EFFICIENCY OF MECHANISMS IN COMPLEX MARKETS

Transcription

1 EFFICIENCY OF MECHANISMS IN COMPLEX MARKETS 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 Vasleos Syrgkans August 2014

2 c 2014 Vasleos Syrgkans ALL RIGHTS RESERVED

3 EFFICIENCY OF MECHANISMS IN COMPLEX MARKETS Vasleos Syrgkans, Ph.D. Cornell Unversty 2014 We provde a unfyng theory for the analyss and desgn of effcent smple mechansms for allocatng resources to strategc players, wth guaranteed good propertes even when players partcpate n many mechansms smultaneously or sequentally and even when they use learnng algorthms to dentfy how to play and have ncomplete nformaton about the parameters of the game. These propertes are essental n large scale markets, such as electronc marketplaces, where mechansms rarely run n solaton and the envronment s too complex to assume that the market wll always converge to the classc economc equlbrum or that the partcpants wll have full knowledge of the competton. We propose the noton of a smooth mechansm, and show that smooth mechansms possess all the aforementoned desderata n large scale markets. We further gve guarantees for smooth mechansms even when players have budget constrants on ther payments. We provde several examples of smooth mechansms and show that many smple mechansms used n practce are smooth (such as formats of poston auctons, unform prce auctons, proportonal bandwdth allocaton mechansms, greedy combnatoral auctons). We gve algorthmc characterzatons of whch resource allocaton algorthms lead to smooth mechansms when accompaned by approprate payment schemes and show a strong connecton wth greedy algorthms on matrods. Last we show how neffcences of mechansms can be allevated when the market grows large n a canoncal manner.

4 BIOGRAPHICAL SKETCH Vasls Syrgkans was born on September 14th, 1986 n Thessalonk, Greece. He receved a dploma n Electrcal Engneerng and Computer Scence from the Natonal Techncal Unversty of Athens n July 2009 and expects to receve a Ph.D. n Computer Scence wth a mnor n Appled Mathematcs from Cornell Unversty n August 2014.

5 To my parents, Chrstos and Evangela. v

6 ACKNOWLEDGEMENTS Frst and foremost, I am truly ndebted to my Ph.D. advsor, Éva Tardos. Her nsghts and advce throughout my Ph.D. years, on all aspects of research, have been nvaluable to me. I would lke to thank her for the countless research dscussons, for her nsprng deas she generously shared wth me, for beng patent wth all my research endeavors, even those that were bound to fal and for the many research opportuntes she strove to make avalable to me. She has made my years at Cornell an awesome research experence whch I wll never forget. I would also lke to thank my professors at Cornell and especally Bobby Klenberg, Jon Klenberg and Davd Shmoys for the many research dscussons and feedback. I would also lke to thank Larry Blume and Emn Gün Srer for servng n my commttee and for ther feedback on my research. I would lke to thank my research collaborators for all ther help, patence and for the great tme we had dong research. Frst I would lke to thank Renato Paes Leme for beng an awesome frend and research collaborator from the very frst years of my Ph.D.. Hs help durng the frst years has been nvaluable and I want to thank hm for sharng wth me many of hs research deas, many of whch resulted n core results of ths thess. I would also lke to thank my collaborators at MSR New England and at MSR Cambrdge, Chrstan Borgs, Jennfer Chayes, Mchal Feldman, Ncole Immorlca and Brendan Lucer, Yoram Bachrach and Mlan Vojnovć for three amazng summers. Before comng to Cornell I benefted from the research gudance of my professors at the Natonal Techncal Unversty of Athens, Staths Zachos, Dmtrs Fotaks and Ars Pagourtzs. I would lke to thank them for ntroducng me to the feld of algorthmc game theory. v

7 I would also lke to deeply thank my parents for encouragng me to pursue a research career and for beng supportve of all my decsons. Fnally, I would lke to thank my wfe Mareta for puttng up wth me and for her support throughout my Ph.D. years, as well as all my frends at Cornell for makng Ithaca a fun place to be. Ths work was supported n part by a Smons Graduate Fellowshp n Theoretcal Computer Scence, ONR grant N and NSF grant CCF and was performed n part whle vstng MSR New England, MSR Cambrdge, UK, and the Hebrew Unversty of Jerusalem. v

8 TABLE OF CONTENTS Bographcal Sketch Dedcaton v Acknowledgements v Table of Contents v Lst of Fgures x I Introducton and Prelmnares 1 1 Introducton Allocatng Resources to Self-Interested Users Desderata n Large-Scale Dstrbuted Markets Thess Goal: Robust Effcency Guarantees n Markets Composed of Smple Mechansms A Concrete Example Approach and Man Conclusons Our Contrbutons Robust Effcency Guarantees for Mechansms Composablty of Mechansms Budget Constrants Algorthmc Characterzatons of Approxmately Effcent Mechansms Effcency n Large Market Lmts Applcatons Comparson to Related Work Bblographc Notes Prelmnares Notatonal Conventons Mechansm Desgn wth Quas-Lnear Preferences Effcency Measure Equlbrum Concepts and the Prce of Anarchy Incomplete Informaton and Bayes-Nash Equlbrum Repeated Games, Learnng and Correlated Equlbra II Theory of Smooth Mechansms 40 3 Smooth Mechansms and Effcency Defnton and Effcency under Complete Informaton Extenson to Incomplete Informaton Extenson to No-Regret under Incomplete Informaton Bayesan Correlated and Coarse Correlated Equlbrum. 51 v

