Revenue in Resource Allocation Games and Applications

Transcription

1 Revenue n Resource Allocaton Games and Applcatons by Thanh Ten Nguyen Ths thess/dssertaton document has been electroncally approved by the followng ndvduals: Tardos,Eva (Charperson) Kozen,Dexter Campbell (Mnor Member) Klenberg,Jon M (Mnor Member) Zabh,Ramn (Mnor Member)

2 REVENUE IN RESOURCE ALLOCATION GAMES AND APPLICATIONS A Dssertaton Presented to the Faculty of the Graduate School of Cornell Unversty n Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy by Thanh Ten Nguyen August 2010

3 c 2010 Thanh Ten Nguyen ALL RIGHTS RESERVED

4 REVENUE IN RESOURCE ALLOCATION GAMES AND APPLICATIONS Thanh Ten Nguyen, Ph.D. Cornell Unversty 2010 Ths dssertaton studes a general class of resource allocaton games n computer systems. The applcatons of these games nclude sharng network bandwdth, schedulng jobs n data centers and dstrbutng clck-through resources n sponsored search. The man focus of the dssertaton s the revenue that can be obtaned by provders. We nvestgate the revenue of proportonal sharng under a symmetry condton among users, and show how to modfy ths mechansm to get a compettve revenue wthout the symmetry condton. We study the weghted proportonal sharng mechansm as a natural extenson of far sharng to capture the ncentves of revenue maxmzng provders.

5 BIOGRAPHICAL SKETCH Thành was born on 28 December, 1979 n làng Bùng, a small vllage located n the Red Rver Delta of the northern part of Vetnam. Làng Bùng s a remote vllage n Bac Nnh provnce, they dd not have electrcty untl the late eghtes. Thành s mother was one of the few teachers n the vllage, most other people were farmers growng rce and sweet potatoes. At the age of fourteen Thành was sent to Hano, the captal cty, for a better school. That was the begnnng of hs journey away from hs famly. At the age of eghteen Thành traveled to Budapest to study at Eötvös Unversty. He studed Hungaran and mathematcs n Budapest durng academc years and worked at Lake Balaton n summers. After sx years n Hungary, Thành contnued hs journey to the Unted States. Snce then, he has been lvng n Ithaca, NY, dong research n appled mathematcs at Cornell Unversty. When wrtng these lnes Thành s close to fnshng hs Ph.D. and preparng to start a postdoctoral poston at Northwestern Unversty.

6 To my famly. v

7 ACKNOWLEDGEMENTS Frst and foremost, I would lke to thank my thess advsor, Éva Tardos. Wthout her gudance, advces and endless support ths thess would not exst. I would also lke to thank the members of my commttee: Jon Klenberg, Dexter Kozen and Ramn Zabh for valuable advces durng my tme at Cornell. I thank my Vetnamese frends at Cornell for beng a source of support and encouragement. They have been wth me n both dffcult and happy tmes. I also thank my fellow graduate students at CAM and CS department, especally Yog Sharma, Muthu Venktasubramanam, Tudor Maran, Hu Fu, Joel Nshmura and Tm Novkoff for beng my great frends and colleagues. I have saved the best for last. I dedcate ths thess to my famly for ther love and support. v

8 TABLE OF CONTENTS Bographcal Sketch Dedcaton v Acknowledgements v Table of Contents v Lst of Fgures v 1 Introducton Illustratng Examples The Questons and Contrbutons of the Thess Related Lterature Prelmnares: Basc notatons and settngs Basc Notatons Polyhedral Envronments and Applcatons Proportonal Sharng for Polyhedral Envronments Proportonal Sharng n General Polyhedral Envronments Revenue and Effcency of Proportonal Sharng The Prmal Dual Approach Bound on the Revenue Bound on the Effcency Related Lterature Revenue Maxmzaton for General Sngle-parameter Auctons Wost-case Revenue Benchmarks The Mechansms Truthful Mechansm for Dgtal Good Auctons Nash Implementaton for the General Settng Related Lterature Weghted Proportonal Sharng and Keyword Auctons Sponsored Search Applcatons Weghted Proportonal Mechansm Revenue Prce of Anarchy Multple Provders Related Lterature Concluson Summary Future Research Bblography 93 v

9 LIST OF FIGURES 1.1 Network bandwdth sharng and downward close set system Network bandwdth sharng General aucton settng The shape of the utlty functons New utlty functons General aucton settng descrbed by a set system A revenue benchmark for downward closed set systems Convexty of the revenue Reducton to a smple constrant Geometrc nterpretatons of δ-utltes U (x) concave The functon (1 α)α 1 α α versus α Upper-boundng U (x) by an affne functon V (x) v

10 CHAPTER 1 INTRODUCTION New nternet technologes over the past decade have been changng economes and socetes around the world. These nnovatons, varyng from nformaton retreval, socal networks to electronc commerce, are creatng a trend n nformaton technology, namely, the mergng of human collectve behavor and technologes to create knowledge on a global scale. Computer scence, the man scence behnd ths technology trend, s facng many great challenges. The tradtonal computng models of Turng machnes, whch assume that the desgner has a full control on the nput nformaton and the executon of the program, are unrealstc n many modern applcatons. Prmary examples of these applcatons nclude the nternet routng networks that consst of multple routers makng ndependent decsons based on local nformaton and several applcatons n electronc commerce where nformaton s held by self-nterested agents. In the past decade an mportant lne of research n theoretcal computer scence, known as algorthmc game theory, has emerged as an nterdscplnary research area between algorthms and game theory. Ths s a subfeld of theoretcal computer scence that evolved from studyng computer programs executed by stand-alone machnes to complex systems nvolvng a large number of agents, who pursue ther own nterests. Algorthmc game theory has become a natural research area that uses game theoretcal approaches to nvestgate many problems n algorthm desgn and studes several concepts of game theory usng algorthmc methods. Algorthmc game theory formulates new problems and develops novel solutons for relevant modern computer scence applcatons. 1

