15-451/651: Design & Analysis of Algorithms December 3, 2013 Lecture #28 last changed: November 28, 2013

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1 15-451/651: Design & nlysis of lgorithms Deemer 3, 2013 Leture #28 lst hnged: Novemer 28, 2013 Lst time we strted tlking out mehnism design: how to llote n item to the person who hs the mximum vlue for it, so tht we re inentive omptile no one hs n inentive to misreport their vlutions (.k.. lie). Then we sw the very generl VG mehnism whih extends to ritrry llotion prolems, with the sme inentive-omptiility property. ut VG required s input the vlutions of ll plyers for ll outomes. In this leture, we give different kind of mehnism, more nturl one, for mthing mrkets, where we wnt mthing etween uyers nd goods. 1 1 Mthing Mrkets Think of the setting of ssigning dorm rooms to students, houses to uyers, et. Eh person wnts one of eh item. Eh item n e given to t most one person. Formlly, onsider the setting where there is set I of n items on sle, nd set of n uyers who my uy some of them. The vriles j will refer to generi uyer nd vriles i will refer to items. Eh uyer wnts to uy t most one item, nd hs vlution v j (i) Z for eing lloted item i. We ssume tht v j (i) 0. n exmple with n = 3, items on the left, uyers on the right, uyer vlues items,, t 10, 9, 0 respetively. Items uyers If the prie for item i is p i, the utility tht j gets from eing ssigned item i is u j (i) := v j (i) p i. Nturlly, the uyer seeks to mximize his utility. Moreover, we wnt to ensure individul rtionlity, tht the ssignment results in every uyer hieving non-negtive utility; hene if item i is ssigned to uyer j t prie p i, it must e tht v j (i) p i. The soil welfre hieved y n ssignment of items to uyers is the sum of vlutions of the uyers for their llotions. Not the sum of the utilities, the tul vlutions. (If you sum utilities, you must inlude the utility of the sellers in the summtion.) In the exmple ove, t pries p = 6, p = 5, p = 3, ssigning,, would hieve soil welfre = 22. The utility of the plyers under this ssignment would e 10 6 = 4, 7 5 = 2, 5 3 = 2 respetively. We wnt to find n llotion tht mximizes the soil welfre. Lst time we sw the VG mehnism tht finds n llotion (nd priing sheme) whih mximizes the soil welfre, nd lso gives inentive omptiility. Tht is, no plyer hs ny reson to misreport their vlutions. nd wht would VG do in this setting of mthing mrket? It would ompute the mx-weight mthing in the iprtite grph where the weight of n edge (i, j) is the vlution v j (i). This would give us the llotion. Then the pyment tht uyer j mkes is the externlity tht j uses: the 1 gentle nd very redle introdution to mthing mrkets (with more exmples) n e found in hpter 10 of the Esley-Kleinerg ook: Networks, rowds, nd Mrkets: Resoning out Highly onneted World. 1

2 differene etween the vlutions of ll the other plyers when i prtiiptes, nd when it does not. In the exmple ove, mx-weight mthing is M V G = {,, }. The orresponding VG pries re given elow. $2 (Wht is the prie VG hrges to? The vlutions of the other plyers under M V G is 7+5 = 12. Now if does not id, then the optiml llotion for the others is, with totl vlution 14. Hene the externlity used is = 2, whih is the prie p VG sets for.) ut the VG mehnism required us to sumit our vlutions to the enter, whih then finds the llotion nd pries in this entrlized wy. Tody we will give different, distriuted mehnism tht ts more like rel mrkets seem to do: it will grdully rise pries until they stilize, nd eh uyer will uy their fvorite item (whih mximizes their own utility), nd tht will e the optiml llotion, mximizing soil welfre. Importnt note: for ll of Setion 1, we will ssume everyone ts truthfully. It will e just n lgorithm design prolem. We ome k to inentive-omptiility in Setion Preferred Items nd the Preferred Grph Given pries p 1, p 2,..., p n for the items, the preferred items for uyer j re ll the items tht mximize his utility (nd for whih the utility is non-negtive). I.e., if u j := mx i I (v j(i) p i ) (1) is the mximum utility tht j n hieve t these prie, nd if u j these pries) re 0, the preferred items (t Else if u j < 0 then uyer j hs no preferred items (nd S j = ). S j := {i I v j (i) p i = u j}. (2) Now you n rete the preferred grph H y hving node for eh item i I, one for eh uyer j [n], nd n edge (i, j) if item i is preferred item for j, i.e., i S j. (ll this is with respet to the urrent set of pries, of ourse.) Here re the preferred grphs with respet to two different prie settings. $2 1.2 Mrket-lering Pries set of pries p 1, p 2,..., p n, one for eh item, is lled mrket-lering if 2

