Optimal Mechansim Design and Money Burning

Optimal Mec hanism Desi gn and Money Burning Jason D. Hartline ∗ Tim Roughgarden † First Draft: Jan uary 2007 ; this draft: April 2008 Abstract Mechanism design is no w a standard to ol in co mputer science for alig ning the incentiv es of s e lf- interested ag ent s with the o b jectives of a system designer. There is, how ever, a fun- damental disconnect b etw een the traditional application domains o f mechanism design (such as auctions) and those arising in computer science (such a s netw or ks): while monetar y tr ans- fers (i.e., paymen ts) ar e essential for most of the known positive results in mech anism design, they a re undesira ble or even technologically infea s ible in many co mputer s ystems. Classical impo ssibility results imply that the r each of mechanisms without transfers is sev erely limited. Computer systems typically do hav e the abilit y to reduce serv ice qualit y—routing systems can drop or dela y tr a ffic, scheduling proto co ls can delay the relea s e of jobs, and computa- tional pa yment schemes can require co mputational paymen ts fr o m user s (e.g., in spam-fighting systems). Service deg r adation is tan tamount to r equiring that users bu r n money , and suc h “paymen ts” can b e used to influence the prefer e nces of the agents a t a cost of degr ading the so cial sur plus. W e dev elop a framework for the design and ana lysis of money-burning me chanisms to maxi- mize the residua l sur plus—the total v alue of the chosen outcome minus the paymen ts requir e d. Our prima ry contributions are the follo wing. • W e define a general template for prio r-free optimal mechanism design that explicitly con- nects Bayesian optimal mechanism design, the dominan t par a digm in economics, with worst-case ana lysis. In particular, we establish a genera l and princ iple d wa y to identif y appropria te perfo rmance b enchmarks for prio r-free optimal mec ha nism design. • F or genera l sing le -parameter agent settings, we characterize the Bay esian optimal money- burning mec hanism. • F or multi-unit auctions, we design a near -optimal prio r -free money-burning mechanism: for ev ery v alua tion profile, its exp ected residual surplus is within a constant factor of our benchmark, the residual surplus of the b est Bay esian optimal mec hanism fo r this profile. • F or m ulti-unit auctions, w e quantify the benefit of general transfers over money-burning: optimal money-burning mechanisms always obtain a logarithmic fraction o f the full so cial surplus, and this b ound is tigh t. ∗ Electrical Engineering and Computer Science, N orthw estern Universit y , Ev anston, IL. This w ork w as done while author w as at Microsof t R esearc h, Silicon V alley . Email: hartline@eecs.north western.edu . † Department o f Computer Science, Stanford Universit y , 462 Gates Building, 353 Serra Mall, Stanford, CA 94305. Supp orted in part by N SF CAREER A w ard CCF-0448664, an O N R Y oung Inv estigator Award, and an Alfred P . Sloan F ello wship. Email: tim@cs.stanford .edu . 1 1 In tro duction Mec hanism d esign is no w a standard to ol in compu ter science for designing resource allocation proto cols (a.k.a. mec h anisms) in computer systems u sed b y agen ts with diverse and selfish in terests. The goal of mec hanism design is to ac hieve non-trivial op timization eve n when the un d erlying data—the preferences of p articipan ts—are unkno w n a priori. F undamenta l for most p ositiv e results in mec hanism design are monetary transf ers (i.e., pa ymen ts) b et w een participan ts. F or example, in the surplus-maximizing V CG mec hanism [34, 7, 20], suc h transfers enable the mec hanism designer to align fully the incentiv es of th e agen ts with the system’s ob jectiv e. Most computer systems differ f r om classical en vironment s for mec h anism design, su c h as tradi- tional m arkets and au ctions, in that monetary transfers are u np opu lar, undesirable, or tec hn olog- ically infeasible. It is sometimes p ossible to d esign mec hanisms that eschew transfer s completely; see [32] for classical results in economics and [15, 24] for recen t app lications in inte rdomain routing. Unfortunately , negativ e r esults der ived from Arrow’s Theorem [3, 17, 31] imply that the reac h of mec hanisms without transfers is seve r ely limited. The follo win g observ ation m otiv ates our w ork: c omputer systems typic al ly have the ability to arbitr arily r e duc e servic e qu ality . F or example, routing systems can dr op or dela y traffic (e.g. [8]), sc hedu ling proto cols can dela y the r elease of jobs (e.g. [6 ]), and computational pa ymen t sc hemes allo w a mec hanism to demand computational pa yments from agen ts (e.g., in spam-figh ting sys- tems [11, 10, 25]). 1 Suc h service degradation can b e used to align the p references of the agen ts with the so cial ob jectiv e, at a cost: these “p ayments” also de gr ade the so cial surplus. W e dev elop a fr amew ork f or the design and analysis of money-burning me chanisms —mec hanisms that can emplo y arb itrary pa yments and seek to m aximize the r esidual surplus , defin ed as th e total v alue to the participants of the chosen outcome min u s the su m of the (“burn t”) paymen ts. 2 Suc h mec hanisms must trade off the so cial cost of imp osing pa ym en ts with the abilit y to elicit priv ate information from participan ts and thereb y enable accurate surplu s-maximization. F or example, supp ose we in tend to aw ard one of t w o participant s acce ss to a netw ork. Assu me that the t wo agen ts h a ve v aluations (i.e., maximum willingness to pa y) v 1 and v 2 ≤ v 1 for acquiring access, and that these v aluations are p riv ate (i.e. , u nknown to th e mec hanism d esigner). The Vickr ey or se c ond-pric e auction [34] w ould aw ard access to agent 1, charge a pa ymen t of v 2 , and thereby obtain residual sur plus v 1 − v 2 . A lottery w ould a ward access to an agent c hosen at random, c harge nothing, and ac hiev e a (residual) surp lus of ( v 1 + v 2 ) / 2, a b etter r esult if and only if v 1 < 3 v 2 . Even in this trivial scenario, it is not clear ho w to define (let alone d esign) an optimal money-burn ing mec hanism. 3 Our goal is to rigorously answer the follo wing t wo questions: 1. What is the optimal m oney-burning mec hanism ? 1 Computational payment schemes do not need the infrastructure req uired by micropa yment sc h emes. On e can interpret our results as analyzing th e p ow er of computational paymen t s, whic h w ere first devised for sp am- fighting, in a general mechanis m design setting. 2 W e assume that v aluations and burnt pa yments are measured in the same units. In oth er w ords, there is a know n mapping b etw een decreased service qu alit y (e.g., additional d ela y ) and lost va lue (e.g., dollars). This mapping can b e differen t for different participan ts, b ut it must be pub licly known and map onto [0 , ∞ ). See Section 6 for furth er discussion. 3 Indeed, it follo ws from our results that in some settings lotteries are optimal (i.e., money- burning is useless); in others, V ic krey auctions are optimal; and sometimes, neith er is optimal. 