Diskrete Mathematik und Optimierung

Transcription

1 Diskrete Mathematik ud Optimierug Wifried Hochstättler, Robert Nickel ad David Schiess: Mixed Matchig Markets Techical Report feu-dmo00.08 Cotact: wifried.hochsta FerUiversität i Hage Lehrgebiet Mathematik Lehrstuhl für Diskrete Mathematik ud Optimierug D Hage

2 2000 Mathematics Subject Classificatio: 05D59A2,9B26 Keywords: stable marriage, assigmet game, core

3 Mixed Matchig Markets Wifried Hochstättler Departmet of Mathematics FerUiversität i Hage D Hage Robert Nickel Departmet of Mathematics FerUiversität i Hage D Hage David Schiess Departmet of Mathematics ad Statistics Uiversity of St. Galle CH-9000 St. Galle Jauary 24, 2008 Abstract We itroduce a ew model for two-sided markets that geeralizes stable marriages as well as assigmet games. Our model is a further geeralizatio of the model itroduced by Eriksso ad Karlader [2]. We prove that the core of our model is always o-empty by providig a algorithm that determies a stable solutio i O( 4 ). Itroductio The stable marriage problem itroduced by Gale ad Shapley [3] is quite well kow to scietists from differet fields such as game theory, ecoomics, computer sciece, ad combiatorial optimizatio. There are at least three moographs, by Kuth [6], Gusfield ad Irvig [4] ad Roth ad Sotomayor [8] devoted to it. The problem reads as follows: give two disjoit groups of players (me-wome or workers-firms etc.), where each player is edowed with a preferece list o the other group, the objective is to match the players from oe group to players from the other group such that there is o pair which is ot matched but prefers each other over their parters. The existece of such a stable matchig is proved algorithmically usig the so called me propose wome dispose -algorithm give i [3]. I their moograph [8] Roth ad Sotomayor observed that the set of stable solutios from aother game o bipartite matchig, amely the assigmet game [], has several structural similarities with the set of stable matchigs. They challeged the readers to fid a uifyig theory for the two games. I the assigmet game we are give a weighted bipartite graph. A solutio cosists of a matchig ad a allocatio of its weight to the players. A solutio

4 is stable if o pair gets allocated less tha the weight of its coectig edge. Shapley ad Shubik [] observed that this coditio is idetical to the dual costraits of the liear programmig model for weighted bipartite matchig, thus the dual variables i a optimal solutio coicide with the stable allocatios. Roth ad Sotomayor [9] themselves preseted a first model uifyig stable marriage ad the assigmet game ad showed that its set of stable solutios, if it is o-empty, has the desired structural properties. Eriksso ad Karlader [2] modified this model ad gave a algorithmic proof of the existece of a stable solutio. For the classical special cases, their algorithm coicides with me propose wome dispose, respectively with the exact auctio procedure of []. As implemeted, this algorithm is ot polyomial time but pseudopolyomial. A careful aalysis of the algorithm (see [5]), though, reveals that a proper implemetatio solves the problem i O( 4 ). The purpose of the preset paper is to preset a further atural geeralizatio of these games. While i the model of Eriksso ad Karlader the players are partitioed ito rigid ad flexible players ad the distributio of the value of a edge is flexible oly if both players are flexible, i our model each possible pair of players may choose whether it closes a flexible or a rigid cotract. This mirrors the situatio o labor markets owadays. I additio the iput data becomes simpler ad the assigmet game is o loger iterpreted as a stable marriage game with side paymets (see [2]). We geeralize the algorithm from [5] to the ew model ad show that it computes a stable solutio i O( 4 ). I the ext sectio we itroduce the model ad discuss its special cases. I Sectio 3 we preset our algorithm. Sectios 4 ad 5 are devoted to the correctess proof of the algorithm ad its ruig time aalysis. We assume some familiarity with bipartite matchig ad combiatorial optimizatio. Our otatio should be fairly stadard. 2 The Model Give three o-egative real square matrices A = (a ij ), B = (b ij ), C = (c ij ) R + a outcome is a matchig permutatio σ S, a flexibility map f : {,..., } {0, } ad two payoff fuctios u : {,..., } R +, v : {,..., } R + such that i : ( if f(i) = the u i + v σ(i) = c iσ(i) ) ad ( if f(i) = 0 the u i = a iσ(i) ad v σ(i) = b iσ(i) ). The outcome is stable if i additio i j : u i + v j c ij () i j : u i a ij or v j b ij. (2) We write a outcome as the quadruple (σ, f, u, v). Sice i our algorithm preseted later we use a alteratig path techique from matchig theory we 2