3 There is mthing in the preferred grph H (t these pries) tht mthes ll the uyers to items, nd if n item i is not mthed, then its prie p i = 0. (This is irrelevnt when the numer of items equls the numer of uyers, sine if ll uyers re mthed, so re the items. ut if you hd more items, you need this ondition.) This mens t these pries, eh uyer n ome y nd pik some item tht mximizes her utility 2, nd the mrket will ler. (Of ourse, items tht re not desirle might e left ehind, ut they hve no vlue in this mrket nd hene hve zero prie.) The mthing given y the definition of mrket-lering pries gives us the orresponding llotion. In the previous figure on the right p = 2, p = 1, p = 0 were mrket-lering pries, the grph hs perfet mthing. ut p = 1, p = 0, p = 0 re not mrket-lering pries. Why not? There is no perfet mthing in the preferred grph H (on the left). How do you prove the sene of perfet mthing? Look t the set S = {,, }. The neighorhood hs size N(S) = {, } = 2 wheres this set hs size S = 3. There is no wy we n mth S to its neighors. Hll s theorem 3 sys tht iprtite grph with equl numer of verties on either side hs no perfet mthing if nd only if there is set S with neighorhood size N(S) < S. The set N(S) is lled onstrited set or Hll set. Here is nother exmple of iprtite grph with n = 8 where there is no perfet mthing, s shown y onstrited set, the four red verties on the right with only three (lue) neighors on the left. Tke wy messge: if there is no perfet mthing in iprtite grph, we n lwys find onstrited set. This will e useful lter. 1.3 Mrket-lering Pries nd Soil Welfre Mximiztion Why re we interested in mrket-lering pries, prt from them seeming nturl? nd do they lwys exist? The first nswer is short nd sweet. Theorem 1 If there re mrket-lering pries, then the orresponding llotion mximizes soilwelfre. Proof: Fix ny mrket-lering pries p i. Let M e the mthing from items to uyers. y the definition, eh plyer is given preferred item, nd hene is (individully) mximizing her utility. So the mximum utility tht n e hieved (y ny mthing) with respet to these pries is preisely u j (i) = (v j (i) p i ). (i,j) M (i,j) M 2 Due to ties, they n t pik n ritrry item tht mximizes their utility; we might hve to give them utilitymximizing item reking ties refully 3 You might hve seen this theorem in In ny se, this is yet nother onsequene of the mxflow-minut theorem, you should prove it for yourself. 3

4 Now look t the items. Some of them re mthed to uyers nd their pries re inluded in the sum. Others re not mthed, nd y the definition gin, their pries must e zero. Hene the ove expressions re equl to v j (i) p i. i (i,j) M ut the ltter sum does not depend on the mthing, so in finding the mthing mximizing this expression, we hve found mthing tht mximizes the soil welfre. QED. ool: if we hve mrket-lering pries, they utomtilly mximize the vlue to the uyers. nd why should these mrket-lering pries exist? Here s n lgorithm tht will terminte with these pries. 1.4 The sending-prie Mehnism We wnt to find mrket-lering pries. The ide will e nturl one: if set of k items hs more thn k people preferring it t the urrent pries, it seems likely tht the pries of these items should rise. nd tht is preisely wht we will do. Strt with ll pries p i = 0. Our pries will lwys e integers. Rell tht the vlutions v j (i) re ll non-negtive integers. Now do the following: 1. hek if the urrent pries re mrket-lering. (I.e., uild the preferred grph, hek if there is mthing tht mthes ll the uyers to items.) If so, we re done. 2. Else, there must e some set S of uyers suh tht the set of neighors of S in the grph H (denoted y N(S)) hve stritly smller rdinlity. I.e., S > N(S). (Rell we lled N(S) onstrited set). Suh set is over-susried, the demnd for it (i.e., S ) is more thn the supply (i.e., N(S) ). So suh set seems like nturl ndidte to see prie inreses. There my e multiple onstrited sets N(S). So hoose miniml onstrited set: i.e., N(S) does not ontin nother onstrited set N(T ). Then for eh item i N(S), inrement the pries p i p i + 1. Go k to Step 1. Here s run of the lgorithm (with the onstrited sets shown in red). $2 Esy enough. ut does this lwys give us mrket-lering pries? Hmm... wht if the pries keep rising forever? It n t relly e forever, eventully some item will e too expensive for everyone nd would not e lloted. Sine we hve n equl numer of uyers nd items, we would then hve people who re not ssigned ny item t ll. Equivlently, we would my eventully get some set S with N(S) =, nd then Step 2 would not mke progress t ll. This would violte the definition of mrket lering pries, nd would e d news. No worries. Here s the good news out the lgorithm. We first show tht eh person lwys hs t lest one preferred item, nd then we show tht this mens the lgorithm dereses some potentil in eh step nd hene must eventully stop. 4

