CS 286r: Matchng and Market Desgn Lecture 2 Combnatoral Markets, Walrasan Equlbrum, Tâtonnement Matchng and Money Recall: Last tme we descrbed the Hungaran Method for computng a maxmumweght bpartte matchng. In prncple, we could use ths algorthm to determne outcomes for a market. For example: Housng markets Assgnment of workers to jobs Note: The algorthm wll fnd an effcent matchng, but t mght be that not every partcpant s happy wth the outcome. If we were to resolve real economc markets usng the Hungaran method (or any other algorthm), the partcpants mght beneft by lyng about ther preferences. Goal: Instead of smply choosng a market outcome, desgn a market where the partcpants are ndvdually ncentvzed to select the optmal matchng. Idea: Use payments! Combnatoral Markets In a combnatoral market, there s a set N of n agents (buyers) and a set M of m goods (or tems). There s one ndvsble copy of each good. Thnk: houses. Def: A valuaton functon assgns a nonnegatve value to each set of goods: v : 2 M R 0. Each agent has a valuaton functon v. We wll assume valuaton functons are monotone: v(s) v(t ) S T, and normalzed so that v( ) = 0. Def: An allocaton x s a partton of the goods among the agents, wth possbly some goods left unallocated. We wrte x for the set of goods allocated to agent. Def: The socal welfare of an allocaton x s v (x ). An allocaton that maxmzes socal welfare s sad to be effcent. Goal: Fnd an allocaton x that maxmzes socal welfare. We wll call ths the allocaton problem. Specal Case: Matchngs Note: In the specal case that each agent can be allocated at most one tem, an allocaton s precsely a matchng. In ths case, a valuaton functon smply assgns a value to each tem. The allocaton problem s equvalent to fndng a maxmum-weght bpartte matchng. To see that ths s a specal case of a combnatoral market, suppose that each valuaton functon s unt-demand. A valuaton v s unt-demand f t assgns a value v(j) to each tem j M, and then for any set of tems S we have v(s) = max j S v(j). 1
That s, each agent gets value from at most one of the tems allocated to hm. When valuaton functons are unt-demand, t s wthout loss of generalty to allocate at most one tem to each agent. Example: We wll use ths runnng example throughout. There are 3 tems, {a, b, c}, and 3 agents: Alce: v(a) = $2, v(b) = $3, v(c) = $0 Bob: v(a) = $0, v(b) = $2, v(c) = $4 Charle: v(a) = $0, v(b) = $4, v(c) = $5 Alce and Bob are both unt-demand agents. (Alce {a}, Bob {c}, Charle {b}) s an allocaton, wth socal welfare 10. Walrasan Equlbrum Left to ther own devces, each agent would naturally want to take the tem they value the most, but ths mght cause some tems to be overdemanded. To coordnate the agents preferences, we ntroduce prces. Imagne that every good j M has a prce p j 0. If agent s allocated a set of goods x, he must pay j x p j n exchange for recevng those goods. Def: the utlty of agent, gven that he s allocated set x, s v (x ) j x p j. Each agent wants to maxmze hs or her own utlty. Def: the demand correspondence of agent at prces p, D (p), s the set of utltymaxmzng sets of goods. That s, D (p) equals {S : v (S) j S p j v (T ) j T p j T M}. Def: a Walrasan Equlbrum s a choce of tem prces p, plus an assgnment x, such that every agent s allocated a demanded set: x D (p) for all. every tem wth postve prce s sold: f j x for all, then p j = 0. Example: Consder our runnng example. Prce vector (p a = 0, p b = 1, p b = 2), along wth allocaton (Alce {a}, Bob {c}, Charle {b}), together form a Walrasan Equlbrum. Theorem 0.1 If (x, p) s a Walrasan equlbrum, then x maxmzes socal welfare. Proof: Say (x, p) s a WE, and let y be any other allocaton. For each agent, we know v (x ) j x p j v (y ) j y p j. Take a sum over all agents to get v (x ) p j v (y ) j x j y p j. Snce all tems wth postve prce are allocated under x, we actually have that j x p j s equal to j M p j, whch cannot be less than j y p j. So p x p y, j x j y from whch we conclude v (x ) so x must be optmal. v (y ), 2
