Algorithmic Game Theory. Edited by Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay Vazirani

Transcription

1 Algorthmc Game Theory Edted by Noam Nsan, Tm Roughgarden, Éva Tardos, and Vjay Vazran

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3 Contents 1 Computatonally-Effcent Approxmaton Mechansms R. Lav page 4 3

4 1 Computatonally-Effcent Approxmaton Mechansms Ron Lav Abstract We study the ntegraton of game theoretc and computatonal consderatons. Four representatve subjects are dscussed: () The monotoncty propertes that decouple the game-theoretc ssues from the algorthmc ssues, () Sngle-Dmensonal schedulng, a problem doman that exemplfes the possbltes of monotone approxmaton algorthms, () Incentves n Combnatoral Auctons, where the clash between CS and GT s notable and dffcult, and (v) The mpossbltes of domnant-strategy mplementablty, where we pn-pont the strong mplcatons of ths soluton concept. Addtonal ssues and open questons are brefly dscussed. 1.1 Introducton Algorthms n Computer Scence, and Mechansms n Game Theory, are remarkably smlar objects. 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 n order 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 4

5 Computatonally-Effcent Approxmaton Mechansms 5 the game theoretc needs. There seems to be a basc clash between classc algorthmc technques and classc mechansm desgn technques. Ths rases many ntrgung questons: n 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. As was motvated n prevous chapters, the game-theoretc quest should start wth the soluton concept of mplementaton n domnant strateges. To desgn computatonally-effcent domnant-strategy mechansms, the algorthmc mplcatons of ths noton should be better understood. Secton 1.2 studes ths subject, descrbng two smple monotoncty condtons that enable the decouplng of algorthmc desgn from game-theoretc desgn. Wth these at hand, we can concentrate on algorthmc desgn, nvestgatng the algorthmc mplcatons of monotoncty, wth no need to mplctly check for the game-theoretc propertes. As a frst example to the constructon of computatonally-effcent approxmaton mechansms we study (Secton 1.3) the classc machne schedulng problem, from a game-theoretc pont of vew. Ths problem demonstrates a key dfference between Economcs and Computer Scence the goals of algorthms vs. socal choce functons. 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, most exstng algorthmc solutons do not yeld the desred ncentves, and exstng technques for ncentve-compatble mechansm desgn do not ft such an objectve. Ths clash was partally resolved n the recent lterature, and nsghtful technques have evolved. Combnatoral Auctons (CAs) capture many of the dffcult cases where the clash occurs between the ncentves and computatonal effcency, even though the goal s the classc economc goal of welfare maxmzaton. Chapter 11 studes ths model mostly from a computatonal pont of vew, and so our openng motvaton bols down to a concrete task n ths case: desgnng computatonally effcent and ncentve compatble CAs. Ths s the subject of Secton 1.4. We frst descrbe a technque to plug-n truthfulness n a certan class of algorthms, usng lnear programmng and randomzaton. Ths s used to gve tght bounds for the most general case, showng that the clash can be resolved f one s allowed to use randomness. We then attempt to avod randomzaton, and n addton to answer some crtcsms on drect revelaton mechansms. We descrbe an ascendng aucton for strategc players that uses a new novel soluton concept. A broader vew of addtonal mechansms and open questons concludes ths secton.

6 6 R. Lav In Secton 1.5 we study the mpossbltes assocated wth domnantstrategy mplementablty. To demonstrate postve results for Combnatoral Auctons, we have used ether randomzed technques or alternatve soluton concepts. Does the three requrements: () determnstc truthfulness, () computatonal-effcency, and () non-trval approxmaton guarantees, clash n a fundamental way when one consders a rch enough doman lke general Combnatoral Auctons? The answer to ths queston s, to date, unknown. But n ths secton we descrbe another, rcher doman for whch mpossblty results do exst. 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 fundamentally dfferent structure, whch causes a 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, wth enlghtenng dfferences. See Chapter 13 for examples. 1.2 Monotoncty and Implementablty When approachng mechansm desgn wth tradtonal algorthmc tools, t would be helpful to obtan a decouplng of the game theoretc ssues from the algorthmc ssues. Ths could be acheved by formulatng algorthmc condtons that mply the desred game-theoretc propertes. If that was at hand, one could focus on algorthmc desgn, leavng the game-theoretc type of analyss behnd. We next descrbe two propertes, cyclc monotoncty and weak monotoncty, that acheve that A socal choce settng The secton bulds on the basc notons ntroduced n Chapter 9. As dscussed there, t wll be convenent to use the abstract socal choce settng: 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. A drect mplcaton of ths mplementablty