2 2. Ho w m u c h more p o werful are mec hanisms with m onetary transfers than money-bu rning mec h- anisms? Our Results. Our fir st con tribu tion is to iden tify a general template f or prior-free (i.e., worst- case) optimal mec h anism design. The basic idea is to c haracterize the set of mec h anisms that are Ba y esian optimal for some i.i.d. distribution on v aluations, and then define a prior-free p erformance b enchmark that corresp onds to comp eting sim ultaneously w ith all of these on a fixed (w orst-case) v aluation profile. The template, whic h we detail b elow, is general and we exp ect it to apply in man y mechanism design settings b ey ond money-burn ing mec hanisms. Second, we c haracterize Ba yesian optimal money-bur ning mec hanisms—the incen tiv e-compatible mec hanisms with maxim um-p ossible exp ected residual su rplus. O ur charac terization applies to gen- eral single-parameter agen ts, meaning th at the pr eferences of eac h agen t is n aturally summarized b y a single r eal-v alued v aluation, w ith in dep end en t bu t not necessarily iden tically distrib uted v al- uations. The c haracterization u nifies resu lts in the economics literature [5, 26] and also extends them in t wo imp ortant directions. First, the results in [5, 26] concern only multi-unit au ctions, where k ident ical units of an item can b e allocated to agent s who eac h d esire at most one unit. Our c haracterization applies to the general, p ossibly asymmetric, setting of single-parameter agen ts; for example, agen ts could b e s eeking disjoint paths in a m u lticommod it y netw ork. 4 In addition, for m u lti-unit auctions, w e giv e a simple description of the optimal mec hanism ev en when the “hazard rate” of the v aluation d istribution is not monotone in either direction. T h is imp ortant case is the most tec h nically interesting and c hallenging one, and it has n ot b een considered in d etail in the literature. Third, for multi-unit auctions, we d esign a mec hanism that is app ro ximately optimal in the w orst case. W e derive our b enc h mark using our charac terization of Ba yesian optimal m ec hanisms restricted to i.i.d. v aluations and symmetric mechanisms. W e p ro ve that suc h m ec hanisms are alw a ys w ell approxi mated b y a k -u ni t p -lottery , defined as follo w s: order the agen ts rand omly , sequen tially mak e eac h agen t a take- it-or-lea v e-it offer of p , and stop after either k items h av e b een allo cated or all agen ts h a ve b een considered. This result reduces the design of a constan t- appro x im ation prior-free money-bur n ing mec hanism to the prob lem of appro ximating th e residual surplu s ac hieve d by the optimal k -unit p -lottery . Our pr ior-free mec hanism obtains a constan t appro x im ation of this b enchmark us in g rand om sampling to select a goo d v alue of p . Surprisin gly , w e accomplish th is eve n when k is very small (e.g., k = 1). 5 Our b enchmark definition ensures that suc h a guaran tee is strong: for exa mple, if v aluations are dra wn from some unkno wn i.i.d. distribu- tion F , our mechanism obtains a constant fraction of the exp ected residu al su rplus of an optimal mec hanism tailored sp ecifically for F . Finally , for multi-unit auctions, we pro vide a p rice-of-anarc h y-t yp e analysis that measur es the so cial cost of bu rnt p aymen ts. Recall that the full s u rplus is ac hiev able w ith monetary transfers using the Vic krey-Clark e-Gro v es (V CG) mec hanism. W e p ro ve th at the largest-p ossib le relativ e loss in su rplus due to money-bur ning is pr ecisely logarithmic in the num b er of participan ts, in b oth the Ba yesia n and worst-c ase settings. In deed, our near-optimal mon ey-bu rning mec hanism alw a ys obtains residu al surp lus within a loga rithmic factor of the f u ll surplu s. T his result suggests that the 4 Multi-unit auctions mo del symmetric situations, as when eac h agen t seeks a path from a common source s to a common d estination t ; here, the num b er k of units equals th e number of edges in a minimum s - t cut. 5 Previous exp erience with p rior-free mechanism design, e.g. , for digital goo ds, suggests that conditions like “tw o or more winners” might b e necessary to achiev e a constan t appro ximation. S ee Section 6 for further discussion. 3 cost of implement ing money-bur ning (e.g., computational pa ym en ts) rather than general tran s fers (e.g., micropaymen ts) in a system is relativ ely mo dest. F urther, our p ositiv e result con trasts w ith the linear lo wer b oun d that w e pr o ve on the fraction of the fu ll su rplus obtainable b y mec hanisms without an y kind of pa ym en ts. A T emplate for Prior-F ree Auction Design. The f ollo wing template forges an explicit con- nection b et w een the Ba yesia n analysis of Ba y esian optimal mechanism design, the d ominan t ap- proac h in economics, and the w orst-case analysis of prior-free optimal mec hanism design, the ubiq- uitous approac h in theoretical computer science. Its goal is to fill a fu ndamenta l gap in p rior-free optimal mec hanism design metho dology: the selection of an appropriate p erform ance b enc h mark. 1. Characterize the Ba y esian optimal mechanism for ev ery i.i.d. v aluation distribution. 2. In terp ret the b ehavio r of the sy m metric, ex p ost incentiv e compatible, Ba yesian optimal mec hanism for ev ery i.i.d. distribution on an arbitrary v aluation pr ofile to giv e a distribu tion- indep end en t b enchmark . 3. Design a single ex p ost incenti v e compatible m ec hanism that appro ximates the abov e ben c h- mark on ev ery v aluation profile; the p erformance ratio of suc h a mec hanism pro vides an up p er b ound on that of the optimal prior-free mec hanism. 4. Obtain low er b ounds on the b est p erformance ratio p ossible in this fr amew ork b y exhibiting a distribution ov er v aluations su c h that the r atio b et w een the exp ected v alue of the b enc hm ark and the p erformance of the Ba y esian optimal mec hanism f or th e giv en distribution is large. In h indsight, this appr oac h has b een employ ed imp licitly in the con text of (profit-maximizing) digital goo d auctions [19, 18]. Ho we ver, the simplicit y of th e d igital go o d auction p roblem obs cures the imp ortance of the first t wo steps, as the Ba y esian optimal digital go o d auction is trivial: offer a p osted price. F or money-burnin g mec hanisms, the b enc h mark w e ident ify in Step 2 is not a priori ob vious. F urther Relat ed W ork. McAfee and McMillan s tudy collusion among bidd ers in m u lti-unit auctions [26]. In a we ak c artel , wh ere the agents w ish to m aximize the cartel’s total utilit y b ut are n ot able to mak e sid e paymen ts amongst themselv es, paymen ts made to the auctioneer are effectiv ely burnt. T he optimization and incen tive problem faced b y th e grand coalition in a m ulti- unit auction is similar to the auctio neer’s pr ob lem in our m on ey-bu rning setting; therefore, results for w eak cartels follo w from similar analyses to ours [26, 9]. Our c haracterization of Ba y esian optimal money-burn in g mechanisms b uilds on analysis to ols dev elop ed for profit maximization in Ba yesia n settings (see Myerson [28] and Riley and Samuel- son [29]) th at apply in general single parameter settings (see, e.g., [22]). Ind ep endently from our w ork, Chakra v art y and Kaplan [5] describ e the optimal Ba yesia n auction in multi -unit mon ey- burn in g settings. Our work extends this analysis to general s ingle-parameter agent settings with explicit fo cus on the case where the hazard rate is not monotone in either direction. Our p ap er is the first to study the relativ e p o wer of money-burning mec hanisms and mec hanisms with or with ou t transfers. It is also the fir st to consider prior-free money-bur n ing mec hanisms. Our results that quan tify the b en efi t of transfer s ha v e analog s in the pr ice of anarc hy literature, sp ecifically in the standard (nonatomic) mo del of selfish routing (e.g. [30]). Namely , full efficiency is 4 ac hiev able in this mo del with general transfers, in the form of “congestion pr ices”; without transfers the ou tcome is a Nash equilibr ium, with inefficiency m easur ed by the price of anarc hy; and with burnt transfers (“sp eed bum ps” or other artifi cial delays) it is generally p ossible to reco ver some but not all of the efficiency loss at equilibrium [8]. There are seve ral other s tudies that view transfers to an auctioneer as un desirable; how ev er, these works are tec hnically unr elated to ours. W e already noted recen t w ork on incen tive -compatible in terd omain routing without pa yments [15, 24 ]. Moulin [27] and Guo and Con itzer [21] indep en- den tly stud ied ho w to redistrib u te the paymen ts of th e VCG mec hanism in a multi-unit auction among the participants (using general transfer s ) to minimize the total pa ymen t to the auctioneer. Finally , as already mentio n ed, our prior-free tec hniques are related to r ecen t work on profit max- imization (e.g., [19, 16, 4]) and there is a related literature on the problem of cost m inimization, a.k.a. frugalit y (e.g., [2, 33, 12, 23]). 