Incentives in Dynamic Markets

Transcription

1 Incentves n Dynamc Markets A dssertaton submtted towards the degree Doctor of Natural Scence of the Faculty of Mathematcs and Computer Scence of Saarland Unversty by Bojana Kodrc Saarbrücken / 2017

2 Day of Colloquum: 23. July 2018 Dean of the Faculty: Prof. Dr. Sebastan Hack Char of the Commttee: Prof. Dr. Markus Bläser Reporters Frst revewer: Prof. Dr. Martn Hoefer Second revewer: Prof. Dr. Dr. h.c. mult. Kurt Mehlhorn Academc Assstant: Dr. Marvn Künnemann

3 Abstract Abstract In ths thess, we consder a varety of combnatoral optmzaton problems wthn a common theme of uncertanty and selfsh behavor. In our frst scenaro, the nput s collected from selfsh players. Here, we study extensons of the so-called smoothness framework for mechansms, a very useful technque for boundng the neffcency of equlbra, to the cases of varyng mechansm avalablty and partcpaton of rsk-averse players. In both of these cases, our man results are general theorems for the class of (λ, µ)-smooth mechansms. We show that these mechansms guarantee at most a (small) constant factor performance loss n the extended settngs. In our second scenaro, we do not have access to the exact numercal nput. Wthn ths context, we explore combnatoral extensons of the well-known secretary problem under the assumpton that the ncomng elements only reveal ther ordnal poston wthn the set of prevously arrved elements. We frst observe that many exstng algorthms for specal matrod structures mantan ther compettve rato n the ordnal model. In contrast, we provde a lower bound for algorthms that are oblvous to the matrod structure. Fnally, we desgn new algorthms that obtan constant compettve ratos for a varety of combnatoral problems. Zusammenfassung In deser Dssertaton betrachten wr ene Auswahl kombnatorscher Optmerungsprobleme, denen allen das Thema der Unscherhet und des egostschen Verhaltens zugrunde legt. In unserem ersten Szenaro wrd de Engabe von egostschen Spelern empfangen. Her studeren wr Erweterungen des sogenannten Smoothness-Frameworks für Mechansmen, ener sehr nützlchen Technk um de Ineffzenz von Glechgewchten zu beschränken. Wr betrachten den Fall von varerender Mechansmen-Verfügbarket und de Telnahme von rskoaversen Spelern. In beden Fällen snd unsere Hauptresultate allgemene Sätze für Mechansmen, de (λ, µ)-smooth snd. Wr zegen, dass dese Mechansmen höchstens enen (klenen) konstanten Faktor Effzenzverlust n den verallgemenerten Stuatonen garanteren. In unserem zweten Szenaro haben wr kenen Zugang zur exakten (numerschen) Engabe. In desem Kontext untersuchen wr kombnatorsche Erweterungen des wohlbekannten Sekretärnnenproblems unter der Annahme, dass de engehenden Elemente nur hren Rang n der Ordnung der bsher empfangenen Elemente verraten. Wr beobachten zunächst, dass vele exsterende Algorthmen für spezelle Matrodstrukturen hre Kompettvtät n desem Ordnungsmodell behalten. Im Gegensatz herzu geben wr ene untere Schranke für Algorthmen an, de kene Matrodstrukturen erkennen können. Schleßlch entwerfen wr neue Algorthmen mt konstanter Kompettvtät für ene Auswahl von kombnatorschen Problemen.

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5 Acknowledgments Frst of all, I would lke to thank my advsor Martn Hoefer for hs support throughout my Ph.D., both on and off duty. I never expected the advsor-student relatonshp to be such an accurate reenactment of the customary growng up and I do hope I wasn t too naughty n my teenage years. I am also very grateful to Thomas Kesselhem for hs knd gudance. It was a pleasure experencng an exceedngly clear way of thnkng and posng questons, n a scentfc world that s often cluttered wth nformaton and deprved of common sense. Fnally, I would lke to thank Kurt Mehlhorn for acceptng me to hs group n the frst place, for provdng a lovely work envronment, for always beng there not just for me but for all of us.

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7 Contents 1 Overvew Part I: Extensons of the Smoothness Framework for Mechansms Part II: Combnatoral Secretary Problems wth Ordnal Informaton.. 5 I Extensons of the Smoothness Framework for Mechansms 7 2 Introducton to Part I Our Contrbuton Related Work Notaton and Prelmnares Settng Soluton Concepts and Benchmarks Smoothness Framework Rsk-Averse Agents Model Sngle-Item Auctons n the Quaslnear Settng Smoothness Beyond Quaslnear Utltes Quaslnear Often Imples Rsk-Averse Smoothness Unbounded Prce of Anarchy for All-Pay Auctons Varance-Averson Model Smultaneous Composton of Mechansms wth Admsson Model Composton wth Independent Admsson Composton wth Everybody-or-Nobody Admsson A Lower Bound for General XOS Functons Applcatons beyond Auctons Extenson to Changng Unt-Demand Functons II Combnatoral Secretary Problems wth Ordnal Informaton 59 6 Introducton to Part II Our Contrbuton Related Work Notaton and Prelmnares

8 7 Matrods Submodular Matrods A Lower Bound Matchng, Packng and Independent Set Bpartte Matchng Packng Matchng n General Graphs Independent Set and Local Independence Concluson and Open Problems 81 Lst of Fgures 83 Lst of Tables 85 Lst of Algorthms 87 Lst of Acronyms 89 Bblography 91 v