9 3.3.2 Convergence to Bayes-Coarse Correlated Equlbra Effcency of Bayes-Coarse Correlated Equlbra Smultaneous Composablty Smultaneous Composton Framework Composablty of Smooth Mechansms Complement Free Valuatons across Mechansms Fractonally Subaddtve XOS across Mechansms Herarchy of Valuatons across Mechansms Partally Ordered Allocaton Spaces Lattce Allocaton Spaces Example: Smultaneous Frst Prce Auctons Composablty under Restrcted Complements Maxmum over Postve Hypergraph Set Functons Restrcted Complements across Mechansms Composablty Theorem wth Complements Example: Smultaneous Frst Prce Auctons wth Complements Example: Smultaneous Poston Auctons Sequental Composablty Smoothness va Swap Devatons Extenson to Incomplete Informaton and Bluffng Sequental Composablty of Smooth Mechansms Example: Sequental Frst Prce Auctons Necessty of Unt-Demand Assumpton Weak Smoothness and No-Overbddng Effcency under the No-Overbddng Assumpton On the No-Overbddng Assumpton Example: Smultaneous Second Prce Auctons Budget Constrants The Effectve Welfare Benchmark Smoothness va Conservatve Devatons Smultaneous Composablty wth Budget Constrants Example: Smultaneous Item Auctons wth Budgets Algorthmc Characterzatons of Smoothness Combnatoral Allocaton Spaces and Greedy Mechansms Smoothness for Matrod Feasblty Constrants Acton Space Restrctons and Extenson to Polymatrods Smoothness for Matrod Intersectons Matchngs and Greedy Allocaton Intersectons of Matrods and Optmal Algorthm v

10 9 Smoothness n Large Markets Smoothness n the Lmt Smultaneous Unform Prce Auctons wth Nosy Arrval Full Effcency n the Lmt III Applcatons Sngle Item Auctons All-Pay Aucton Hybrd Aucton Poston Auctons General Monotone Valuatons Per-Clck Valuatons Varable Clck Value Poston Independent Value-per-Clck Drect Mechansms Crtcal Payments and Smooth Drect Mechansms Greedy Drect Combnatoral Auctons Bandwdth Allocaton Kelly s Proportonal Bandwdth Allocaton Mechansm Mult-Unt Auctons Margnal Bd Mult-Unt Auctons Unform Bd Mult-Unt Aucton Combnatoral Publc Projects Item Bddng Mechansm for Publc Projects Local Publc Good Auctons n Networks Bblography 213 A Collecton of Lower Bounds 220 A.1 93% Lower Bound for Bayes-Nash Equlbra of the Asymmetrc the Frst Prce Aucton e A.2 Tght Lower Bound for Bayes-Nash of Frst Prce Aucton e 1 wth Correlated Values e A.3 Tght Lower Bound for CCE of Frst Prce Aucton e 1 A.4 4/3 Lower bound for Bayes-Nash of All-Pay Aucton wth Correlated Values A.5 8/7 Lower bound for mxed Nash of All-Pay Aucton A.6 Ineffcency of GSP wth per-mpresson values grows wth slots. 225 x

11 LIST OF FIGURES 2.1 Comparson among soluton concepts, wth respect to robustness of guarantees Comparson among statc soluton concepts n the ncomplete nformaton settng and connecton to no-regret learnng under ncomplete nformaton The left fgure depcts a spectrum aucton nspred hypergraph valuaton wth postve edges and negatve hyperedges, whch can be expressed as the maxmum over the postve graphcal valuatons on the rght Sequental Mult-unt Aucton generatng POA 3/2: there are 4 players {a, b, c, d} and three tems that are auctoned frst A, then B and then C. The optmal allocaton s b A, c C, d B wth value 3α ϵ. There s a subgame perfect equlbrum that has value 2α + ϵ. In the lmt when ϵ goes to 0 we get POA = 3/ Effcency bound for the hybrd aucton x

12 Part I Introducton and Prelmnares 1

13 1 INTRODUCTION 1.1 Allocatng Resources to Self-Interested Users How would you allocate resources n a system so as to maxmze the total value of the users? If you knew how much each user values each possble allocaton of resources then ths would be purely an optmzaton problem (most probably a hard one n each generc formulaton) and thereby desgnng the approprate algorthm for the approxmately effcent allocaton of the resources, would be the way to go. However, n most stuatons, the value that users have for each allocaton of resources s prvate nformaton. Hence, the system also has the task to elct these parameters or some good approxmaton of them, takng nto account that users wll behave strategcally and selfshly. The standard approach to solvng the ncentve problem s to ntroduce payments. Roughly, the combnaton of an allocaton and a payment rule s what we wll refer to as a mechansm. Such mechansms are ubqutous n both economc systems and large-scale computer systems. From the computer scence perspectve, they can capture most modern electronc markets such as aucton marketplaces (e.g. ebay), onlne advertsement markets (e.g. Google AdWords etc.), crowdousrcng contests (e.g. topcoder), where payments are mplct n the form of costly effort, bandwdth allocaton mechansms (e.g. Kelly s proportonal allocaton mechansm) and computng resource sharng mechansms n the cloud. From the purely economc perspectve they can capture settngs such as spectrum auctons, government auctons for natural resources (e.g. tmber auctons), art auctons (e.g. Sotheby s) and auctons for fnancal dervatves (e.g. government bonds). 2