7 Computatonally-Effcent Approxmaton Mechansms 7 requrement s the exstence of a prce functon p : V A R, for every player, wth the followng property. Fx any v V,andanyv,v V. Suppose that f(v,v )=a and f(v,v ) =b. Then t s the case that: v (a) p (a, v ) v (b) p (b, v ) (1.1) In other words, player s utlty from declarng hs true v s no less than hs utlty from declarng some le, v, no matter what the other players declare. Gven a socal choce functon f, we ask f there exst prce functons that satsfy 1.1. In ths secton we provde condtons on f that guarantee ths Two monotoncty condtons Fx a player, and fx the declaratons of the others to v. Let us assume, wthout loss of generalty, that f s onto A (or, alternatvely, defne A to be the range of f(,v ), and replace A wth A for the dscusson below). Snce the prces of Eq. 1.1 now become constant, we smply seek an assgnment to the varables {p a } a A such that v (a) v (b) p a p b for every a, b A and v V wth f(v,v )=a. Ths motvates the followng defnton: δ a,b. =nf{v (a) v (b) v V,f(v,v )=a} (1.2) Wth ths we can rephrase the above assgnment problem, as follows. We seek an assgnment to the varables {p a } a A that satsfes: p a p b δ a,b a, b A (1.3) Ths s easly solved by lookng at the representaton graph: Defnton 1.1 The representaton graph of a socal choce functon f s a drected weghted graph G =(V,E) wherev = A and E = A A. The weght w a,b of an edge a b (for any a, b A) sδ a,b. In other words, the graph nodes are exactly the alternatves of A, andfor any a, b A we have a drected edge a b wth weght w a,b = δ a,b.wecan now apply a standard basc result of graph theory: Proposton 1.2 There exsts a feasble assgnment to 1.3 f and only f the representaton graph has no negatve-length cycles. Furthermore, f all cycles are non-negatve, the feasble assgnment s as follows: set p a to the length of the shortest path from a to some arbtrary fxed node a A. See Chapter 9 for full detals.

8 8 R. Lav Proof Suppose frst that there exsts a negatve-length cycle a 1,a 2,..., a k,.e. a 1 = a k and k =1 δ a,a +1 < 0. Consder the followng nequaltes: p a1 p a2 δ a1,a 2. p ak 1 p ak δ ak 1,a k Summng these k 1 nequaltes, the left-hand sde sums to zero, whle the rght-hand sde s the cycle s length, whch s negatve. Thus no assgnment of p a1,..., p ak 1 satsfes all these nequaltes. Ths shows one drecton. Now suppose that every cycle s non-negatve, fx some arbtrary a A, andset p a to the length of the shortest path from a to a (snce there are no negatve cycles, the shortest path s well defned). Let us verfy that p a p b δ a,b for any a, b A. The shortest path from a to a s not longer than w a,b plus the shortest path b to a.i.e.p a δ a,b + p b. Wth ths we can easly state a condton for mplementablty: Defnton 1.3 (Cyclc monotoncty) A socal choce functon f satsfes cyclc monotoncty f for every player, v V, some nteger k A, and v 1,..., vk V, k [v j (a j) v j (a j+1)] 0 j=1 where a j = f(v j,v ) for1 j k, anda k+1 = a 1. Proposton 1.4 f satsfes cyclc monotoncty f and only f the representaton graph of f has no negatve cycles. Proof If the graph has no negatve cycles then k j=1 [vj (a j) v j (a j+1)] k j=1 δ a j,a j+1 0. In the other drecton, suppose that the cyclc monotoncty condton s volated wth v 1,..., vk. Therefore k j=1 δ a j,a j+1 k j=1 [vj (a j) v j (a j+1)] < 0andsothecyclea 1,..., a k,a 1 s negatve. Corollary 1.5 A socal choce functon f s domnant-strategy mplementable f and only f t satsfes cyclc monotoncty. Cyclc monotoncty satsfes our motvatng goal: a condton on f that nvolves only the propertes of f, wthout exstental prce qualfers. However, t s qute complex. k could be large, and a shorter condton would have been ncer. Weak monotoncty (W-MON) s exactly that: Defnton 1.6 (Weak monotoncty) A socal choce functon f satsfes W-MON f for every player, every v, and every v,v V wth f(v,v )=a and f(v,v ) =b, v (b) v (b) v (a) v (a),