9 CHAPTER 1 Overvew Algorthmc mechansm desgn strves towards desgnng games that have both good game-theoretcal and algorthmc propertes. It became a popular topc n the computer scence communty only n the last two decades, whle tradtonally mechansm desgn was extensvely studed n economcs. The algorthmc and computatonal aspects stem from a mechansm smply beng an algorthm: upon recevng players prvate nformaton as nput, t outputs an outcome for all of the partcpants. The players, on the other hand, are assumed to be selfsh ratonal agents that are nterested only n achevng the best possble outcome for themselves. To ths end, they mght msreport ther prvate nformaton. The goal of a mechansm, as opposed to a regular algorthm, s to algn the ncentves of the players such as to reach a desred objectve n an equlbrum. An equlbrum s generally defned as a state n whch no player can mprove hs outcome by changng hs part of the nput and players preferences are modeled va valuaton functons over the possble outcomes. It s commonly assumed that the utlty a player experences from an outcome s equal to hs valuaton for the outcome mnus the payment mposed by the mechansm (f any). Payments allow the mechansm to encourage players to report truthfully, through maxmzng ther utltes for truthful actons. A smple example of a truthful mechansm s the second prce aucton. 1 In a secondprce aucton, the hghest bdder wns the tem and pays the second hghest bd. It s easy to check that submttng a bd equal to hs valuaton s each player s utlty maxmzng strategy. Second prce auctons stay truthful also f the auctoneer n addton sets a reserve prce such that the tem s sold only f the bd of the hghest bdder exceeds the reserve. The hghest bdder has to, n ths case, pay the maxmum of the second hghest bd and the reserve prce. 2 An example of a non-truthful mechansm s the frst prce aucton, where the hghest bdder wns the tem and pays hs bd. If a player reports hs valuaton for obtanng the tem correctly, hs utlty s zero, ndependently of whether he wns or loses. The two commonly studed objectves n mechansm desgn are socal welfare maxmzaton and revenue maxmzaton. Socal welfare maxmzaton s concerned wth the sum of all the valuatons n an equlbrum outcome, whle revenue maxmzaton ams at maxmzng the sum of the payments. The second-prce aucton s maxmzng socal welfare, whle a second-prce aucton wth an accordngly set reserve maxmzes revenue. The doman of algorthmc mechansm desgn has a wde range of applcatons. Some of the most promnent real world examples are onlne auctons and exchanges, onlne advertsng, and search engne s page rankng. These applcatons occur on a daly bass 1 In fact, the second prce aucton s a specal case of the more general Vckrey-Clark-Groves mechansm (VCG), appled to the case of assgnng one tem. 2 Second-prce aucton wth reserve prce s the applcaton of the more general optmal mechansm by Myerson [74 to the case of a sngle tem aucton.

10 Chapter 1. Overvew and consttute a mult-bllon-dollar ndustry. In tradtonal mechansm desgn, one of the most fundamental prncples s the revelaton prncple. It states that f there exsts a mechansm that mplements a socal choce functon, then t s possble to mplement the same functon by an ncentvecompatble mechansm (.e., by a mechansm that allows every partcpant to acheve the best outcome for themselves by truthfully reportng ther prvate nformaton). The two mechansms have the same equlbrum outcome and the same equlbrum payoffs. Ths postulate enabled researchers to restrct ther attenton solely to truthful mechansms and made ncentve compatblty nto somethng of a standard requrement. In the recent years, t has become apparent that truthful mechansm desgn s nherently complcated from an algorthmc pont of vew. The queston of desgnng a mechansm can ndeed be vewed as a standard optmzaton problem, where truthfulness s mposed as an addtonal constrant. The optmzaton tself, however, s very often computatonally ntractable and the resultng mechansm complex and mpractcal. Truthful mechansms are, n fact, very rarely used n practce even n the tractable cases. For nstance, t would be possble to use the celebrated VCG for the sponsored search applcaton. Instead, smple and non-truthful procedures are used to allocate ads on search result pages. A very mportant example of a non-truthful mechansm that s wdely used s the generalzed second-prce aucton (GSP). It s employed manly n the context of keyword auctons, where sponsored search slots are sold on an aucton bass. In GSP each bdder places a sngle bd. The hghest bdder then wns the frst sponsored search slot and pays the second hghest bd, the second hghest bdder wns the second slot and pays the thrd hghest bd, and so on. Ths mechansm s used by Google s AdWords technology and has evolved nto Google s man source of revenue, whch hnts to the fact that we are ndeed dealng wth a mult-bllon aucton market. A recent trend n algorthmc mechansm desgn s, therefore, to study non-truthful and conceptually smple mechansms for allocaton n markets and ther nherent loss n system performance. If one, n addton, wants to capture dynamc aspects, e.g., when agents arrve and depart over tme, as s often the case wth practcally used mechansms and specfcally also wth the mult-bllon-dollar example of sponsored search, one needs to resort to onlne mechansm desgn. Once the onlne aspect s ntroduced, however, problems usually become much harder. For nstance, f we consder the onlne verson of a sngle-tem aucton, we already arrve at an algorthmc problem of selectng the maxmum element n a worst case sequence that admts no non-trval onlne approxmaton algorthm. The classcal truthfulness objectve s, not surprsngly, also harder to reach n the onlne settng. Both of these dffcultes become less present when one assumes that the agents are arrvng n a random order. In ths case, the algorthmc problem correspondng to the sngle-tem aucton translates drectly nto the well-known secretary problem. Assumng random arrval, we are left wth the problem of choosng the maxmum element n a randomly ordered sequence. The (optmal) soluton, rejectng all the elements n a samplng phase and then acceptng the frst one that has a hgher value than any element seen so far [35, can be nterpreted as a posted-prce mechansm that both gves a constant approxmaton to the optmal socal welfare and s truthful. In fact, there s a rch nterplay between secretary problems and onlne mechansm desgn. Algorthms for secretary problems can be drectly transformed nto truthful 2