14 1.1.1 Desderata n Large-Scale Dstrbuted Markets The large-scale nature of modern markets, especally those enabled by computer systems, such as electronc marketplaces, ntroduces new challenges n the theoretcal desgn and analyss of mechansms. Though mechansm desgn s a feld wth a long and dstngushed hstory, startng from the early works of Vckrey, Clarke and Groves [68, 16, 33] n the 60 s and 70 s, many of the challenges we lst below have not been at the forefront of the feld. Composablty n the Presence of Multple Mechansms. In most markets lsted n the prevous secton, resources are owned by dfferent enttes and many mechansms are runnng at the same tme, wth players smultaneously or sequentally partcpatng n many of them (e.g. dfferent sellers on ebay, dfferent onlne advertsement platforms). Even f the resources are owned by a sngle entty (e.g. a sngle onlne advertsement platform),t s almost nfeasble or mpractcal to run a global centralzed mechansm for all the resources and a more decentralzed market structure, where small groups of resources are sold va a separate mechansm, s preferred (e.g. the advertsement slots n each search query are sold separately va the means of an aucton, called the Generalzed Second Prce aucton). In these stuatons, t s crucal that the market as a whole performs reasonably well,.e. the global allocaton of resources s approxmately effcent. Thus the local mechansms used must satsfy some composablty property: local propertes that mply local effcency guarantees for each mechansm n solaton, also drectly mply global effcency guarantees. 3

15 Robustness to Learnng Behavor and Incomplete Informaton. In most large-scale markets, the decson problem that each partcpant s facng s far too complex to assume wth certanty that the market wll arrve at the classc economc equlbrum,.e. a state where no partcpant wants to unlaterally devate, aka a Nash equlbrum. Rather we need more robust guarantees even f players use learnng algorthms to dentfy how to play n the game. Such learnng behavor wll not necessarly lead to a Nash equlbrum and could potentally also lead to correlatons n the behavor of partcpants. Any effcency guarantee of a mechansm should extend to generc enough models of learnng behavor. Moreover, we cannot expect the partcpants to know all the parameters of the game (e.g. valuatons of opponents). Therefore, the mechansm should also be robust wth respect to nformatonal assumptons and should be approxmately effcent even when players have only partal nformaton (e.g. dstrbutonal belefs) about these parameters. Smple Rules wth Fast Implementaton. The Internet envronment allows for runnng mllons of mechansms, whch necesstates the use of very smple and ntutve allocaton and payment schemes wth a fast mplementaton. As an example, approxmately seven thousand search queres happen on Google s search engne every second and each of these search queres trggers an aucton among advertsers for the allocaton of the specal advertsement slots that wll appear together wth the organc search results. Hence, the aucton rule that s used, should be able to be computed n a matter of mllseconds. 4

16 1.1.2 Thess Goal: Robust Effcency Guarantees n Markets Composed of Smple Mechansms The goal of ths thess s to provde a theoretcal framework for the desgn and analyss of smple mechansms for allocatng resources to self-nterested and strategc users, wth guaranteed good propertes even when the users partcpate n multple dfferent mechansms smultaneously or sequentally and even when players use learnng algorthms and have ncomplete nformaton of the market. One of the key questons we wll address s: What propertes of local mechansms guarantee global effcency n a market composed of such mechansms and even under learnng behavor and ncomplete nformaton? Tradtonal mechansm desgn consdered mechansms only n solaton, an assumpton not so realstc n many large-scale markets, where players can cover ther needs from multple dfferent mechansms. As perfectly summarzed by two leadng economsts of the feld n the concludng remarks of ther semnal paper on compettve bddng: Most analyses of compettve bddng stuatons are based on the assumpton that each aucton can be treated n solaton. Ths assumpton s sometmes unreasonable., Mlgrom and Weber, 1982 Moreover, mechansm desgn has mostly focused on truthful mechansms, where players are ncentvzed to truthfully reveal all ther prvate parameters 5

17 to the mechansm. In an envronment wth several auctons runnng smultaneously or sequentally, truthfulness of each ndvdual aucton loses ts appeal, as the global mechansm s no longer truthful, even f each ndvdual part s. The lterature s focus on truthful mechansms s based on the revelaton prncple, showng that f there are better non-truthful solutons, the mechansm desgner can run ths alternate soluton on the players behalf. However, the revelaton prncple s lmted to mechansms runnng n solaton: wth multple mechansms run by dfferent partes, there s no global coordnator to mplement the soluton. Requrng global coordnaton between mechansms s not vable and could lead to complcated coordnaton problems, such as agreeng on ways to dvde the global revenue. From the analyss perspectve, a handful of papers n the economc lterature have analyzed propertes of strategc outcomes of games arsng from sellng a set of tems va auctons smultaneously or sequentally [55, 22, 53, 9, 30, 6, 58] (a specal case of our general setup). However, most of the economcs lterature has made several smplfyng assumptons, such as symmetrc user propertes or small number of users or complete nformaton of the parameters of the market. The man hurdle n extendng the analyss to more realstc settngs s analytcally solvng for the equlbrum. In the general setup that we study, analytcally solvng for the equlbrum s an mpossble task and even more mportantly t s not true that the equlbrum of the market s always unque. Instead we wll follow the prce of anarchy analyss from the computer scence lterature [15, 57, 46, 8, 36] that attempts to analyze the effcency wthout solvng for the equlbrum, as we wll descrbe n subsequent sectons. 6