9 Computatonally-Effcent Approxmaton Mechansms 9 In other words, f the outcome changes from a to b when changes her type from v to v then s value for b has ncreased at least as s value for a n the transton v to v. Note that W-MON s equvalent to cyclc monotoncty wth k = 2, or, alternatvely, to the requrement that the representaton graph has no negatve two-edge cycles. Hence t s necessary for truthfulness. As t turns out, many tmes t s also suffcent: Theorem 1.7 If the doman V s convex then any socal choce functon f that satsfes W-MON s domnant-strategy mplementable. We next gve a proof for the specal case of order-based domans. Recall that we fx a player and some v V. Snce we are nterested only n, w.l.o.g. for any a, b A there exsts v V such that v (a) v (b),.e. a and b are not nherently the same (otherwse we remove b from A). Defnton 1.8 (Order-based doman) A doman V s order-based f there exsts a partal order over the set A such that for any v R A, v V f and only f v (a) v (b) for every a, b A wth a b. Note that the f and only f statement mples that every v R A that respects must belong to V. Thus for example V s unbounded. CAs (see Chapter 11) are an example for an ordered-based doman. An alternatve s smply a subset of tems (snce we are n a sngle player settng), and we naturally get the partal order snce for any S T, v (S) v (T ). Theorem 1.9 If the doman V s ordered-based then any socal choce functon f that satsfes W-MON s domnant-strategy mplementable. Proof We need several clams to prove the theorem. Clam 1.10 If f(v )=a then v (x) v (a) δ xa for any x A. Proof W-MON mples that every two-edge cycle s non-negatve,.e. δ xa + δ ax 0. By defnton, we get v (a) v (x) δ ax δ xa. Clam 1.11 If x a then for any c A, δ cx δax. Proof v V, v (a) v (x), hence v (c) v (x) v (c) v (a). Let c A be some maxmal alternatve w.r.t.,.e. for any a c, a c. Thus, for any v V and any Δ > 0, the valuaton v that s defned to be v (a) =v (a) fora c and v (c) =v (c)+δmustbelongtov. Clam 1.12 For any a, b A, δ ab δ cb δ ca. However, specal cases of CAs, e.g. bdders wth submodular valuatons, are not ordered based snce a partal order by tself cannot capture the addtonal structure mposed there.

10 10 R. Lav Proof It suffces to show that for every v wth f(v )=a, v (a) v (b) δ cb δ ca. Suppose by contradcton that v (a) v (b) <δ cb δ ca. Rearrangng, and usng Clam 1.10 wth x = c, wegetthatv (c) v (a)+δ ca <v (b)+δ cb. Let X = {x A x a and v (x) = v (a)}. If we wsh to ncrease a s value by ɛ, we need to ncrease the value of every alternatves n X by ɛ, to mantan the order. Fx some ɛ > 0andΔ> 0 such that v (a)+ɛ + δ ca <v (c)+δ<v (b)+δ cb,andletv be: v (x)+δ x = c v (x) = v (x)+ɛ x X {a}\{c} v (x) otherwse By the choce of c and X, v respects and therefore v V. Snce v (a) v (a) =ɛ>0, W-MON mples that f(v ) X {a, c}. For every x X {a} \{c}, wehavethatv (x) =v (a) (by constructon of X and v )andδ cx δ ca (by clam 1.11). Thus v (x) +δ cx v (a) +δ ca <v (c). Rearrangng, we get v (x) v (c) < δ cx δ xc, mplyng that f(v ) x. Thus f(v )=c. Butv (c) v (b) <δ cb, a contradcton. By Clam 1.12, we get an assgnment that satsfes 1.3: For any x A set p x = δ cx. Ths concludes the proof of the theorem. W-MON s suffcent for mplementablty for every convex doman, but ths does not hold for every doman. An exact characterzaton s mssng: Open Queston Exactly characterze the domans for whch W-MON s suffcent for mplementablty. 1.3 Sngle-Dmensonal Domans and 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 exst: Schedulng related machnes. In ths doman, n jobsaretobeas- sgned 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 on machne. Our schedule ams to mnmzes the term max l /s, (the makespan). Each machne s a