11 1.1. Part I: Extensons of the Smoothness Framework for Mechansms onlne mechansms whch are constant-compettve for agents wth random arrval order. At the same tme, the goal of desgnng onlne mechansms wth varous combnatoral constrants has led to the formulaton and soluton of new secretary problems that are nterestng n ther own rght. In ths thess, we consder a varety of combnatoral optmzaton problems wthn a common theme of uncertanty and selfsh behavor. All of the consdered problems can be seen from the perspectve of studyng ncentves n dynamc markets, where the objectve s to maxmze socal welfare ether va desgnng truthful mechansms or by pontng to non-truthful mechansms wth good performance guarantees. Our work can be grouped nto two scenaros: (1) In the frst scenaro, the nput s collected from selfsh agents who mght msreport ther prvate nformaton n order to acheve a better outcome for themselves. Alternatvely, the format of the nteracton wth the agents does not even allow them to fully express ther preferences. Ths scenaro obvously falls wthn the doman of Mechansm Desgn. More precsely, we nvestgate extensons of the smoothness framework for mechansms, a very useful technque for boundng the neffcency of equlbra of the nduced games, to the cases of varyng mechansm avalablty and partcpaton of rsk-averse players. (2) In the second scenaro, we assume that access to the exact numercal nput s not possble. We nevertheless want to desgn algorthms wth provable performance guarantees wth respect to the optmal soluton. Ths s motvated by problems where t s ether generally dffcult or even mpossble to assgn exact numercal values to the elements n the nput or we beleve there mght be mprecsons n the values that we are provded wth. Wthn ths Onlne Mechansm Desgn scenaro, we explore varous extensons of the well-known secretary problem under the assumpton that the ncomng elements only reveal ther ordnal poston wth respect to all elements seen so far, nstead of ther numercal value. In what follows, we expand upon the specfc problems consdered n ths thess and gve a hgh-level overvew of the results. For more detals, see Chapter 2 and Chapter Part I: Extensons of the Smoothness Framework for Mechansms The study of truthful mechansms s a classc branch of mcroeconomcs and has resulted n a varety of fundamental results, such as VCG for socal welfare maxmzaton or Myerson s revenue-optmal auctons. Strkngly, these technques are only very rarely used n practce, as they often nvolve heavy algorthmc machnery, complcated allocaton technques, or other hurdles to easy and transparent mplementaton. Ths has led to the study of non-truthful and conceptually smple mechansms, n whch bdders mght have the opportunty to gan from non-truthful bds. The approach of formally analyzng smple mechansms s to study the nduced game among the bdders and bound the qualty of (possbly manpulated) outcomes n equlbrum. In a semnal paper, Syrgkans and Tardos [85 propose a general framework for boundng socal welfare of these equlbra, based on a so-called smoothness 3

12 Chapter 1. Overvew technque. In Part I of ths thess we generalze ths framework nto two drectons n Chapters 4 and 5. We start by gvng further ntroducton to non-truthful mechansm desgn n Chapter 2 and ntroducng the necessary notaton and prelmnares n Chapter Rsk Averson A standard assumpton n Algorthmc Game Theory s that players are rsk neutral, meanng that they do not dstngush between dfferent strateges that gve them the same expected utlty. Ths s n turn modeled by defnng utlty to be the dfference between the valuaton and payment for any gven outcome. So, an agent havng a value of 1 for an tem would be ndfferent between gettng ths tem wth probablty 0.1 for free and gettng t all the tme, payng 0.9. However, there are many reasons to beleve that agents are not rsk neutral. For nstance, n the above example the agent mght favor the certan outcome to the uncertan one. Therefore, n Chapter 4, we rase the followng queston: What smple aucton mechansms preserve good performance guarantees n the presence of rsk-averse agents? Rsk averseness can be formalzed n varous ways. The two most common models are (1) defnng utlty as a concave functon of the dfference between the valuaton and the payment, or (2) takng nto account the standard devaton when computng the utlty. These two are also the ones nspected n Chapter 4. We gve bounds on the prce of anarchy for Bayes-Nash and (coarse) correlated equlbra of mechansms n the presence of rsk-averse agents and expose how the two models lead to dfferent results. More specfcally, our man postve result states that the loss of performance n model (1) compared to the rsk neutral settng s bounded by a small constant f a slghtly stronger smoothness condton s fulflled. We also prove that ths condton s necessary by showng that the second prce aucton has an unbounded prce of anarchy n the presence of rsk averse players. Ths s algned wth the ntuton that players should be more unwllng to partcpate n an all-pay aucton than n, say, a frst prce aucton. The results n Chapter 4 gve the frst theoretcal backup to ths observaton. In model (2), we arrve at qute dfferent results: frst prce and all-pay auctons do not sgnfcantly dffer. Furthermore, (λ, µ)-smoothness of a mechansm does not brng any guarantees n the presence of rsk-averse players. These results mply that the varanceaverson model s not necessarly the most natural model for rsk averson n the settng studed here Smultaneous Composton wth Varyng Avalablty In Chapter 5, we study a varant of smultaneous composton of mechansms. Our scenaro s motvated by lmted avalablty or admsson: Suppose bdders try to acqure tems n a repeated onlne market, n whch m tems are sold smultaneously va, say, frst-prce auctons. However, n each round only some of the tems are actually avalable for purchase. More specfcally, n each round each tem s avalable for each bdder only wth a certan probablty. Our scenaro s an elementary case of smple mechansm desgn wth ncomplete nformaton, where avalabltes are bdder types. It captures natural applcatons n onlne markets wth lmted supply and can be used to model access of unrelable channels n wreless networks. The man queston that we pose n Chapter 5 4