11 Ths dssertaton studes a general class of resource allocaton games n computng systems, an mportant topc of algorthmc game theory. The focus of ths study s on the desgn of decentralzed mechansms to allocate resources to self-nterested agents. Game theoretcal approaches for resource allocaton problems overcome the drawbacks of centralzed schedulng algorthms, where prvate nformaton on the needs of agents s assumed to be avalable. The man startng pont of ths dssertaton s to study the revenue that can be obtaned by the provders. Ths s an mportant queston because n many cases the mechansms are desgned by provders who care about proft. Even when provders can only use a certan type of mechansms, t s often the case that ths class of mechansms has a flexblty for provders to choose some parameters. In these scenaros provder wll also adjust the parameters to maxmze ther revenue. There s a large lterature n resource allocaton games and proportonal sharng mechansm that we study n ths dssertaton. However, most of the works along ths lne focus on the socal welfare of the system. Another dfference of our studes, compared wth the lterature, s the soluton concept that we use. In tradtonal mechansm desgn, revenue maxmzng auctons are studed n Bayesan settngs, where the type of each player s drawn form commonly known dstrbutons. In ths thess we use the concept of Nash equlbrum n full nformaton settngs, where provders do not know users prvate nformaton or they need to use smple and natural mechansms. Mathematcally, we study an abstract resource allocaton game, where the resource constrants can be captured by a general polyhedron. Ths class of games captures a wde range of applcatons n computer scence, varyng from 2

12 sharng network bandwdth to schedulng jobs n data centers and dstrbutng clck-through resources n sponsored search. By takng a general approach, we can nvestgate dfferent problems n a unfed framework and use technques developed n one problem for another. We now start wth some smple examples to llustrate the content of the dssertaton. 1.1 Illustratng Examples In our resource allocaton games, we assume that there are n users, and the goal of the game s to determne a real valued outcome x 0 for each player, whch we thnk of as the player s level of actvty or allocaton. Each user has a non negatve, monotone ncreasng and concave utlty functon U (x ). Each user maxmzes hs pay-off whch s assumed to be the dfference between the utlty and hs payment U (x ) p. Far Sharng for Sngle Resource and Equlbrum Prce The smplest example n resource allocaton games s the case of sharng an nfntely dvsble resource of capacty 1 to a set of n users, each submts a non negatve number (bd) b. The allocaton x to user s set proportonal to b as follows: x = b j b j, and the payment that user wll need to pay s b. There are many other alternatve mechansms for ths smple aucton, such as the the frst and second prce aucton, but the proportonal sharng s a natural and smple mechansm. Most mportantly, proportonal sharng s scalable for 3

13 a wde class of users utlty functons, for example, any concave functon U (x ), a scenaro, where n the frst or second prce auctons users need to report the whole utlty functon. The far sharng mechansm above also provdes a framework of recoverng the market cleanng prce va a decentralzed mplementaton. The market cleanng prce n ths settng s an unt prce p, such that when each user chooses to buy an x fracton of the resource to maxmze hs payoff U (x ) px, (equvalently, U (x ) = p f x > 0) the total demand x s equal to the avalable resource, n our case x = 1. Now, n proportonal sharng, each user chooses a bd b and because of the proportonal sharng rule, we always have x = 1. Furthermore, when the mechansm reaches a Nash equlbrum, then the prce per unt that each user pays s the same and equal to b x = b, whch s not exactly the market cleanng prce. However, t can be shown that when the number of users ncreases, ths value approaches to the market cleanng prce. The proportonal sharng mechansm also captures the effect of each ndvdual user on the equlbrum prce. Classcal economc theory sometmes explans the market equlbrum by models consstng of nfntely many buyers where ndvdual s strategy does not affect the prce of the whole market. Ths assumpton s not a reasonable n many settngs. The proportonal sharng mechansm overcomes ths crtcsm by the fact that users strateges do affect the unt prce of resources. In other words, the users are prce antcpatng. More General Settngs Another advantage of the far sharng mechansm s that t can be extended to a much more general settng. Consder the followng example. The servce provder can ether serve a sngle user (numbered 0) or a 4

14 set of other users (numbered from 1 to n). If randomzaton s allowed then ths settng can be captured by the followng nequalty system x 0 + x 1 [1,.., n]. Ths nequalty system captures exactly the followng network bandwdth sharng game. User 0 s nterested n a path of bandwdth x 0 contanng n dfferent edges e 1,.., e n, each wth a capacty of 1. User, 1 n, s only nterested n a path contanng sngle edge e. See Fgure 1.1. x 1 x 2 x 3 x 4 x 0 PSfrag replacements x 1 x 2 x 3 x 4 x 0 Fgure 1.1: Network bandwdth sharng and downward close set system. The proportonal sharng mechansm can be extended for ths settng as follows. User 0 bds a non negatve number b 0 on each edge of the graph and user, 1 n, only bds b on the edge e. The mechansm wll use the far sharng on each lnk. User, 1 n pays b and gets x = b. b + b 0 User 0 pays b 0 and gets x0 = mn b 0. b + b 0 Smlar to the case of a sngle resource dscussed above, the sum of the bds on each lnk can be seen as the prce of each resource, whch s determned 5

15 by the demand of the users. It s also well understood that n the computer network settng, these prces have an nterpretaton of the average delay on each lnk [30, 29]. Proportonal Sharng for a General Polyhedron In ths thess, we study ths proportonal sharng and extend t for an even more general settng that we call polyhedral envronment. Ths wll be defned more formally n Chapter 3. Intutvely, each edge n the example of the network above corresponds to a lnear constrant of a general polyhedron. We wll see later n Chapter 4 that, n the aucton settng one can thnk of ths generalzed proportonal sharng as a way to desgn competton among users. Moreover, ths general problem captures many other applcatons, ncludng the sponsored search aucton. Our approach gves a rch model for ths applcaton because t can model complex externaltes among advertsers. 1.2 The Questons and Contrbutons of the Thess Our frst goal s to analyze the revenue of Nash equlbrum of the proportonal sharng mechansm for general polyhedral envronments. We consder a stuaton where the system conssts of many users havng smlar demands and utltes. Ths s a natural scenaro n many systems, such as the nternet routng network and many nternet auctons. Queston 1: Wth symmetry among competng users, what s the revenue and effcency of the proportonal sharng? 6

