜   ,      obtained before, so it is obvious that    󰇛   󰇜      󰇛   󰇜  . Subsequen tly, ref erring to [8], a s known               , the optimal solu tion happens when the total rates (no more than fixed capacity 1) with all the  users existin g are only allocate d to user 1:     ,     ,     ,  ,     , and     . Consequently , the maxim al aggregate surplus is     , when        󰇛    󰇜        󰇛    󰇜     , thus the optimal solution of functi on ( 3) is obta ined. Next, the w orst cas e occurs w hen the util ity functions of th e users are line ar ( Assu mption 3 ). We then o ptim ize o ver all games w ith linear utility functions to determine the worst case efficiency loss. The worst case of gam e   is t o s olve the optimizati on pro blem as foll ows: min imize    󰇛    󰇜        󰇛    󰇜    , (10) where    is a Nash equilibrium for Game   . Because the surplus   is concave for fixed   , a v ector    is a Nash equilibrium if and only if the following first-order condition s are satisf ied f or each  ,   󰇛      󰇜      󰇛   󰇜      󰇛   󰇜        󰇛  󰇜     󰇛  󰇜     , (1 1)     󰇛  󰇜     󰇛  󰇜            ,     , (1 2)     󰇛  󰇜     ,     . (1 3) Given     󰇛  󰇜    before, then     is conclude d. To find the soluti on to (10 ), none of the user can be allocated the total capacity. We use these conditions to investiga te the efficiency loss when users are price anticipating. Specific ally , we ar e interested in comparin g the aggregate surplu s achieve d at a Nash equili brium with the aggregate surplus achieve d at a optimal solution as (8). As a result, from (1 2) and the given condition               , it indicates th at              . T h erefore, according to the Chebysh ev's sum inequality , w e have            󰇛       󰇜󰇛       󰇜 , (1 4) where              owin g to (12), and it is dependent on the total rates through the single link. T heref ore, we obtain the aggregate surplus under the worst case of Nash Equilibria for gam e   :    󰇛    󰇜       󰇛    󰇜          . Hence, the POA, i .e., th e eff iciency bounda ry is                                               . (1 5) For               , when    , the POA appro aches    , as required. T he worst case efficiency loss is always exactly 50% for any AC S price fu nction b y an appropriate choice of utility and price functi ons. Furthe rmore, the correspon ding n um erical results on POA f or 100 random ly sharing use rs w ith  approachi ng to 1 a re sh own in Fig . 3. ■ III. ACS P R I CING M ECHANI SM ---- N ETWORK C ODING A ND R OUTING G AME We also concentra te on the impacts on th e selfish users with the ACS pricing mechanism when inter-session network coding is applied . Without loss of simplicity and generality, w e only have considered two users performing the inter-sess i on netw ork coding as show n in Fi g. 2 . The network model is similar to that in Fig . 1, except two aspects: 1) the two dir ect side l inks: 󰇛      󰇜 from sour ce node   to destination node   , and 󰇛      󰇜 from source node   to destinati on node   ; 2) shared single link divi ded into two independent single links from the view o f t wo data transmis sion techniques (network coding and routing). In this scenari o, the first and the last users (i.e., users 1 and  ) can perform inter-session network coding . Let   and   denote the packets transmitte d from sender n odes   and   , respective ly . T he intermediat e node  can encode packets   and   together , and then send the encoded p acket , denoted by      , tow ards no de  (and f rom there towards   and   ) through the network coding link. Given the remedy data   from the sid e lin k 󰇛      󰇜 and the remedy d ata   from the side link 󰇛      󰇜 , nodes   and   can decode the encoded packets that they receive. In fact, nodes   and   can both d ec ode   and   . What  s more, the packet   w ithout encoded is simply transmitte d from node  to node  throu gh the r outin g link. Here   and   denote the data rates of source n odes   and   , respective ly . We have      by independent users 1 and  in general. Figure 2. A single link shared by two independ ent flow s, i.e., networ k coding and routing. Use rs 1 and  perform inter-se ssion netwo rk coding. Assumption 4 : The si de l inks 󰇛      󰇜 and 󰇛      󰇜 in Fig . 