13 1.2. Part II: Combnatoral Secretary Problems wth Ordnal Informaton s: What bddng strateges gve good performance guarantees n a market composed of multple mechansms and where bdders are facng lmted avalablty? To avod the drawbacks of exstng results n terms of plausblty and computatonal complexty, we assume that the players learn wth no-regret strateges n a way that s oblvous to ther own and all other bdders avalabltes. Thereby, bdders arrve at what we term an avalablty-oblvous coarse-correlated equlbrum a bd dstrbuton not talored to the specfc avalabltes of bdders, whch can be computed (approxmately) n polynomal tme. Our man result s that for a large class of valuaton functons, we can apply smoothness deas n ths framework and prove bounds that mrror the known guarantees for compostons of smooth mechansms. In more detal, we prove general composton theorems for smooth mechansms when valuaton functons of bdders are lattce-submodular. They rely on an nterestng connecton to the noton of correlaton gap of submodular functons over product lattces. Our results hold for ndependent and fully correlated bdder avalabltes. In addton, we gve an almost logarthmc prce of anarchy lower bound for general fractonally subaddtve (XOS) valuaton functons. 1.2 Part II: Combnatoral Secretary Problems wth Ordnal Informaton The secretary problem s a classc model for onlne decson makng. Recently, combnatoral extensons such as matrod or matchng secretary problems have become an mportant tool to study algorthmc problems n dynamc markets. Here the decson maker must know the numercal value of each arrvng element, whch can be a demandng nformatonal assumpton. In Part II of ths thess, we ntate the study of algorthms for combnatoral secretary problems that rely only on ordnal nformaton. We assume that there s an unknown value for each element, but our algorthms only have access to the total order of the elements arrved so far, whch s consstent wth ther values. We term ths the ordnal model; as opposed to the cardnal model, n whch the algorthm learns the exact values. We show bounds on the compettve rato,.e., we compare the qualty of the computed solutons to the optma n terms of the exact underlyng but unknown numercal values. Consequently, compettve ratos for our algorthms are robust guarantees aganst uncertanty n the nput. The gudng queston of Part II of ths thess s: How can we desgn onlne algorthms wth small compettve ratos for combnatoral secretary problems n the ordnal model? In Chapter 6, we gve an ntroducton and motvaton to the problems we consder, together wth prelmnares and related work. For the matrod secretary problem, we observe that many exstng algorthms for specal matrod structures mantan ther compettve ratos even n the ordnal model. In these cases, the restrcton to ordnal nformaton does not represent any addtonal obstacle. In Chapter 7 we show that ordnal varants of the submodular matrod secretary problems can be solved usng algorthms for the lnear versons by extendng the results from [39. In contrast, we also provde a lower bound of Ω( n/(log n)) for algorthms that are oblvous to the matrod structure, where n s the total number of elements. Ths contrasts an upper bound of O(log n) n the cardnal model, and t shows that the 5

14 Chapter 1. Overvew technque of thresholdng s not suffcent for good algorthms n the ordnal model. In Chapter 8, we desgn new algorthms that obtan constant compettve ratos for a varety of combnatoral problems, such as bpartte matchng, general packng LPs and ndependent set wth bounded local ndependence. 6

15 PART I Extensons of the Smoothness Framework for Mechansms Ths part s the result of close collaboraton wth Martn Hoefer and Thomas Kesselhem. It s based on an artcle that appeared n Proceedngs of the Web and Internet Economcs - 12th Internatonal Conference (WINE) 2016, pages , n December 2016 [50, and an artcle that appeared n Proceedngs of the 45th Internatonal Colloquum on Automata, Languages, and Programmng (ICALP) 2018, pages 155:1 155:14, n July 2018 [57. Full versons are avalable at and https: //arxv.org/abs/ , respectvely.

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17 CHAPTER 2 Introducton to Part I A common way to understand the effects of strategc behavor s to analyze the nduced game among the bdders and bound the qualty of (possbly manpulated) outcomes n equlbrum. That s, one compares the socal welfare that s acheved at the (worst) equlbrum of the nduced game to the maxmum possble welfare. Typcal equlbrum concepts are Bayes-Nash equlbra and (coarse) correlated equlbra, whch extend mxed Nash equlbra toward ncomplete nformaton or learnng settngs respectvely. In a semnal paper, Syrgkans and Tardos [85 propose a general technque for boundng socal welfare of equlbra of smple mechansms, based on a so-called smoothness technque. These guarantees apply even to mxed Bayes-Nash equlbra n envronments wth composton of mechansms. For example, n a combnatoral aucton we mght not sell all tems va a complcated truthful mechansm, but nstead sell each tem smultaneously va smple ndvdual sngle-tem auctons. Such a mechansm s obvously not truthful, snce bdders are not even able to express ther valuatons for all subsets of tems. However, f bdders have complement-free fractonally subaddtve (XOS) valuatons, the (expected) socal welfare of allocatons n a mxed Bayes-Nash equlbrum turns out to be a constant-factor approxmaton of the optmal socal welfare. Whle ths s a fundamental nsght nto non-truthful mechansms, t s not wellunderstood how ths result extends under more realstc condtons. In partcular, there has been recent concern about the plausblty and computatonal complexty of exact and approxmate Bayes-Nash equlbra [18. For more general Bayesan concepts based on no-regret learnng strateges n repeated games, there are two natural approaches ether bdder types are drawn newly wth bds, or types are drawn only once ntally. Whle the latter s not really n lne wth the dea of ncomplete nformaton (bdders could communcate ther type n the course of learnng [18), the former s n general hard to obtan. Also, the composton theorem apples only f bdders types are drawn ndependently. In Chapter 5 of ths thess, we study a varant of smultaneous composton of mechansms and show how to avod the drawbacks of the Bayesan approach. Our scenaro s motvated by lmted avalablty or admsson: Suppose bdders try to acqure tems n a repeated onlne market, n whch m tems are sold smultaneously va, say, frst-prce auctons. However, n each round only some of the tems are actually avalable for purchase. Ths scenaro can be phrased n the Bayesan framework when bdder s type s gven by the set of tems avalable to hm. To obtan an equlbrum n the Bayesan sense, each bdder would have to consder a complcated bd vector and satsfy an equlbrum condton for each of the possble 2 m subsets of tems. In contrast, we assume that bdders do not even get to know (or are not able to account for) ther own avalabltes before makng bds n each round. We assume they learn wth no-regret strateges n a way that s oblvous to ther own and all other bdders