16 In Theorem 3.2, we show that both the effcency and revenue converges to the optmal f the number of competng users ncreases. The bounds on the effcency and revenue n Theorem 3.2 are qute strong. But the case when there s no symmetry among users remans an mportant queston. Although as we wll see n Chapter 3, the proportonal sharng always gves a near effcent allocaton, the revenue can be very poor. An smple example s the case of bandwdth sharng game n Fgure 1.1, where user 0 has a lnear utlty U 0 (x) = ɛ x for a small ɛ and user s utlty s U (x) = x. It s not hard to see that the proportonal sharng descrbed there only creates drect competton between users, 1 n and user 0, thus user, 1 n do not have ncentve to pay hgh and therefore the revenue s low. For nstance, f ɛ = 0, then at Nash equlbrum, the revenue s 0. Thus, we come to the followng queston. Queston 2: How much revenue should we expect to get and how should we desgn a mechansm to get a compettve revenue when there s a lack of symmetry among competng users? In Chapter 4, we answer ths queston by ntroducng a new revenue benchmark for the general aucton settng (Defnton 4.2) and show that one can combne the proportonal sharng mechansm wth a reserve prce scheme to obtan a constant factor of ths revenue benchmark (Theorem 4.4). Questons 1 and 2 are concerned wth the desgn of mechansms that do not have any nformaton on the valuaton (utlty) of the users. It has been recognzed that n practce, provders try to learn the market demand and charge dfferent prces for dfferent market segments. Ths s commonly called prce 7

17 dscrmnaton [61]. Prce dscrmnaton s studed n many settngs, ncludng the full nformaton games [59, 60] and asymmetrc nformaton games [5]. Prce dscrmnaton s usually used to ncrease the seller s revenue, the effects of prce dscrmnaton on socal welfare, however, are unclear. We would lke to understand ths effect of dscrmnaton n proportonal sharng n a full nformaton settng. To study ths queston, we ntroduce a generalzaton of proportonal sharng mechansm, whch we call the weghted proportonal sharng mechansm. In ths mechansm, each user s allocated x = b j b j C, where the values C are decded by strategc provders to ncrease the revenue. Our thrd queston s Queston 3: When the provder uses the weghted proportonal sharng mechansm to dscrmnate among users, how much revenue can the provder get and what s the effcency loss? We show that the revenue of the weghted proportonal allocaton s nearly as good as the revenue under standard prce-dscrmnaton, where provder can charge dfferent unt prces for dfferent users. For lnear user utlty functons, the socal welfare at Nash equlbra s at least 1/(1+2/ 3) fracton of the maxmum socal welfare, and ths bound s tght (Theorem 5.4). We extend ths result to a broader class of utlty functons and to the case of many provders (Theorem 5.9). In the applcaton to sponsored search, our framework gves a dfferent approach from the the General Second Prce (GSP) auctons that s n common use by search engnes. GSP s an algorthm for placng ads to ad-slots, where the bds of advertsers are multpled by weghts that can be dfferent for dfferent 8

18 advertsers and such weghted bds are used for placng the ads. The larger the weghted bd, the better the poston that the ad gets. The reason to ntroduce these weghts s explaned by the term clck-through rates r j, whch s the probablty that users clck on ad when t s placed at poston j. It s commonly assumed that r j = α β j, where α s the qualty of an ad capturng how relevant the ad s to the search keyword, and β j s the qualty of the poston: a large value of β j s assocated to a good poston among the sponsored lnks. The frst dsadvantage of ths approach s that t s possble that n addton to the ad s qualty, other ads that appear on the same page can affect ts clckthrough rate (externaltes). Thus, α cannot capture the real clck-through rates. Second, the values α are gven by search engnes, t mght be the case that these parameters are also desgned strategcally to maxmze revenue. The weghted proportonal mechansm s studed for general polyhedral envronment, a model that captures an applcaton of sponsored search wth complex externaltes among ads. Also, n weghted proportonal mechansms, the weghts are decded by proft maxmzer provders as an analog to the structure of the General Second Prce auctons. 1.3 Related Lterature Optmal Mechansm n Bayesan Settngs Proft maxmzaton n mechansm desgn has an extensve hstory begnnng, prmarly, wth the semnal paper of Myerson [40] and smlar results by Rley and Samuelson[50]. These papers study optmal mechansm desgn n Bayesan settngs and the soluton concept the Bayes-Nash equlbra. In ths settng, players types are assumed to 9

19 be drawn from commonly known dstrbutons, and each player only knows about hs own type. A Bayesan Nash equlbrum s a strategy profle that maps each player s type to an actons such that, gven ths strategy each player maxmzes ther expected payoff over other players dstrbutons. In the optmal aucton of Myerson, Rley and Samuelson, players are the bdders and the goal of the auctoneer s to desgn a mechansm to maxmzes the expected revenue. Ths materal s by now standard and can be found n basc texts on aucton theory [37, 25]. Pror-free Truthful Mechansm Desgn The optmal mechansm n Bayesan settngs hghly depends on the dstrbutons of bdders type. In many applcatons, these dstrbutons are hard or mpossble to obtan. Pror-free auctons have recently been of much research focus because of the need for more robust auctons that do not depend on the underlyng dstrbutons of bdders valuatons.the man constrant n ths lne of work s to requre the mechansm to be truthful, that s, t s best for bdders report ther true type regardless what other bdders do. Ths approach s consdered n economcs lterature as detalfree or robust mechansm desgn [8]. In computer scence the approach was frst consdered by [12] and followed by a large lterature [14, 16, 15]. The truthful condton s strong, furthermore, ths framework does not provde a nce characterzaton for the optmal revenue as n the Bayesan settng. The works n [12, 14, 16, 15] defne revenue benchmarks and desgn mechansms that approxmate these benchmarks. Nash Implementaton n Full Informaton Settngs Ths thess takes a dfferent approach from the two lnes of research descrbed above. We use the theory 10

