Computationally Efficient Approximation Mechanisms

Transcription

1 CHAPTER 12 Computatonally Effcent Approxmaton Mechansms Ron Lav Abstract We study the ntegraton of game theoretc and computatonal consderatons. In partcular, we study the desgn of computatonally effcent and ncentve compatble mechansms, for several dfferent problem domans. Issues lke the dmensonalty of the doman, and the goal of the algorthm desgner, are examned by provdng a techncal dscusson on four results: () approxmaton mechansms for sngle-dmensonal schedulng, where truthfulness reduces to a smple monotoncty condton; () randomness as a tool to resolve the computatonal vs. ncentves clash for Combnatoral Auctons, a central multdmensonal doman where ths clash s notable; () the mpossbltes of determnstc domnant-strategy mplementablty n multdmensonal domans; and (v) alternatve soluton concepts that ft worst-case analyss, and am to resolve the above mpossbltes Introducton Algorthms n computer scence, and Mechansms n game theory, are very close n nature. Both dscplnes am to mplement desrable propertes, drawn from real-lfe needs and lmtatons, but the resultng two sets of propertes are completely dfferent. A natural need s then to merge them to smultaneously exhbt good game theoretc propertes as well as good computatonal propertes. The growng mportance of the Internet as a platform for computatonal nteractons only strengthens the motvaton for ths. However, ths ntegraton task poses many dffcult challenges. The two dscplnes clash and contradct n several dfferent ways, and new understandngs must be obtaned to acheve ths hybrdzaton. The classc Mechansm Desgn lterature s rch and contans many techncal solutons when ncentve ssues are the key goal. Qute nterestngly, most of these are not computatonally effcent. In parallel, most exstng algorthmc technques, answerng the computatonal questons at hand, do not yeld the game theoretc needs. There seems to be a certan clash between classc algorthmc technques and classc mechansm desgn technques. Ths rases many ntrgung 301

2 302 computatonally effcent approxmaton mechansms questons: In what cases ths clash s fundamental a mathematcal mpossblty? Alternatvely, can we fx ths clash by applyng new technques? We wll try to gve a feel for these ssues. The possblty of constructng mechansms wth desrable computatonal propertes turns out to be strongly related to the dmensonalty of the problem doman. In sngle-dmensonal domans, the requrement for game-theoretc truthfulness reduces to a convenent algorthmc monotoncty condton that leaves ample flexblty for the algorthm desgner. We demonstrate ths n Secton 12.2, were we study the constructon of computatonally effcent approxmaton mechansms for the classc machne schedulng problem. Although there exsts a rch lterature on approxmaton algorthms for ths problem doman, qute remarkably none of these classc results satsfy the desred game-theoretc propertes. We show that when the schedulng problem s sngle-dmensonal, then ths clash s not fundamental, and can be successfully resolved. The problem doman of job schedulng has one addtonal nterestng aspect that makes t worth studyng: t demonstrates a key dfference between economcs and computer scence, namely the goals of algorthms vs. the goals of classc mechansms. Whle the economcs lterature manly studes welfare and/or revenue maxmzaton, computatonal models rase the need for completely dfferent objectves. In schedulng problems, a common objectve s to mnmze the load on the most loaded machne. As s usually the case, exstng technques for ncentve-compatble mechansm desgn do not ft such an objectve (and, on the other hand, most exstng algorthmc solutons do not yeld the desred ncentves). The resoluton of these clashes has led to nsghtful technques, and the techncal exploraton of Secton 12.2 serves as an example. As opposed to sngle-dmensonal domans, mult-dmensonalty seems to pose much harder obstacles. In Chapter 9, the monotoncty condtons that characterze truthfulness for multdmensonal domans were dscussed, but t seems that these condtons do not translate well to algorthmc constructons. Ths ssue wll be handled n the rest of the chapter, and wll be approached n three dfferent ways: we wll explore the nherent mpossbltes that the requred monotoncty condtons cast on determnstc algorthmc constructons, we wll ntroduce randomness to solve these dffcultes, and we wll consder alternatve notons to the soluton concept of truthfulness. Our man example for a multdmensonal doman wll be the doman of combnatoral auctons (CAs). Chapter 11 studes CAs mostly from a computatonal pont of vew, and n contrast our focus s on desgnng computatonally effcent and ncentve compatble CAs. Ths demonstrates a second key dfference between economcs and computer scence, namely the requrement for computatonal effcency. Even f our goal s the classc economc goal of welfare maxmzaton, we cannot use Vckrey Clarke Groves mechansms (whch classcally mplement ths goal) snce n many cases they are computatonally neffcent. The doman of CAs captures exactly ths pont, and the need for computatonally effcent technques that translate algorthms to mechansms s central. In Secton 12.3 we wll see how randomness can help. We descrbe a rather general technque that uses randomness and lnear programmng n order to convert algorthms to truthful-n-expectaton mechansms. Thus we get a postve answer to the computatonal clash, by ntroducng randomness.

3 sngle-dmensonal domans: job schedulng 303 In Secton 12.4 we return to determnstc settngs and to the classc defnton of determnstc truthfulness, and study the mpossbltes assocated wth t. Our motvatng queston s whether the three requrements () determnstc truthfulness, () computatonal effcency, and () nontrval approxmaton guarantees, clash n a fundamental and well-defned way. We already know that sngle dmensonalty does not exhbt such a clash, and n ths secton we descrbe the other extreme. If a doman has full dmensonalty (n a certan formal sense, to be dscussed n the secton body), then any truthful mechansm must be VCG. It s mportant to remark that ths result further emphaszes our lack of knowledge about the state of affars for all the ntermedate range of multdmensonal domans, to whch CAs and ts dfferent varants belong. As was motvated n prevous chapters, the game-theoretc quest should start wth the soluton concept of mplementaton n domnant strateges, and ndeed most of ths chapter follows ths lne of thought. However, to avod the mpossbltes mentoned earler, we have to deepen our understandngs about the alternatves at hand. Studes n economcs usually turn to the soluton concept of Bayesan Nash that requres strong dstrbutonal assumptons, namely that the nput dstrbutons are known, and, furthermore, that they are commonly known, and agreed upon. Such assumptons seem too strong for CS settngs, and crtcsm about these assumptons have been also rased by economsts (e.g., Wlson s doctrne ). We have already seen that randomzaton, and truthful-n-expectaton n partcular, can provde a good alternatve. We conclude the chapter by provdng an addtonal example, of a determnstc alternatve soluton concept, and descrbe a determnstc CA that uses ths noton to provde nontrval approxmaton guarantees. Let us menton two other types of GT-versus-CS clashes, not studed n ths chapter, to complete the pcture. Dfferent models: Some CS models have a sgnfcantly dfferent structure, whch causes the above-mentoned clash even when tradtonal objectves are consdered. In onlne computaton, for example, players arrve over tme, a fundamentally dfferent assumpton than classc mechansm desgn. The dffcultes that emerge, and the novel solutons proposed, are dscussed n Chapter 16. Dfferent analyss conventons: CS usually employs worst-case analyss, avodng strong dstrbutonal assumptons, whle n economcs, the underlyng dstrbuton s usually assumed. Ths greatly affects the character of results, and the reader s referred to, e.g., Chapter 13 for a broader dscusson Sngle-Dmensonal Domans: Job Schedulng As a frst example for the nteracton between game theory and algorthmc theory, we consder sngle-dmensonal domans. Smple sngle-dmensonal domans were ntroduced n Chapter 9, where every alternatve s ether a wnnng or a losng alternatve for each player. Here we dscuss a more general case. Intutvely, sngle dmensonalty mples that a sngle parameter determnes the player s valuaton vector. In Chapter 9, ths was smply the value for wnnng, but less straght-forward cases also make sense: Schedulng related machnes. In ths doman, n jobs are to be assgned to m machnes, where job j consumes p j tme-unts, and machne has speed s. Thus machne requres p j /s tme-unts to complete job j. Let l = j j s assgned to p j be the load