5 Lemm 2 t most n 1 items will hve stritly positive pries. Proof: Just for the ske of nlysis, let us keep tenttive mthing M round, where M ontins suset of edges in the preferred grph. We mintin the invrint tht if n item hs non-zero ost, tht item is tenttively mthed to some uyer. In mth, p i > 0 = j : (i, j) M. Initlly M is empty, whih stisfies the invrint. Suppose you hve pries nd tenttive mthing, nd now you rised pries for items in N(S). Tenttively mth ll the items in N(S) to uyers in S. (If either these items, or the uyers they re mthed to, were tenttively mthed erlier, drop those tenttive mthing edges.) Sine S > N(S) nd we hose miniml N(S), we know tht this tenttive mthing n e done. (Why? Hll s theorem gin!) So the items in N(S), they hve non-zero pries nd re tenttively mthed. nd the items outside N(S) must e tenttively mthed to uyers outside S (sine ll neighors of S re in N(S)), nd we did not drop ny suh edge. Hene, the invrint is mintined. For ske of ontrdition, if we ever wnt to rise the prie for the n th item, we would get tenttive mthing of size n, whih would men we would hve hd mthing of ll uyers to items, nd hene would not wnt to rise pries t this step. Hene, t most n 1 items would every hve their pries rised. orollry 3 Eh uyer hs degree t lest 1 in the preferred grph (t ll times). Proof: y Lemm 2, t lest one item stys t prie 0. The utility of every uyer for this item remins non-negtive, nd hene the preferred set for ny uyer n never e empty. Now we re redy to prove the min result. lerly if the lgorithm termintes it will output mrketlering pries; sine every uyer hs degree t lest 1 in this finl preferred grph, ll items will e lloted. So we show tht the lgorithm termintes. Theorem 4 The lgorithm termintes in finite time. Proof: We use potentil funtion. Given pries p 1, p 2,..., p n, define the potentil to e Φ = p i + u j. (3) items i uyers j where u j is defined s in (1). Note tht ll the terms here re non-negtive. t the eginning, the pries re zero, nd the u j = mx i I v j (i). So Φ 0 = j mx i I v j (i). It remins to show tht every time we exeute Step 2 of the lgorithm, the potentil dereses. Why? When we exeute Step 2, we rise the pries on some onstrited set N(S) of items. y orollry 3, eh uyer hs t lest one edge out of it, so N(S) 1. (The following rgument does not sy nything if N(S) =.) Now we rise the pries on N(S) nodes. This rises the potentil y N(S). However, the vlue of u j dereses for ll nodes in S, nd hene dereses the potentil y S. Sine S > N(S), the potentil dereses y t lest 1. So the lgorithm stops in t most Φ 0 steps. TW, this lgorithm is not polynomil-time, if the mximum vlution is V this proess ould tke Ω(V ) time to finish. nything interesting to sy out this? 2 Reltionship to Vikery nd VG The nending-prie ution we just sw is diret generliztion of the Vikery ution from the previous leture. In the Vikery ution there ws single item we were utioning mong n 5