20 of Nash mplementaton n full nformaton settngs. In ths settng, players have the complete nformaton about each other. Ths does not mean that the desgner knows ths nformaton. In Chapter 3, 4, we assume that the desgner does not have any nformaton about the users utlty, he only knows the set of possble outcomes. The lterature on Nash mplementaton of full nformaton games s large, ntated wth the semnal work Maskn [33], for whch he won the Nobel prze n For more on related works n the area, see the surveys [33, 34, 48, 32]. Ths lterature, however, s mostly concerned about mplementaton for the goal of maxmzng socal welfare. Ths s where ths dssertaton dffers from prevous work. We focus on the revenue can be obtaned n Nash equlbra. Proportonal Sharng Mechansm The classcal proportonal sharng mechansm s ntroduced and studed by Kelly [24]. There s a large lterature studyng varous aspects of the proportonal sharng mechansm, ncludng robustness, convergence of response dynamcs, effcency and practcablty [62, 13, 55, 22, 20, 18, 30, 49]. Johar and Tstskls [20] show that, when the utltes U are concave, then at Nash equlbra the socal welfare s at least 3/4 tmes the socal welfare of the most effcent allocaton. The revenue of proportonal sharng s studed by the author wth Éva Tardos and Mlan Vojnovć n [42, 43, 44, 41]. Sponsored Search Auctons Sponsored search s a form of advertsng, typcally sold at auctons where merchants bd for postonng alongsde web search results. Ths s one of the fastest growng, most effectve and proftable forms of advertsng, that has attracted researchers n both computer scence and economcs [28, 10, 6, 58, 3]. Our work connects the basc proportonal sharng 11

21 mechansms to the applcatons of sponsored search. Our framework captures complex externaltes, an mportant feature of sponsored search auctons. Mechansm Desgn wth Many Sellers One of our results n ths thess s for the case of multple provders. Ths s an exctng drecton n mechansm desgn. Mechansm desgn for multple provders s complex and not very well understood. Many standard technques such as revelaton prncple fals n ths envronment. For more detals on recent development of ths area see the survey of D. Martnmort [31] and recent works of M. Pa [45, 46]. Structure of the Thess The dssertaton has 6 chapters. In Chapter 2 we gve some basc notatons and concepts that wll be used throughout the thess, we also descrbe applcatons of the general polyhedral envronments. Chapter 3, 4 and 5 gve answers to the three questons dscussed at the begnnng of ths chapter. Chapter 6 concludes the dssertaton wth future research drectons. 12

22 CHAPTER 2 PRELIMINARIES: BASIC NOTATIONS AND SETTINGS 2.1 Basc Notatons Provders and Users The general resource allocaton games we study consst of multple provders and users. Provders own the resources and use some types of mechansms to allocate the resources to the users. In ths thess, dependng on the context, we sometmes use sellers, auctoneers for provders or buyers, bdders for users. In Chapter 3 and Chapter 4, we nvestgate the case of a sngle provder. The general case of many provders s consdered n Chapter 5. We denote by n the number of users n our system. Allocaton Vectors The resource that a user gets s expressed as a non negatve real value x ndcatng the user s level of actvty or allocaton. We call x an allocaton vector. Usually, provders have lmted resources, and the allocaton vectors need to satsfy some constrants. In ths thess, we assume that the provder knows the set of all possble allocaton vectors x. Users Utltes Each user has an utlty functon U (x ) on the amount of resource x that he gets. We wll assume that all U are non negatve, monotone ncreasng and concave, and U (0) = 0. The concavty condton s a tradtonal assumpton n the lterature to capture the dmnshng returns property of utltes. Ths s one of the most common assumptons used n economcs lterature. 13

23 Sngle Parameter Settng We sometmes focus on a specal case of utlty functons, namely, lnear utltes: U (x ) = v x. In ths stuaton, we call the settng as sngle parameter, because each utlty can be represented as a sngle number v 0. We call v the (prvate) valuaton of user. In desgnng a mechansm, we assume that the provder does not know the valuatons of users. Mechansms In a mechansm, each user has a message space M to report or sgnal to the provder and other users about hs type. Based on the reported messages (m 1,..., m n ), m M from all the users, the provder allocates the resource x and asks for a payment p from the user. Thus, x, p are functons on the doman n =1 M. We assume that there s also an opton for each user not to partcpate n the mechansm. Ths can be encoded as a specal message n each M. Quas-lnear Payoff In ths thess, we assume users have quas-lnear payoff, whch s the dfference between the utlty and the payment: U (x) p. Nash Equlbrum Nash equlbrum s the soluton concept mostly consdered n ths thess. We assume a vector m to be a Nash f assumng no other user want to change ther message, t s best for user to keep hs m to maxmze the payoff, whch s U (x ) p. Because there s a not partcpate opton for each user, at Nash equlbrum, for every user, we have U (x ) p. Domnant Strategy Equlbrum A Nash equlbrum s called domnant strategy f t s best for each user to keep hs m no matter how other users report ther messages. Domnant strategy s a stronger soluton concept than 14