4 304 computatonally effcent approxmaton mechansms on machne. Our schedule ams to mnmzes the term max l /s, (the makespan). Each machne s a selfsh entty, ncurrng a constant cost for every consumed tme unt (and w.l.o.g. assume ths cost s 1). Thus the utlty of a machne from a load l and a payment P s l /s P. The mechansm desgner knows the processng tmes of the jobs and constructs a schedulng mechansm. Although here the set of alternatves cannot be parttoned to wns and loses, ths s clearly a sngle-dmensonal doman. Defnton 12.1 (sngle-dmensonal lnear domans) A doman V of player s sngle-dmensonal and lnear f there exst nonnegatve real constants (the loads ) {q,a } a A such that, for any v V, there exsts c R (the cost ) such that v (a) = q,a c. In other words, the type of a player s smply her cost c, as dsclosng t gves us the entre valuaton vector. Note that the schedulng doman s ndeed sngle-dmensonal and lnear: the parameter c s equal to 1/s, and the constant q,a for alternatve a s the load assgned to accordng to a. A natural symmetrc defnton exsts for value-maxmzaton (as opposed to costmnmzaton) problems, where the types are nonnegatve. We am to desgn a computatonally effcent approxmaton algorthm, that s also mplementable. As the socal goal s a certan mn max crteron, and not to mnmze the sum of costs, we cannot use the general VCG technque. Snce we have a convex doman, Chapter 9 tells us that we need a weakly monotone algorthm. But what exactly does ths mean? Luckly, the formulaton of weak monotoncty can be much smplfed for sngle-dmensonal domans. If we fx the costs c declared by the other players, an algorthm for a sngledmensonal lnear doman determnes the load q (c) of player as a functon of her reported cost c. Take two possble types c and c, and suppose c >c. Then the weak monotoncty condton from Chapter 9 reduces to q (c )(c c) q (c)(c c), whch holds ff q (c ) q (c). Hence from Chapter 9 we know that such an algorthm s mplementable f and only f ts load functons are monotone nonncreasng. Fgure 12.1 descrbes ths, and wll help us fgure out the requred prces for mplementablty. Fgure A monotone load curve.

5 sngle-dmensonal domans: job schedulng 305 Suppose that we charge a payment of P (c) = c 0 [q (x) q (c)] dx from player f he declares a cost of c. Usng Fgure 12.1, we can easly verfy that these prces lead to ncentve compatblty: Suppose that player s true cost s c. If he reports the truth, hs utlty s the entre area below the load curve up to c. Now f he declares some c >c, hs utlty wll decrease by exactly the area marked by A: hs cost from the resultng load wll ndeed decrease to c q (c ), but hs payment wll ncrease to be the area between the lne q (c ) and the load curve. On the other hand, f the player wll report c <c, hs utlty wll decrease by exactly the area marked by B, snce hs cost from the resultng load wll ncrease to c q (c ). Thus these prces satsfy the ncentve-compatblty nequaltes, and n fact ths s a smple drect proof for the suffcency of load monotoncty for ths case. The above prces do not satsfy ndvdual ratonalty, snce a player always ncurs a negatve utlty f we use these prces. To overcome ths, the usual exercse s to add a large enough constant to the prces, whch n our case can be 0 q (x) dx. Note that f we add ths to the above prces we get that a player that does not receve any load (.e., declares a cost of nfnty) wll have a zero utlty, and n general the utlty of a truthful player wll be nonnegatve, exactly c q (x) dx. From all the above we get the followng theorem. Theorem 12.2 An algorthm for a sngle-dmensonal lnear doman s mplementable f and only f ts load functons are nonncreasng. Furthermore, f ths s the case then chargng from every player a prce P (c) = c 0 [q (x) q (c)] dx c q (x) dx wll result n an ndvdually ratonal domnant strategy mplementaton. In the applcaton to schedulng, we wll construct a randomzed mechansm, as well as a determnstc one. In the randomzed case, we wll employ truthfulness n expectaton (see Chapter 9, Defnton 9.27). One should observe that, from the dscusson above, t follows that truthfulness n expectaton s equvalent to the monotoncty of the expected load A Monotone Algorthm for the Job Schedulng Problem Now that we understand the exact form of an mplementable algorthm, we can construct one that approxmates the optmal outcome. In fact, the optmum tself s mplementable, snce t can satsfy weak monotoncty (see the exercses for more detals), but the computaton of the optmal outcome s NP-hard. We wsh to construct effcently computable mechansms, and hence desgn a monotone and polynomal-tme approxmaton algorthm. Note that we face a classc algorthmc problem no game-theoretc ssues are left for us to handle. Before we start, let us assume that jobs and machnes are reordered so that s 1 s 2 s m and p 1 p 2 p n. For the algorthmc constructon, we frst need to estmate the optmal makespan of a gven nstance. Estmatng the optmal makespan. Fx a job-ndex j, and some target makespan T. If a schedule has makespan at most T, then t must assgn any job out of 1,...,j to a