2 alw ays have zero cost and im pose ze ro pric es. A. Discri minatory prices In contrast to the ro ut ing-only situation in section II , we establish two d isc rimin atory prices  and  for network coding and routing users for each indepen dent single li nk, respective ly, as displayed in Fig. 2. Both the two prices are based on the ACS pri cing m ech anism in propositi on 1. Under su ch congesti on network setting in Fig. 2, the routing group is com bined by    (assume there exist    us ers) routing-only users and the one b etw een 1 and  with rate  󰇛      󰇜 (i.e., user 1, for the assumption               ). In additi on, they transm it data through the shared routing link at rat e   for each user     󰇝   󰇞 and at 󰇟  󰇛      󰇜   󰇛      󰇜 󰇠 for user 1, r esp ectively . As a result, the price for routing users follow ing th e ACS pricing m echanism is  󰇛  󰇜    󰇡     󰇛     󰇜     󰇛     󰇜  󰇢 , (1 6) where      󰇛      󰇜       󰇛      󰇜 is the sum of rates through the routing link as show n in Fig . 2 , and t he variable    is the number o f us ers forwarding data on the routing lin k, i .e., 󰇛      󰇜 . Afterw ards, the price for network coded users f ollow ing the same A CS pricing m echanism is defined as  󰇛  󰇜    󰇡  󰇛     󰇜  󰇢   󰇡    󰇢 , (1 7) where  󰇛      󰇜 is the sum of rates through the netw ork coding link, the constant 2 is the num ber of users transmitti ng data on the netw ork co ding lin k, i.e., u sers 1 and  . In particular, the aggregate data rate is no more than the fixed capacity 1 over the routin g link o f the single link (  ,  ) , i.e.,      󰇛      󰇜         as a whole. Clearly , the benefit of the network coding is to reduce the traffic load on the shared link 󰇛    󰇜 (thus r educin g the link cost) and save the resources w hile achie ving th e sam e rates. Moreover , we want to investigate the interactions betw een the two discrim inatory price s depending on the chang es of rates and the t otal num ber of users . Then w e define    󰇛  󰇜  󰇛  󰇜  󰇛  󰇜    󰇛    󰇜 . (18) From the equation, we can see that  is only affected by the total number  of u sers, the total rates  through the whole lin k and the minim al net w ork coding rate   . Owin g to the assumption that                  , we can conclude tha t       when    , and      when    . T he refore, compared with [5 ] there exist wider ranges for  because of the num ber o f sharing users and their choices b etw een netw ork codin g and rou ting. B. Ga me and POA There are enough conditions to defin e the resource allocati on game   among network coding and routing flow s as foll ows:       󰇝   󰇞  󰇝   󰇛       󰇜 󰇞  , ( 19 ) where   󰇝     󰇞 is the set of players (sharing users ) ,   is the strateg y of each p lay er    , and 󰇝   󰇛       󰇜 󰇞  󰇝   󰇛       󰇜    󰇛       󰇜      󰇛       󰇜 󰇞 is the set of surplus f uncti ons that each play er    wish es to max imize. The netw ork codin g users 1 an d  have   󰇛       󰇜    󰇛   󰇜  󰇛      󰇜  󰇛  󰇜     󰇛  󰇜 , (2 0)   󰇛       󰇜    󰇛   󰇜     󰇛  󰇜 . (2 1) While each routing user     󰇝   󰇞 has   󰇛       󰇜    󰇛   󰇜     󰇛  󰇜 . (2 2) We also analy ze the pe rform ance of tw o discrimin atory prices by effici ent allo cation, and m axim ize the aggr egate surplus as an optimal soluti on similar in (3). According to the similar proof in section II , the optimal solution o f game   occurs w hen the vari able   󰇛   󰇜 is all ocated to al l the users. Th at m eans    󰇛   󰇜           󰇛    󰇜         . (2 3) If      , the m aximal value of    󰇛   󰇜    is    . It impli es the optimal rate allocati on that          and           , wh ich means the shared link o nly occupied by netw ork codin g users t ransm itting their dat a. If       , the m axim al value of    󰇛   󰇜    is   󰇛󰇜  󰇛󰇜 when        . B y ( 11) and (12) i n II , th e worst case is identified by findin g the worst Nash Equili bri um Routing:       󰇛      󰇜     󰇛      󰇜             󰇛      󰇜                 Network coding:  󰇛      󰇜 Price:  Price:                           .   .    