18 1.1.3 A Concrete Example Smultaneous Item Auctons. Consder an example wth two sellers A, B, each havng one tem for sale. For smplcty, the market has two partcpants α, β and each partcpant wants only one tem (.e. s unt-demand). The tems that are for sale are not completely dentcal and the partcpants exhbt some slght preference: player α has value 2 for tem A, value 1 for tem B and 2 for the bundle of the two tems (snce he wll only use one of them). Player β prefers tem B, havng value 2 for tem B, value 1 for tem A and 2 for the bundle. Obvously the optmal allocaton n ths market s for each player to wn hs most preferred tem, yeldng a total value of 4. What would happen n ths market f each seller was usng a second-prce aucton to sell hs tem (.e. the hghest bdder wns and pays the second hghest bd)? The two partcpants are playng a game where ther strategy s to submt a bd on each of the two tems. For smplcty, assume that a Nash equlbrum of the game wll arse,.e. a profle of bds such that no player can gan by devatng. Assumng that the utlty that the player derves from the nteracton s hs value for hs allocaton mnus hs total payment, then t s easy to see that the followng s one equlbrum: player α bds 1 on tem B and 0 on tem A and player β bds 1 on tem A and 0 on tem B. Both players derve a utlty of 1 and t s easy to see that no unlateral devaton of a player can lead to a better utlty. Thus at the equlbrum, the allocaton s suboptmal and the total value s only half of the value of the optmal allocaton. One of the man take-away messages of ths example s that the nce propertes of a sngle-tem second-prce aucton n solaton, break the moment there are several mechansms runnng smultaneously: n a sngle-tem second prce 7

19 aucton t s a domnant strategy for the player to bd hs true value for the tem rrespectve of what the opponents are dong and under such truthful behavor the tem wll go the hghest value player. In the smultaneous aucton settng, not only players don t have domnant strateges, but even the concept of truthfulness does not make sense, as the players can no longer express ther whole valuaton functon through ther bds. Another observaton, n the above example s that f a frst-prce aucton (.e. wnner pays hs bd) was used nstead of a second prce, then every Nash equlbrum wth determnstc bds would have resulted n the optmal allocaton. Thus mechansms that seem nferor when studed n solaton mght perform better n an envronment where multple mechansms occur at the same tme Approach and Man Conclusons We wll approach the problem usng technques from the computer scence lterature and more specfcally, the work on the prce of anarchy, ntated by the semnal papers of [42, 62]. The prce of anarchy lterature attempts to quantfy the effcency of all possble strategc outcomes wthout analytcally solvng for the equlbrum, but rather smply from the fact that f an outcome s an equlbrum then every devaton of a user must lead to lower utlty, a.k.a. the best-response property. Our work develops a unfyng theory of how to analyze mechansms va such best-response arguments. Specal cases of our theory ncludes some earler work on the prce of anarchy n specfc aucton settngs [15, 57, 46, 8, 36]. Our work wll unfy and heavly extend the results n these papers n a sngle theory on the prce of anarchy of mechansms. Apart from the applcatons we present n the thess, our theory has been used subsequent to 8

20 our work, n quantfyng the effcency of mechansms at equlbrum [17, 14, 5]. More formally, we defne the noton of a (λ, µ)-smooth mechansm and show that smooth mechansms are approxmately effcent and possess all the desred propertes of composablty and robustness under learnng behavor and ncomplete nformaton. The defnton of a smooth mechansm s based on the exstence of a well-behaved best response acton for each player. Intutvely, the mechansm must admt for each player an optmal acton, such that no matter what the other players are dong, ths acton guarantees her a good fracton of her optmal allocaton and at a prce that s comparable to what s currently beng pad for that allocaton. Our noton of smoothness s focused on mechansms where players have quaslnear utltes and s closely related to the noton of smooth games ntroduced by Roughgarden [59]. Our man result s to show that smooth mechansms compose well and are robust to ncomplete nformaton and learnng behavor: A market composed of smooth mechansms runnng smultaneously s approxmately as effcent as each ndvdual mechansm would have been f run n solaton, when players have complement-free valuatons across mechansms. Effcency s acheved even n learnng outcomes, as well as n Bayesan settngs wth uncertanty about partcpants. We present several other robustness propertes of smooth mechansms, such as composablty when mechansms are run sequentally rather than smultaneously, effcency propertes when players have budget constrants on the payments they can make and how the neffcences of some smooth mechansms can be allevated f the market becomes large n a canoncal manner. 9

21 We further show that many well-studed and used mechansms are smooth, such as several forms of sngle-tem auctons such as frst prce and all-pay, some formats of ad-auctons, Kelly s [41, 40] proportonal bandwdth allocaton mechansm, unform prce auctons, as well as a number of other mechansms. In that respect, we also present algorthmc characterzatons of what algorthms for allocatng resources, lead to smooth mechansms when accompaned wth approprate payment schemes and show a strong connecton between smoothness and greedy algorthms under well-behaved resource allocaton constrants. 1.2 Our Contrbutons Robust Effcency Guarantees for Mechansms We defne the noton of a (λ, µ)-smooth mechansm and show that any such mechansm acheves at least a λ max(1,µ) fracton of the maxmum possble socal welfare at every Nash equlbrum. Moreover, ths guarantee extends drectly to any coarse correlated equlbrum, whch s a superset of Nash equlbra. No-Regret Learnng (Sectons 2.4 and 3.1). As s known coarse correlated equlbra have a strong connecton to no-regret learnng n games. Suppose that the mechansm s played repeatedly wth the parameters of every player remanng fxed and the players use some update rule to learn how to play the game. All we assume s that the learnng rule satsfes the property that n the long run the player doesn t regret havng played a fxed strategy n all perods. Then t s known that the emprcal dstrbuton of players actons of any such 10