2 Ba y esian Optimal Money Burning In th is section w e s tudy optimal money-bu rning mec hanism design from a standard economics viewp oint , where agen t v aluations are dra w n fr om a known prior distribution . Th is will complete the first step of our template for prior-free optimal mec hanism design. Mec hanism design basics. W e consider mec h anisms th at p ro vide a go o d or service to a subset of n agen ts. The outcome of such a mec hanism is an al lo c ation ve ctor , x = ( x 1 , . . . , x n ), where x i is 1 if agen t i is ser ved and 0 otherwise, and a p ayment ve ctor , p = ( p 1 , . . . , p n ). In this p ap er, the p aymen t p i is the amount of money that agen t i must “bur n ”. W e allo w the set of feasible allocation v ectors, X , to b e constr ained arbitrarily; f or example, in a m ulti-un it au ction with k iden tical units of an item, the f easible allocation v ectors are th ose x ∈ X with P i x i ≤ k . W e assume that eac h agen t i is risk-neutr al , h as a p r iv ately kn o wn v aluation v i for receiving service, and aims to maximize their (quasi-linear) utilit y , defined as u i = v i x i − p i . W e denote the valuation pr ofile b y v = ( v 1 , . . . , v n ). Our mec hanism design ob jectiv e is to maximize the r esidual surplus , defined as X i ( v i x i − p i ) for a v aluation pr ofile v , a feasible allo cation x , and pa ymen ts p . If th e pa yments were transferred to the s eller then the r esulting so cial surplus wo uld b e P i v i x i ; ho we v er, in our setting the pa yments are burnt and the social surplus is equal to the residu al surplus. Ba yesian mec hanism design basics. In this section, w e assume that the agen t v aluations are dra w n i.i.d. from a p ublicly kno wn distribu tion with cumulativ e distrib ution fu nction F ( z ) and probabilit y densit y function f ( z ). W e let F denote the joint (pro duct) distrib ution of agen t v alues. See Section 6 for a generalization to general pro du ct d istributions. W e consider the problem of implemen tation in Ba yes-N ash equilibrium. Agen t i ’s strategy is a mapping fr om th eir priv ate v alue v i to a course of actions in the m echanism. T he distribution on v aluations F and a s tr ategy profi le induce a distribution on agen t actions. These agent actions are in Bayes-Nash e q u ilibrium if no agen t, giv en their o wn v aluation and the distribution on other agen ts’ actions, can improv e its exp ected pay off via alternativ e actions. By the r evelation principle [28], 5 w e can restrict our atten tion to s ingle roun d, sealed bid, dir e ct mec hanisms in which truthtel ling , i.e., submitting a bid b i equal to the p riv ate v alue v i , is a Ba y es-Nash equ ilibrium. It will turn out that th ere is alw a ys an optimal mec h anism that is n ot on ly Ba y esian incentiv e compatible b ut also dominant str ate gy inc entive c omp atible , meaning tru thtelling is an optimal agen t strateg y for ev ery strategy profile of the other agen ts. An al lo c ation rule , x ( v ), is th e mapping (in the tru th telling equilibrium) from age n t v alua- tions to the outcome of the mec han ism . Similarly the p ayment rule , p ( v ), is the mapp ing from v aluations to p a yments. Giv en an allo cation rule x ( v ), let x i ( v i ) b e the pr obabilit y that agen t i is al- lo cated when its v aluation is v i (o v er th e probabilit y distribution on the other agen ts’ v aluations): x i ( v i ) = E v − i [ x i ( v i , v − i )] . S imilarly define p i ( v i ). Po sitiv e transfers from the m ec hanism to the agen ts are n ot allo w ed and w e requ ire ex in terim individual rationalit y (i.e., that non -p articipation in the mec h an ism is an allo wable agen t strategy). The follo wing lemma is the standard charact eri- zation of the allo cation rules implemen table by Ba y esian incen tiv e-compatible mec hanism s and the accompan ying (uniquely defined) pa y m en t rule. Lemma 2.1 [28] Every Bayesian inc entive c omp atible me chanism satisfies: 1. Al lo c ation monotonicity: for al l i and v i > v i ′ , x i ( v i ) ≥ x i ( v i ′ ) . 2. P ayment identity: for al l i and v i , p i ( v i ) = v i x i ( v i ) − R v i 0 x i ( v ) dv . Virtual v aluations. Ass ume for simplicit y that th e d istribution F has sup p ort [ a, b ] and p ositiv e densit y th roughout this inte rv al. My erson [28] defin ed “virtual v aluations” and sho w ed that they c haracterize the exp ected pa ymen t of an agen t in a Ba yesia n incen tive compatible mec h anism. Definition 2.2 (virtual v aluation for paymen t [28]) If agent i ’s valuation is distribute d ac- c or ding to F , then its virtual v aluation for paymen t is ϕ ( v i ) = v i − 1 − F ( v i ) f ( v i ) . Lemma 2.3 [28] In a B ayesian inc entive-c omp atible me chanism with al lo c ation rule x ( · ) , the ex- p e cte d p ayment of agent i satisfies E v [ p i ( v )] = E v [ ϕ ( v i ) x i ( v )] . My erson uses this corresp ondence to d esign optimal mec hanism s for profit-maximization. Th e optimal mec hanism for a giv en distribution is the one th at maximizes the v i rtual surplus (for pa yment). Definition 2.4 (virtual surplus) F or virtual valuation function ϕ ( · ) and valuations v , the vir- tual surp lus of al lo c ation x is X i ϕ ( v i ) x i . Our ob jectiv e is to maximize the residual surplus, P i ( v i x i ( v ) − p i ( v )), whic h w e can do quite easily us ing virtual v aluations. T o justify our termin ology , b elo w, notice that an age n t’s u tilit y is u i ( v ) = v i x i ( v ) − p i ( v ), and our ob jectiv e of residu al surplu s maximization is simply that of maximizing the exp ected utilit y of the agen ts, E v [ P i u i ( v )] . W e defin e a virtual valuation for utility b y s imply p lugging in the virtual v aluation for paymen ts in to th e equ ation that d efines utilit y . 6 Definition 2.5 (virtual v aluation for ut ilit y) If agent i ’s valuation is distribute d ac c or ding to F , then its virtual v aluation for u tilit y is ϑ ( v i ) = 1 − F ( v i ) f ( v i ) . This quantit y is also kno w n as th e “information r en t” or “in verse hazard rate function”. T reating it as a v ir tual v aluation of sorts, we can generalize the theory of optimization by virtual v aluations, b eginning with the follo win g lemma. Lemma 2.6 In a Bayesian inc entive-c omp atible me chanism with al lo c ation rule x , the exp e cte d utility of agent i satisfies E v [ u i ( v )] = E v [ ϑ ( v i ) x i ( v )] . W e can conclude from this that th e Bay esian optimal mec han ism s for residu al surplu s are precisely those that maximize the exp ected virtual su rplus (for utilit y) sub ject to feasibilit y and monotonicit y of the allo cation rule. In other words, w e should choose a feasible allo cation v ector x ( v ) to maximize P i ϑ ( v i ) x i ( v ) for eac h v , sub ject to m on otonicit y of x i ( v i ). I t is easy to see that if ϑ ( · ) is monotone non -d ecreasing in v i , then c h o osing x ( v ) ∈ argmax x ′ ∈X X i ϑ ( v i ) x i ′ results in a monotone allo cation rule. Un fortunately ϑ ( · ) is often not mon otone non -d ecreasing; indeed, under the standard “monotone hazard rate” assum ption, d iscussed f u rther b elo w, ϑ ( · ) is monotone in the wr ong dir e ction . Ironing. W e n ext generalize an “ironing” pro cedure of Myerson [28] that transforms a p ossib ly non-monotone virtual v aluation function in to an i r one d virtual valuation function that is monotone; the optimization approac h of the p revious p aragraph can then b e app lied to these ironed f u nctions to obtain a monotone allo cation rule. F urth er, the ironin g pro cedure preserve s the target ob jectiv e, so that an optimal allo cation for the ironed vir tual v aluations is equal to the optimal monotone allocation for the original virtu al v aluations. Definition 2.7 (ironed virtual v aluations [28]) Given a distribution function F ( · ) with virtual valuation (for utility) function ϑ ( · ) , the ironed vir tual v aluation fu nction , ¯ ϑ ( · ) , is c onstructe d as fol lows: 1. F or q ∈ [0 , 1] , define h ( q ) = ϑ ( F − 1 ( q )) . 2. D e fine H ( q ) = R q 0 h ( r ) dr . 3. D e fine G as the c onvex hul l of H — the lar gest c onvex function b ounde d ab ove by H for al l q ∈ [0 , 1] . 