11 Computatonally-Effcent Approxmaton Mechansms 11 q(x) q(c'') q(c) q(c') B A c'' c c' x Fg A monotone load curve. selfsh entty, ncurng 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 1.13 (sngle-dmensonal lnear domans) A doman V of player s sngle-dmensonal and lnear f there exst non-negatve real constants (the loads ) {q,a } a A such that, for any v V,thereexstsc R (the cost ) such that v (a) =q,a c. In other words, the type of a player s smply her cost c 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 cost-mnmzaton) problems, where the types are non-negatve. We am to desgn a computatonally-effcent approxmaton algorthm, that s also mplementable. As the socal goal s a certan mn-max crtera, and not to mnmze the sum of costs, we cannot use the general VCG technque. Snce we have a convex doman, we already know that we need a weakl monotone algorthm. But what exactly does ths mean? Luckly, the formulaton of weak monotoncty can be much smplfed for ths case. If we fx the costs c declared by the other players, an algorthm for a sngle-dmensonal lnear doman determnes the load q (c) ofplayer as a functon of her reported cost c. Take two possble types c and c, and suppose c >c. Then weak monotoncty reduces to q (c )(c c) q (c)(c c),

12 12 R. Lav whch holds ff q (c ) q (c). Hence by theorem 1.7 we get that such an algorthm s mplementable f and only f ts load functons are monotone non-ncreasng. Fgure 1.1 descrbes ths, and wll help us to fgure out the requred prces for mplementablty. 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 Fg. 1.1, we can easly verfy that these prces lead to ncentve compatblty: Suppose 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, sncehscostfrom 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 non-negatve, exactly c q (x)dx. From all the above we get: Theorem 1.14 An algorthm for a sngle-dmensonal lnear doman s mplementable f and only f ts load functons are non-ncreasng. Furthermore, f ths s the case then chargng from every player aprce 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 use a randomzed mechansm, and 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 acheves a reasonable approxmaton. The frst attempt could be to check f exstng approxmaton algorthms for the problem are mplementable, but the answer s unfortunately no. We next show

13 Computatonally-Effcent Approxmaton Mechansms 13 how ths clash can be partally resolved by desgnng a monotone approxmaton algorthm. Note that we face a classc algorthmc problem no game-theoretc ssues are left for us to handle. Before we start, we 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, andsometarget makespan T. If a schedule has makespan at most T then t must assgn any job out of 1,..., j to a 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: j k=1 T p k (j,t ). (1.4) l=1 s l Now defne T j =mnmax{ p j j k=1, p k s l=1 s } (1.5) l Lemma 1.15 For any job-ndex j, the optmal makespan s at least T j. Proof Fx any T<T j.weprovethatt volates 1.4, 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 1.4 for each case seperately. Case 1 (P j k=1 p k P j l=1 s l T j s the mn-max, we get T j and T<T j = P j k=1 p k P j l=1 s l Case 2 (P j k=1 p k P j l=1 s l p j s j ): For j + 1 the max term s receved by P j k=1 p k P (j,t ) < p j s j ): T j p j s j +1,Snce p j s j +1. Snce T<T j,wehave(j, T ) j, P j k=1 p k P j 1 l=1 s l l=1 s l. Hence T volates 1.4, as clamed. snce T j s the mn-max, and the max for j 1 s receved at the rght. Addtonally, (j, T ) < j snce T j = p j and T<T j.thust<t j P j k=1 p k P j 1 l=1 s l P j k=1 p k P (j,t ) l=1 s l, as we need. Wth ths, we get a good lower bound estmate of the optmal makespan: T LB =max j T j (1.6) The optmal makespan s at least T j for any j, hence t s at least T LB. s j