24 that of Nash equlbrum. It has been showed that every mechansm wth domnant strategy equlbrum can be mplemented by a mechansm, where each user report drectly ther utlty, or n the sngle parameter settng to report ther valuaton. Ths mechansm s called truthful mechansm. Revenue The revenue of a mechansm s the total payment of all users p. Dependng on the soluton concepts, one can talk about the revenue of a Nash equlbrum or of a truthful mechansm. Socal Welfare, Prce of Anarchy The socal welfare s defned as the total of users utlty: U (x ). In quas lnear-payoff model, the socal welfare s the sum of users payoffs and the total revenue obtaned by the provders. In many cases, we would lke to compare the socal welfare at Nash equlbrum wth the optmal socal welfare. The rato between the worst socal welfare of a Nash and the optmal socal welfare s call the prce of anarchy. 2.2 Polyhedral Envronments and Applcatons In the followng we wll descrbe a general envronment that we call polyhedral envronment. Ths s a general type of constrants on the resources, that capture a wde range of applcatons n computer scence. The provder has polyhedral constrants on the resource. That s, the allocaton vector x that the provder can allocate needs to be n a convex set of a form { x IR n + : A x 1}, where A s a non negatve matrx. 15

25 Note that any polyhedron of the form {A x c, x 0}, where A s a non negatve matrx and c s a non negatve vector, can be normalzed to the form of {A x 1, x 0}. Network Bandwdth Sharng The most natural example s the bandwdth sharng game, where each provder owns a network of capactated lnks, each user s sendng traffc along a path P and x s the data transfer rate for user. In ths case we have a resource constrant assocated to each lnk e: :e P x c e where c e s the capacty of lnk e. ce x PSfrag replacements Fgure 2.1: Network bandwdth sharng. Keyword Auctons The general convex constrants can also capture a general model of keyword auctons. Ths s the man applcaton to be consdered n Chapter 5. The aucton s for a sngle keyword, and there are n advertsers bddng to have ther ad appear as a sponsored search result. There are fnte set of outcomes, dependng on whch bdder gets dsplayed n whch poston. We descrbe each of these outcomes as a n dmensonal vector whose coordnates are the expected number of clcks that the correspondng advertser gets. More precsely, let x 1,..., x N be all the possble outcome vectors, and x k = (x k 1,..., xk n), where x k s the expected number of clcks that advertser receves at outcome k. To thnk of keyword aucton as a convex resource allo- 16

26 caton, we need to allow randomzaton n the allocaton of bdders to postons. Choosng between the determnstc allocatons by the probablty dstrbuton p = (p 1,..., p N ), we have that j p jx j s the vector whose coordnates correspond to the expected number of clcks of an advertser. Now the set of expected allocaton vectors obtaned ths way s exactly the convex hull conv( x 1,..., x N ) = { x : x = j p jx j, p j [0, 1] for every j and j p j = 1}. Ths way of modelng keyword auctons wll be dscussed n more detals n Chapter 5. We wll show that the convex hull of x k can be seen as a specal case of our polyhedral envronment under a natural assumpton. Sngle Parameter Aucton Represented by a Downward-closed Set System Ths s applcaton wll be dscussed n more detal n Secton 4. In ths settng each agent has a prvate valuaton for recevng servce and there s a set system representng feasble sets. A feasble set s a set of agents that can be served smultaneously. For example n aucton for sngle tem the feasble set system Sngle tem Dgtal goods PSfrag replacements General settng Fgure 2.2: General aucton settng. contans sngleton. We focus on the typcal case of downward-closed envronment where every subset of a feasble set s agan feasble. Another example of such an envronment s a combnatoral aucton wth sngle-mnded bdders, 17

27 where feasble sets correspond to subsets of bdders seekng dsjont bundles of goods. A more general example s a combnatoral aucton wth sngle-value bdders, each of them has an utlty of a sngle value, v, when he obtans one of many possble sets. It wll be shown latter that the randomzed outcomes of ths envronment can be captured by our general polyhedral settng. Schedulng Jobs n Data Centers Ths s a problem of allocatng data center resources to users. In ths applcaton, typcally each user needs to fnsh a job whch requres readng many dfferent blocks of data across machnes n a data center. Let D j be the amount of data of type j that job needs to process and s j be the speed that job can process data of type j. Thus, the tme to read ths data s D j /s j. The fnshng tme of job s t, whch s the maxmum processng tme of the job across all types of data that t requests. One can consder the model when each job tres to maxmze the utlty U (1/t ). Typcally, data centers are complex systems consstng of many clusters of machnes and data has many copes across the clusters. The constrants on s j are complex, but n many cases t can be captured by convex constrants. Therefore, the allocaton vector x can also be captured by convex constrants. In ths example, t s unrealstc to desgn a mechansm that requres every job to know exactly the complex constrants on x. Smple mechansms are crucal n these applcatons. 18

28 CHAPTER 3 PROPORTIONAL SHARING FOR POLYHEDRAL ENVIRONMENTS The far sharng mechansm was motvated by the need for a smple and easy to mplement mechansm for the resource sharng problem on the Internet. Ths mechansm s now qute well studed and has been used to mplement many nternet routng protocols. The desgn of nternet congeston control protocols s based on several deas varyng from usng auctons to smple prcng. But these proposal share the basc goal of maxmzng socal welfare. The dea s to mplement a smple lghtweght mechansm that helps arrange the socally optmal sharng of resources. Congeston prcng [23, 53], has emerged as a natural way to decde how to share bandwdth n a congested Internet. Whle maxmzng socal welfare s mportant to keep customers subscrbed to the system, we beleve that revenue should also be consdered. Once a mechansm gets mplemented, the network managers wll try to take advantage of the users, and am to maxmze ncome, and wll no longer only thnk of the mechansm as a way to arrange the best use of the network by maxmzng socal welfare. As a result, t s mportant that we also understand the revenue generatng propertes of the proposed mechansms. In ths chapter we nvestgate ths queston n the context of a proportonal sharng mechansm of Johar and Tstskls [20] that generalzes the far sharng for general polyhedral envronments. Our man motvaton s to study the performance of ths mechansm n settng where there s a hgh symmetry among competng users. Ths s a natural assumpton, especally n the networkng scenaro where users are often classfed nto few categores : small or heavy 19