Tâtonnement The followng smple procedure s useful for constructng a Walrasan Equlbrum. It s called Tâtonnement, whch translates roughly to graspng around blndly. The basc dea s to rase prces on over-demanded tems untl the market reaches equlbrum. We wll begn by presentng a smple verson that modfes prces n dscrete jumps, but obtans only an approxmate equlbrum. Later we wll see how to turn ths nto a method for fndng exact Walrasan equlbra. Def: the ɛ-approxmate demand correspondence of agent gven prces p, D(p), ɛ s the set of sets of goods that maxmze agent s utlty, up to an addtve ɛ error. That s, D(p) ɛ s equal to {S : v (S) j S p j v (T ) j T Def: an ɛ-approxmate Walrasan equlbrum s an allocaton x and prce vector p such that x D(p) ɛ for all, and all unsold tems have prce 0. Algorthm: 1. Let δ > 0 be arbtrarly small. 2. Start wth p j = 0 for all j M, and x = for all N. 3. If x D δ m (p) for all, stop and return (x, p). 4. Suppose x D δ m (p). Pck y D (p). 5. For each tem j y, ncrease p j by δ. 6. Allocate y to agent : set x = y, and set x k = x k \y for all k. 7. Contnue on lne 3. Tâtonnement does not always succeed n fndng an equlbrum; we wll see an example of that later. However, for the specal case of matchng markets, t turns out that Tâtonnement does always converge to a Walrasan equlbrum. Theorem 0.2 The Tâtonnement process must termnate. In a matchng market, t termnates at a (δ m)-approxmate Walrasan equlbrum. Proof: Prces only rse, and some prce rses on each teraton of the algorthm. The algorthm must therefore termnate eventually, snce an tem whose prce s suffcently hgh s not ncluded n any agent s demanded set(s). 3 By lne 3, f the algorthm termnates, t must be that x D δ m for each agent. So t p j ɛ T M} only remans to show that each unallocated tem has prce 0. We clam that once an tem s allocated, t never agan becomes unallocated. Ths wll complete the proof, snce then the only unallocated tems are tems that were never allocated, and thus never had ther prces ncremented on lne 5. Suppose for contradcton that the algorthm unallocates some tem j on some teraton of lnes 3-7; say teraton t. Ths can only occur f an agent wth j x s selected n lne 4, but j y. In ths case, tem j would be unallocated on lne 6. However, n a matchng market, allocatons are sngletons; t must therefore be that x = {j}. Moreover, t must be that {j} was the allocaton assgned to agent on a prevous teraton, say t < t. So, on teraton t, t must have been that {j} D (p). Snce the prces of other tems can only ncrease, and p j ncreased only by δ on teraton t, {j} must be n D δ (p) on teraton t. Ths contradcts the fact that agent was selected on lne 4 n teraton t.
Bonus Materal: Exact Walrasan Equlbrum The Tâtonnement algorthm descrbed n the prevous secton only fnds an approxmate Walrasan equlbrum. We can make ths approxmaton arbtrarly good by takng δ as small as desred. In the lmt as δ 0, ths has the net effect of allowng prces to rse contnuously. In ths lmt, the algorthm fnds an exact Walrasan equlbrum for matchng markets. To formalze ths, suppose (x δ, p δ ) s the output of the Tâtonnement process for a gven choce of δ, say restrcted to δ (0, 1]. Consder the lmt δ 0. As there are only fntely many choces of allocatons, some allocaton x must occur for nfntely many δ. Restrct attenton to only those values of δ for whch x δ = x; ths produces a subsequence of values of δ wth lmt 0. For each of these nfntely many δ s there s a prce vector p δ ; moreover, these prces vectors le n a bounded range (snce no prce can be greater than all agents values for all tems). There must therefore be a subsequence of δ s for whch the assocated prce vectors converge to some vector of lmt prces, say p, as δ 0. But ths then mples that (x, p) s an ɛ-approxmate Walrasan equlbrum for all suffcently small ɛ (snce, as δ 0, t can be made arbtrarly close to a (m δ)- approxmate equlbrum). Therefore (x, p) s an exact Walrasan equlbrum as well. Faster Tâtonnement The Tâtonnement method nvolves contnuously rasng prces, so t may take a long tme to run. We