14 14 R. Lav The 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 1.16 (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. Lemma 1.17 There s enough space to fractonally assgn all jobs, and f job j s fractonally assgned to machne then p j /s T LB. Proof Let j be the ndex that determnes T j. Snce T LB T j P j k=1 p k P j l=1 s l we can fractonally assgn jobs 1,.., j up to machne j.sncet j p j /s j we get the second part of the clam, and settng j = n gves the frst part. Lemma 1.18 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. Snces l s l for all l we have that T LB T LB.Now, f l = T LB s,.e. was full, then l T LB s T LB s = l. Otherwse l <T LB s, hence s the last non-empty 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). Thus we resort to randomzaton: Defnton 1.19 (Roundng the schedule) 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 1.20 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. Ifj 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,

15 Computatonally-Effcent Approxmaton Mechansms 15 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 1.17 we have that max{p j,p k } T LB and snce T LB s not larger than the otpmal maxmal makespan, the approxmaton clam follows. For truthfulness, we only need that the expected load s monotone. Note that machne 1getsjobj wth probablty α, so gets t wth probablty 1 α, and gets k wth probablty α. So the expected load of machne s exactly ts fractonal load. The clam now follows from Lemma A note about prce computaton s n place. A polynomal-tme mechansm must compute the prces n polynomal tme. To compute the prces for our mechansm we need to ntegrate over the load functon of a player, fxng the others speeds. 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 have verfed that prce computaton s polynomal-tme n our case. 1.4 Incentves n Combnatoral Auctons Combnatoral Auctons (CAs) are a central model wth theoretcal mportance and practcal relevancy. 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 Combnatoral Auctons, provdng a comprehensve dscusson on the model and ts varous computatonal aspects. Here, we touch on a dfferent, mportant pont: 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 (Ω) to n players. Players value subsets of tems, and v (S) denotes s value of a bundle S Ω. 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 )thatmaxmzes v (S ). Snce a general valuaton has sze exponental nn and m, the representaton ssue must be dscussed. Chapter 11 examnes two models. In the bddng languages model, the bd of a player represents hs valuaton s a concse way. For ths model t s NP-hard to obtan an approxmaton of Ω(m 1/2 ɛ ), for any ɛ>0 (f sngle-mnded bds are allowed). In the query

16 16 R. Lav 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 of Ω(m 1/2 ɛ ) for any ɛ>0. These bounds are tght, as there exst 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. We next detal two mechansms that resolve ths. The frst s a randomzed mechansm for the general case, whch develops an LP-based technque to convert algorthms to truthful mechansms. The second s a determnstc mechansm for a specal case of sngle-value players, that takes the useful form of an ascendng aucton, and uses a novel soluton concept to analyze the strategc propertes. A short survey of other mechansms and open problems concludes the secton A randomzed mechansm for the general case We descrbe a rather general technque to convert approxmaton algorthms to truthful mechansms, by usng randomzaton: gven any algorthm to the 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 x,s v (S) (CA-P) subject to,s x,s 1 for each player (1.7) S S:j S x,s 1 for each tem j (1.8) x,s 0, S.

17 Computatonally-Effcent Approxmaton Mechansms 17 By constrant 1.7, a player receves at most one ntegral subset, and constrant 1.8 ensures that each tem s not over-allocated. 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 (for example 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 non-zero 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, wecanusevcg(seechapter9)toget: Proposton 1.21 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 1.22 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 Lemma 1.23 (The decomposton lemma) Suppose A verfes a c-ntegraltygap 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 1.24 (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).,s

18 18 R. Lav 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): VCG s ndvdually ratonal, hence p F (v) v (x ). Thus p R (v) v (x l ) for any l I. Lemma 1.25 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)] = l I λ l [v (x l )/v (x )] p F (v) = (1.9) [p F (v)/v (x )] l I λ l v (x l )=[p F (v)/v (x )] v (x /c) =p F (v)/c 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 v (x ) p F (v,v ) v (z ) p F (v,v ) (1.10) By 1.9 and by the decomposton, dvdng 1.10 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 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 a weakenng of the soluton concept to 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: 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