5 will frequetly use P for the idex set of the rows ad Q for the set of idices of the colums of the matrices. We will idetify the data with a complete bipartite graph K P Q which has two copies of each edge. For oe of the copies we have the weight fuctio c ij ad for the other the pair of weight fuctios (a ij, b ij ). The permutatio together with the flexibility fuctio f the correspods to a perfect matchig, where we choose the edge weight c iσ(i) if f(i) = ad (a iσ(i), b iσ(i) ) if f(i) = 0. Remark. I our matchig games we assume that the graphs are complete ad that both color classes have the same cardiality. This ca always be achieved by addig a appropriate umber of dummy vertices ad correspodig edges of weight zero. 2. Stable Marriages The iput for a stable marriage game cosists of a complete bipartite graph K = K P Q ad for each vertex i P resp. j Q of a total order i o the vertices of Q respectively of a total order j o the vertices of P. We call these total orders preferece lists. A perfect matchig, also called a marriage, which we may idetify with a permutatio σ S, is stable, if i P j Q : σ(i) i j or σ (j) j i, i.e. if there is o umatched pair that prefers each other to their matchig parters. Cosider our model i the case that C = 0 is the zero-matrix ad (σ, f, u, v) are a stable outcome. If for some i P we have f(i) =, the u i + v σ(i) = c iσ(i) = 0 which implies u i = v σ(i) = 0 sice the payoffs are o-egative. The outcome is stable ad hece 0 = u i a iσ(i) or 0 = v σ(i) b iσ(i). If oly oe of a iσ(i) ad b iσ(i) is zero, say a iσ(i) > 0 the we may chage u i to a iσ(i) ad set f(i) = 0 without chagig the stability of the outcome. I additio the total payoff of the game possibly icreases. Possibly applyig this several times we may assume that f is costatly zero. Now iterpretig the umbers a ij for fixed i as priority o the matchig parters j by defiig j i k a ij a ik ad similarly for the b ij we derive a iput example for the stable marriage game ad the permutatio σ clearly gives a stable marriage. O the other had, give a iput of a stable marriage game we may represet the preferece lists by two strictly positive matrices A, B R + ad set C = 0. The the stable outcomes of our model are i oe-oe correspodece with the stable marriages of the give game. 2.2 Assigmet Games I the assigmet model we are give a complete bipartite graph K = K P Q with o-egative weights C = (c ij ). A perfect matchig σ S together with two payoff fuctios u : P R +, v : Q R + such that i P : u i + v σ(i) = c iσ(i) 3