according to the equations (20), ( 21) and (22) in game   . Furtherm ore, the utility parameters for a Nash Equilibrium are     󰇛     󰇜  ,      for netw ork coding user 1 and user  respectiv ely , and         󰇛   󰇜   for ea ch routing user     󰇝   󰇞 . Due to the a bove equa tions of   for each user    , we can infer that        󰇛   󰇜  . The f irst- order deriva tive equation    󰇛   󰇜              is non-decre asing because of the concave function   󰇛   󰇜 . To obtain the worst case of game   , the same objective problem as in (10) is required to be solved. In this pa per, only tw o excep tional cases at    and    ar e analy zed in d etail. First, we consider the situation at    , only user 1 and user 2, f or            , the aggregate surplus is :    󰇛   󰇜       󰇛     󰇜  . We obtain      , and     owing to the given assumpti ons. Finally, the worst case is      w hen     . Then we consider the other situati on at    . We also suppose                  , and then                   f or th e concave utility function , so we can get the follow ing range        󰇛  󰇜  󰇛   󰇜  . With Cheby shev' s sum inequality in (14), w e can s olve the aggregate su rplus :    󰇛   󰇜     󰇛   󰇜 󰇛  󰇜    󰇛   󰇜   . Thus we can see that the worst case is a quadric equation of action   . For simplicity, let       , 󰇛  󰇜        , we denote  󰇛  󰇜   󰇛   󰇜      . W hen the number of sharing users    ,  󰇛  󰇜 is non-decreas ing, it indicates that  󰇛  󰇜   󰇛  󰇜   󰇡 󰇛   󰇜    󰇢 . Then we can obtain the w orst case of game   is the aggrega te surplus    󰇛   󰇜      when    . As mentioned above, we summarize the POA situations o f different numbers o f users sharing the single link when both netw ork coding and routing are applied. The proof in secti on II can be as referenc es. When    ,        , then     . (24) When    ,          , then     . (25) The above results extend the POA for routing-only flows in section II . I n addition, th e POA boundary is decreasing from   to   as the sharing users increasing from 2 to a quite lar ge number with proper choices of p ric e and utility param eters,  and  , respectively . Compared with the routing -only boundary   in section II , it suggests that network coding cannot improve the POA rem arkably w ith the A CS. IV. C ONCLUSIO N This p aper considers the ACS p ricing model dealing with netw ork resou rce allocati on problem betw een t wo independe nt techniques for data t ransm ission, i.e., routing and n etwork coding. Without loss of generality , we focus o n the case where two out of    users perform ing inter-sessi on ne tw ork coding in a sharing single link . T he results are dramatica lly different from the case when the AMC is applied . T he ACS pricing mechanism can be utilized to represent both the non-discrim inato ry pricing with routing-only users and discrim inatory pricing with b oth routing and network coding simu ltaneously . W e show that the choice o f diffe rent transmis sion techniques and the num ber o f sharing users can affect on the exact value o f POA, b ut actually network coding can not improve the POA obviously . Therefore, for mo re efficient use of limited resources, it indicat es the sharing users have a hig her ten dency to choose netw ork coding . Figur e 3. POA of 100 r an domly ro uting -only s hari ng users . R EFERENCES [1]                           1216, Apr. 2000. [2]        multicas t networks: maximu m        Theory , vol. 52, pp. 2433  2466, J une 2006. [3]      -           Phoenix, AZ , Apr. 2008. [4]         ame theoretic framew ork for wireless           China, May 2008. [5] A. Hamed Mohsenia n-Rad, Jianwe i Huang, and   A Game-T heoretic Analysis of Inter-Session Netw ork Coding  , IEEE ICC , Dresden, G ermany, June, 2009 [6] Amir- Hamed Mohsenian-R ad, Jianw ei Huang, Vin cent W.S. Wong, and Robert Scho ber,  Repeated I nter-Session Netw ork Coding G ames:   -   , unp ublished. [7] Amir- Hamed Mohsenian-Rad, Jianwe i Huang, Vincent W.S. Wong,  Inter-Se ssion Network Coding with Strategic User s: A Game-Theoretic   , un published. [8] Ramesh Johari          Allocation Mechanism With Bo    IEEE Journal On Sele cted Areas I n Communications, Vol . 24, NO. 5, M ay 2006. [9] Vasilis Ntranos   Cost Shar ing Networ k Routing Game  , www -bcf.usc.edu/~sha nghua/teach ing/Fall 2010/cost_sharing. pdf. 10 20 30 40 50 60 70 80 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 th e Nu mber of Sha ri ng u sers POA of Routing -only Ga me 1