6 306 computatonally effcent approxmaton mechansms machne such that T p j /s. Let (j,t ) = max{ T p j /s }. Thus any schedule wth makespan at most T assgns jobs 1,...,j to machnes 1,...,(j,T ). From space consderatons, t mmedately follows that Now defne j k=1 T p k (j,t ). (12.1) l=1 s l { j p j k=1 T j = mn max, p } k s l=1 s l (12.2) Lemma 12.3 For any job-ndex j, the optmal makespan s at least T j. proof Fx any T<T j. We prove that T volates 12.1, hence cannot be any feasble makespan, and the clam follows. Let j be the ndex that determnes T j. The left expresson n the max term s ncreasng wth, whle the rght term s decreasng. Thus j s ether the last where the rght term s larger than the left one, or the frst for whch the left term s larger than the rght one. We prove that T volates 12.1 for each case separately. j k=1 Case 1 ( p k j l=1 s l s the mn-max, we get T j p j T<T j = j k=1 p k j l=1 s l p j s j ): For j + 1 the max term s receved by p j s j +1, Snce T j s j +1. Snce T<T j,wehave(j,t ) j, and j k=1 p k (j,t ) l=1 s l. Hence T volates 12.1, as clamed. j k=1 Case 2 ( p k j < p j j k=1 l=1 s s j ): T j p k j 1 snce T j s the mn max, and the max for l l=1 s l j 1 s receved at the rght. In addton, (j,t ) < j snce T j = p j Thus T<T j j k=1 p k j 1 l=1 s l j k=1 p k (j,t ) l=1 s l, as we need. Wth ths, we get a good lower bound estmate of the optmal makespan: s j and T<T j. T LB = max j T j (12.3) The optmal makespan s at least T j for any j, hence t s at least T LB. A fractonal algorthm. We start wth a fractonal schedule. If machne gets an α fracton of job j then the resultng load s assumed to be (α p j )/s. Ths s of course not a vald schedule, and we later round t to an ntegral one. Defnton 12.4 (The fractonal allocaton) Let j be the frst job such that j k=1 p k >T LB s 1. Assgn to machne 1 jobs 1,...,j 1, plus a fracton of j n order to equate l 1 = T LB s 1. Contnue recursvely wth the unassgned fractons of jobs and wth machnes 2,...,m.

7 sngle-dmensonal domans: job schedulng 307 Lemma 12.5 There s enough space to fractonally assgn all jobs, and f job j s fractonally assgned to machne then p j /s T LB. j k=1 p k proof Let j be the ndex that determnes T j. Snce T LB T j j,we l=1 s l can fractonally assgn jobs 1,.., j up to machne j. Snce T j p j /s j we get the second part of the clam, and settng j = n gves the frst part. Lemma 12.6 The fractonal load functon s monotone. proof We show that f s ncreases to s = α s (for α>1) then l l. Let T LB denote the new estmate of the optmal makespan. We frst clam that T LB α T LB. For an nstance s 1,...,s m such that s l = α s l for all machnes l we have that T LB = α T LB snce both terms n the max expresson of T j were multpled by α. Snce s l s l for all l we have that T LB T LB.Now,fl = T LB s,.e. was full, then l T LB s T LB s = l. Otherwse l <T LB s, hence s the last nonempty machne. Snce T LB T LB, all prevous machnes now get at least the same load as before, hence machne cannot get more load. We now round to an ntegral schedule. The natural roundng, of ntegrally placng each job on one of the machnes that got some fracton of t, provdes a 2-approxmaton, but volates the requred monotoncty (see the exercses). We offer two types of roundng, a randomzed roundng and a determnstc one. The former s smpler, and results n a better approxmaton rato, but uses the weaker soluton concept of truthfulness n expectaton. The latter s slghtly more nvolved, and uses determnstc truthfulness, but results n an nferor approxmaton rato. Defnton 12.7 (A randomzed roundng) Choose α [0, 1] unformly at random. For every job j that was fractonally assgned to and + 1, f j s fracton on s at least α, assgn j to n full, otherwse assgn j to + 1. Theorem 12.8 The randomzed schedulng algorthm s truthful n expectaton, and obtans a 2-approx. to the optmal makespan n polynomal-tme. proof Let us check the approxmaton frst. A machne may get, n addton to ts full jobs, two more jobs. One, j, s shared wth machne 1, and the other, k, s shared wth machne + 1. If j was rounded to then ntally has at least 1 α fracton of j, hence the addtonal load caused by j s at most α p j. Smlarly, If k was rounded to then ntally has at least α fracton of k, hence the addtonal load caused by k s at most (1 α) p k. Thus the maxmal total addtonal load that gets s α p j + (1 α) p k. By Lemma 12.5 we have that max{p j,p k } T LB and snce T LB s not larger than the optmal maxmal makespan, the approxmaton clam follows. For truthfulness, we only need that the expected load s monotone. Note that machne 1 gets job j wth probablty α, so gets t wth probablty 1 α,

8 308 computatonally effcent approxmaton mechansms and gets k wth probablty α. So the expected load of machne s exactly ts fractonal load. The clam now follows from Lemma An ntegral determnstc algorthm. To be accurate, what follows s not exactly a roundng of the fractonal assgnment we obtaned above, but a smlar-n-sprt determnstc assgnment. We set vrtual speeds, where the fastest machne s set to be slghtly faster, and the others are set to be slghtly slower, we fnd a fractonal assgnment accordng to these vrtual speeds, and then use the natural roundng of placng each job fully on the frst machne t s fractonally assgned to. Wth these vrtual speeds, the roundng that prevously faled to be monotone, now succeeds: Defnton 12.9 (A determnstc algorthm) Gven the bds s 1,...,s m, perform: () Set new (vrtual) speeds d 1,...,d m,asfollows.letd 1 = 8 5 s 1, and for 2, let d be the the closest value of the breakponts s 1 (for = 1, 2,...) such that 2.5 d s. () Compute T LB accordng to the vrtual speeds,.e. T LB = T LB (d,d ). () Assgn jobs to machnes, startng from the largest job and the fastest machne. Move to the next machne when the current machne,, holds jobs wth total processng tme larger or equal to T LB d. Note that f the fastest machne changes ts speed, then all the d s may change. Also note that step 3 manages to assgn all jobs, snce what we are dong s exactly the determnstc natural roundng descrbed above for the fractonal assgnment, usng the d s nstead of the s s. As we shall see, ths crucal dfference enables monotoncty, n the cost of a certan loss n the approxmaton. To exactly see the approxmaton loss, frst note that T LB (d) 2.5T LB (s), snce speeds are made slower by at most ths factor. For the fastest machne, snce s 1 s lower than d 1, the actual load up to T LB (d) may be 1.6T LB (d) 4T LB (s). As we may ntegrally place on machne 1 one job that s partally assgned also to machne 2, observe () that d 1 4d 2, and () by the fractonal rules the added job has load at most T LB (d)d 2. Thus get that the load on machne 1 s at most T LB(d) 5T LB (s). For any other machne, d s, and so after we ntegrally place the one extra partal job the load can be at most 2T LB (d)d 2 2.5T LB (s)s = 5T LB (s)s. Snce T LB (s) lower bounds the optmal makespan for s the approxmaton follows. To understand why monotoncty holds, we frst need few observatons that easly follow from our knowledge on the fractonal assgnment. For any >1 and β<d, T LB (β,d ) 5 4 T LB(d,d ). Consder the followng modfcaton to the fractonal assgnment for (d,d ): machne does not get any job, and each machne 1 <gets the jobs that were prevously assgned to machne + 1. Snce s faster than + 1, any machne 2 <does not cross the T LB (d,d ) lmt. As for machne 1, note that t s always the case that d 1 4d 2, hence the new load on machne 1 s at most 5 4 T LB(d,d ).