can mprove the runtme for matchng markets by beng more careful about the order n whch we rase prces, and by groupng together many prce-ncrement operatons. We wll thnk of a matchng market as beng represented by a weghted bpartte graph, wth agents on one sde and goods on the other. The weght of an edge between agent and good j wll be v (j), the value of agent for good j. Def: Gven a weghted bpartte graph between agents and goods, plus a prce vector p over the goods, the utlty-maxmzng graph, E(p), conssts of all edges (, j) for whch {j} D (p). That s, E(p) contans all edges between agents and utlty-maxmzng goods for those agents. An observaton about the Tâtonnement process s that each agent s tryng to obtan a utlty-maxmzng good. If an agent s assgned good s taken away on some teraton of the algorthm, then that agent wll swtch to a dfferent utlty-maxmzng good f there s one; otherwse he wll reaqure the prevous good and ncrease ts prce. Ths process wll contnue untl a new good becomes utltymaxmzng for some agent, whch must occur eventually f prces keep ncreasng. We can therefore thnk of the Tâtonnement process as attemptng to match every agent to a utlty-maxmzng good, and f ths s not possble then the prces of the currently desred tems are rased untl new tems become utlty-maxmzng (that s, new edges are added to E(p)). Wth ths nterpretaton n mnd, we can speed up the Tâtonnement process by lookng ahead to see how much we would need to ncrease prces n order for new edges to be added to the utlty-maxmzng graph. Ths leads to the followng alternatve mplementaton of Tâtonnement, whch runs n polynomal tme. 4
Algorthm: 1. Start wth all prces set to 0. 2. Let M be a maxmal matchng n E(p). If M s perfect, stop and return matchng M and prces p. 3. Otherwse, fnd an agent not matched n M. 4. Drect E(p): all edges n M pont toward the agents, and all other edges pont toward the tems. 5. Rase the prces of all tems reachable from n ths drected graph, untl a new edge s added to E(p). 6. Go to lne 2. Example: Consder our runnng example. When all prces are set to 0, E(p) contans edges (A, b), (B, c), and (C, c). Let M = {(A, b), (C, c)}. Agent B s not matched. Item c s reachable from B, so we wll rase prce p c. Once p c s rased to 1, edge (C, b) becomes part of E(p). The maxmal matchng n E(p) becomes {(B, c), (C, b)}. Agent A s not matched. Items b and c are both reachable from A, so we wll rase p b and p c n tandem. When the prce vector becomes (0, 1, 2) (.e., p b and p c are rased by 1), edge (A, a) becomes utltymaxmzng. So E(p) becomes {(A, a), (A, b), (B, c), (C, b), (C, c)}. The maxmal matchng n E(p) becomes {(A, a), (B, c), (C, b)}. Ths s a perfect matchng, so the algorthm termnates. The fnal output of the algorthm s allocaton (Alce a, Bob c, Charle b) at prces (0, 1, 2). Clam: Ths procedure termnates n at most nm teratons, and fnds a Walrasan equlbrum. Proof: (sketch) Each teraton causes at least one edge to be added to E(p), and once an edge s added to E(p) t never stops beng n E(p). So there can be at most nm teratons. That the algorthm fnds a Walrasan equlbrum follows from equvalence wth Tâtonnement. Observaton: Ths algorthm s precsely the Hungaran method! The prces correspond to the dual varables on the goods sde, and the agent utltes correspond to the dual varables on the agent sde. So, n fact, the Hungaran method mplements the optmal weghted matchng as a market outcome, when the dual varables are used as tem prces. Beyond Matchng The Tâtonnement algorthm s defned for arbtrary combnatoral markets, not just matchng markets. However, n general, a WE s not always guaranteed to exst. Why does Tâtonnement fal? Recall our proof that Tâtonnement fnds an approxmaton Walrasan equlbrum for matchng markets: we needed to show that once an tem s allocated, t s never unallocated. Ths s true for matchng markets, but s not true n general. Example: Suppose our market has two goods {L, R}, a left shoe and a rght shoe. 5