18 Chapter 2. Introducton to Part I avalabltes. Thereby, bdders arrve at what one mght term an avalablty-oblvous coarse-correlated equlbrum a bd dstrbuton not talored to the specfc avalabltes of bdders, whch can be computed (approxmately) n polynomal tme. Our man result n Chapter 5 s that for a large class of valuaton functons, we can apply smoothness deas n ths framework and prove bounds that mrror the guarantees from [85. The guarantees apply even f some bdders learn oblvously and others follow a Bayes-Nash bddng strategy. In partcular, we cover a broad doman wth smultaneous composton of weakly smooth mechansms n the sense of [85 when bdders have lattce-submodular valuatons. Our study covers cases where avalabltes are correlated among bdders and provdes lower bounds for combnatoral auctons wth tem-bddng and XOS valuatons. As a part of our analyss, we use the concept of correlaton gap from [2 for submodular functons over product lattces. Another key assumpton n analyses of smple mechansms s that agents are rsk neutral: Agents are assumed to maxmze ther expected quaslnear utlty, whch s defned to be the dfference of the value assocated to the outcome and payment mposed to the agent. As already dscussed, ths would mply that an agent havng a value of 1 for an tem would be ndfferent between gettng ths tem wth probablty 10% for free and gettng t all the tme, payng 0.9. However, there are many reasons to beleve that agents are not rsk neutral. For nstance, n the above example the agent mght favor the certan outcome to the uncertan one. Therefore, n Chapter 4 of ths thess, we characterze smple aucton mechansms that preserve good performance guarantees n the presence of rsk-averse agents. The standard model of rsk averson n economcs (see, e.g., [69) s to apply a (weakly) concave functon to the quaslnear term. That s, f agent s outcome s x and hs payment s p, hs utlty s gven as u (x, p ) = h (v (x ) p ), where h : R R s a weakly concave, monotone functon. Agent s rsk neutral, f and only f h s a lnear functon. If the functon s strctly concave, ths has the effect that, by Jensen s nequalty, the utlty for fxed x and p s hgher than for a randomzed x and p wth the same expected v (x ) p. We compare outcomes based on ther socal welfare, whch s defned to be the sum of utltes of all nvolved partes ncludng the auctoneer. That s, t s the sum of agents utltes and ther payments SW(x, p) = u (x, p ) + p. In the quaslnear settng ths defnton of socal welfare concdes wth the sum of values v (x ). Wth rsk-averse utltes these two quanttes usually dffer. However, all our results bound the sum of values and therefore also hold for ths benchmark. We assume that the mechansms are oblvous to the h -functons and work lke n the quaslnear model. Only the ndvdual agent s percepton changes. Ths makes t necessary to normalze the h -functons as they could be on dfferent scales. 1 Therefore, we wll assume that u (x, p ) = v (x) f p = 0 and that u (x, p ) = 0 f p = v. That s, for the two cases that p s ether 0 or the full value, the utlty matches exactly the quaslnear one. However, due to rsk averson, the agents mght be less senstve to payments. We note here that ths wll not n turn allow the mechansm to arrve at huge utlty gans, as compared to the quaslnear model, by just cleverly splttng the quaslnear utlty among the players. Indeed, Lemma 4.1 n Secton 4.1 wll show that 1 E.g., the functons could be such that h 1(y) = y and h 2(y) = 1000 y, whch would be mpossble for the mechansm to cope wth wthout addtonal nformaton. 10

19 2.1. Our Contrbuton the dfference between the two optma s bounded by at most a factor of 2. For sake of smplcty, after provdng the notaton and prelmnares n Chapter 3, we start frst wth rsk averson n Chapter 4. Then, we present the techncally more demandng materal dealng wth mechansm avalablty n Chapter Our Contrbuton Rsk Averson We gve bounds on the prce of anarchy for Bayes-Nash and (coarse) correlated equlbra of mechansms n the presence of rsk-averse agents. Our postve results are stated wthn the smoothness framework, whch was ntroduced by [81. We use the verson that s talored to quaslnear utltes by [85, whch we extend to mechansm settngs wth general utltes (for a formal defnton see Secton 4.3). Our man postve result n Chapter 4 states that the loss of performance compared to the quaslnear settng s bounded by a constant f a slghtly stronger smoothness condton s fulflled. Man Result 1. Gven a mechansm wth prce of anarchy α n the quaslnear model provable va smoothness such that the devaton guarantees non-negatve utlty, then ths mechansm has prce of anarchy at most 2α n the rsk-averse model. Ths result reles on the fact that the devaton acton to establsh smoothness guarantees agents to have non-negatve utlty. A suffcent condton s that all undomnated strateges never have negatve utlty. Frst-prce and second-prce auctons satsfy ths condton, we thus get constant prce-of-anarchy bounds for both of these aucton formats. In an all-pay aucton every postve bd can lead to negatve utlty. Therefore, the postve result does not apply. As a matter of fact, ths s not a concdence because, as we show, equlbra can be arbtrarly bad. Man Result 2. The sngle-tem all-pay aucton has unbounded prce of anarchy for Bayes-Nash equlbra, even wth only three agents. Ths means that although equlbra of frst-prce and all-pay auctons have very smlar propertes wth quaslnear utltes, n the rsk-averse settng they dffer by a lot. We feel that ths to some extent matches the ntuton that agents should be more reluctant to partcpate n an all-pay aucton compared to a frst-prce aucton. In our constructon, we gve a symmetrc Bayes-Nash equlbrum for two agents. The equlbrum s desgned n such a way that a thrd agent of much hgher value would lose wth some probablty wth every possble bd. Losng n an all-pay aucton means that the agent has to pay wthout gettng anythng, resultng n negatve utlty. In the quaslnear settng, ths negatve contrbuton to the utlty would be compensated by respectve postve amounts when wnnng. For the rsk-averse agent n our example, ths s not true. Because of the rsk of negatve utlty, he prefers to opt out of the aucton entrely. We also consder a dfferent model of averson to uncertanty, n whch soluton concepts are modfed. Instead of evaluatng a dstrbuton over utltes n terms of ther expectaton, agents evaluate them based on the expectaton mnus a second-order term. We fnd that ths model has entrely dfferent consequences on the prce of anarchy. For 11