19 Computatonally-Effcent Approxmaton Mechansms 19 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 c x,s w,s + z (D) (,S) E s.t. l I s.t. l λ l x l,s = x,s c (, S) E (,S) E x l,sw,s + z 1 l I (1.11) (1.12) λ l 1 z 0 l λ l 0 l I w,s unconstraned (, S) E. Constrants 1.11 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 1.11 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 1.26 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. We have (,S) E xl,s w,s 1 c (,S) E x,sw,s > 1 z, where the frst nequalty follows snce A s a c-approxmaton to the fractonal optmum, and the second nequalty s the volated nequalty of the clam. Thus constrant 1.12 s volated (for x l ). Corollary 1.27 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 1.26 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,

20 20 R. Lav 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 1.28 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 c (snce w,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,setx +,T =1forT =argmax 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 xl as follows: set x l,s = x +,S f w,s 0and0otherwse. Clearly, (,S) E xl,s w,s = (,S) E x+,s w+,s, and x l 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 sngle-mnded players satsfes the requrement, and a natural varant verfes a 2 m ntegralty-gap for CA-P:

21 Computatonally-Effcent Approxmaton Mechansms 21 Defnton 1.29 (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 1.30 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 > 0and(,S) was removed from E when (, S ) was added }. We show that (,S) Y y,sw,s ( 2 m)w,s, whch proves the clam. We frst have: (,S) Y y,sw,s = w (,S) Y y,s,s S (1.13) S w,s S (,S) Y y,s S w,s ( S (,S) Y y,s)( (,S) Y y,s S ) 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 1+1 S +1 (1.14) (,S) Y (,S) Y,j S (,S) Y j S 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 (1.15) (,S) Y j Ω (,S) Y,j S Combnng 1.13, 1.14, and 1.15, 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: Theorem 1.31 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. Perhaps the most notable case s mult-unt 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 decompostonbased mechansm s the only truthful mechansm wth ths tght guarantee.

22 22 R. Lav 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 submodular 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 worstcase 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 Algorthmc mplementaton n undomnated strateges As mentoned, randomzaton has problematc aspects for mechansm desgn, and the search for determnstc mechansms s an mportant challenge. In addton, most computatonally effcent mechansms are drect revelaton, but n practce, ndrect mechansms (players compete by rasng prces and wnners pay ther last bd) are much preferred. We next dscuss an nterestng attempt to handle these crtcsms. Sngle-Value players. The mechansms of ths secton ft the specal case of players that desre several dfferent bundles, all for the same value: Player s sngle-valued f there exsts v 1 such that for any bundle s, v (s) {0, v }.I.e.desres any one bundle out of a collecton S of bundles, for a value v. We denote such a player by ( v, S ). v and S )areprvate nformaton of the player. Snce S may be of sze exponental n m, we assume the query access model, as detaled below. An teratve wrapper. We start wth a wrapper to a gven algorthmc sub-procedure, whch wll eventually convert algorthms to a mechansms, wth a small approxmaton loss. It operates n teratons, wth teraton ndex j, and mantans the tentatve wnners W j,thesure-losersl j,and a wnnng bundle s j for every. In each teraton, the sub-procedure s

23 Computatonally-Effcent Approxmaton Mechansms 23 nvoked to update the set of wnners to W j+1 and the wnnng bundles to s j+1. Every actve non-wnner then chooses to double hs bd (v j )orto permanently retre. Ths s terated untl all non-wnners retre: Defnton 1.32 (The Wrapper) Intalze j =0,W j = L j =, andfor every player, v 0 =1ands0 = Ω. Whle W j L j all players, perform: 1. (W j+1,s j+1 ) PROC(v j,s j,w j ). 2. / W j+1 L j, chooses whether to double hs value (v j+1 2 v j ) or to permanently retre (v j+1 0). For all others set v j+1 v j. 3. Update L j+1 = { N v j+1 =0} and j j + 1, and reterate. Outcome: Let J = j (total number of teratons). Every W J gets s J and pays v J. All others lose (get nothng, pay 0). For feasblty, PROC must mantan:, W j+1, s j+1 s j+1 =. We need to analyze the strategc choces of the players, and the approxmaton loss (relatve to PROC). Ths wll be done gradually. We frst worry about mnmzng the number of teratons: Defnton 1.33 (Proper procedure) PROC s proper f (1) Pareto: / W j+1 L j, s j+1 ( l Wj+1 s j+1 l ), and(2)shrnkng-sets:, s j+1 s j. A reasonable player wll not ncrease v j above v, otherwse hs utlty wll be non-postve (ths strategc ssue s formally dscussed below). Assumng ths, there wll clearly be at most n log(v max ) teratons, where v max = max v. Wth a proper procedure ths bound becomes ndependent of n: Lemma 1.34 If every player never ncreases v j above v, then any proper procedure performs at most 2 log(v max )+1 teratons. Proof Consder teraton j =2 log(v max ) + 1, and some 1 / W j+1 L j that (by contradcton) doubles hs value. By Pareto, there exsts 2 W j+1 such that s j+1 1 s j+1 2. By shrnkng-sets, n every j <jther wnnng bundles ntersect, hence at least one of them was not a wnner, and doubled hs value. But then v j 1 v max, a contradcton. Ths affects the approxmaton guarantee, as shown below, and also mples that the Wrapper adds only a polynomal-tme overhead to PROC. A warm-up. Consder the case of known sngle mnded players (KSM), where a player desres one specfc bundle, s, whch s publc nformaton (she can only le about her value). Ths allows for a smple analyss: the wrapper converts any gven c-approx. to a domnant-strategy mechansm wth O(log(v max ) c) approxmaton. Thus, we get a determnstc technque to convert algorthms to mechansms, wth a small approxmaton loss.