9 sngle-dmensonal domans: job schedulng 309 If a machne >1 slows down then the total work assgned to the faster machnes does not decrease, whch follows mmedately from the fact that T LB (d,d ) T LB (d,d ), for d d. If the fastest machne slows down, yet remans the fastest, then ts assgned work does not ncrease. Let s 1 = c s 1 for some c<1. Therefore all breakponts shft by a factor of c. If no speed s moves to a new breakpont then all d s move by a factor of c, the resultng T LB wll therefore also move by a factor of c, meanng that machne 1 wll get the same set of jobs as before. If addtonally some s s move to a new breakpont ths mples that the respectve d s decrease, and by the monotoncty of T LB t also decreases, whch means that machne 1 wll not get more work. Lemma The determnstc algorthm s monotone. proof Suppose that machne slows down from s to s <s. We need to show that t does not get more work. Assume that the vector d has ndeed changed because of s change. If s the fastest machne and t remans the fastest then the above observaton s what we need. If the fastest machne changes to, then we add an artfcal breakpont to the slowdown decrease, where and s speeds are dentcal, and the ttle of the fastest machne moves from to. Note that the same threshold, T,s computed when the ttle goes from to. s work when t s the fastest machne s at least 8 5 s T, whle s work when s the fastest s at most 2 s T< 8 5 s T, hence decreases. If s not the fastest, but stll full, then d <d (snce the breakponts reman fxed), and therefore T LB (d,d ) 5 4 T LB(d,d ). Wth s, s work s at least T d (where T = T LB (d,d )), and wth s 5 ts work s at most 2 4 T d 2.5 = T d, hence s load does not ncrease. Fnally, note that f s s not full then by the thrd observaton, snce the work of the prevous machnes does not decrease, then s work does not ncrease. By the above arguments we mmedately get the followng theorem. Theorem There exsts a truthful determnstc mechansm for schedulng related machnes, that approxmates the makespan by a factor of 5. A note about prce computaton s n place. A polynomal-tme mechansm must compute the prces n polynomal tme. To compute the prces for both the randomzed and the determnstc mechansms, we need to ntegrate over the load functon of a player, fxng the others speeds. In both cases ths s a step functon, wth polynomal number of steps (when a player declares a large enough speed she wll get all jobs, and as she decreases her speed more and more jobs wll be assgned elsewhere, where the set of assgned jobs wll decrease monotoncally). Thus we can see that prce computaton s polynomal-tme. Wthout the monotoncty requrement, a PTAS for related machnes exsts. The queston whether one can ncorporate truthfulness s stll open.

10 310 computatonally effcent approxmaton mechansms Open Queston Does there exst a truthful PTAS for related machnes? The techncal dscusson of ths secton ams to demonstrate that, for sngledmensonal domans, the algorthmc mplcatons of the game-theoretc requrement are manageable, and leave ample flexblty for the algorthmc desgner. Multdmensonalty, on the other hand, does not exhbt ths easy structure, and the rest of ths chapter s concerned wth exactly ths ssue Multdmensonal Domans: Combnatoral Auctons As opposed to sngle-dmensonal domans, the monotoncty condtons that characterze mplementablty n multdmensonal domans are far more complex (see the dscusson n Chapter 9), hence desgnng mplementable approxmaton algorthms s harder. As dscussed n the Introducton, ths chapter examnes three aspects of ths ssue, and n ths secton we wll utlze randomness to overcome the dffcultes of mplementablty n multdmensonal domans. We study ths for the representatve and central problem doman of Combnatoral Auctons. Combnatoral Auctons (CAs) are a central model wth theoretcal mportance and practcal relevance. It generalzes many theoretcal algorthmc settngs, lke job schedulng and network routng, and s evdent n many real-lfe stuatons. Chapter 11 s exclusvely devoted to CAs, provdng a comprehensve dscusson on the model and ts varous computatonal aspects. Our focus here s dfferent: how to desgn CAs that are, smultaneously, computatonally effcent and ncentve-compatble. Whle each aspect s mportant on ts own, obvously only the ntegraton of the two provdes an acceptable soluton. Let us shortly restate the essentals. In a CA, we allocate m tems ( ) ton players. Players value subsets of tems, and v (S) denotes s value of a bundle S. Valuatons addtonally satsfy () monotoncty,.e., v (S) v (T )fors T, and () normalzaton,.e., v ( ) = 0. In ths secton we consder the goal of maxmzng the socal welfare: fnd an allocaton (S 1,...,S n ) that maxmzes v (S ). Snce a general valuaton has sze exponental n n and m, the representaton ssue must be taken nto account. Chapter 11 examnes two models. In the bddng languages model, the bd of a player represents hs valuaton n a concse way. For ths model t s NP-hard to approxmate the socal welfare wthn a rato of (m 1/2 ɛ ), for any ɛ>0(f sngle-mnded bds are allowed). In the query access model, the mechansm teratvely queres the players n the course of computaton. For ths model, any algorthm wth polynomal communcaton cannot obtan an approxmaton rato of (m 1/2 ɛ )for any ɛ>0. These bounds are tght, as there exsts a determnstc m-approxmaton wth polynomal computaton and communcaton. Thus, for the general case, the computatonal status by tself s well-understood. The basc ncentves ssue s agan well-understood: wth VCG (whch requres the exact optmum) we can obtan truthfulness. The two consderatons therefore clash f we attempt to use classc technques, and our am s to develop a new technque that wll combne the two desrable aspects of effcent computaton and ncentve compatblty. We descrbe a rather general LP-based technque to convert approxmaton algorthms to truthful mechansms, by usng randomzaton: gven any algorthm to the