4. D e fine g ( q ) as the derivative of G ( q ) , wher e define d, and extend to al l of [0 , 1] by right- c ontinuity. 5. Final ly, ¯ ϑ ( z ) = g ( F ( z )) . 7 Step 4 of Definition 2.7 mak es sense b ecause G is con vex fun ction. Conv exit y of G also implies that g , and hence ¯ ϑ , is a monotone non -d ecreasing function. The pro of My erson giv es for ironing virtu al v aluations for paymen ts extends simply to any other k in d of virtual v aluation in clud ing our virtu al v aluations for utilit y . W e summ arize this in Lemma 2.8 with a pr o of in App endix A. Lemma 2.8 L et F b e a distribution function with virtual valuation function ϑ ( · ) and x ( v ) a mono- tone al lo c ation rule. Define G , H , and ¯ ϑ as in D efinition 2.7. Then E v [ ϑ ( v i ) x i ( v )] ≤ E v  ¯ ϑ ( v i ) x i ( v )  , (1) with e qu ality holding if and only if d dv x i ( v ) = 0 whenever G ( F ( v )) < H ( F ( v )) . Our main theorem now follo ws easily . Theorem 2.9 L e t F b e a distribution function with vi rtual v aluation function ϑ ( · ) . De fine G , H , and ¯ ϑ as in D efinition 2.7. F or valuation pr ofiles dr awn fr om distribution F , the me chanisms that maximize the exp e cte d r esidual surplus ar e pr e cisely those satisfying 1. x ( v ) ∈ argmax x ′ ∈X P i ¯ ϑ ( v i ) x i ′ for every v ; and 2. for al l i , d dv x i ( v ) = 0 whenever G ( F ( v )) < H ( F ( v )) . Pro of: Fi rst, there exists a mec h anism that satisfies b oth of the desired prop erties. T o see this, consider an allo cation rule th at maximizes P i ¯ ϑ ( v i ) x i ( v ) for ev ery v . Su c h a r ule can without loss of generalit y b e a f unction only of ¯ ϑ ( v i ) an d not of v i directly . A t p oin ts v where G ( F ( v )) < H ( F ( v )), G is locally linear (since it is the con ve x h u ll of H ) and hence ¯ ϑ ( v ) is lo cally constan t. Thus suc h an allocation rule will satisfy d dv x i ( v ) = 0 at all s u c h p oin ts (for all i ). A mechanism that meets b oth conditions sim u ltaneously maximizes the righ t-hand side of (1) while satisfying the inequ ality with equality . Lemmas 2.6 and 2.8 imply that su c h a mechanism maximizes the exp ected residual surp lus and , con versely , that all optimal mec hanisms m ust meet b oth conditions. ✷ Theorem 2.9 shows that maximizing the ironed virtual surp lus (for utilit y) is equiv alen t to maximizing exp ected resid u al surplu s s u b ject to incentiv e-compatibilit y . Differen t tie-breaking rules can yield different optimal mec h an ism s. With sym m etric participan ts (that is, i.i.d. v aluations) and a sy m metric feasible region (e.g., k -item auctions), it is natural to consider symmetric mec hanism s, and these will p la y a crucial role in ou r b enc hm ark for prior-free mon ey-bu rning mec h anisms (see Definition 3.1). In t e rpre t ation. T o in terpret Th eorem 2.9, r ecall that the hazar d r ate of d istribution F at v is defin ed as f ( v ) 1 − F ( v ) . The monotone hazar d r ate (MHR) assu m ption is th at the hazard r ate is monotone non-decreasing an d is a standard assumption in mec h anism design (e.g. [28]). W e will analyze this stand ard setting (MHR), the setting in whic h the hazard r ate is monotone in the opp osite sense (an ti-MHR), and the setti ng where it is n either monotone increasing n or d ecreasing (non-MHR). Notice that the hazard rate function is pr ecisely th e recipro cal virtual v aluation (for utilit y) function. Our int erpretation is su mmarized b y Figur e 1. When the v aluation distribu tion satisfies the MHR condition, the ironed virtual v aluations (for utilit y) ha ve a sp ecial form: they are constan t w ith v alue equ al to their exp ectation. 8 MHR nonMHR an tiMHR (e.g., uniform) (e.g., bimo dal) (e.g., sup er-exp onentia l) ¯ ϑ ( v ) ¯ ϑ ( v ) ¯ ϑ ( v ) Lottery is optimal. indirect Vic kr ey is optimal. Vic krey is optimal. Figure 1: Ironed virtual r esidual surplu s in the three cases. Lemma 2.10 F or every distribution F that satisfies the monotone hazar d r ate c ondition, the ir one d virtual valuation (for utility) fu nction is c onstant with ¯ ϑ ( z ) = µ , wher e µ denotes the exp e cte d value of the distribution. Pro of: Apply the ironing pro cedure from Definition 2.7 to ϑ ( z ). The monotone hazard rate condition implies that ϑ ( z ) is monotone non-increasing. Since F ( z ) is m onotone non -d ecreasing so is F − 1 ( q ) for q ∈ [0 , 1]. Thus, h ( q ) = ϑ ( F − 1 ( q )) is monotone n on-increasing. The integral H ( q ) of the monotone non-increasing fu nction h ( q ) is conca ve. Th e con v ex hull G ( q ) of the conca ve function H ( q ) is a str aight line. In particular, H ( q ) is defined on the range [0 , 1], so G ( q ) is the straigh t line b etw een (0 , H (0)) and (1 , H (1)). Th us, g ( q ) is the deriv ativ e of a straight line and is therefore constan t with v alue equal to the line’s slop e, n amely H (1 ). Thus, ¯ ϑ ( z ) = H (1). It remains to sho w that H (1) = µ . By definition, H (1) = Z 1 0 ϑ ( F − 1 ( q )) dq . Substituting q = F ( z ), dq = f ( z ) dz , and the supp ort of F as ( a, b ), w e ha v e H (1) = Z b a ϑ ( z ) f ( z ) dz . Using the definition of ϑ ( · ) and the definition of exp ectation for non-n egativ e random v ariables giv es H (1) = Z b a (1 − F ( z )) dz = µ. ✷ Therefore, u nder MHR the mec hanism that maximizes the ironed virtual surplus is the one that maximizes the ex ante exp e cte d surplus , without asking for bids and without any transfers. F or example, in a m ulti-unit auction with i.i.d. bidders, all agen ts are equal ex ante , and th us any allocation ru le th at ignores the bid s and alw a ys allocates all k units (c harging nothing) is optimal. 9 Corollary 2.11 F or agents with i.i.d. valuations satisfying the MHR c ondition, an optimal (sym- metric) money-burning me chanism for al lo c ating k uni ts is a k -unit lottery. Supp ose the distrib ution satisfies the ant i-MHR condition whic h implies that the virtual v al- uation (for utilit y) fun ctions are monotone non-decreasing. T he ironed virtual v aluation fun ction is then identica l to the virtual v aluation fu nction. The i.i.d. assumption implies that all agent s ha ve the same virtu al v aluation function, s o the agen ts with the highest virtu al v aluations are also the agents with the highest v aluations. Therefore, an optimal money-burnin g mec hanism for allocating k u nits assigns the units to the k agen ts with the h ighest v aluations. 6 This is precisely the allo cation rule used b y the k -unit Vic krey auctio n [34], so the truth telling pa yment rule is that all winners pa y the k + 1st highest v aluation. Corollary 2.12 F or agents with i.i.d. valuations satisfying the anti-MHR c ondition, an optimal (symmetric) money-burning me chanism for al lo c ating k units is a k - unit V ickr ey auction. T o optimally allocate k u nits of an item in the non-MHR case, we simp ly a w ard the items to the agen ts with the largest ironed virtual v aluations (for utilit y). Ironed virtual v aluations are constan t o ve r regions in w h ic h non-trivial ironing tak es place, resulting in potentia l ties among pla y ers with distinct v aluations. The allo cation rule of an optimal m ec hanism cannot change ov er ironed regions (Lemma 2.8), so w e cannot break ties among iron ed virtu al v aluations in fa vor of agen ts with higher v aluations. W e can break these ties arb itrarily (e.g., based on a p redetermined total ordering of the agen ts) or ran d omly . In either case the optimal mec hanism can b e describ ed su ccinctly as an indir e ct generalization of the k -unit Vickrey auction where the bid space is restricted to b e in terv als in whic h the ironed virtu al v aluation fu nction is strictly increasing. T he k agen ts with the h ighest bids win and ties are b r ok en in a predetermined wa y . Pa ymen ts in this mechanism are giv en by Lemma 2.1 and are describ ed in more detail for this case in the next section. Corollary 2.13 F or agents with i.i.d. non-M HR valuations, an optimal (symmetric) money-burning me chanism for al lo c ating k units is an indir e ct k -u nit Vickr ey auction: for valuations in the r ange R = [ a, b ] and subr ange R ′ ⊂ R on which ¯ ϑ ( v ) has p ositive slop e, i t is the indir e ct me chanism wher e agents b i d b i ∈ R ′ and the k agents with the highest bids win, with ties br oken uniformly at r andom. 