22 no-regret sequence of play wll converge to a coarse correlated equlbrum of the statc game [12]. Thus the effcency guarantee of a smooth mechansm drectly extends to the average welfare of any such no-regret sequence. Bayesan Incomplete Informaton (Secton 3.2). In addton, we show that ths guarantee extends drectly to Bayesan settngs of ncomplete nformaton, where each player s prvate parameters are drawn ndependently from some commonly known dstrbuton. In that settng we defne a noton of a Bayesan coarse correlated equlbrum and we show that the expected welfare of any such equlbrum s at least λ max{1,µ} of the expected optmal welfare (n expectaton over player parameters). Bayesan coarse correlated equlbra are a superset of Bayes-Nash equlbra [34] and smlar to coarse correlated equlbra have a strong connecton wth no-regret dynamcs as we explan below. No-Regret Learnng wth Stochastc Parameters (Secton 3.3). We consder a stuaton where the game s played repeatedly and at each teraton each player s prvate parameters are re-sampled ndependently from some dstrbuton (unlke n the prevous repeated game verson, where they were fxed). Equvalently, one can vew each player as a populaton, where each atom n the populaton has some fxed parameter and at each tme step one player from each populaton s pcked to play n the mechansm. We show that f each player acheves the no-regret property for each possble parameter nstantaton (or equvalently each atom n the populaton acheves the no-regret property), then the lmt emprcal dstrbuton of play converges almost surely to the set of Bayes coarse correlated equlbra that we defned. Therefore, the effcency guarantee drectly extends to the average welfare of any no-regret sequence n 11

23 the above Bayesan repeated game settng. Bandt Learnng. The no-regret learnng guarantees have the extra robustness propertes that for a player to acheve the no-regret property he doesn t need to be aware of any parameters of the game, nether the dstrbutons from whch the parameters are re-drawn. There are update rules that the player can nvoke (e.g. multplcatve weght updates [3]), that only requre for the player to be able to calculate hs utlty from the acton he took at each tme step. Thus t suffces to know just hs value for the allocaton he receved and hs payment. Effcency wth the No-overbddng Refnement (Chapter 6). For some mechansms, such as the second prce aucton, good performance requres that partcpants do not bd above ther value. It s easy to see that even n a sngletem second prce aucton, there exst Nash equlbra where players overbd and whose welfare s arbtrarly worse than the optmal. However, f we consder only the subset of equlbra where players don t bd above ther value for the tem, then every Nash equlbrum s effcent. For such second-prce type mechansms, we dentfy the noton of a weakly smooth mechansm. Weakly smooth mechansms acheve hgh welfare at any equlbrum that satsfes a generalzaton of the non-overbddng assumpton that we descrbed above for the case of a sngle-tem second prce aucton. Moreover, ths guarantee s equally robust to the guarantees of smooth mechansms, n the sense that t extends to learnng outcomes and Bayesan ncomplete nformaton. 12

24 1.2.2 Composablty of Mechansms Smultaneous Composablty of Smooth Mechansms (Secton 4.2). We show that smooth mechansms compose well n parallel: f we run any number of (λ, µ)-smooth mechansms smultaneously and players valuatons over outcomes of dfferent mechansms satsfy a complement-free condton that we explan n the next paragraph, then the global market can also be vewed as a (λ, µ)-smooth mechansm, and hence acheves a λ/ max(1, µ) fracton of the maxmum socal welfare n all Bayesan coarse correlated equlbra and for any ndependent dstrbutons of player parameters. Complement-Free Valuatons Across Mechansms (Secton 4.3). For our smultaneous composablty results, we need to assume that user s valuatons have no complements across the dfferent mechansms. At a hgh-level all we need to assume for the composablty property s that the margnal valuaton of a player for an allocaton from a specfc mechansm can only decrease f more mechansms come nto the market and gve hm some non-empty allocaton. In more detal, we develop a herarchy of valuatons on outcomes that have no complements across mechansms. Exstng valuaton herarches consder only valuatons on sets of tems. We dentfy analogs of complement-free valuatons across mechansms, wthout makng any assumpton about the valuatons of players for outcomes wthn a mechansm. We defne natural generalzatons of submodular, fractonally subaddtve, XOS and subaddtve valuatons over outcomes of dfferent mechansms. In the context of valuatons on sets of tems, fractonally subaddtve s a superset 13

25 of submodular valuatons, and s known to be equvalent to the class of XOS valuatons. We show an equvalent connecton among the generalzed versons of these functons extendng the results of Fege [23] and Lehmann et al. [45]. If smoothness of each local mechansm holds only for some restrcted class of local valuatons, then we wll need to make roughly the same assumpton for the component-wse margnals of the valuaton of a player across mechansms. For nstance, f the allocaton space of a mechansm s partally ordered and the smoothness property holds only when the valuatons of players are monotone wth respect to the partal order, then we wll also need to assume that f we fx the allocaton from other mechansms, the valuaton of the player across mechansms s also monotone wth respect to the allocaton from the specfc mechansm. Smlarly, f the allocaton space forms a lattce and local smoothness holds only for submodular valuatons over the lattce, then we wll need to assume that the valuaton across mechansms s submodular wth respect to the product lattce of allocatons of dfferent mechansms. Approxmate Composablty wth Restrcted Complements (Secton 4.5). We also show that n the presence of complementary valuatons, the smoothness of the global market degrades smoothly wth the sze of the complements. For the case of set functons, a natural class of complementary relatons are those defned va the means of a postvely weghted hypergraph, where the value for a set of tems s the total weght of hyperedges contaned n the set. Then the sze of the complements can be defned as the cardnalty of the largest hyperedge. Based on ths ntuton we defne a novel measure of complementarty of a set functon and more generally of a valuaton functon over outcomes of mech- 14