24 24 R. Lav Here, we ntalze s 0 = s,andsets j+1 = s j, whch trvally satsfes the shrnkng-sets property. In addton, pareto s satsfed w.l.o.g. snce f not, add wnnng players n an arbtrary order untl pareto holds. For KSM players ths takes O(n m) tme. Thrd, we need one more property: Defnton 1.35 (Improvement) W j+1 v j W j v j. Ths s agan wthout loss of generalty: f the wnners outputted by PROC volate ths, smply output W j as the new wnners. To summarze, we use: Defnton 1.36 (The KSM-PROC) Gven a c-approx. A forksmplayers, KSM-PROC nvokes A wth s j (the desred bundles) and v j (player values). Then, t post processes the output to verfy pareto and mprovement. Proposton 1.37 Under domnant strateges, retres ff v /2 v j v. (the smple proof s omtted). For the approxmaton, the followng analyss carres through to the sngle-value case. Let S s j = {s S s s j },and R j ( v, S)={ (v,s s j ) retred at teraton j }, (1.16).e. for every player that retred at teraton j the set R j ( v, S)contansa sngle-value player, wth value v (gven as a parameter), and desred bundles S s j (where S s gven as a parameter). For the KSM case, R j ( v, S) s exactly all retred players n teraton j, as the operator s j hasno effect. Hence, to prove the approxmaton, we need to bound the value of the optmal allocaton to the players n R = J j=1 R j( v, S). For an nstance X of sngle-value players, let OPT(X) be the value of the optmal allocaton to the players n X. In partcular: OPT(R j ( v, S)) = max all allocatons (s1,...,s n) s.t.s S { j : s v }. s Defnton 1.38 (Local approxmaton) A proper procedure s a c-localapproxmaton w.r.t a strategy set D f t satsfes mprovement, and, for any combnaton of strateges n D and any teraton j, Algorthmc approxmaton OPT(R j (v j, S)) c W j v j Value bounds v j v (s j ), and, f retres at j then vj v /2. Clam 1.39 Gven a c-approxmaton for sngle mnded players, KSM-PROC s a c-local-approxmaton for the set D of domnant strateges. (Ths follows from all the above). We next translate local approxmaton to global approxmaton (ths s vald also for the sngle-value case): Clam 1.40 A c-local-approxmaton satsfes OPT( R) 5 log(v max ) c W J v whenever players play strateges n D.