3 Prior-F ree Money-Bu rning Mec h anism Design W e n o w depart from the Ba y esian sett ing and design near-optimal prior-free mec hanisms for multi- unit auctions. Section 3.1 corresp onds to the second step in our pr ior-free mechanism design template and lev erages our c haracterization of Ba yesia n optimal mec h anisms to identify a simple, tigh t, and d istribution-indep enden t p erformance b enchmark. Section 3.3 gives a p rior-free mec h- anism that, for ev ery v aluation pr ofile, obtains exp ected r esidual surplu s within a constan t factor of this b enc hmark. Th is mec hanism implemen ts the third step of our d esign template. W e con- sider lo wer b oun d s on the approximati on ratio of all prior-free mec h anisms (the fi nal s tep of the template) in Section 4. 6 Virtual val uations n eed not be strictly increasing, so tw o bidd ers with different va luations may hav e identi cal virtual v aluations. In the anti-MHR case, it is permissible to break ties in fa vor o f the agent with the highest v aluation. In the notation of Lemma 2.8, G = H throughout [0 , 1], so the tie-breaking ru le d oes not affect the exp ected residual surplus. 10 F or ease of discussion the p a yment rules we describ e in this section are f or mec hanism im p le- men tations that are domin an t strategy in cen tive compatible f or agen ts th at are r isk-neutral with resp ect to randomization in the mec hanism, i.e., m echanisms that are truthful in exp e ctation . All of these mec hanisms ha ve natural implementa tions with paymen t rules that make them dominant strategy incentiv e compatible for any fixed outcome of the mec han ism ’s rand om decisions, i.e., mec hanisms that are truthful al l the time . In the computer science literature, discussion of these distinctions can b e found in [1]. 3.1 A Performance Benchma rk for Prior -F ree Mecha nisms In tuitiv ely , our p erformance b enc hmark f or a v aluation profile is the maximum residual surplus ac hiev ed by a symmetric mec hanism that is optimal for some i.i.d. distrib u tion. The next definition formalizes the class of mechanisms that define the b enchmark. Definition 3.1 ( Opt F ) F or an i.i.d. distribution F with ir one d vi rtual valuation (for utility) func- tion ¯ ϑ , the me chanism Opt F is define d as fol lows. 1. Given v , cho ose a fe asible al lo c ation maximizing P i ¯ ϑ ( v i ) x i . If ther e ar e multiple such al lo- c ations, cho ose one uni f ormly at r andom. 2. L et x denote the c orr esp onding al lo c ation rule, with x i ( v ) denoting the pr ob ability that player i r e c eives an item given the valuation pr ofile v . L et p denote the (unique) p ayment rule dictate d by L emma 2.1. 3. Given valuations v and the r andom choic e of al lo c ation in the first step, char ge e ach winner i the pric e p i ( v ) /x i ( v ) and e ach loser 0. By Theorem 2.9, Opt F maximizes the exp ected r esidual sur plus for v aluations dra wn from F . Using Lemma 2.1, it is also incen tiv e-compatible and ex p ost ind ividually r ational. It is symmetric pro vid ed the set of feasible allocations is symmetric (i.e., is a k -item auction). In this case, the fir st step a w ard s the k items to the bidders with the top k ironed virtual v aluations (for u tilit y) with resp ect to the distr ib ution F , breaking ties un iformly at random. Our b enchmark is then: G ( v ) = sup F Opt F ( v ) , (2) where Op t F ( v ) d enotes the exp ected residu al surp lu s (o v er the c hoice of rand om allo cation) ob - tained by the mec h anism Opt F on the v aluation profile v . This b enchmark is, b y definition, distribution-indep endent . As suc h , it provides a yardstic k b y which w e can measur e prior-fr ee mec hanisms: we sa y that a (randomized) mec hanism β -appr oximates the b enchmark G if, for ev ery v aluation profi le v , its exp ected residual surplus is at least G ( v ) /β . Note the strength of this guar- an tee: for example, if a mec han ism β -appro ximates the b enchmark G , then on an y i.i.d. distribution it achiev es at lea s t a β fraction of the exp ected residual sur plus of ev ery mec h anism. Naturally , no prior-free mec hanism is b etter than 1-appr o ximate; we giv e s tr onger lo wer b ound s in Section 4. Remark. Restricting atten tion in Defin ition 3.1 to optimal mechanisms that use s y m metric tie- breaking ru les is crucial for obtaining a tracta ble b enchmark. F or example, when F is an i.i.d. distribution satisfying the MHR assump tion, Theorem 2.9 imp lies that every constant allo cation 11 rule that allocates all items (with zero p a yments) is optimal (recall Corollary 2.11). F or a single- item auction and a v aluation p rofile v , sa y w ith the fir st bid d er ha ving the highest v aluation, the mec h anism that alwa ys aw ards the go o d to the fi rst bidd er and charges nothing achiev es the full surplus. (Of course, this mec hanism h as extremel y p o or p erformance on man y other v aluation profiles.) As no incen tiv e-compatible money-bur ning mec h anism alw a ys ac hieves a constan t fraction of the full surp lus (see Prop osition 5.1), allo w ing arbitrary asymm etric optimal m echanisms to participate in (2) wo uld yield an u nac hiev able b enc h mark. 3.2 Multi-Unit Auctions and Tw o-Pr ice Lott eries The defin ition of G in (2 ) is meaningfu l in general single-parameter s ettings, bu t app ears to b e analyticall y tractable only in pr oblems with additional stru cture, symmetry in particular. W e next giv e a sim p le description of this b enc h m ark, and an ev en simp ler approxima tion of it, for multi -unit auctions. What d o es O p t F lo ok lik e f or suc h problems? When the distribution on v aluations satisfies the MHR assu mption, O pt F is a k -unit lottery (cf., Corollary 2.11). Under the an ti-MHR assumption, Opt F is a k -un it Vic krey auction (cf., Corollary 2.12). W e can view the k -unit Vic krey auction, ex p ost, as a k -unit v ( k +1) -lottery , wh ere v ( k +1) is the k + 1st highest v aluation, in the f ollo wing sense. Definition 3.2 ( k -unit p -lottery) The k -unit p -lottery , denote d Lot p , al lo c ates to agents with value at le ast p at pric e p . If ther e ar e mor e than k such agents, the winning agents ar e sele cte d uniformly at r andom. One natural conjecture is that, ex p ost, the outcome of eve r y mechanism of th e form Opt F on a v aluation profile v lo oks lik e a k -un it p -lottery for some v alue of p . F or non-MHR distributions F , ho wev er, O pt F can assume the more complex form of a t wo -p rice lottery , ex p ost. Definition 3.3 ( k -unit ( p, q ) -lottery) A k -unit ( p, q )-lottery , denote d Lot p,q , is the f ol lowing me chanism. L et s and t denote the numb er of agents with bid in the r ange ( p, ∞ ) and ( q , p ] , r e sp e ctively. 1. If s ≥ k , run a k -unit p -lottery on the top s agents. 2. If s + t ≤ k , sel l to the top s + t agents at pric e q . 3. O therwise, run a ( k − s ) -unit q -lottery on the agents with bid in ( q , p ] and al lo c ate e ach of the top s agents a go o d at the pric e dictate d b y L emma 2.1: k − s +1 t +1 q + s + t − k t +1 p. W e n o w p r o ve that for ev ery i.i.d. distrib ution F and ev ery v aluation pr ofile v , the mec h anism Opt F results in an outcome and p a yments that, ex p ost, are iden tical to those of a k -unit ( p, q )- lottery . Lemma 3.4 F or every valuation pr ofile v , ther e is a k - unit ( p, q ) -lottery with exp e cte d r esidual surplus G ( v ) . Pro of: By defin ition (2), w e only need to sho w that, for ev ery i.i.d. distribution F and v aluation profile v , O pt F ( v ) has the same outcome as a k -unit ( p, q )-lott ery . 