26 ansms and we show an approxmate composablty property that degrades smoothly wth ths measure. Such restrcted complement valuatons fnd good applcaton n spectrum auctons where bands n neghborng geographc regon exhbt complementary relatons (e.g. have extra value when acqured n conjuncton). They also fnd applcatons n onlne advertsement auctons where slots n dfferent parts of a webpage can have a complementary effect, as they create an mpresson effect when acqured n conjuncton. Sequental Composablty of Smooth Mechansms (Chapter 5). We also show that smooth mechansms compose well sequentally, though for a more restrctve assumpton on valuatons : f we run any number of (λ, µ)-smooth mechansms sequentally and a player s value s the maxmum valued allocaton she got among all mechansms then the global mechansm acheves welfare at least λ/(µ + 1) of the optmal socal welfare at every Bayes correlated equlbrum (not coarse correlated equlbrum). To show ths theorem we defne a more relaxed smoothness condton, denoted as smoothness va swap devatons and show that the global mechansm satsfes ths relaxed (λ, µ+1)-smoothness condton. We then show that smooth mechansms va swap devatons guarantee good effcency at every correlated equlbrum, hence no-swap regret dynamcs (.e. dynamcs where n the longrun players don t regret swappng some acton wth some other) and even under ncomplete nformaton. Our effcency proof for the ncomplete nformaton settng uses a bluffng technque to handle the fact that n a sequental mechansm, past actons mght reveal nformaton about the prvate value of 15

27 a player Budget Constrants The results dscussed so far, assume that a partcpants utlty from the mechansm s equal to hs value for hs allocaton mnus hs payment,.e. utltes are quas-lnear wth respect to money. The most common non-quas-lnear valuaton s when players have budget constrants on ther payments. We extend our results to settngs where partcpants have budget constrants. Wth budget constrants, maxmzng welfare s not an achevable goal, as we cannot expect a low budget partcpant to be effectve at maxmzng her contrbuton to welfare. We defne a new benchmark n ths settng, whch we call the optmal effectve welfare ; cappng the contrbuton of each player to the welfare by ther budget. We show that, subject to a mnor strengthenng of the smoothness defnton, dubbed conservatve smoothness (whch all the known and presented smooth mechansms satsfy), all our results about effcency for the case of smultaneous mechansms carry over to bounds for ths benchmark when players have budget constrants. For more detals see Chapter 7. 16

28 1.2.4 Algorthmc Characterzatons of Approxmately Effcent Mechansms The defnton of a smooth mechansm s a semantc one, based on an exstental property of a best-response acton. It does not drectly gve algorthmc gudelnes about what mechansms are smooth. An analogue s truthfulness, whch s also a semantc property; t s useful to have descrptve algorthmc condtons for truthfulness, such as optmal algorthms (as n the VCG mechansm). Can we gve analogous, useful characterzatons of algorthmc condtons that guarantee smoothness? A common feature n many of the mechansms that we show are smooth s the greedness of the allocaton rule. Indeed, an ntuton that arses from the lne of work on approxmately effcent mechansms s that greedy algorthms lend themselves well to mechansm desgn, n the sense that they generate auctons wth good performance at equlbrum. We formalze ths ntuton and provde algorthmc characterzatons of smoothness. Specfcally, we show that f a greedy allocaton rule s used to allocate resources subject to a matrod constrant, and players have submodular 1 preferences over the resources, then the resultng mechansm s smooth and wll acheve a constant fracton of the optmal welfare at every Bayes coarse correlated equlbrum. We also provde smlar characterzatons for greedy and optmal algorthms when the feasblty constrants are ntersectons of matrods. For more detals see Chapter 8. 1 Our results actually hold for the more general class of fractonally subaddtve preferences. 17

29 1.2.5 Effcency n Large Market Lmts We address the queston of whether the effcency guarantees of a mechansm mprove as the game grows large n a canoncal way. The ntuton s that f a player has a neglgble effect n the outcome of the market then any strategc manpulaton that he mght employ, wll not run socal welfare by much. Hence, t s reasonable to expect that as the market grows large the neffcency of a mechansm wll mprove. We propose a smoothness n the lmt framework and show a very general full effcency n the lmt result for the case of smultaneous unform prce auctons, wth multple goods and arbtrary monotone combnatoral valuatons, assumng that the supply of each good grows as the number of players grows and that each player fals to arrve n the market wth some probablty δ. For more detals see Chapter Applcatons We show that many well-known auctons are smooth and can be analyzed n our framework. We lst a few representatve examples below, and note that our composton result apples when runnng any set of such auctons smultaneously or sequentally. Sngle Item Auctons (Chapter 10). We show that the frst prce sngle tem aucton s ( 1 1 e, 1) -smooth, the all-pay aucton s ( 1 2, 1) -smooth and the second prce aucton s weakly and (1, 0, 1)-smooth. We also gve a smoothness 18

30 proof for the hybrd aucton n whch the wnner pays a convex combnaton of her own bd and the second hghest bd. Our framework mples that runnng m smultaneous frst prce auctons and bdders have fractonally subaddtve valuatons and budget constrants acheves effcency at least 1 1 e of the optmal effectve welfare. All-pay auctons acheve a guarantee of 1. Second prce 2 auctons acheve a guarantee of 1 2 under the no-overbddng assumpton. For sequental auctons wth unt-demand bdders and no budget constrants the frst prce, all-pay and second prce auctons gve guarantees of 1 2 (1 1 e ), 1 4 and 1 respectvely. 2 Poston Auctons (Chapter 11). We analyze poston auctons for more general valuaton spaces than what has been typcally consdered [21, 13]. We use the model of Abrams et al [2], where each player has an arbtrary valuaton v j for appearng at poston j, that s monotone n the poston. Most of the lterature n poston auctons has consdered valuatons of the form v j = a j γ v,.e. players have only value per clck v and ther clck-through-rate s dependent n a separable way on ther qualty and on the poston. The more general class of valuatons can capture settngs where players have value both for clck and for the mpresson tself, and settngs where the clck-through-rates are not separable. We show that the followng very smple frst prce analog of the aucton of [2] s ( 1, 1)-smooth: solct bds from the players, allocate postons n order 2 of bds and charge each player hs bd. The mpled guarantee of 1 2 holds for smultaneous composton when players have monotone fractonally subaddtve valuatons and budget constrants. Such valuatons capture, for nstance, settngs where bdders have value v only for the frst k clcks, or settngs where the margnal value per-clck of a player decreases wth the number of clcks he 19