25 Computatonally-Effcent Approxmaton Mechansms 25 Proof By the value bounds, OPT(R j ( v, S)) 2 OPT(R j (v j, S)). Snce OPT(R j (v j, S)) c W j v j by algorthmc approxmaton, W j v j W j+1 v j+1 by mprovement, and v J v (agan by the value bounds), we get OPT(R j ( v, S)) 2 c W J v. Hence OPT( R) J j=1 OPT(R j( v, S)) J 2 c W J v.sncej 2 log(v max ) + 1, the clam follows. For sngle-mnded players, R s the set of losng players, hence we conclude: Theorem 1.41 Gven any c-approx. for KSM players, the Wrapper wth KSM-PROC mplements an O(log(v max ) c) approx. n domnant strateges. A sub-procedure for sngle-value players. Two assumptons are relaxed: players are now mult-mnded, and ther desred bundles are unknown. Here, we defne the followng specfc sub-procedure. For a set of players X, let Free(X, s j+1 ) denote the tems not n X s j. Defnton 1.42 (1-CA-PROC) Let M j =argmax N {v j }, GREEDY j =. For every player wth v j > 0, n descendng order of values, perform: Shrnkng the wnnng set: If / W j allow hm to pck a bundle s j+1 Free(GREEDY j,s j+1 ) s j such that sj+1 m. In any other case ( W j or does not pck) set s j+1 = s j. Updatng the current wnners: If s j+1 m,addto any of the allocatons W {W j,m j,greedy j } for whch s j+1 Free(W, s j+1 ). Output s j+1 and W {W j,m j,greedy j } that maxmzes W vj. Recall that the non-wnners then ether double ther value or retre, and we reterate. Ths s the man conceptual dfference from regular drect revelaton mechansms: here, the players themselves gradually determne ther wnnng set (focusng on one of ther desred bundles), and ther prce. Intutvely, t s not clear how should a reasonable player shrnk hs wnnng set, when approached. Ideally, a player should focus on a desred bundle that ntersects few, low-value compettors. But n early teratons ths nformaton s not avalable. Thus there s no clear-cut on how to shrnk the wnnng set, and the resultng mechansm does not contan a domnant strategy. Ths brngs up the need for a new soluton concept. Algorthmc mplementaton n undomnated strateges. We would lke to allow the mechansm desgner to leave n several strategc choces, but to requre the approxmaton n each of these choces. Defnton 1.43 (Algorthmc mplementaton) A mechansm M s an

26 26 R. Lav algorthmc mplementaton of a c-approxmaton n undomnated strateges f there exsts a set of strateges, D, such that () M obtans a c- approxmaton for any combnaton of strateges from D, n polynomal tme, and () For any strategy not n D, there exsts a strategy n D that weakly domnates t, and ths transton s polynomal-tme computable. The mportant ngredents of domnant-strateges mplementaton are here: the only assumpton s that a player s wllng to replace any chosen strategy wth a strategy that domnates t. Indeed, ths guarantees at least the same utlty, even n the worst-case, and by defnton can be done n polynomal tme. In addton, agan as n domnant-strategy mplementablty, ths noton does not requre any form of coordnaton among the players (unlke Nash equlbrum), or that players have any assumptons on the ratonalty of the others (as n teratve deleton of domnated strateges ). However, two dfferences from domnant-strateges mplementaton are worth mentonng: (I) A player mght regret hs chosen strategy, realzng n retrospect that another strategy from D would have performed better, and (II) Decdng how to play s not straght-forward. Whle a player wll not end up playng a strategy that does not belong to D, t s not clear how wll he choose one of the strateges of D. Ths may depend for example on the player s own belefs about the other players, or on the computatonal power of the player. Another remark, about the connecton to mplementaton n undomnated strateges, s n place. The defnton of D does not mply that all undomnated strateges belong to D, but rather that for every undomnated strategy, there s an equvalent strategy nsde D (.e. a strategy that yelds the same utlty, no matter what the others play). The same problem occurs wth domnant-strategy mplementatons, e.g. VCG, where t s not requred that truthfulness should be the only domnant strategy, just a domnant strategy. Analyss. We proceed by characterzng the requred set D of strateges. We say that player s loser-f-slent at teraton j f, when asked to shrnk her bundle by 1-CA-PROC, v j v /2 (retres f losng), / W j and / M j (not a wnner), and s j ( W j s j+1 ) and s j ( M j s j+1 ) (remans a loser after pareto). In other words, a loser-f-slent loses (regardless of the others actons) unless she shrnks her wnnng set. Let D be all strateges that satsfy, n every teraton j: () v j v (s j ), and, f retres at j then vj v /2. () If s loser-f-slent then she declares a vald desred bundle s j+1, f such a bundle exsts. There clearly exsts a (poly-tme) algorthm to fnd a strategy st D that