11 multdmensonal domans: combnatoral auctons 311 general CA problem that outputs a c-approxmaton to the optmal fractonal socal welfare, one can construct a randomzed c-approxmaton mechansm that s truthful n expectaton. Thus, the same approxmaton guarantee s mantaned. The constructon and proof are descrbed n three steps. We frst dscuss the fractonal doman, where we allocate fractons of tems. We then show how to move back to the orgnal doman whle mantanng truthfulness, by usng randomzaton. Ths uses an nterestng decomposton technque, whch we then descrbe. The fractonal doman. Let x,s denote the fracton of subset S that player receves n allocaton x. Assume that her value for that fracton s x,s v (S). The welfare maxmzaton becomes an LP: max subject to x,s v (S),S (CA-P) x,s 1 for each player (12.4) S x,s 1 for each tem j (12.5) S:j S x,s 0, S. By constrant 12.4, a player receves at most one ntegral subset, and constrant 12.5 ensures that each tem s not overallocated. The empty set s excluded for techncal reasons that wll become clear below. Ths LP s solvable n tme polynomal n ts sze by usng, e.g., the ellpsod method. Its sze s related to our representaton assumpton. If we assume the bddng languages model, where the LP has sze polynomal n the sze of the bd (e.g., k-mnded players), then we have a polynomal-tme algorthm. If we assume general valuatons and a query-access, ths LP s solvable wth a polynomal number of demand queres (see Chapter 11). Note that, n ether case, the number of nonzero x,s coordnates s polynomal, snce we obtan x n polynomal-tme (ths wll become mportant below). In addton, snce we obtan the optmal allocaton, we can use VCG (see Chapter 9) to get: Proposton In the fractonal case, there exsts a truthful optmal mechansm wth effcent computaton and communcaton, for both the bddng languages model and the query-access model. The transton to the ntegral case. The followng techncal lemma allows for an elegant transton, by usng randomzaton. Defnton Algorthm A verfes a c-ntegralty-gap (for the lnear program CA-P) f t receves as nput real numbers w,s, and outputs an ntegral pont x whch s feasble for CA-P, and c w,s x,s max w,s x,s feasble x s,s,s

12 312 computatonally effcent approxmaton mechansms Lemma (The decomposton lemma) Suppose that A verfes a c- ntegralty-gap for CA-P (n polynomal tme), and x s any feasble pont of CA-P. Then one can decompose x/c to a convex combnaton of ntegral feasble ponts. Furthermore, ths can be done n polynomal-tme. Let {x l } l I be all ntegral allocatons. The proof wll fnd {λ l } l I such that () l I, λ l 0, () l I λ l = 1, and () l I λ l x l = x/c. We wll also need to provde the ntegralty gap verfer. But frst we show how to use all ths to move back to the ntegral case, whle mantanng truthfulness. Defnton (The decomposton-based mechansm) () Compute an optmal fractonal soluton, x, and VCG prces p F (v). () Obtan a decomposton x /c = l I λ l x l. () Wth probablty λ l : () choose allocaton x l, () set prces p R(v) = [v (x l )/v (x )]p F (v). The strategc propertes of ths mechansm hold whenever the expected prce equals the fractonal prce over c. The specfc prces chosen satsfy, n addton to that, strong ndvdual ratonalty (.e., truth-tellng ensures a nonnegatve utlty, regardless of the randomzed choce) 1 : VCG s ndvdually ratonal, hence p F (v) v (x ). Thus p R (v) v (x l ) for any l I. Lemma The decomposton-based mechansm s truthful n expectaton, and obtans a c-approxmaton to the socal welfare. proof The expected socal welfare of the mechansm s (1/c) v (x ), and snce x s the optmal fractonal allocaton, the approxmaton guarantee follows. For truthfulness, we frst need that the expected prce of a player equals her fractonal prce over c,.e., E λl [p R(v)] = pf (v)/c: [ E {λl } l I p R (v) ] = λ l [v (x l )/v (x )] p F (v) l I = [ p F (v)/v (x ) ] λ l v (x l ) = [ p F (v)/v (x ) ] v (x /c) = p F (v)/c (12.6) Fx any v V. Suppose that when declares v, the fractonal optmum s x, and when she declares v, the fractonal optmum s z. The VCG fractonal prces are truthful, hence l I v (x ) p F (v,v ) v (z ) p F (v,v ) (12.7) By 12.6 and by the decomposton, dvdng 12.7 by c yelds [ ] [ λ l v (x l ) E λl p R (v,v ) ] [ ] [ λ l v (z l ) E λl p R (v,v ) ] l I l I 1 See Chapter 9 for defntons and a dscusson on randomzed mechansms.

13 multdmensonal domans: combnatoral auctons 313 The left-hand sde s the expected utlty for declarng v and the rght-hand sde s the expected utlty for declarng v, and the lemma follows. The above analyss s for one-shot mechansms, where a player declares hs valuaton up-front (the bddng languages model). For the query-access model, where players are beng quered teratvely, the above analyss leads to the weaker soluton concept of ex-post Nash: f all other players are truthful, player wll maxmze hs expected utlty by beng truthful. For example, consder the followng sngle tem aucton for two players: player I bds frst, player II observes I s bd and then bds. The hghest bdder wns and pays the second hghest value. Here, truthfulness fals to be a domnant strategy. Suppose II chooses the strategy f I bds above 5, I bd 20, otherwse I bd 2. If I s true value s 6, hs best response s to declare 5. However, truthfulness s an ex-post Nash equlbrum: f II fxes any value and bds that, then, regardless of II s bd, I s best response s the truth. In our case, f all others answer queres truthfully, the analyss carry through as s, and so truth-tellng maxmzes s the expected utlty. The decomposton-based mechansm thus has truthfulness-n-expectaton as an ex-post Nash equlbrum for the query-access model. Puttng t dfferently, even f a player was told beforehand the types of the other players, he would have no ncentve to devate from truth-tellng. The decomposton technque. We now decompose x/c = l I λ l x l, for any x feasble to CA-P. We frst wrte the LP P and ts dual D. Let E ={(, S) x,s > 0}. Recall that E s of polynomal sze. mn λ l (P) max 1 x,s w,s + z (D) c s.t. l I s.t. (,S) E λ l x,s l = x,s (, S) E (12.8) x,s l w,s + z 1 l I (12.9) c (,S) E l λ l 1 z 0 l λ l 0 l I w,s unconstraned (, S) E. Constrants 12.8 of P descrbe the decomposton; hence, f the optmum satsfes l I λ l = 1, we are almost done. P has exponentally many varables, so we need to show how to solve t n polynomal tme. The dual D wll help. It has varables w,s for each constrant 12.8 of P, so t has polynomally many varables but exponentally many constrants. We use the ellpsod method to solve t, and construct a separaton oracle usng our verfer A. Clam If w, z s feasble for D then 1 c (,S) E x,sw,s + z 1. Furthermore, f ths nequalty s reversed, one can use A to fnd a volated constrant of D n polynomal-tme. proof Suppose 1 c (,S) E x,sw,s + z>1. Let A receve w as nput and suppose that the ntegral allocaton that A outputs s x l.wehave (,S) E xl,s w,s (,S) E x,sw,s > 1 z, where the frst nequalty follows snce A s a 1 c