20 Chapter 2. Introducton to Part I example, the all-pay aucton has a constant prce of anarchy n correlated and Bayes-Nash equlbra, whereas the second-prce aucton can have an unbounded prce of anarchy n correlated equlbra Smultaneous Composton wth Varyng Avalablty In the smultaneous composton settng, we assume that every mechansm satsfes a weak smoothness bound (for more detals see Secton 4.1) wth parameters λ, µ 1, µ 2 0. It s known that for each ndvdual mechansm, ths mples an upper bound of (max(1, µ 1 ) + µ 2 )/λ on the prce of anarchy for no-regret learnng outcomes and Bayes- Nash equlbra. Furthermore, the same bound also apples for outcomes of multple smultaneous mechansms that are talored to avalabltes,.e., not oblvous. In Secton 5.2 we consder smoothness for oblvous learnng and composton wth ndependent avalabltes, where n each round t, each mechansm j s avalable to each bdder ndependently wth probablty q,j. Our smoothness bound nvolves the above parameters and the correlaton gap of the class of valuaton functons. In partcular, f valuatons v come from a class V wth a correlaton gap of γ(v), the prce of anarchy becomes γ(v) (max(1, µ 1 ) + µ 2 )/λ. Man Result 3. The prce of anarchy for oblvous learnng wth monotone valuatons that come from a class V wth a correlaton gap of γ(v) and fully ndependent admsson s at most γ(v) (µ 2 + max(1, µ 1 ))/λ. Our constructon uses smoothness of smultaneous composton from [85. However, snce learnng s oblvous, the devatons establshng smoothness must be ndependent of avalablty. Here we use correlaton gap to relate the value for ndependent devatons to that of type-dependent Bayesan devatons. Correlaton gap s a noton orgnally defned for submodular set functons n [3. It captures the worst-case rato between the expected value of ndependent and correlated dstrbutons over elements wth the same margnals. We use an extenson of ths noton from [2 to Cartesan products of outcome spaces such as product lattces. For the class V of monotone lattce-submodular valuatons, we prove a correlaton gap of γ(v) = e/(e 1), whch smplfes and slghtly extends prevous results. In Secton 5.3, we analyze oblvous learnng for composton wth correlated avalabltes n the form of everybody-or-nobody each mechansm s ether avalable to all bdders or to no bdder. The probablty for avalablty of mechansm j s q j, and avalabltes are ndependent among mechansms. In ths case, we smulate ndependence by assumng that each bdder draws random types and outcomes for hmself. We also consder dstrbutons where outcomes are drawn ndependently accordng to the margnals from the optmal correlated dstrbuton over outcomes. Whle these two dstrbutons are drectly related va correlaton gap, the techncal challenge s to show that there s a connecton to the value obtaned by the bdder. For lattce-submodular functons, we show a smoothness bound that mples a prce of anarchy of 4e/(e 1) (max(1, µ 1 ) + µ 2 )/λ 2. Man Result 4. The prce of anarchy for oblvous learnng wth monotone lattcesubmodular valuatons and everybody-or-nobody admsson s at most 4e/(e 1) (µ 2 + max(1, µ 1 ))/λ 2. 12

21 2.2. Related Work For nether of the results s t necessary that all bdders follow our oblvous-learnng approach. We only requre that bdders have no regret compared to ths strategy. Ths s also fulflled f some or all bdders determne ther bds based on the actually avalable tems rather than n the oblvous way. Fnally, n Secton 5.4 we show a lower bound for smultaneous composton of sngletem frst-prce auctons wth general XOS valuaton functons. The correlaton gap for such functons s known to be large [3, but ths does not drectly mply a lower bound on the prce of anarchy for oblvous learnng. We provde a class of nstances where the prce of anarchy for oblvous learnng becomes Ω((log m)/(log m log m)). Ths shows that for XOS functons t s mpossble to generalze the constant prce of anarchy for sngle-tem frst-prce auctons. Man Result 5. The prce of anarchy for pure Nash equlbra wth oblvous bddng can be as large as Ω((log m)/(log log m)) f we allow XOS valuaton functons. Our results have addtonal mplcatons beyond auctons for the analyss of regret learnng n wreless networks. We dscuss these n Secton Related Work The smoothness framework was ntroduced by [81, 79 to analyze correlated and Bayes- Nash equlbra of general games. In [85 t was adjusted to the quaslnear case of mechansms, and t was shown that smultaneous or sequental composton of smooth mechansms s agan smooth. Combnatoral auctons wth tem bddng are an example of a smultaneous composton. To show smoothness of the combned mechansm, t s thus enough to show smoothness of each sngle aucton. Other examples of smooth mechansms are poston auctons wth generalzed second prce [19, 77 and greedy auctons [65. The smoothness approach for mxed Bayes-Nash equlbra shown n [85 s, n fact, slghtly more general and contnues to hold for varants of Bayesan correlated equlbrum [48. Closely related to our work on mechansm avalablty are combnatoral auctons wth tem bddng, where multple tems are beng sold n separate auctons. Bdders are generally nterested n multple tems. However, dependng on the bdder, some tems may be substtutes for others. As the auctons work ndependently, bdders have to strategze n order to buy not too many tems smultaneously. In a number of papers [23, 16, 49, 36 the effcency of Nash and Bayes-Nash equlbra has been studed. It has been shown that, f the sngle tems are sold n frst or second prce auctons and f the valuaton functons are XOS or subaddtve, the prce of anarchy s constant. Lmtatons of ths approach are shown n [24, 80. The complexty of fndng Bayes-Nash equlbra and Bayesan correlated equlbra has been studed only very recently. It has been shown n [18, 31 that equlbra are hard to fnd n some settngs. In contrast, n [28 a dfferent aucton format s studed that yelds good bounds on socal welfare for equlbra that can be found more easly. Although smlar n sprt, our approach s dfferent t shows that n some scenaros agents can reduce the computatonal effort and stll obtan reasonably good states wth exstng mechansms. 13