12 Fix F and v , and assume that v 1 ≥ · · · ≥ v n . Thus, ¯ ϑ ( v 1 ) ≥ · · · ≥ ¯ ϑ ( v n ). Recall by Definition 3.1 that Opt F maximizes P i ¯ ϑ ( v i ) x i and breaks ties randomly . Define S = { i : ¯ ϑ ( v i ) > ¯ ϑ ( v k +1 ) } , T = { i : ¯ ϑ ( v i ) = ¯ ϑ ( v k +1 ) } , s = | S | , and t = | T | . Assume we are in the more tec hnical case th at 0 < s < k < s + t (the other cases follo w from similar arguments). It is easy to see that Op t F assigns a unit to eac h bidder in S and allocates the remaining k − s units randomly to bidders in T . Let q = inf { v : ¯ ϑ ( v ) = ¯ ϑ ( v k +1 ) } and p = inf { v : ¯ ϑ ( v ) > ¯ ϑ ( v k +1 ) } . The allo cation is thus iden tical to a k -unit ( p, q )-lottery . It remains to show th at the pa yments are correct. Let x i ( · ) b e as in Defin ition 3.1. Consider agen t i ∈ T . If i bids b elo w q then i loses, wh ile if i bids at least q then i wins with the same p robabilit y as when i bids v i . Therefore, x i ( v ) for v ≤ v i is step fu nction at v = q . Thus, p i ( v i ) = v i x i ( v i ) − R v i 0 x i ( v ) dv = q x i ( v i ) and i ’s pa ym ent on winning is p i ( v i ) /x i ( v i ) = q , as in the k -un it ( p, q )-lottery . Now consider an agen t i ∈ S . If i we re to bid v < q , i w ould lose, i.e., x i ( v ) = 0. If i w ere to bid v ∈ [ q , p ) then i w ould lea ve the set S of agen ts guaran teed a unit, and wo uld join the set T , making t + 1 age n ts who would share s − k + 1 remaining items by lottery . In this case, x i ( v ) = s − k +1 t +1 . Of course, x i ( v ) = 1 w h en v > p . As x i ( · ) is identic al to the allo cation f u nction for agen t i in the k -u nit ( p, q )-lottery , th e pa ymen ts are also iden tical. ✷ As w e hav e seen, mec hanisms of the form Opt F can pro du ce outcomes not equiv alen t to that of a single-price lottery . Our next lemma sho ws that k -unit p -lotteries giv e 2-appro xim ations to k -u nit ( p, q )-lotte ries. This allo ws us to relate the p erformance of single-price lotteries to our b enc hmark (Corollary 3.6), whic h w ill b e u seful in our construction of an appro ximately optimal pr ior-fr ee mec hanism in the next section. Lemma 3.5 F or every valuation pr ofile v and p ar ameters k , p , and q , ther e is a p ′ such that the k -unit p ′ -lottery obtains at le ast half of the exp e cte d r esidual surplus of the k -unit ( p, q ) - lottery. Pro of: W e p ro ve the lemma by sh o wing that Lot p,q ( v ) ≤ Lot p ( v ) + Lot q ( v ). W e argue the stronger statemen t that eac h agen t enjo ys at least as large a com bined exp ected utilit y in Lot p ( v ) and Lot q ( v ) as in Lot p,q ( v ). Let S and T denote the agen ts with v alues in the ranges ( p, ∞ ) and ( q , p ], resp ectiv ely . Let s = | S | and t = | T | . Assu m e that 0 < s < k < s + t as otherwise the k -unit ( p, q ) lottery is a single-price lottery . Each age n t in T participates in a k -u nit q -lottery in Lot q and only a ( k − s )-unit q -lottery in Lot p,q ; its exp ected u tilit y can only b e smaller in the second case. No w consider i ∈ S . W riting r = ( k − s + 1) / ( t + 1), we can upp er b ound the utilit y of an agen t i in Lot p,q b y v i − r q − (1 − r ) p = (1 − r )( v i − p ) + r ( v i − q ) ≤ ( v i − p ) + k s + t · ( v i − q ) , whic h is the com bined exp ected utilit y that the agen t obtains f rom participating in b oth a k -unit p -lottery (with s < k ) and a k -unit q -lottery . ✷ Corollary 3.6 F or every valuation pr ofile v , ther e is a k -unit p -lottery with exp e c te d r esidual surplus at le ast G ( v ) / 2 . 3.3 A N ear-Optimal Prior-F ree Money-Burning Mec hanism W e n o w giv e a prior-free mec hanism that O (1)-a pproximat es the b enc hmark G . Th is mechanism is motiv ated by the follo wing observ ations. First, by Corollary 3.6, our mechanism only needs to comp ete with k -u n it p -lotteries. Second, if many agen ts make significant con tribu tions to the 13 optimal r esid ual surp lu s, then we can use r andom sampling tec hniques to approximate th e optimal k -unit p -lottery . Third, if a few agen ts are single-handedly resp onsible f or the residual surp lu s obtained b y the optimal k -unit p -lottery , then th e k -un it Vic krey auction obtains a constan t fraction of the optimal residual s urplus. The precise mec hanism is as follo ws. Definition 3.7 (Random Sampling Opt ima l Lottery ( RSO L )) With a set S = { 1 , . . . , n } of n agents and a supply of k identic al units of an item, the Rand om Samplin g Optimal Lottery (RSOL) is the fol lowing me chanism. 1. Cho ose a subse t S 1 ⊂ S of the agents uniformly at r andom , and let S 2 denote the r est of the agents. L et p 2 denote the pric e char ge d by the optimal k -unit p -lottery for S 2 . 2. Wi th 50% pr ob ability, run a k -unit p 2 -lottery on S 1 . 3. O therwise, run a k - u nit V ickr ey auction on S 1 . W e ha v e delib erately a vo ided optimizing this m echanism in order to keep its description and analysis as simple as p ossible. Theorem 3.8 RSOL O (1) -app r oximates the b enchmark G . In our pro of of Theorem 3.8, w e use the follo wing “Ba lanced Samp ling Lemma” of F eige et al. [13] to con trol the similarit y b et ween the random sample S 1 c hosen b y RSOL and its complemen t S 2 . Lemma 3.9 (Balanced Sampling Lemma [13]) L et S b e a r andom subset of { 1 , 2 , . . . , n } . L et n i denote | S ∩ { 1 , 2 , . . . , i }| . Then Pr  n i ≤ 3 4 i for al l i ∈ { 1 , 2 , . . . , n }   n 1 = 0  ≥ 9 10 . Pro of: (of T heorem 3.8). Fix a v aluation p rofile v with v 1 ≥ · · · ≥ v n and a supply k ≥ 1. F or clarit y , w e m ak e n o attempt to optimize the constan ts in the follo wing analysis. W e analyze th e p erform ance of RSOL only when certain sampling ev en ts o ccur. F or i = 1 , 2, let E i denote the ev en t that agen t i is includ ed in the set S i . Clearly , Pr [ E 1 ∩ E 2 ] = 1 / 4. Conditioning on E 1 ∩ E 2 , let E 3 denote the ev en t that the Bala n ced Samplin g Lemma holds for the sample S 1 \{ 1 } wh en viewed as a s ubset of { 2 , 3 , . . . , n } . Similarly , let E 4 denote the ev ent that the Balanced Sampling Lemma holds for the samp le S 2 \{ 2 } w hen viewed as a sub set of { 1 , 3 , . . . , n } . By the Principle of Deferred Decisions and the Union Bound, Pr [ E 3 ∩ E 4 |E 1 ∩ E 2 ] ≥ 4 / 5. Hence, Pr  ∩ 4 i =1 E i  ≥ 1 / 5. W e prov e a b ound on the appro ximation ratio conditioned on the ev ent ∩ 4 i =1 E i ; since the mec hanism alwa ys has nonnegativ e residual surp lu s, its unconditional appro ximation r atio is at most 5 times as large. Let n i and ¯ n i denote | S 1 ∩ { 1 , 2 , . . . , i }| and | S 2 ∩ { 1 , 2 , . . . , i }| , r esp ectiv ely . Since the ev ent ∩ 4 i =1 E i holds, we h a ve n i , ¯ n i ∈  1 6 i, 5 6 i  (3) for ev ery i ∈ { 2 , 3 , . . . , n } , and also n 1 = 1 and ¯ n 1 = 0. By Corollary 3.6, w e only need to sh o w that the exp ected residual surplus of the mec hanism is at least a constan t fraction of that of the optimal k -un it p -lottery for v . F or a subset T of agent s and a price p , let W ( T , p ) denote the resid u al sur plus of the k -unit p -lottery f or T . Letting n T i 14 denote | T ∩ { 1 , 2 , . . . , i }| and d i denote v i − v i +1 for i ∈ { 1 , 2 , . . . , n } (interpreting v n +1 = 0), for ev ery ℓ w e obtain the follo wing u seful ident it y: W ( T , v ℓ +1 ) = min { k , n T ℓ } n T ℓ   X i ∈ T ∩{ 1 ,...,ℓ } v i   − min { k , n T ℓ } · v ℓ +1 = min { k, n T ℓ } n T ℓ ℓ X i =1 n T i d i . ( 4) Let v ℓ ∗ +1 denote the optimal price for a k -unit p -lottery for v , and note that ℓ ∗ ≥ k . By (4), the residual sur plus of this optimal lottery is W ( S, v ℓ ∗ +1 ) = k ℓ ∗ ℓ X i =1 id i . T o analyze the expected residu al s urplus of RS OL, firs t supp ose that it executes a k -unit p 2 -lottery where p 2 = v m +1 for some m . W e then ha v e W ( S 2 , p 2 ) ≥ W ( S 2 , v ℓ ∗ +1 ) = min { k , ¯ n ℓ ∗ } ¯ n ℓ ∗ ℓ ∗ X i =1 ¯ n i d i ≥ k ℓ ∗ ℓ ∗ X i =2 i 6 d i ≥ W ( S, v ℓ ∗ +1 ) 6 − d 1 , where the first inequ alit y follo ws from the optimalit y of p 2 for S 2 , the fir st equalit y follo ws from (4), and th e second inequalit y follo ws from (3 ). O n the other hand , inequality (3) and a similar deriv a- tion sho ws that the p rice p 2 is nearly as effectiv e for S 1 : W ( S 1 , p 2 ) = min { k, n m } n m m X i =1 n i d i ≥  1 5 · min { k, ¯ n m } ¯ n m  m X i =1 ¯ n i 5 d i = W ( S 2 , p 2 ) 25 ≥ W ( S, v ℓ ∗ +1 ) 150 − d 1 . Finally , if the mec hanism executes a k -unit Vic krey auction for S 1 , then it obtains r esidual surplu s at least v 1 − v 2 = d 1 (since the fi rst agen t is in S 1 ). Ave