14 314 computatonally effcent approxmaton mechansms c-approxmaton to the fractonal optmum, and the second nequalty s the volated nequalty of the clam. Thus constrant 12.9 s volated (for x l ). Corollary The optmum of D s 1, and the decomposton x/c = l I λ l x l s polynomal-tme computable. proof z = 1,w,S = 0 (, S) E s feasble; hence, the optmum s at least 1. By clam t s at most 1. To solve P, we frst solve D wth the followng separaton oracle: gven w, z,f 1 c (,S) E x,sw,s + z 1, return the separatng hyperplane 1 c (,S) E x,sw,s + z = 1. Otherwse, fnd the volated constrant, whch mples the separatng hyperplane. The ellpsod method uses polynomal number of constrants; thus, there s an equvalent program wth only those constrants. Its dual s a program that s equvalent to P but wth polynomal number of varables. We solve that to get the decomposton. Verfyng the ntegralty gap. We now construct the ntegralty gap verfer for CA-P. Recall that t receves as nput weghts w,s, and outputs an ntegral allocaton x l whch s a c-approxmaton to the socal welfare w.r.t. w,s. Two requrements dfferentate t from a regular c-approxmaton for CAs: () t cannot assume any structure on the weghts w,s (unlke CA, where we have non-negatvty and monotoncty), and () the obtaned welfare must be compared to the fractonal optmum (usually we care for the ntegral optmum). The frst property s not a problem. Clam Gven a c-approxmaton for general CAs, A, where the approxmaton s wth respect to the fractonal optmum, one can obtan an algorthm A that verfes a c-ntegralty-gap for the lnear program CA-P, wth a polynomal tme overhead on top of A. proof Gven w ={w,s } (,S) E, defne w + by w +,S = max(w,s, 0), and w by w,s = max T S,(,T ) E w +,T (where the maxmum s 0 f no T S has (, T ) E. w s a vald valuaton, and can be succnctly represented wth sze E. Let O = max x s feasble for CA-P (,S) E x,sw,s. Feed w to A to get x such that,s x,s w,s O (snce w c,s w,s for every (, S)). Note that t s possble that (,S) E x,sw,s <,S x,s w,s, snce () the left hand sum only consders coordnates n E and () some w,s coordnates mght be negatve. To fx the frst problem defne x + as follows: for any (, S) such that x,s = 1, set x +,T = 1forT = arg max T S:(,T ) E w +,T (set all other coordnates of x + to 0). By constructon,,s x,s w,s = (,S) E x+,s w+,s. To fx the second problem, defne x l as follows: set x,s l = x+,s f w,s 0 and 0 otherwse. Clearly, (,S) E xl,s w,s = (,S) E x+,s w+,s, and xl s feasble for CA-P. The requrement to approxmate the fractonal optmum does affect generalty. However, one can use the many algorthms that use the prmal-dual method, or a derandomzaton of an LP randomzed roundng. Smple combnatoral algorthms may also satsfy ths property. In fact, the greedy algorthm from Chapter 11 for

15 multdmensonal domans: combnatoral auctons 315 sngle-mnded players satsfes the requrement, and a natural varant verfes a 2 m ntegralty-gap for CA-P. Defnton (Greedy (revsted)) Fx {w,s } (,S) E as the nput. Construct x as follows. Let (, S) = arg max (,S ) E(w,S / S ). Set x,s = 1. Remove from E all (,S ) wth = or S S.IfE, reterate. Lemma Greedy s a ( 2m)-approxmaton to the fractonal optmum. proof Let y ={y,s } (,S) E be the optmal fractonal allocaton. For every player wth x,s = 1 (for some S ), let Y ={(,S) E y,s > 0 and (,S) was removed from E when (, S ) was added }. We show that (,S) Y y,s w,s ( 2 m)w,s, whch proves the clam. We frst have w y,sw,s = y,s,s S (,S) Y (,S) Y S w,s S y,s S (,S) Y w,s S y,s y,s S (12.10) (,S) Y (,S) Y The frst nequalty follows snce (, S ) was chosen by greedy when (,S)was n E, and the second nequalty s a smple algebrac fact. We also have: y,s y,s + y,s S +1 (12.11) (,S) Y j S (,S) Y,j S (,S) Y j S where the frst nequalty holds snce every (,S) Y has ether S S or =, and the second nequalty follows from the feasblty constrants of CA-P, and, y,s S y,s m (12.12) (,S) Y j (,S) Y,j S Combnng 12.10, 12.11, and 12.12, we get what we need: y,sw,s w,s S S +1 m 2 m w,s (,S) Y Greedy s not truthful, but wth the decomposton-based mechansm, we use randomness n order to plug-n truthfulness. We get the followng theorem. Theorem The decomposton-based mechansm wth Greedy as the ntegralty-gap verfer s ndvdually ratonal and truthful-n-expectaton, and obtans an approxmaton of 2 m to the socal welfare. Remarks. The decomposton-based technque s qute general, and can be used n other cases, f an ntegralty-gap verfer exsts for the LP formulaton of the problem.