22 Chapter 2. Introducton to Part I As such, our approach s closer to recent work [27 that shows hardness results for learnng full-nformaton coarse-correlated equlbra n smultaneous sngle-tem second-prce auctons wth unt-demand bdders. As a remedy, a form of so-called no-envy learnng s proposed, n whch bdders use a dfferent form of bddng that enables convergence n polynomal tme. Whle achevng a general no-regret guarantee aganst all possble bd vectors s hard, we note here that our approach based on smoothness requres only a guarantee wth respect to bds that are derved drectly from the XOS representaton of the bdder valuaton. As such, bdders can obtan the guarantees requred for our results n polynomal tme. Conceptually, we here treat a dfferent problem the mpact of avalabltes, and more generally, dfferent bdder types on learnng outcomes n repeated mechansm desgn. A model wth dynamc populatons n games has recently been consdered n [66. Each round a small porton of players are replaced by others wth dfferent utlty functons. When players use algorthms that mnmze a noton called adaptve regret, smoothness condtons and the resultng bounds on the prce of anarchy contnue to hold f there are solutons whch reman near-optmal over tme wth a small number of structural changes. Usng tools from dfferental prvacy, these condtons are shown for some specal classes of games, ncludng frst-prce auctons wth unt-demand or gross-substtutes valuatons. In contrast, our scenaro s orthogonal, snce we consder much more general classes of mechansms and allow changes n each round for possbly all players. However, our model of change captures the noton of avalablty and therefore s much more specfc than the adversaral approach of [66. The noton of correlaton gap was defned and analyzed for stochastc optmzaton n [3, 2. It was used n [87 for analyzng revenue maxmzaton wth sequental auctons, whch s very dfferent from our approach. Studyng the mpact of rsk-averseness s a regularly reoccurrng theme n the lterature. A proposal to dstngush between money and the utlty of money, and to model rsk averson by a utlty functon that s concave frst appeared n [14. The expected utlty theory, whch bascally states that the agent s behavor can be captured by a utlty functon and the agent behaves as a maxmzer of the expectaton of hs utlty, was postulated n [86. Ths theory does not capture models that are standardly used n portfolo theory, expectaton mnus varance or expectaton mnus standard devaton [67, the latter of whch we also consder n Secton 4.6. In the context of mechansms, one usually models rsk averson by concave utlty functons. One research drecton n ths area s to understand the effects of rsk averson on a gven mechansm. For example, n [42 the authors study symmetrc equlbra n all-pay auctons wth a homogeneous populaton of rsk-averse players. They compare the bddng behavor to the rsk-neutral case. In [71 a smlar analyss for auctons wth a buyout opton s performed; n [54 customers wth heterogeneous rsk atttudes n mechansms for cloud resources are consdered. In [34 t s shown that for certan classes of mechansms the correlated equlbrum s unque and has a certan structure. One consequence of ths result s that rsk averson does not nfluence the form of the equlbra or the revenue. Another drecton s to desgn mechansms for the rsk-averse settng. For example, the optmal revenue s hgher because buyers are less senstve to payments. In a number of papers, mechansms for revenue maxmzaton are proposed [72, 70, 84, 55, 15,

23 2.2. Related Work Furthermore, randomzed mechansms that are truthful n expectaton lose ther ncentve propertes f agents are not rsk neutral. Black-box transformatons from truthful-nexpectaton mechansms nto ones that fulfll stronger propertes are gven n [33 and [51. Studyng the effects of rsk averson also has a long hstory n game theory, where dfferent models of agents atttudes towards rsk are analyzed. One major queston s, for example, f equlbra stll exst and f they can be computed [41, 53. Prce of anarchy analyses have so far only been carred out for congeston games. Tght bounds on the prce of anarchy for atomc congeston games wth affne cost functons under a range of rsk-averse decson models are gven n [78. It s mportant to remark here that our approach to rsk s dfferent from the one taken by [73. They use the smoothness framework to prove generalzed prce of anarchy bounds for games n whch players have based utlty functons. They assume that players are playng the wrong game and ther pont of comparson s the true optmal socal welfare, meanng that the bases only determne the equlbra but do not affect the socal welfare. We take the utlty functons as they are, ncludng the rsk averson, to evaluate socal welfare n equlbra and also to determne the optmum, whch makes our models ncomparable. For precse relaton of von Neumann-Morgenstern preferences to mean-varance preferences, see for nstance [68. Mean-varance preferences were explored for congeston games n [75, 76, whle the authors n [62 study the bddng behavor n an all-pay aucton dependng on the level of varance-averseness. 15

24 Chapter 2. Introducton to Part I 16

25 3.1 Settng CHAPTER 3 Notaton and Prelmnares We consder the followng settng: There s a set N of n players and X s the set of possble outcomes. Each player has a utlty functon u θ, whch s parameterzed by hs type θ Θ. Gven a type θ, an outcome x X, and a payment p 0, hs utlty s u θ (x, p ). The tradtonally most studed case are quaslnear utltes, n whch types are valuaton functons v V, v : X R 0 and u v (x, p ) = v (x) p. For fxed utlty functons and types, the socal welfare of an outcome x X and payments (p ) N s defned as SW θ (x, p) := N u θ (x, p ) + N p. (3.1) In the quaslnear case, ths smplfes to SW θ (x, p) = N v (x). (3.2) Unless noted otherwse, by OPT(θ), we wll refer to the optmal socal welfare under type profle θ,.e., OPT(θ) = max x,p SWθ (x, p). (3.3) A mechansm M s a trple (A, f, p), where A = A s the set of actons and A s the set of actons for each player, f : A X s an allocaton functon that maps actons to outcomes and p: A R n 0 s a payment functon that maps actons to payments p for each player. Gven an acton profle a A, we wll use the short-hand notaton u θ (a), or sometmes even u (a), to denote u θ (f(a), p ). We assume that players always have the possblty of not partcpatng, hence any ratonal outcome has non-negatve utlty n expectaton over the non-avalable nformaton and the randomness of other players and the mechansm Smultaneous Composton In Chapter 5 we wll focus on the followng generalzed settng: There are, as before, n players but they are partcpatng n m mechansms, where m 1. The mechansms are not runnng n solaton, but rather take place smultaneously. Each mechansm M j has ts own outcome space X j and conssts of a trple (A j, f j, p j ) as descrbed prevously,.e., A j = A,j s the acton space, f j : A j X j s the allocaton functon and p j : A j R n 0 the payment functon.