Alce would lke to purchase the shoes only as a par. Her valuaton s v A ({L, R}) = 5, but v A (L) = v A (R) = 0. Bob s nterested n any sngle shoe (perhaps Bob s a dog). Hs valuaton s v B (L) = v B (R) = v B ({L, R}) = 3. In ths example, Tâtonnement does not fnd a Walrasan equlbrum. Consder the smple Tâtonnement process, wth a small prce ncrement δ. Startng at prces 0, Alce wll choose both shoes on every teraton, and Bob wll choose whchever shoe s cheaper. Ths wll contnue untl the prce of each shoe rses above $2.50. At ths pont, Bob wll take one shoe from Alce (say the left), rasng ts prce further, and then Alce wll dscard her remanng shoe and take her demanded set whch s. The resultng allocaton s (Alce, Bob {L}) at prces ($2.50+, $2.50+). Ths s not an approxmate Walrasan equlbrum, snce the rght shoe s unallocated but has a postve prce. Provng a Walrasan Equlbrum does not Exst Queston: How do we prove that a combnatoral market does not have a WE? One approach uses the theory of lnear relaxatons. The assgnment problem can be expressed as the followng mathematcal program. The varables are x,s for N and S M, whch we nterpret as ndcator varables: x,s = 1 means that agent s allocated set S, and x,s = 0 means that S s not the set allocated to agent. Allocaton Program: max v (S) x,s s.t.,s x,s 1 for every N S M x,s 1 for every j M,S j x,s {0, 1} for every N, S M The program descrbes the space of all vald assgnments. The frst constrant guarantees that every agent s only allocated one set; the second constrant guarantees that every tem s only allocated once. Consder relaxng the program to allow fractonal assgnments. Ths corresponds to a lnear program, known as the confguraton LP. The dfference between ths program and the prevous one s that the varables x,s are allowed to take on any fractonal value n [0, 1]. Confguraton LP: max v (S) x,s s.t.,s S M,S j x,s x,s x,s 1 for every N 1 for every j M [0, 1] for every N, S M Snce the lnear relaxaton allows more possble solutons, ts maxmum value can only be larger than the optmal welfare of the (ntegral) allocaton problem. We already saw that a Walrasan equlbrum (x, p) acheves the optmal socal welfare among all ntegral allocatons, but how does ts socal welfare compare to the (possbly larger) optmal socal welfare among all fractonal allocatons? As t turns out, a Walrasan equlbrum s also optmal for the LP relaxaton. 6
Clam: A Walrasan Equlbrum maxmzes socal welfare among all fractonal assgnments. Proof: The proof s very smlar to the proof of optmalty over ntegral assgnments. See the assgned readng. Ths clam s somewhat surprsng, snce the assgnment of a Walrasan equlbrum s not fractonal! So a corollary s that a Walrasan equlbrum can exst only f the optmal fractonal assgnment s, n fact, ntegral. Example: Modfy the two-shoes example so that Alce has value 7 for the par of shoes, and Bob stll has value 3 for any one shoe. In ths case, the allocaton (Alce {L, R}, Bob ) wth prces (p L = 3, p R = 3) s a Walrasan equlbrum. Ths allocaton generates welfare 7. Ths s the optmal welfare even among all fractonal assgnments. Corollary 0.3 A combnatoral market has a Walrasan Equlbrum only f the ntegral optmal socal welfare and the fractonal optmal socal welfare are the same. Note: In fact, t turns out that ths s an f and only f condton. We wll not cover the proof (whch uses the theory of dualty), but you can fnd t n the assgned readng. Example: Consder the two-shoes example from before. We wll prove that t does not have a Walrasan equlbrum. The welfare-optmal assgnment s to gve both shoes to Alce; ths acheves socal welfare 5. Consder the followng fractonal assgnement: x A,{L,R} = 1 2, x B,{L} = 1 2, x B,{R} = 1 2. Ths s a vald fractonal assgnment: the total weght of all allocatons that nclude L s at most 1, and smlarly for R; and the total weght of all allocatons to Alce s at most 1, and smlarly for Bob. The value of the fractonal assgnment s 1 2 v A({L, R}) + 1 2 v B(L) + 1 2 v B(R) = 11 2 > 5. Snce the optmal fractonal assgnment generates more value than the optmal ntegral assgnment, a Walrasan equlbrum does not exst. 7