16 316 computatonally effcent approxmaton mechansms Perhaps the most notable case s multunt CAs, where there exst B copes of each tem, and any player desres at most one copy from each tem. In ths case, one can verfy a O(m 1 B+1 ) ntegralty gap, and ths s the best possble n polynomal tme. To date, the decomposton-based mechansm s the only truthful mechansm wth ths tght guarantee. Nevertheless, ths method s not completely general, as VCG s. One drawback s for specal cases of CAs, where low approxmaton ratos exst, but the ntegralty gap of the LP remans the same. For example, wth sub-modular valuatons, the ntegralty gap of CA-P s the same (the constrants do not change), but lower-than-2 approxmatons exst. To date, no truthful mechansm wth constant approxmaton guarantees s known for ths case. One could, n prncple, construct a dfferent LP formulaton for ths case, wth a smaller ntegralty gap, but these attempts were unsuccessful so far. Whle truthfulness-n-expectaton s a natural modfcaton of (determnstc) truthfulness, and although ths noton ndeed contnues to be a worst-case noton, stll t s nferor to truthfulness. Players are assumed to only care about ther expected utlty, and not about the varance, for example. A stronger noton s that of unversal truthfulness, were players maxmze ther utlty for every con toss. But even ths s stll weaker. Whle n classc algorthmc settngs one can use the law of large numbers to approach the expected performance, n mechansm desgn one cannot repeat the executon and choose the best outcome as ths affects the strategc propertes. Determnstc mechansms are stll a better choce A General Overvew of Truthful Combnatoral Auctons The search for truthful CAs s an actve feld of research. Roughly speakng, two technques have proved useful for constructng truthful CAs. In Maxmal-n-Range mechansms, the range of possble allocatons s restrcted, and the optmal-n-thsrange allocaton s chosen. Ths acheves determnstc truthfulness wth an O( m)- m log m )- approxmaton for subaddtve valuatons (Dobznsk et al., 2005), an O( approxmaton for general valuatons (Holzman et al., 2004), and a 2-approxmaton. when all tems are dentcal ( mult-unt auctons ) (Dobznsk and Nsan, 2006). A second technque s to partton the set of players, sample statstcs from one set, and use t to obtan a good approxmaton for the other. See Chapter 13 for detals. Ths technque obtans an O( m)-approxmaton. for general valuatons, and an O(log 2 m)for XOS valuatons (Dobznsk et al., 2006). The truthfulness here s unversal,.e., for any con toss a stronger noton than truthfulness n expectaton. Bartal et al. (2003) use a smlar dea to obtan a truthful and determnstc O(B m 1 B 2 )-approxmaton for multunt CAs wth B 3 copes of each tem. For specal cases of CAs, these technques do not yet manage to obtan constant-factor truthful approxmatons (Dobznsk and Nsan, 2006 prove ths mpossblty for Maxmal-In-Range mechansms). Due to the mportance of constant-factor approxmatons, explanng ths gap s challengng: Open Queston Does there exst truthful constant-factor approxmatons for specal cases of CAs that are NP-hard and yet constant algorthmc approxmatons are known? For example, does there exst a truthful constant-factor approxmaton for CAs wth submodular valuatons?

17 mpossbltes of domnant strategy mplementablty 317 For general valuatons, the above shows a sgnfcant gap n the power of randomzed vs. determnstc technques. It s not known f ths gap s essental. A possble argument for ths gap s that, for general valuatons, every determnstc mechansm s VCG-based, and these have no power. Lav et al. (2003) have ntated an nvestgaton for the frst part of the argument, obtanng only partal results. Dobznsk and Nsan (2006) have studed the other part of the argument, agan wth only partal results. Open Queston What are the lmtatons of determnstc truthful CAs? Does approxmaton and domnant-strateges clash n some fundamental and well-defned way for CAs? Ths secton was devoted to welfare maxmzaton. Revenue maxmzaton s another mportant goal for CA desgn. The mechansm of Bartal et al. (2003) obtans the same guarantees wth respect to the optmal revenue. More tght results for mult-unt auctons wth budget constraned players are gven by Borgs et al. (2005), and for unlmtedsupply CAs by Balcan et al. (2005). It should be noted that these are prelmnary results for specal cases; ths ssue s stll qute unexplored Impossbltes of Domnant Strategy Implementablty In the prevous sectons we saw an nterestng contrast between determnstc and randomzed truthfulness, where the key dfference seems to be the dmensonalty of the doman. We now ask whether the source of ths dffculty can be rgorously dentfed and characterzed. What exactly do we mean by an mpossblty, especally snce we know that VCG mechansms are possble, n every doman? Well, we mean that nothng besdes VCG s possble. Such a stuaton should be vewed as an mpossblty, snce () many tmes VCG s computatonally ntractable (as we saw for CAs), and () many tmes we seek goals dfferent from welfare maxmzaton (as we saw for schedulng domans). The monotoncty characterzatons of Chapter 9 almost readly provde few easy mpossbltes for some specal domans (see the exercses at the end of ths chapter), and n ths secton we wll study a more fundamental case. To formalze our exact queston, t wll be convenent to use the abstract socal choce settng ntroduced n Chapter 9: there s a fnte set A of alternatves, and each player has a type (valuaton functon) v : A Rthat assgns a real number to every possble alternatve. v (a) should be nterpreted as s value for alternatve a. The valuaton functon v ( ) belongs to the doman V of all possble valuaton functons. Our goal s to mplement n domnant strateges the socal choce functon f : V 1 V n A (where w.l.o.g. assume that f : V A s onto A). From chapter 9 we know that VCG mplements welfare maxmzaton, for any doman, and that affne maxmzers are also always mplementable. Defnton (Affne maxmzer) f s an affne maxmzer f there exst weghts k 1,...,k n and {C x } x A such that, for all v V, f (v) argmax x A { n =1 k v (x) + C x }.

18 318 computatonally effcent approxmaton mechansms The fundamental queston s what other functon forms are mplementable. Ths queston has remaned mostly unexplored, wth few exceptons. In partcular, f the doman s unrestrcted, the answer s sharp. Theorem Suppose A 3 and V =R A for all. Then f s domnantstrategy mplementable ff t s an affne maxmzer. We wll prove here a slghtly easer verson of the suffcency drecton. The proof s smplfed by addng an extra requrement, but the essental structure s kept. The exercses gve gudelnes to complete the full proof. Defnton (Neutralty) f s neutral f for all v V, f there exsts an alternatve x such that v (x) >v (y), for all and y x, then f (v) = x. Neutralty essentally mples that f a functon s ndeed an affne maxmzer then the addtve constants C x are all zero. Theorem Suppose A 3 and for every, V =R A.Iff s domnantstrategy mplementable and neutral then t must be an affne maxmzer. For the proof, we start wth two monotoncty condtons. Recall that Chapter 9 portrayed the strong connecton between mplementablty and certan monotoncty propertes. The monotoncty condtons that we consder here are stronger, and are not necessary for all domans. However, for an unrestrcted doman, ther mportance wll soon become clear. Defnton (Postve assocaton of dfferences (PAD)) f satsfes PAD f the followng holds for any v, v V. Suppose f (v) = x, and for any y x, and any, v (x) v (x) >v (y) v (y). Then f (v ) = x. Clam Any mplementable functon f, on any doman, satsfes PAD. proof Let v = (v 1,...,v,v +1,...,v n ),.e., players up to declare accordng to v ; the rest declare accordng to v. Thus v 0 = v, v n = v, and f (v 0 ) = x. Suppose f (v 1 ) = x for some 1 n. For every alternatve y x we have v (y) v 1 (y) <v (x) v 1 (x), and n addton v 1 = v. Thus, W-MON mples that f (v ) = x. By nducton, f (v n ) = x. In an unrestrcted doman, weak monotoncty can be generalzed as follows. Defnton (Generalzed-WMON) For every v, v V wth f (v) = x and f (v ) = y there exsts a player such that v (y) v (y) v (x) v (x). Wth weak monotoncty, we fx a player and fx the declaratons of the others. Here, ths qualfer s dropped. Another way of lookng at ths property s the followng: If