6 uyers. To mke it fit our model here, just introdue n 1 dummy items (numered 2 through n), nd numer the rel item 1. nd if the uyer j hd vlue v j for the rel item, define v j (1) = v j nd v j (i) = 0 for i 2. ssuming tht eh v j > 0, the preferred grph strts off with edges from item 1 to ll uyers, nd no other uyers. So {1} is miniml onstrited set, nd we rise its prie until ll exept one uyer hs preferred edge to t lest one item from {2, 3,..., n}. This will hppen preisely when p 1 rehes the seond-highest vlution, nd the person mthed to the rel item will e the one with the highest vlution. lso, for the se when we hve K identil items to sell, the sme rgument shows tht our lgorithm ove will sell to the K highest idders, t prie equl to the vlution of the K + 1 st - highest idder. Is this just oinidene? Or ould it e the se tht the mthing nd pries found y our lgorithm ove re the VG llotion nd pries? Tht would e gret, euse this would sy tht the sending-prie mehnism is lso inentive omptile, tht it is dominnt strtegy for the plyers to ply truthfully. Well, lerly the mthing we found is the mx-weight mthing (y Theorem 1), nd hene the sme s the VG llotion. ut wht out the pries? This is little more triky, ut it is true: the pries we find re extly the VG pries! We ll not over this here, see hpter 15.9 of the Esley-Kleinerg ook for more detils. 3 Wlrsin equilirium The sme ides work for more generl mrkets where the plyers need not just wnt one item, they my e interested in uying multiple items. The vlution funtions my hve different hrteristis. For instne, you my wnt hotel room nd flight tiket nd r rentl (nd ny strit suset of these might e useless to you). Or you my wnt multiple omputers, ut eh dditionl omputer gets less vlule to you the more you hve ( diminishing returns ). Or you my vlue set t $50, or set t 00, ut ll other sets re worthless to you. One n imgine very generl vlution funtions. 4 Formlly, suppose you hve set of m items I, set of n uyers. Eh plyer j now hs vlutions v j (S) for eh suset S I of items. Let us define some useful shorthnd: given pries p 1, p 2,..., p m, define p(s) := i S p i. (4) The utility of set S for uyer j is u j (S) := v j (S) p(s). We imgine tht v j ( ) = 0, nd hene u j ( ) = 0 p( ) = 0. Definition 5 (Preferred Set) t pries p 1, p 2,..., p m, set S is lled preferred for plyer j if u j (S) 0, nd lso for ll T I, u j (T ) u j (S). Definition 6 (Wlrsin Equilirium) The pries p 1, p 2,..., p m nd llotions S 1, S 2,..., S n re t Wlrsin equilirium if () for eh j the set S j is preferred for plyer j, nd () if item i is not lloted (i.e., i / j S j ), the prie p i = 0. This is the nturl wy to extend mrket-lering pries to the more generl setting, nd proof similr to Theorem 1 shows tht if {p i } i I, {S j } j re Wlrsin equilirium, then the llotion {S j } mximizes soil welfre. 4 For more detils on this setion s mteril, see the survey y lumrosen nd Nisn from the lgorithmi Gme Theory ook, whih hs lods of other ool topis. (Link to downlod entire ook, 5M.) 6

7 3.1 Existene No Longer Gurnteed Unfortuntely, Wlrsin equiliri don t lwys exist, when the vlution funtions re llowed to e so generl. There re 2 items {, } nd 2 uyers {, }. uyer vlues the entire set t 3, nd ll other sets hve vlue 0. vlues every non-empty suset t 2. The welfre mximizing llotion is to give oth items to, with soil welfre 3. For this to e Wlrsin equilirium, wht pries should we hoose? The pries of eh of the items must e t lest 2 (for s preferred set to e empty). ut then the prie of the entire set is t lest 4, so s preferred set nnot e {, }. 3.2 n sending-prie Mehnism for Nie Vlutions The prolem in the exmple ove omes from omplementrity. Items nd omplement eh other, nd the whole {, } set is worth more to thn the sum of its prts. For exmple, think of s eing left shoe nd the right shoe. Or hotel room nd flight tiket. Or red nd utter. ut if the vlution funtions re nie, n sending-prie ution find n (pproximte) Wlrsin equilirium. Wht re these nie vlutions? These re lled sustitutes, nd is one wy of ensuring the lk of omplements. Loosely, it sys tht if you rise the prie of set of items, your preferene or demnd for some other item does not go down. (This lerly exludes omplements, sine if you were to rise the prie of left shoes, not only would the demnd for left shoes go down, ut the demnd for right shoes would lso go down despite their pries not hnging.) Definition 7 (Sustitutes) Formlly, vlution v j stisfies the sustitutes property if the following holds. For ny two pries p 1, p 2,..., p m nd q 1, q 2,..., q m where p i q i, if P is preferred set for uyer j t pries p, nd P = := {i P p i = q i } is the suset of P ontining those items whose pries did not hnge, then there exists preferred set Q for j t pries q with P = Q. Strt with tenttive ssignments T j =, nd pries p i = Look t the preferred set P j for uyer j under the following pries: eh item i T j osts p i, nd eh item i / T j osts p i + ɛ. (Think of this s sying tht quiring n item not in our tenttive set hs some ɛ extr ost, you hve to work for it.) 2. If eh suh preferred set P j = T j, stop nd return the pries p i. 3. Else, pik some uyer with P j T j, who is not stisfied. Set T j = P j, tht is, give him his preferred set. Now inrese the pries of ll items newly ssiged to j, nmely those in P j \ T j, to p i + ɛ. Finlly, remove these items in P j from other tenttive sets: T j T j \P j. Go k to Step 1. gin, nturl sending-prie mehnism. If the vlutions stisfy the sustitutes property, this gives us n ɛ-pproximte Wlrsin equilirium. The preise definition of this pproximte equilirium is not importnt, wht is interesting is how this simple lgorithm n give mrketlering pries. (Sdly, now the pries re no longer VG pries, nd plyers hve n inentives to lie. One n get round this y priing sets of items, ut tht is story for nother dy.) 7