29 users, uploaders or downloaders. Our man queston s: Under a symmetry assumpton how the far sharng mechansm acheve both goals of revenue and effcency? Results We show that wth few assumptons, whch we wll explan more formally n Secton 3.2, we can obtan good bounds on both effcency and socal welfare. We develop a new technque for analyzng such allocaton games, and bound the revenue. Our technque for boundng the revenue uses the smlarty between the condton of Nash equlbrums of the game and the dual of a certan lnear program. We show that the game approxmately maxmzes the revenue of the auctoneer, wth the approxmaton rato tendng to 1 f players utltes are lnear and the number of dentcal players ncreases. In a more general class of utltes satsfyng U(2x) > αu(x) for some constant α > 1, the approxmaton rato of the revenue wll tend to α 1. We also strengthen the effcency result to show f there are k users of every type than the effcency s at least (1 1 ) tmes the socal welfare of the most effcent allocaton,.e., the 4k effcency tends to the optmal as the number of dentcal players ncreases. Our man theorem can be clamed more precsely as follows: MAIN THEOREM Gven a constant α > 1, and an nteger k 2, under the assumpton that each player s utlty satsfes U(2x) > αu(x) and for each player type, there are at least k players (defned formally n secton 3.2), the far sharng mechansm (defned n secton 3.1) obtans both approxmately maxmum effcency, and approxmately maxmum revenue. The effcency s at least (1 1 ) tmes the optmal effcency and the revenue 4k s at least (α 1)(1 1 ) the optmal revenue. k 20

30 Remark Note that ths bound s very strong when utlty s lnear (and so α = 2). For ths case we have the revenue of the mechansm s at least f (k) = (1 1) k tmes the optmal. Already when there are 2 players of each type (when k = 2) the mechansm acheves 7 tmes the optmal effcency and half of the maxmum 8 revenue. Organzaton of the Chapter In Secton 3.1 we descrbe the mechansm n the polyhedral envronment. Secton 3.2 dscusses the bound on the revenue and the effcency of ths game. Secton 3.3 dscusses the related lterature. 3.1 Proportonal Sharng n General Polyhedral Envronments In ths secton we descrbe the far sharng mechansm for the general class of games ntroduced n Chapter 2. The mechansm s an extenson of the mechansms ntroduced by Kelly [23], Johar and Tstskls [20]. Let E denote the set of constrants (the rows of A). For smplcty of notaton, we assume that u e = 1 for each e E by normalzng each row. We wll use α e to denote the row e of matrx A, whch we wll also call constrant e. We now have the followng descrpton of the set of feasble allocatons: α e x 1 for all e E, (3.1) x 0. The Mechansm When sharng a sngle resource wth constrant x 1 the far sharng [23] mechansm requres that each player j submts a bd b j, the amount of money she wants to pay, and the resource s allocated proportonal 21

31 to the bds, as x j = b j / b. We can thnk of b as the unt prce p of the good. The allocaton s derved from ths unt prce, as user j gets x j = b j /p amount for the cost w j = b j at ths prce. To extend ths mechansm to a sngle constrant wth coeffcents α x 1, we agan requre that each player j submt a bd b j, her wllngness to pay, and vew p = b as the unt prce of the good. Recall that α e j s the rate at whch user j uses resource e, so user j needs α j x j allocaton for a value x j. At the unt prce of p she gets α j x j = b j /p allocaton, and hence we need to set x j = b j /(α j p) = and she wll have to pay w j = b j = α j x j p. b j α j b, For envronments wth more constrants, Johar and Tstskls [20] extends the far sharng mechansm by requrng that users submt bds b e j separately on each resource e. As before, we can vew the sum of bds p e = b e as the unt prce of resource e, and allocate the resource at ths prce. Ths allocaton lmts the value x j for user j to at most x e j = be j /(αe j pe ). The dea s to ask users to submt bds b e j for each resource e, allocate the resources separately, make user j pay w j = e b e j, and then set x j = mn {e:α e j 0} x e j. We need to extend ths mechansm to be able to deal wth resources that are under-utlzed. Some constrants e may not be bndng at any soluton, and the far sharng method does not deal well wth such constrants: users wll want to bd arbtrary small amounts as there s too much of the resource. To deal wth such constrants, we allow each player to request an amount r e j wthout any monetary bd. For each resource e f the prce s 0 (that s p e = b e = 0) and α e r e 1 (the requested rates can all be satsfed) then we settng x e j = re j for all j. 22

32 The mechansm can be descrbed formally as follows: DEFINITION 3.1 (Generalzed Proportonal Sharng) Each player j submt a bd b e j and a request r e j for each resource e. For resource e we use the followng allocaton: If b e > 0 then x e j = b e j α e j ( b e ) for j If b e = 0 and α e re 1 then x e j = re j for j Else, set x e j = 0 for j. For each player j, the amount of money that she needs to pay s w j = e b e j and the fnal allocated x j = mn {e:α e j 0} x e j. Prce Takng Strategy Kelly [23] has consdered a verson of ths game when prces are assgned by the network, and users are prce takers n the sense that they act to optmze ther value at the gven prces. We can also vew our farsharng game as a prcng game, but n our game the prces are determned as part of the game. However, t s useful to compare the mechansm above wth a game where players behave as prce takers. Consder an equlbrum of the game, t must be the case that x e j = x j for all resources e that costs money, or otherwse player j can reduce her bd b e j wthout affectng her allocaton. One way to thnk about the mechansm above s the followng: Players decde on each resource (constrant) a prce p e = j b e j ; now players have to pay for each resource e at ts unt prce p e. To make sure a player gets enough of resource e to have a share of x, he needs α e x of the 23