r aging the residu al su rplus from the t wo cases prov es that RSOL O (1)-appro ximates G . ✷ W e can impro ve th e ap p ro ximation factor in T heorem 3.8 by more than an order of magnitud e b y mo difying RSOL and optimizing the pro of. O b taining an appro ximation f actor less than 10, sa y , app ears to require a different approac h. 4 Lo w er Bound s for Prior-F ree Mon ey-Burning Mec hanisms This sectio n esta blishes a lo wer b ound of 4 / 3 on the appro x im ation r atio of eve ry prior-fr ee money- burn in g mec hanism. This implements the fourth step of the prior-free mec hanism design template outlined in the Intro d uction. Ou r pro of follo ws f rom sho win g th at there is a i.i.d. d istr ibution F for whic h the expected v alue of our b enc hmark G is a constan t factor larger than the exp ected residual surplu s of an optimal mechanism for the d istribution, such as Opt F . This sh o ws an inheren t gap in the prior-free analysis framework that will manifest itself in the appr oximati on factor of every prior-free mec hanism . Prop osition 4.1 No prior-fr e e money-burning me chanism has appr oximation r atio b etter than 4 / 3 with r esp e ct to the b enchmark G , even for the sp e cial c ase of two agents and one unit of an item. 15 Pro of: Our plan to exhibit a distribution o ver v aluations suc h that the exp ected residual su rplus of the Ba ye sian optimal mec hanism is at most 3 / 4 times th at of the exp ected v alue of the b enc hmark G . It follo w s that, for ev ery r andomized m ec hanism, there exists a v aluation p rofile v f or wh ic h its exp ected residu al surplus is at most 3 / 4 times G ( v ). Supp ose there are tw o agents with v aluations dr a wn i.i.d. from a standard exp onent ial distr i- bution with d ensit y f ( x ) = e − x on [0 , ∞ ). Th ere is a s in gle unit of an item. This d istribution has constan t hazard r ate, so a lottery is an optimal mec hanism (as is ev ery mechanism that alw ays allocates th e item and c harges pa yments according to L emm a 2.1). The exp ected (residual) surplus of this mec h anism is 1. T o calculate the exp ected v alue of G ( v ), firs t n ote that for a v aluation pr ofile ( v 1 , v 2 ) with v 1 ≥ v 2 , the optimal ( p, q )-lottery either c ho oses p = q = 0 or p = v 2 and q = 0. Thus, G ( v ) = max  v 1 + v 2 2 , v 1 − v 2 2  . Next, note that ( v 1 + v 2 ) / 2 ≥ v 1 − ( v 2 / 2) if and only if v 1 ≤ 2 v 2 . No w condition on th e smaller v aluation v 2 and write v 1 = v 2 + x for x ≥ 0. Since th e exp onen tial distribution is memoryless, x is exp onen tially d istributed. Th us, E [ G ( v 1 , v 2 ) | v 2 ] can b e computed as follo ws (in tegrating o v er p ossible v alues for x ∈ [0 , ∞ )): E [ G ( v 1 , v 2 ) | v 2 ] = Z v 2 0  v 2 + x 2  e − x dx + Z ∞ v 2  v 2 2 + x  e − x dx = v 2 (1 − e − v 2 ) + 1 2  1 − ( v 2 + 1) e − v 2  + v 2 2 e − v 2 + ( v 2 + 1) e − v 2 = v 2 + 1 2  1 + e − v 2  . The smaller v alue v 2 is distributed according to an exp onentia l distrib u tion with rate 2. Inte - grating out yields E [ G ( v 1 , v 2 )] = Z ∞ 0 (2 e − 2 x )  x + 1 2 + 1 2 e − x  dx = 1 2 + 1 2 + Z ∞ 0 e − 3 x dx = 4 3 . ✷ F or the sp ecial case of tw o agen ts and a single go o d, an appropriate mixtu r e of a lottery and the Vic krey aucti on is a 3 / 2-a pproximat ion of the b enc h mark G ( v ). Determining the b est-p ossible appro x im ation ratio is an op en question, even in the t wo agen t, one unit sp ecial case. Prop osition 4.2 F or two bidders and a single u ni t of an item, ther e is a prior -fr e e me chanism that 3 / 2 -app r oximates the b enchmark G . Pro of: Consider a v aluation profile with v 1 ≥ v 2 . If we run a Vic kr ey auction with probability 1 / 3 and a lottery with pr ob ab ility 2 / 3 , then th e exp ected r esid ual surplus is 1 3 ( v 1 − v 2 ) + 2 3  v 1 + v 2 2  = 2 3 v 1 ≥ 2 3 max  v 1 + v 2 2 , v 1 − v 2 2  = 2 3 G ( v ) . ✷ 16 5 Quan tifying the P o w er of T ransfers and M oney-Burning F or the ob jectiv e of surplus maximization, mechanisms with general transfers are clearly as p o w- erful as money-burning mec hanisms, which in turn are as p o werful as mechanisms without m on ey . This section quan tifies the distance b etw een the lev els of this hierarch y by studyin g sur plus approx- imation in multi- unit auctions. Precisely , w e call a class of mec hanisms α - su rplus maximizers if, for ev ery m u lti-unit auctio n problem, there is a mec h anism in the class that obtains at least a 1 /α fraction of the full surplus for every v aluatio n p r ofile. F or example, mec h anisms with transfers are 1-surplus maximizers, b ecause th e V CG mechanism ac hiev es full surp lus in ev ery m ulti-unit auction problem. Mec h an ism s without transfers are ( n/k )-surplu s maximizers, since the exp ected s urplus of a k -unit lottery is k /n times the full surp lus. One can sho w (details omitted) th at mec hanism s without transfers are not significantl y b etter than Θ( n/k )-sur plus maximizers. The inte r esting q u estion is to iden tify the exact lo cation of m oney-burning m ec hanisms b et ween these tw o extremes: w hat is the p otentia l b enefit of im p lemen ting monetary transfers in a system that initially only supp orts money burn ing? W e giv e a lo w er b ound and a matc hing upp er b ou n d, for all k and n . Prop osition 5.1 Money-burning me chanisms ar e Ω(1 + log n k ) -surplus maximizers in k -unit auc- tions. Pro of: By Y ao’s Minimax Theorem, we on ly need to lo w er b ound the su rplus appr o ximation ac hiev ed by an optimal mec h anism on a worst-ca se d istribution o ver v aluation profiles. Fix k and dra w n v aluations i.i.d. from an exp onen tial d istribution (with densit y e − x on [0 , ∞ )). This d istribution has constant h azard rate and so, b y our results in Section 2, the k -unit lottery maximizes the exp ected residu al sur plus. Since the exp ected v aluation of every bid d er is 1, the exp ected (residual) s u rplus of this mec h an ism is k . The exp ected v alue of the full surplu s is that of th e sum of the top k out of n i.i.d. samples of an exp onent ial distrib ution. A calculatio n shows that this exp ectation equ als Θ( k (1 + log n k )), completing the pro of. ✷ Theorem 5.2 Money-burning me chanisms ar e O (1 + log n k ) -surplus maximizers in k -unit auctions. Pro of: Fix k and a v aluation profile v w ith v 1 ≥ · · · ≥ v n . Assu me for simplicit y that b oth k and n are p o w ers of 2. Our simple mec hanism is as follo ws. Firs t, choose a nonnegativ e integ er j uniformly at random, sub ject to k ≤ 2 j ≤ n . No te that there are 1 + log 2 ( n/k ) p ossib le c hoices for j . Second, run a k -un it v 2 j +1 -lottery , where we interpret v n +1 as zero. W rite V ∗ = P k i =1 v i for the full surplu s. F or j ∈ { log 2 k , . . . , log 2 n } , let R j denote the residual surplu s obtained by th e mec hanism for a giv en v alue of j . W e claim that E [ R j | j is chosen] ≥  V ∗ 2 − k 2 v k +1 if j = log 2 k k 2 ( v 2 j − 1 +1 − v 2 j +1 ) otherwise . When j = log 2 k , the residu al surplus is exactly V ∗ − kv k +1 ≥ ( V ∗ − kv k +1 ) / 2. T o ju stify the second case, note that k u n its will b e ran d omly allocated amongst the top 2 j bidders at price v 2 j +1 . Eac h of th ese go o d s is allo cated to one of the top 2 j − 1 of th ese bid ders with 50% probabilit y , and the residual surp lu s con tributed by suc h an allo cation is at least v 2 j − 1 − v 2 j +1 ≥ v 2 j − 1 +1 − v 2 j +1 . 17 Let R denote the resid u al su rplus obtained by our mec h anism. Th e follo w ing deriv ation com- pletes the pr o of: E [ R ] = X log 2 n j =log 2 k E [ R j | j is c hosen ] · Pr [ j is c h osen] ≥ 1 1+log 2 ( n/k )  V ∗ 2 − k 2 v k +1 + X log 2 n j =1+log 2 k k 2 ( v 2 j − 1 +1 − v 2 j +1 )  = V ∗ 2(1+log 2 ( n/k )) . ✷ Since the mec h anism in Th eorem 5.2 is p rior-free, we obtain the s ame (tight) guarante e for ev ery Ba y esian optimal mechanism. Corollary 5.3 F or every i.i.d. distribution F , the exp e cte d r esidual surplus of the Bayesian optima l me chanism for F obtains an Ω(1 / (1 + lo g ( n/k ))) fr action of the exp e cte d fu l l surplus. Theorem 5.2 and Corollary 5.3 suggest that the cost of implemen ting money-burning pa ymen ts instead of (p ossib ly exp ensiv e or infeasible) general tran s fers is relativ ely mo dest, p ro vided an optimal money-burnin g mec hanism is used . 