31 gets. In addton, a bound of 1 s mpled for the sequental composton when 4 bdders value s the maxmum value among all mpressons he got. In contrast, [2] consder the second prce analog of ths aucton, and show that t always has an effcent Nash equlbrum, but do not consder the prce of anarchy. We show that the second prce verson s weakly ( 1, 0, 1)-smooth, mplyng an effcency 2 guarantee of 1 4 for smultaneous and sequental composton of such auctons under the no-overbddng assumpton. We also consder other varatons of the well-studed GFP and GSP mechansms for the case when players have only values per clck. Greedy Drect Auctons (Chapter 12). Lucer and Borodn [46] consders combnatoral auctons, whose allocaton functon s based on a restrcted class of greedy c-approxmaton algorthms. When a frst prce payment s used, they show that such a greedy aucton has a c + O(log(c)) effcency guarantee. We mprove ths bound, by showng that ths mechansm s (1 e 1/c, c)-smooth 1 mplyng an effcency guarantee of at least. Ths bound extends to the c+0.58 smultaneous composton of such mechansms when bdders have fractonally subaddtve valuatons across auctons and budget constrants. For example, when each auctons sells only a small number of tems, greedy algorthms can do qute well (gvng a k-approxmaton for arbtrary valuatons, f each aucton sells at most k tems). Observe, that fractonally subaddtve valuatons across auctons allow for complements wthn the tems of a sngle greedy aucton, hence s more general than just assumng that players have fractonally subaddtve valuatons over the whole unverse of tems. We also show that the above analyss s a specal case of a more general property of drect auctons,.e. auctons where players can report ther whole valuaton over allocatons. 20

32 Bandwdth Allocaton Mechansms (Chapter 13). We consder the sngle-lnk bandwdth sharng verson of the settng studed by Johar and Tstskls [38] where a set of players want to share a resource: an edge wth bandwdth C. Each player has a concave valuaton v (x ) for gettng x unts of bandwdth. The mechansm studed n [38] s the well-known Kelly Mechansm [41, 40]: solct bds b, allocate to each player bandwdth proportonal to hs bd x = b j b j, charge each player b. We show that ths mechansm s (2 3, 1)-smooth, mplyng an effcency guarantee of approxmately 1/4 for Bayes coarse correlated eqlbra. We note that [38] consdered only Nash equlbra of the complete nformaton settng. Hence, we extend the analyss to ncomplete nformaton. Moreover, the same effcency guarantee extends to the case when we run such mechansms smultaneously and players have budget constrants and monotone, lattce-submodular valuatons on the lattce defned on R m by the coordnate-wse orderng. If the valuatons are twce dfferentable, beng monotone and lattce-submodular translates to: every partal dervatve s nonnegatve and every cross-dervatve s non-postve. Mult-Unt Auctons (Chapter 14). For the settng of mult-unt auctons (.e. all tems are dentcal) where players have concave utltes n the amount of unts they get, we gve two smooth mechansms. Recently, Markaks et al. [50] studed a greedy mechansm and showed a O(log(m)) approxmaton for the case of mxed Bayes-Nash equlbra under a no-overbddng assumpton. We show that a frst prce verson of ther mechansm where each player s charged hs declared margnal bds for the unts he acqured s ( ( ) e), 1 -smooth, ( whle the unform prce verson of [50] s weakly ( e), 0, 1)-smooth. Therefore our smooth analyss mproves the O(log(m)) bound of [50] to a constant 21

33 1 4 ( ) ( 1 1 e and to e) when a frst prce payment rule s used. It also extends the analyss to the case of smultaneous unform prce auctons, where players have submodular valuatons on the product lattce N m of vectors of allocated unts of each good. 1.3 Comparson to Related Work In ths secton we provde an overvew of the man work that s related to the general drecton of the thess. Snce our thess touches several subjects, more specfc related work s mentoned n each correspondng secton, whenever approprate. There has been a long lne of research on quantfyng neffcency of equlbra startng from Koutsoupas and Papadmtrou [42] who ntroduced the noton of the prce of anarchy. More recently, ths analyss technque has also been used to quantfy the neffcency of aucton games, ncludng games of ncomplete nformaton. A seres of papers, Bkhchandan [9], Chrstodoulou et al [15], Bhawalkar and Roughgarden [8], Hassdm et al [36], Paes Leme et al [56], Syrgkans and Tardos [66] studed the effcency of equlbra of non-truthful combnatoral auctons that are based on runnng separate tem auctons (smultaneously or sequentally) for each tem. Lucer and Borodn [46] studed Bayes- Nash Equlbra of non-truthful auctons based on greedy allocaton algorthms. Caraganns et al [13] studed the neffcency of Bayes-Nash equlbra of the generalzed second prce aucton. All ths lterature can be thought of as specal cases of our framework and all the proofs can be understood as smoothness proofs gvng the same or even tghter results. A recent excepton s the paper 22