33 resource, and hence needs to pay p e α e x for resource e. In order to get all the needed resources player must pay a unt prce of e α e pe for hs resource. Now, f we assume that the prce p e are gven, then for each player the unt prce s fxed. Therefore to maxmze her utlty, player wll maxmze hs utlty, that s U (x ) e p e α e x. Takng the dervatve n x to determne the optmal value for user we see that user wll choose to buy an x such that: the dervatve U (x ) s equal to the unt prce or n the case U (0) s less than the unt prce, she wll choose not to buy any resource. We rewrte ths as follow: U (x ) = α e pe OR x = 0 f U (0) < α e pe. (3.2) e e Condton for Nash Equlbrums In our mechansm, the prces p e are not fxed. They are the sum of all the bds on each constrant, whch are gven by strategc players. As a result, the Nash condton gven below s slghtly dfferent from (3.2). In the allocaton game, the prce s a functon of the bds, and ths nduces the players to shade ther bd for the resource, gettng a bt less resource at a smaller prce. Johar and Tstskls [20] prove that a Nash equlbrum exsts and gve the followng condtons. In ths condton, observe that the dfferences between the Nash condton (3.3) and the condton (3.2) are 1 the terms. We wll show later that usng the compettveness condton (1 α e j x j) (defned n Secton 3.2), these terms are small. THEOREM 3.1 ([20]) If the utlty functon of each player s ncreasng, dfferentable and concave, then there always exsts a Nash equlbrum. 24

34 An allocaton x a bd and a request vector b, r s a Nash soluton f and only f: α e x 1; x 0 for all e E, U j (x j) = e p e α e j (1 α e j x j), f x j > 0 or (3.3) x j = 0 f U j (0) e p e α e j ; where pe = b e. Proof. To smplfy the presentaton, and wthout loss of generalty, we wll assume that each resource e has at least two dedcated users who only needs resource e, and who have small, but lnear utlty ɛx. These users wll guarantee that no resource s under-utlzed, but wll not change ether the optmal allocaton of the Nash equlbrum substantally. Usng ths assumpton, we can never have b e = 0 for any resource e. To get the result we need to take the lmt as the rate ɛ of the utlty of the extra users tends to 0 (see [20]). Next we analyze the condton for an equlbrum for ths game. We wll use these condtons to show that an equlbrum always exsts. Consder a set of bds b e, and a resultng allocaton x, where player gets allocaton x. When s ths allocaton at equlbrum? For each resource e we use p e = b e, the sum of the bds, as the unt prce of the resource (recall that we normalzed constrants, so there s 1 unt of every resource avalable). Now consder the optmzaton problem of a player j assumng bds b e for all other players are set. The player j s nterested n maxmzng her utlty at U j (x j ) e b e j. At equlbrum, t must be the case that xe j = x j for all resources e that costs money, or otherwse player j can reduce her bd b e j wthout affectng her allocaton. So we can thnk of the player s optmzaton problem as dependent on one varable x j, the allocaton she wll receve. What bd does player j have to submt for a resource e to get allocaton x e j = x j? Bds must satsfy the 25

35 followng condton: If b e j > 0 then: αe j x j = be j b e. Assumng all other bds b e are fxed, we can express the bd be j needed as follows. b e j (x j) = αe j x j j b e 1 α e j x. j Note that ths expresson assumes that α j x j < 1, that s, j s not the only user of the resource at equlbrum. It s not hard to see that ths s guaranteed by havng at least two dedcated users for each resource. User j wll want to choose x j to maxmze her utlty. For ths end, t wll useful to express the dervatve of the bd b e j when vewed as a functon of x j. We get the followng (agan assumng α j x j < 1): b e j x (x j) = αe j j b e j (1 α e j x j). 2 Substtutng j b e = p e (1 α e j x j) and smplfyng we get that x j b e j (x j) = pe α e j 1 α e j x. j Now consder the optmzaton problem of player j. She wants to maxmze her utlty U j (x j ) e b e j, whch can now be expressed as α e j U j (x j ) x j j b e 1 α e j x, j e as a functon of the sngle varable x j. Note that ths s a concave functon of x j. The maxmum occurs at a value x j, where the dervatve of ths functon 0, or f the dervatve s negatve everywhere, maxmum occurs at x j = 0. Usng the dervatves we computed above, we get the dervatve of user jth utlty as a functon of her allocaton x j to be p e α e U j j(x j ) (1 α e j x j). e 26

36 Ths dervatve s a strctly decreasng functon, so we have the followng Nash condton: α e x 1; x 0 for all e E U j (x j) = e p e α e j (1 α e j x j), f x j > 0 or x j = 0 f U j (0) e p e α e j ; where pe = b e. To see that there s always a Nash equlbrum, observe the game we defne above s a concave n-person game: each payoff functon s contnuous n the composte strategy vector b, and the strategy space of each user s a compact, convex, nonempty subset of R E. Applyng Rosen s exstence theorem [51] (proved usng Kakutan s fxed pont theorem), we conclude that a Nash equlbrum w exsts for ths game. By ths, we fnshed the proof. 3.2 Revenue and Effcency of Proportonal Sharng In ths secton we analyze the revenue and effcency of a Nash equlbrum. In the rest of the secton we wll use the varable x as a soluton of the Nash condton (3.3). To evaluate the outcomes of the game x, we wll compare the socal welfare and the revenue wth the optmal socal welfare, whch can be wrtten as an optmum of the followng a lnear program. In ths program to avod usng x as a soluton of (3.3), we use new varable z for the amount of resource that buyer gets. max OPT = n =1 U (z ) 27