6 is called a outcome. The outcome is stable if additioally i P j Q : u i + v j c ij, i.e. if there is o umatched pair that could idividually improve by leavig their preset parters ad formig a ew matchig edge istead. If i our model A = B = 0 ad for a stable outcome σ, f, u, v we have f(i) = 0 for some i the ecessarily u i = v σ(i) = 0. By the stability assumptio, 0 = u i + v σ(i) c iσ(i) 0 holds. Hece we also have c iσ(i) = 0 ad modifyig f by settig f(i) to maitais stability of the payoff. Similarly, a iput of the assigmet game immediately traslates ito a iput of our model ad - apart from the degeeracy with edges of weight zero described above - we have a oe-oe correspodece betwee stable outcomes i both models. 2.3 The Eriksso-Karlader Model The model of Eriksso ad Karlader was the first geeralizatio of stable matchig ad the assigmet game show to always admit a stable solutio. The iput cosists of two o-egative square matrices A, B R + ad a partitio P Q = R F of the set of vertices ito flexible players F ad rigid players R. We may cosider A, B, R as the iput data of such a game. A outcome is a matchig permutatio σ S ad two payoff fuctios u : P R +, v : Q R + such that i P : if {i, σ(i)} R = the u i + v σ(i) = c iσ(i) otherwise u i = a iσ(i) ad v σ(i) = b iσ(i). The outcome is stable if i additio i P j Q : if {i, σ(i)} R = the u i + v j c ij otherwise u i a ij or v j b ij. Give a istace A, B, R of the Eriksso-Karlader game we defie the matrix C = (c ij ) as { aij + b c ij := ij if {i, j} R = 0 otherwise. ad A = (a ij ) respectively B = (b ij ) as a ij := { aij if {i, j} R 0 otherwise b ij := { bij if {i, j} R 0 otherwise. Now let (σ, f, u, v) be a stable outcome for the istace A, B, C. Assume that for {i, σ(i)} R = we have f(i) = 0. By the defiitio of a outcome we have u i = a iσ(i) = 0 = v σ(i) = b iσ(i). By the stability of the solutio this implies 4

7 u i + v σ(i) = 0 c iσ(i) 0. Hece, c iσ(i) = 0 ad modifyig f to f(i) = maitais stability. If o the other had for {i, σ(i)} R we have f(i) = the u i + v σ(i) = c iσ(i) = 0 ad thus u i = v σ(i) = 0. By the stability of the solutio, this implies a iσ(i) = 0 or b iσ(i) = 0. If oly oe of a iσ(i) ad b iσ(i) is zero, say a iσ(i) > 0, we may chage u i to a iσ(i) ad set f(i) = 0 without chagig the stability of the outcome. I additio the total payoff of the game icreases. Havig dealt with these degeerate situatios, we may assume that f(i) = if ad oly if {i, σ(i)} R = ad hece, our outcome is also a outcome for the Eriksso-Karlader model. Clearly, stability i our model implies stability i the Eriksso-Karlader model. 3 A Algorithm to Fid a Stable Outcome The basic idea of our algorithm is derived from the (pseudopolyomial) auctio procedure of Eriksso ad Karlader. Hochstättler, Nickel ad Ji [5] tured this ito a O( 4 ) algorithm. Schiess [0] geeralized that algorithm to his Decisive Edges Model which is aother special case of our model. It will make the descriptio of the algorithm easier, if we, at least partially, describe it i terms of a ecoomic iterpretatio. We will call the elemets from P firms ad the players from Q workers. A rigid edge betwee a firm ad a worker may be iterpreted as a employmet that is payed accordig to tariffs egotiated by some uios ad flexible edges are payed accordig to idividual cotracts. Durig the algorithm we maitai a (partial) map τ : P Q which we will tur ito a permutatio. If τ(i) = j we say that firm i proposes to worker j ad that j has a proposal. For each firm i P we have a set of possible proposals durig the differet stages of the algorithm. This defies a bipartite graph of feasible proposals which depeds o the curret map v : Q R + of (expected) icome of the workers. Additioally, we maitai a flexibility fuctio f : P {0, } that records whether a proposal refers to the flexible or the rigid edge. The algorithm is a geeralizatio of the classic me propose wome dispose algorithm for the stable marriage problem [3]. I the first stage PlaceProposals i Algorithm, see also Algorithm 2 each firm proposes to a worker who maximizes its expected profit, if it is o-zero, i.e. the firm is solvet. Otherwise, the firm is isolvet, i.e. i all stable solutios its payoff will be zero. We eglect it util the very ed of the algorithm where we map it to a arbitrary worker without proposal. Workers with a rigid proposal, that additioally is the best offer they have got, dispose all other rigid proposals. Firms whose (rigid) proposal has bee disposed propose to the ext best worker. This is iterated util each worker has at most oe rigid proposal. The, usig augmetig path techiques, we try to icrease the set of workers that have a proposal (lie 5-0 of Algorithm ). If the map of the proposals still is ot ijective, we icrease the (expected) icome of workers with a flexible proposal. This is doe i lie 5