34 by Feldman et al. [25] gvng a tghter bound for smultaneous tem-auctons wth subaddtve bdders, than what would follow from our framework. Roughgarden [59] proposed a framework, whch he calls smoothness n games, and showed that a number of classcal prce of anarchy results (such as routng and vald utlty games) can be proved usng ths framework. Further, he showed that such effcency proofs carry over to effcency of coarse correlated equlbra (no-regret learnng outcomes). Nadav and Roughgarden [54] gve the broadest soluton concept for whch smoothness proofs apply. Schoppmann and Roughgarden [61] extend the framework to games wth contnuous strategy spaces, provdng tghter results. However, these papers consder only the full nformaton settng and do not capture several of the auctons descrbed prevously. Our defnton of a smooth mechansm s closely related to the noton of a smooth game. If utltes of the game were always non-negatve (whch we only assume n expectaton) then a (λ, µ)-smooth mechansm can be thought of as a (λ, µ 1)-smooth game. Moreover, our defnton of a smooth mechansm has several techncal dfferences and mposes weaker condtons n some respects, so as to allow us to prove our sequental and smultaneous composablty results and also gve tght effcency results for many of the applcatons descrbed so far. Recent papers offer extensons of the smoothness framework to ncomplete nformaton games. Lucer and Paes Leme [47] ntroduced the concept of semsmoothness (nspred by ther GSP analyss), and showed that effcency results shown va sem-smoothness extend to the ncomplete nformaton verson of the game, even f the types of the players are arbtrarly correlated. Semsmoothness s a much more restrctve property (for nstance, not satsfed by the 23

35 smultaneous tem-bddng auctons) than just requrng that every complete nformaton nstance of the mechansm s smooth n the complete nformaton settng and apples mostly to mechansms where players can express ther whole valuaton profle through ther actons. Moreover, sem-smoothness s a property that does not compose:.e. local sem-smoothness of each mechansm does not mply global sem-smoothness. Independent to our work n Syrgkans [65], Roughgarden [60] also offered a smlar to ours drect extenson theorem of effcency guarantees from complete to ncomplete nformaton. However, the results n [65] and [60] address effcency n general games and not mechansms and for that reason they requre a stronger smoothness property (called unversal smoothness n [65]), whch relates utltes of players wth dfferent types n a sngle nequalty. Moreover, none of the prevous work addresses the ssue of learnng under ncomplete nformaton and provdes guarantees only for the statc game of ncomplete nformaton and only for Bayes-Nash equlbra. Last, the approach used n these papers cannot capture effcency n sequental games, such as sequental tem auctons, whch s acheved by our work. 1.4 Bblographc Notes Large part of the thess appears n publshed or workng research papers [56, 66, 65, 67, 27, 48, 49, 24, 26]. The majorty of the results n Chapters 3, 4, 5, 6, 7 and Part III appeared n [67]. The man theorem n Secton 3.2 s prmarly nspred by the man theorem n [65]. The results of secton 3.3 are novel results. The results n Secton 4.5 appear n [24]. The results n Secton 5.3 frst appeared n 24

36 [56, 66]. The results n subsecton appear n [27]. The results of Chapter 8 appear n [49]. The results of Chapter 9 appear n [26]. 25

37 2 PRELIMINARIES 2.1 Notatonal Conventons We wll use bold letters x to denote a random varable n some probablty space. We wll denote wth (Ω) the space of probablty dstrbutons over a fnte set Ω. Abusng notaton we wll use x to denote both the random varable and ts dstrbuton, snce the dstncton wll be clear from the context. Moreover, we wll wrte SUPP(x) for the support of the dstrbuton of x. We wll typcally use un-ndexed letters x to denote vectors x = (x 1,..., x n ) n some product space. We use R + for non-negatve real numbers. 2.2 Mechansm Desgn wth Quas-Lnear Preferences Most of ths thess wll be consderng the followng generc settng: a set of resources are to be allocated to a set of n players. The allocaton vector x = (x 1,..., x n ) has to le n a set of feasble allocaton vectors X that s a subset of a product space of allocatons X X 1... X n. We assume that players can be asked to pay for ther allocaton and thereby a payment vector p = (p 1,..., p n ) R n + wll also be decded. The par o = (x, p) of an allocaton vector and a payment vector s referred to as an outcome. Each player, has a valuaton functon that maps an allocaton to some nonnegatve real number: v : X R +. We wll denote wth V the set of allowed valuatons for player and wth V = V 1... V n the set of allowed valuaton 26

38 profles. If a player s gven allocaton x and s asked to pay p, then the utlty that she derves s: u (x, p ; v ) = v (x ) p (2.1) In other words, players have quas-lnear preferences wth respect to money. We wll refer to such a mechansm desgn settng wth quas-lnear preferences va the tuple (n, X, V). Some mportant examples captured by ths generc formulaton of a mechansm desgn settng are: 1. Combnatoral auctons: where X s the power set of tems and X s the subset of ths product space such that no tem s assgned to more than one player, 2. Combnatoral publc projects: where X s the power set of potental projects to be bult and X s the subset of the product space such that every coordnate s the same and each coordnate satsfes some constrant based on whch projects can be smultaneously bult (.e. a set of smultaneously feasble projects s bult and shared by the players), 3. Poston auctons: where X s the set of postons and X s the subset of the product space where no two coordnates are assgned the same poston, 4. Bandwdth allocaton mechansms: where X s the porton of the bandwdth assgned to player and X s the subset such that the sum of the coordnates s at most the bandwdth capacty. Defnton (Mechansm). Gven a mechansm desgn settng (n, X, V), a mechansm M s a tuple (A, X, P ), where A = A 1... A n and A s a set of actons avalable to player, X : A (X ) s an allocaton functon that maps each acton profle a = (a 1,..., a n ) to a dstrbuton over feasble allocaton vectors x = (x 1,..., x n ) and 27