37 subject to α e z 1; e E (3.4) z 0. We denote z as a soluton of the program above. We have OPT = U (z ). As already mentoned n the ntroducton, we need to make two assumptons to be able to get a reasonable bound on the revenue. Frst we assume that the players utlty functons grow at a reasonably steady rate. Second, we assume that there are at least k 2 players of each type. DEFINITION 3.2 (ASSUMPTION(α, k)) The two assumptons are Growng Utltes: The utlty functon U j (x) of all users j s non negatve, ncreasng, dfferentable, concave, further, U satsfes: U (2x) αu (x) for some α > 1. Compettveness: We say that the type of a player j s her utlty functon U j (x) and the rate at whch she needs the resources, the coeffcents α e j for all resources e. We assume that there are at least k players of every type: that s for every player j, there are at least (k 1) other players wth the same type. In the context of bandwdth sharng, the second assumpton means that for each player j, there are at least k 1 other players wth the same utlty functon and the same path. The man result of the chapter s the followng: THEOREM 3.2 Under Assumpton(α, k), the mechansm defned n secton 3.1 approxmately maxmzes both effcency and revenue. The loss of effcency s bounded by a fracton of 1 and the revenue s at least (α 1)(1 1 ) tmes the optmal revenue. 4k k 28

38 We prove ths theorem n the rest of the secton (by Theorem 3.7 and Theorem 3.11). We frst ntroduce an Lnear Program Dualty technque and gve some ntutons about ths approach n the next subsecton The Prmal Dual Approach The condton for Nash equlbrums can be ntutvely understood as f there were a common prce p e on each constrants and for each buyer the unt prce that he needs to pay s the weghted sum of these prces wth the coeffcents α e. At Nash equlbrum, x = 0 f U (0) s less than e α e pe, otherwse U (x ) can be approxmated by ths weghted sum. If we consder the prces p e as varables then ths condton on p e s smlar to the complementary slackness condton of a certan lnear program. Usng ths observaton, we consder the followng lnear program and ts dual, where x s a gven an Nash equlbrum. PRIMAL max U (x )z subject to: α e z 1; z 0. (3.5) DUAL mn y e subject to: U (x ) α e y e; y e 0. (3.6) LEMMA 3.3 (Weak Dualty) Gven a z and y e feasble solutons for the Prmal and the Dual program respectvely, we have: e y e U (x )z. To prove the bound on the revenue, we observe that the prce vector p e of a Nash satsfyng the condton (3.3) almost satsfes the condton of the DUAL 29

39 program. The extra terms 1 1 α e x n the Nash condton can be bounded by a constant. Furthermore, the convex program of maxmzng socal welfare s smlar to the PRIMAL program. The dfference between these programs s the objectve functon. And t wll be shown later that usng the growng property of the utlty functons, these two objectve functons are close to each other. Thus, wth the dualty lemma, we can get a connecton between a the revenue of a Nash equlbrum and the optmal socal welfare. To prove the bound on the effcency, based on the Nash condton, we wll ntroduce new game on each constrant. The bound on the effcency of each of these separated games s much easer to check Bound on the Revenue We now prove that the revenue at a Nash equlbrum s at least (α 1)(1 1 k ) of the optmal. As mentoned before we use the optmal socal welfare as an upper bound on the revenue. Recall that we use z as an allocaton maxmzng the socal welfare. To compare the revenue e p e wth U (z ), we frst show that p e 1 1 k s feasble for the Dual program (3.6) and therefore: e p e 1 1 k U (x )z, because z s clearly feasble for the Prmal program (3.6). Next, usng the growng property of the utlty functons, we prove that U (x )z (α 1) U (z ). Combnng these two nequaltes, we obtan: e p e 1 1 k (α 1) U(z ) p e (α 1)(1 1 k )OPT, 30 e

40 whch s what we need to prove. In order to show that p e 1 1 k s feasble, we frst observe that the equlbrum s not know to be unque n the general, however, players of dentcal type must get dentcal allocaton: LEMMA 3.4 If two players and j have the same type, then n any Nash equlbrum, they get the same allocaton. Proof. By the Nash equlbrum condtons (3.3) both x and x j are 0 f U (0) = U j (0) < α e pe and otherwse both are the unque solutons equaton of Nash n (3.3). (The functon on the left hand sde of (3.3): U j (x j) s a decreasng functon, meanwhle the functon on the rght hand sde e functon.) p e α e j 1 α e j x j s an ncreasng We now can prove the followng lemma: LEMMA 3.5 p e 1 1 k s feasble for the Dual program. Proof. Snce for every buyer there are other k 1 buyers of the same type, and due to Lemma 3.4, these buyers get dentcal allocaton. Therefore, for each constrant e, we have 1 j α e j x j kα e x. Thus α e x 1 k, and hence 1 1 α e x k From ths and the Nash condton we have: U (x ) e α e pe 1 1 α e x e α e pe 1. Ths shows that 1 1 k p e 1 1 k s feasble for the Dual program (3.6). We now prove the second nequalty needed for boundng the revenue: LEMMA 3.6 U (x )z (α 1) U (z ) = (α 1)OPT. 31

41 Proof. The objectve functon of the prmal lnear program and the socal maxmzng program are U (x )z and U (z ), respectvely. To compare these two functons at z, we use the tangent lne V (z ) of the utlty functon at x defned as V (z ) = U (x )z + (U (x ) U (x )x ). Ths s a lne gong through (x, U (x )) and s above U (z ) as U s a concave functon. See Fgure 3.1. PSfrag replacements U (x)z + U(x) U (x)x U(x) U(x) U (x)x 2x U (x)x U (x) x U(z) U (x)z Fgure 3.1: The shape of the utlty functons. Observe that for each the functon f (z ) = U (x )z could be smaller than the functon U (z ), but usng the functon V at z we get: U (x )z + U (x ) U (x )x U (z ) U (x )z U (z ) (U (x ) U (x )x ). And summng ths over all : U (x )z U (z ) (U (x ) U (x )x )) = OPT (U (x ) U (x )x ). (3.7) Now, we need to get a bound on (U (x ) U (x )x ). To do that we wll need the growng property of the utlty functons. Because U s concave functon U (x ) s decreasng. We have: U (2x ) U (x ) = 2x x U (t)dt 2x x U (x )dt = U (x )x. Usng the assumpton U (2x) αu (x), we obtan: U (x )x U (2x ) U (x ) (α 1)U (x ). 32