8 of Algorithm, see also Algorithm 3, similarly to the dual update step i the Hugaria Method for the assigmet game [7]. Further proposals become feasible ad we proceed with the first stage omittig the proposal placemets for firms with a proposal. To be more precise: iitially, we set v = 0. I lie 4 of Algorithm 2 for each firm i P we determie a worker j Q who maximizes the firm s expected icome, i.e. such that max {c ik, a ik } {c ij, a ij }, k= ad set f(i) = 0 if max k= {c ik, a ik } = a ij ad f(i) = otherwise. If there are ties we prefer a ik s to c ik s, further ties are broke arbitrarily. We make a proposal oly if the profit is strictly positive. Firms which caot yield ay positive profit remai without proposal util the very ed of the algorithm. If a worker j P has a rigid proposal we set v j = max{b iτ(i) τ(i) = j, f(i) = 0} ad possibly dispose all further rigid proposals to j by udefiig τ(k) ad f(k). Give v, τ ad f we cosider the map u τ,v : {0, } P Q R + defied by c ij v j if g = u τ,v (g, i, j) = a ij if g = 0 ad (v j < b ij or τ(i) = j) 0 otherwise. Note that for fixed v this is the profit that firm i ca expect whe worker j is hired. If v j b ij ad τ(i) j the j will ot accept a rigid proposal that does ot yield a strictly larger icome. For each firm i P we select j ad g {0, } such that u τ,v (g, i, j) = max k= max u τ,v(s, i, k). (3) Agai, if possible we choose j such that u τ,v (0, i, j) = max k= max u τ,v (s, i, k) ad set τ(i) = j. This makes τ a map agai (up to isolvet firms). We iterate this process, takig additioally ito accout that a worker j with v j = b i j where i is the favorite rigidly proposig firm of j ad with at least oe flexible proposal will dispose all rigid proposals, util each worker either has oe or o rigid proposal. The we proceed to the ext stage ad cosider the followig (bipartite) digraph o P Q. We have the backward edges (τ(i), i) for all i P ad forward edges { } D i τ,v := max k= (i, j) a ij = u τ,v (0, i, j) = { (i, j) c ij v j = u τ,v (, i, j) = max u τ,v(s, i, k) > 0, τ(i) j max k= max u τ,v(s, i, k) > 0, τ(i) j Thus, the edges of the digraph correspod to rigid ad flexible proposals (backward arcs) ad to forward arcs that promise maximal profit, from the poit of }. 6

9 view of the firms. If this digraph cotais a directed path from a worker with several proposals that eds i a rigid edge, a isolvet firm, a worker without proposal or to a worker with a rigid proposal, we ivert this path ad modify τ ad f such that the correspodig combiatio of the maps is represeted by the backward edges agai. I the last case we additioally dispose the old rigid proposal (lie 6 i Algorithm ). We iterate this process util such a dipath does o loger exist. If there is still a worker with more tha oe proposal, we eter the third stage (lies ad 2 i Algorithm ). Otherwise, we arbitrarily map umapped firms to umapped workers, set u i = max k= max u τ,v (s, i, k) ad termiate. Algorithm The Mai Loop : v 0 2: PlaceProposals 3: Costruct digraph of feasible proposals 4: while there exists j 0 Q with more tha oe proposal do 5: while there exists a directed path P with at least two edges from j 0 to j Q where P eds i its oly rigid edge or eds i a isolvet firm or j has o proposal or j has a rigid proposal do 6: DisposeRigid(j ) 7: Alterate(P) 8: PlaceProposals 9: Update digraph, τ ad f 0: ed while : HugariaUpdate 2: Update digraph, τ ad f 3: ed while 4: while there exists a firm i P without proposal do 5: Choose j Q without proposal 6: τ(i) j 7: v j b ij 8: f(i) 0 9: ed while 20: for all i P do 2: u i u τ,v (f(i), i, τ(i)) 22: ed for Now, give a worker with more tha oe proposal, assumig that a dipath as required i lie 5 i Algorithm does o loger exist, cosider its compoet i the graph uderlyig the digraph of feasible proposals. Deote by P the firms 7