26 Chapter 3. Notaton and Prelmnares In ths case, we assume that a player has a valuaton over vectors of outcomes from the dfferent mechansms: v : j X j R 0. A player s utlty wll be quaslnear n ths extended settng n the sense that hs utlty from an allocaton vector x = (x 1,..., x m ) j X j and payment vector p = (p,1,..., p,m ) s gven by: u v (x, p ) = v (x 1,..., x m ) m j=1 p,j. (3.4) Players can have valuatons that are complex functons of the outcomes of dfferent mechansms. The man results for smultaneous composton of mechansms n [85, whch we wll use n Chapter 5, hold for the class of valuaton functons knows as XOS Valuaton Functon Classes Defnton 3.1 (XOS). Valuaton v of player s XOS f there exsts a set L of addtve valuatons v l,j : X j R 0, l L, N, j [m, such that: v (x) = max l L j vl,j (x j). Our man results n Chapter 5 wll hold for the class of lattce-submodular valuatons. We frst gve the defnton of a submodular set functon. Defnton 3.2 (Submodular set functon). Let Ω be a fnte set. A functon f : 2 Ω R s set submodular, f S, T Ω such that S T and x Ω \ T Equvalently, S, T Ω f(s {x}) f(s) f(t {x}) f(t ). f(s T ) + f(s T ) f(s) + f(t ). Before delverng the defnton of a lattce-submodular functon, we state the defnton of a lattce. Defnton 3.3 (Lattce). A partally ordered set (L, ) s called a lattce f each twoelement subset {a, b} L has a jon (.e., least upper bound) and a meet (.e., greatest lower bound), denoted by a b and a b, respectvely. Defnton 3.4 (Lattce-submodular valuaton). Suppose for every mechansm j the set X,j of possble outcomes for bdder forms a lattce (X,j,,j ) wth a partal order,j. Bdder has a lattce-submodular valuaton v f and only f t s submodular on the product lattce (X, ) of outcomes for bdder : x, x X : v (x x ) + v (x x ) v (x ) + v ( x ). Lattce-submodular functons generalze submodular set functons but are a strct subclass of XOS functons. A further promnent valuaton class s the class of unt-demand valuatons. Defnton 3.5 (Unt-demand). Valuaton v of player s unt-demand, f there exst valuatons v,j such that v (S) = max j S v,j. 18

27 3.2. Soluton Concepts and Benchmarks 3.2 Soluton Concepts and Benchmarks In the settng of complete nformaton, the type profle θ s fxed. We consder (coarse) correlated equlbra, whch generalze Nash equlbra and are the outcome of (no-regret) learnng dynamcs. Defnton 3.6. A correlated equlbrum (CE) s a dstrbuton a over acton profles from A such that for every player and every strategy a n the support of a and every acton a A, player does not beneft from swtchng to a whenever he was playng a. Formally, E a a [u (a) E a a [u (a, a ), a A,. Defnton 3.7. A coarse correlated equlbrum (CCE) s a dstrbuton a over acton profles from A such that for every player and every acton a A, player does not beneft from swtchng to a. Formally, E a [u (a) E a [ u (a, a ), a A,. In ncomplete nformaton, the type of each player s drawn from a dstrbuton F over hs type space Θ. The dstrbutons are common knowledge and the draws are ndependent among players. The soluton concept we consder n ths settng s the Bayes-Nash equlbrum. Here, the strategy of each player s now a (possbly randomzed) functon s : Θ A. Defnton 3.8. A Bayes-Nash equlbrum (BNE) s a dstrbuton s over functons s such that for every player, every type θ and every strategy a A, player does not beneft from swtchng to a whenever he was playng s (θ ). Formally, E θ θ [u θ (s(θ)) E θ θ [u θ (a, s (θ )), a A, θ Θ,. The measure of effcency s the expected socal welfare over the types of the players: Gven a strategy profle s: Θ A, we consder E θ [SW θ (s(θ)). We compare the effcency of our soluton concept wth respect to the expected optmal socal welfare E θ [OPT(θ). Defnton 3.9. The Prce of Anarchy (PoA) wth respect to an equlbrum concept s the worst possble rato between the optmal expected welfare and the expected welfare at equlbrum. Formally, PoA = max F max D EQ(F ) E θ F [OPT(θ) E θ F,a D [SW θ (a), where by F = F 1 F n we denote the product dstrbuton of the players type dstrbutons and by EQ(F ) the set of all equlbra, whch are probablty dstrbutons over acton profles. Note that PoA generally depends on the set of consdered equlbra and can therefore dffer for dfferent equlbrum concepts. 19

28 Chapter 3. Notaton and Prelmnares 3.3 Smoothness Framework The results n Part I of ths thess wll mostly be based on the smoothness framework for mechansms, as gven n [85. Here we only ntroduce the basc defntons and man theorems. All defntons and theorems n ths secton assume quaslnear utltes. For the sake of smplcty, we defer the defnton and dscusson of further concepts (as, for nstance, weak smoothness ) to Chapter 4 and Chapter 5. Defnton 3.10 (Smooth Mechansm). A mechansm s (λ, µ)-smooth f for any valuaton profle v V and for any acton profle a there exsts a randomzed acton a (v, a ) for each player, s.t.: uv (a (v, a ), a ) λopt(v) µ P (a) (3.5) for some λ, µ 0. If a s a vector of randomzed strateges, u v (a) denotes the expected utlty of a player. The followng theorems from [85 reveal the appeal of checkng the very techncal condton that a mechansm needs to satsfy n order to be (λ, µ)-smooth. Theorem If a mechansm s (λ, µ)-smooth, then any correlated equlbrum n the full nformaton settng and any Bayes-Nash equlbrum n the Bayesan settng acheves λ effcency of at least a fracton of max{1,µ} of OPT(v) or of E v[opt(v), respectvely. Lastly, the next theorem bounds the prce of anarchy of a smultaneous composton of (λ, µ)-smooth mechansms. Theorem 3.12 (Smultaneous Composton). Consder the mechansm defned by the smultaneous composton of m mechansms. Suppose that each mechansm M j s (λ, µ)- smooth when the mechansm restrcted valuatons of the players come from a class of valuatons (V,j ) N. If the valuaton v : X R 0 of each player across mechansms s XOS, meanng t can be expressed by component valuatons v l,j V,j, then the global mechansm s also (λ, µ)-smooth. 20