19 mpossbltes of domnant strategy mplementablty 319 f (v) = x and v (x) v(x) >v (y) v(y) then f (v ) y (a word about notaton: for α, β R n, we use α>βto denote that, α >β ). Clam If the doman s unrestrcted and f s mplementable then f satsfes Generalzed-WMON. proof Fx any v, v. We show that f f (v ) = x and v (y) v(y) >v (x) v(x) for some y A then f (v) y. By contradcton, suppose that f (v) = y. Fx R n such that v (x) v (y) = v(x) v(y), and defne v : mn{v (z),v (z) + v (x) v (x)} z x,y, z A : v (z) = v (x) z = x 2 v (y) z = y. By PAD, the transton v v mples f (v ) = y, and the transton v v mples f (v ) = x, a contradcton. We now get to the man constructon. For any x,y A, defne: P (x,y) ={α R n v V : v(x) v(y) = α, f (v) = x }. (12.13) Lookng at dfferences helps snce we need to show that k [v (x) v (y)] C y C x f f (v) = x. Note that P (x,y) s not empty (by assumpton there exsts v V wth f (v) = x), and that f α P (x,y) then for any δ R n ++ (.e., δ> 0), α + δ P (x,y): take v wth f (v) = x and v(x) v(y) = α, and construct v by ncreasng v(x) byδ, and settng the other coordnates as n v.bypadf (v ) = x, and v (x) v (y) = α + δ. Clam For any α, ɛ R n,ɛ> 0:()α ɛ P (x,y) α/ P (y,x), and () α/ P (x,y) α P (y,x). proof () Suppose by contradcton that α P (y,x). Therefore there exsts v V wth v(y) v(x) = α and f (v) = y. Asα ɛ P (x,y), there also exsts v V wth v (x) v (y) = α ɛ and f (v ) = x. But snce v(x) v(y) = α>v (x) v (y), ths contradcts Generalzed-WMON. () For any z x,y take some β z P (x,z) and fx some ɛ> 0. Fx some v such that v(x) v(y) = α and v(x) v(z) = β z + ɛ for all z x,y. By the above argument, f (v) {x,y}. Snce v(x) v(y) = α/ P (x,y) t follows that f (v) = y. Thus α = v(y) v(x) P (y,x), as needed. Clam Fx α, β, ɛ 1,ɛ 2, R n, ɛ > 0, such that α ɛ 1 P (x,y) and β ɛ 2 P (y,z). Then α + β (ɛ 1 + ɛ 2 )/2 P (x,z). proof For any w x,y,z fx some δ w P (x,w). Choose any v such that v(x) v(y) = α ɛ 1 /2, v(y) v(z) = β ɛ 2 /2, and v(x) v(w) = δ w + ɛ for all w x,y,z (for some ɛ> 0). By Generalzed-WMON, f (v) = x. Thus α + β (ɛ 1 + ɛ 2 )/2 = v(x) v(z) P (x,z).

20 320 computatonally effcent approxmaton mechansms Clam If α s n the nteror of P (x,y) then α s n the nteror of P (x,z), for any z x,y. proof Suppose α ɛ P (x,y) for some ɛ> 0. By neutralty we have that ɛ/4 ɛ/8 = ɛ/8 P (y,z). By Clam we now get that α ɛ/4 P (x,z), whch mples that α s n the nteror of P (x,z). By smlar arguments, we also have that f α s n the nteror of P (x,z) then α s n the nteror of P (w, z). Thus we get that for any x,y,w,z A, not necessarly dstnct, the nteror of P (x,y) s equal to the nteror of P (w, z). Denote the nteror of P (x,y)asp. Clam P s convex. proof We show that α, β P mples (α + β)/2 P. A known fact from convexty theory then mples that P s convex. 2 By Clam 12.32, α + β P.We show that for any α P we have α/2 P as well, whch then mples the Clam. Suppose by contradcton that α/2 / P. Thus by Clam 12.31, α/2 P. Then α/2 = α + ( α/2) P, a contradcton. We now conclude the proof of Theorem Neutralty mples that 0 s on the boundary of any P (x,y); hence, t s not n P. Let P denote the closure of P. By the separaton lemma, there exsts a k R n such that for any α P, k α 0. Suppose that f (v) = x for some v V, and fx any y x. Thus v(x) v(y) P (x,y), and k v(x) v(y) 0. Hence k v(x) k v(y), and the theorem follows. We have just seen a unque example, demonstratng that there exsts a doman for whch affne maxmzers are the only possblty. However, our natural focus s on restrcted domans, as most of the computatonal models that we consder do have some structure (e.g., the two domans we have consdered n ths chapter). Unfortunately, clear-cut mpossbltes for such domans are not known. Open Queston Characterze the class of domans for whch affne maxmzers are the only mplementable functons. Even ths queston does not capture the entre pcture, as, for example, t s known that there exsts an mplementable but not an affne-maxmzer CA. 3 Nevertheless, there do seem to be some nherent dffcultes n desgnng truthful and computatonallyeffcent CAs. 4 The less formal open queston therefore searches for the fundamental ssues that cause the clash. Obvously, these are related to the monotoncty condtons, but an exact quantfcaton of ths s stll unknown. 2 For α, β P and 0 λ 1, buld a seres of ponts that approach λα + (1 λ)β, such that any pont n the seres has a ball of some fxed radus around t that fully belongs to P. 3 See Lav et al. (2003). 4 Note that we have n mnd determnstc CAs.