6 Conclusions W e ph rased our analysis of the Ba y esian setting in terms of feasible allo cations (e.g., x ∈ X if and only if P i x i ≤ k for the k -unit auction problem); how ev er, it app lies more generally to single-parameter agen t problems where the service pr o vider must pa y an arbitrary cost c ( x ) for the allocation x pro d u ced. S tand ard problems in this setting include fixed cost services, non-excludable public go o ds, and m ulticast auctions [14]. The solution to these p roblems is again to maximize the ironed vir tu al surplus, w hic h in this con text is the sum of the agen ts’ ironed virtual v aluations less the cost of pro vid in g the service, P i ¯ ϑ i ( v i ) x i − c ( x ). Th is generalization also applies when the agen ts’ v aluations are ind ep endent but n ot identica lly distributed, i.e., agen t i has ironed virtu al v aluation function ¯ ϑ i ( · ). Theorem 6.1 Given servic e c ost c ( · ) and a valuation pr ofile, v , dr awn fr om distribution F = F 1 × · · · × F n with ir one d vi rtual valuation (for utility) function ¯ ϑ i ( · ) for agent i , every me chanism with al lo c ation rule satisfying 1. x ( v ) ∈ argmax x ′ P i ¯ ϑ i ( v i ) x i − c ( x ′ ) and 2. d dv i ¯ ϑ i ( v i ) = 0 ⇒ d dv i x i ( v i ) = 0 is optimal with r esp e ct to exp e cte d r e si dual surplus. Our results for the Ba yesia n problem also extend b ey on d dominan t strategy mec han ism s. Th e w ell kno wn r evenue e qui v alenc e result [28] is p opu larly stated as: fir st price, s econd price (a.k.a., Vic krey), and all-pa y auctions all ac hieve the same pr ofit. Of course this ap p lies to money bur ning as w ell. While this pap er emphasized the d omin an t strategy “sec ond price” optimal auction, there are also fi r st-price and all-pa y v arian ts that ac hieve the same p erformance. The all-pay v arian t 18 is esp ecially interesting b ecause of its p otent ial us efu lness for net work problems. F or example, in net work routing, all age n ts could attac h a p r o of of a computational pa yment to their pac k ets. T he routing p roto col can then route the appropriate pac ke ts (dep end ing on the amoun t of computational pa yment) and d rop the rest. Th er e is no need f or a round of bidding, a roun d of transmitting the pac ke ts of winning agen ts, and a round of collecting paymen ts. One of our main results is in giving a b enchmark based on the optimal mec h anism f or the symmetric setting of i.i.d. agen ts and k -unit auctions. Another main result is in approximati ng this b enchmark with a p rior-free mec hanism. Can these techniques b e generalized b eyo nd symmetric settings? In particular, the notion that agen ts’ priv ate v aluations m a y b e paired with pub licly observ able attributes allo w ed for pr ior-free mec hanism s to approxi mate Bay esian mec hanisms for digital go o d auctions and non-iden tically distributed v aluations [4]. F urth er, there h as b een some limited s uccess in prior-free optimal mec h an ism design w ith stru ctured costs or f easible allocations (e.g., [16] for multic ast auctions and [23] for path auctions). Our analyses and th e pr ior-fr ee template extend to k -u nit auction pr oblems b eyond our ob jectiv e of residual sur p lus. Imagine th e k -unit auction in an i.i.d. Ba yesia n setting where th e optimal solution is c haracterized by optimizing an ironed virtual v alue for some quanti t y other than utilit y . F or example, the “virtual v aluation for a 8% go v ernment sales tax”, to optimize the v alue of the agen ts and mec hanism less the tax d educted by go v ern men t, would b e ϕ ( v ) = 0 . 92 v − 0 . 08 1 − F ( v ) f ( v ) . The optimal k -unit ( p, q )-lottery is s till the app ropriate b enc hmark. F urthermore, as long as the optimal ( p, q )-lottery mak es us e of pr ices p, q b ounded ab o ve b y the second h ighest bid, v (2) , as in the money-bu r ning con text, then it is likel y that our prior-free mechanism, RSOL , can b e emplo ye d to appro xim ate the b enc hmark. Notic e that when applying this tec hn ique to “virtu al v aluations for pa ymen ts”, wh ic h are the app ropriate notion of virtual v aluations for th e ob jectiv e of profit maximization, the optimal k -unit ( p, q )-lott ery is simply a p osted p rice at p . F u rthermore, the optimal p osted p rice might satisfy p = v (1) ≫ v (2) . As it is not p ossible to approximat e suc h a b enchmark to within an y constan t factor, the prior-free d igital go o ds auction literature exclud es this p ossibilit y by defining the b enc h mark to b e the pr ofit of the optimal p osted pr ice p ≤ v (2) . In our w ork there wa s an implicit, pu blicly kno wn, exc h an ge r ate for mon ey b urnt. In netw ork settings, where b u rnt pa y m en ts corresp ond to degraded service qu alit y or computational pa yments, the designer may not kno w eac h agen t’s relativ e disu tilit y for suc h pa ymen ts. This m otiv ates considering the more general setting wh ere agen ts h av e a pr iv ate v alue for bu r n t money in addition to their pr iv ate v alue for service. This mo ves the prob lem fr om a single-parameter setting to the m uch more chall enging m ulti-parameter setting where optimal mec hanism design has v ery few p ositiv e results. 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J. of Financ e , 16:8–3 7, 1961. 21 A Pro of of Lemma 2.8 Our pro of of Lemma 2.8 is based on the follo wing lemma. Lemma A.1 F or every monoto ne al lo c ation rule x i ( v ) , E v [ ϑ ( v i ) x i ( v )] = E v  ¯ ϑ ( v i ) x i ( v )  − Z b a [ H ( F ( v i )) − G ( F ( v i ))] x i ′ ( v i ) dv i . Pro of: Rec all that x i ( v i ) is the p robabilit y of allo cating to agen t i with their v alue is v i and other agen ts’ v alues are distribu ted according to F : x i ( v i ) = E v − i [ x i ( v i , v − i )] . W e use x i ′ ( v i ) to denote the deriv ativ e of x i ( v i ) with resp ect to v i . By the defi n ition of g and h in Definition 2.7, ϑ ( v i ) = ¯ ϑ ( v i ) + h ( F ( v i )) − g ( F ( v i )) for ev ery v i . Hence, E v [ ϑ ( v i ) x i ( v )] = E v  ¯ ϑ ( v i ) x i ( v )  + E v [( h ( F ( v i )) − g ( F ( v i ))) x i ( v )] . (5) Since F is a pr o duct distribution, the second term satisfies E v [( h ( F ( v i )) − g ( F ( v i ))) x i ( v )] = Z v ( h ( F ( v i )) − g ( F ( v i ))) x i ( v ) f ( v ) d v = Z b a ( h ( F ( v i )) − g ( F ( v i ))) x i ( v i ) f ( v i ) dv i . (6) No w, integrat e b y p arts to obtain E v [( h ( F ( v i )) − g ( F ( v i ))) x i ( v )] = [ H ( F ( v i )) − G ( F ( v i ))] x i ( v i )    b a − Z b a [ H ( F ( v i )) − G ( F ( v i ))] x i ′ ( v i ) dv i = − Z b a [ H ( F ( v i )) − G ( F ( v i ))] x i ′ ( v i ) dv i . (7) Equation (7) follo ws from the fact that, as the con vex hull of H ( · ) on interv al (0 , 1), G ( · ) satisfies G (0) = H (0) and G (1) = H (1 ). Com bining this with equation (5) giv es the lemma. ✷ No w we r estate an d p ro ve our main tec hnical lemma for Ba y esian optimal money-burning mec hanisms. Lemma 2.8 L et F b e a distribution function with virtual valuation function ϑ ( · ) and x ( v ) a monotone al lo c ation rule. Define G , H , and ¯ ϑ as in D efinition 2.7. Then E v [ ϑ ( v i ) x i ( v )] ≤ E v  ¯ ϑ ( v i ) x i ( v )  , with e qu ality holding if and only if d dv x i ( v ) = 0 whenever G ( F ( v )) < H ( F ( v )) . Pro of: Agai n, let x i ′ ( v ) = d dv x i ( v ) b e the deriv ativ e of x i ( v ). F rom Lemma A.1, E v [ ϑ ( v i ) x i ( v )] = E v  ¯ ϑ ( v i ) x i ( v )  − Z b a [ H ( F ( v i )) − G ( F ( v i ))] x i ′ ( v i ) dv i . (8) 22 Since G is the con v ex h u ll of H , G ≤ H on [ a, b ]. Since x is a mon otone allo cation r ule, its deriv ativ e is nonnegativ e. T he in tegral on the right-hand s ide of (8) is therefore nonnegativ e. If x i ′ ( v i ) 6 = 0 only when G ( F ( v i )) = H ( F ( v i )), then the integral v anish es. Conv ersely , since G and H (and hence H − G ) are con tinuous, if x i ′ ( v i ) > 0 at a p oint where G ( F ( v i )) < H ( F ( v i )), then th e int egral is strictly p ositiv e. ✷ 23