10 ad by Q the workers i this compoet. Now compute ad u i := max max u τ,v(s, i, k) i P k= { := mi u i max u τ,v(s, i, j) i P, j Q } 2 := mi { u i u τ,v (0, i, j) i P, j Q } 3 := mi { u i i P } := mi{, 2, 3 } > 0 ad set v j = v j + for all j Q. This way at least oe ew forward arc eters the digraph of proposals (backward arcs) ad feasible proposals (forward arcs) or a firm is marked isolvet. We update the digraph ad proceed with stage oe. Algorithm 2 PlaceProposals : procedure PlaceProposals 2: while there exists a solvet firm i P without proposal do 3: while there exists a solvet firm i P without proposal do 4: Propose(i) 5: ed while 6: for all j Q with a rigid proposal do 7: Let i P be a favorite rigid proposer i τ (j) 8: v j b i j 9: Update flexible proposals 0: if j has o flexible proposal the : Dispose all other rigid proposals 2: else 3: Dispose all rigid proposals 4: ed if 5: ed for 6: ed while 7: ed procedure 4 Correctess of the Algorithm First, ote that all statemets are feasible. I particular i the routie Propose(i) we ca always select j Q as described i (3) if j Q has ay feasible proposal give curret v at all. For a proof of correctess we make the followig observatios. Propositio.. The routie PlaceProposals ever decreases τ(p ). 2. If τ(p ) is decremeted by the statemet DisposeRigid(j ) the it is immediately icremeted agai by Alterate(P). 8

11 Algorithm 3 HugariaUpdate : procedure HugariaUpdate 2: Choose j Q with several proposals 3: Determie the vertices P P ad Q Q i the compoet of j 4: for all i P do 5: u i max k= max u τ,v (s, i, k) 6: ed for 7: mi { u i max u τ,v (s, i, z) i P, z Q \ Q } 8: 2 mi { u i u τ,v (0, i, j) i P, j Q } 9: 3 mi { u i i P } 0: mi{, 2, 3 } : for all j Q do 2: v j v j + 3: ed for 4: ed procedure 3. The fuctio v : Q R + ever decreases durig the algorithm. If v j > 0 the j has a proposal. 4. A disposed rigid proposal will ever be proposed agai. Proof.. I PlaceProposals τ is chaged oly i lies 4, ad 3 of Algorithm 2. Clearly, τ(p ) does decrease i either of these. 2. We dispose a rigid edge to a worker j if we have foud a directed path from a worker with at least two proposals to j. Deote by (i, j ) the last edge i this path. I Alterate(P) we ivert this edge ad thus reistall the old set τ(p ). 3. v j is chaged i lie 8 of Algorithm 2 ad lie 2 of Algorithm 3. I both cases v j does ot decrease ad j has a proposal. A firm chages its proposal oly if either its old rigid proposal has bee disposed or if a path is alterated. Thus, a worker j with v j > 0 will always have a proposer. Whe we modify v j i lie 7 of Algorithm j had o proposal ad thus, v j was zero before. 4. Whe a rigid proposal τ(i) = j is disposed i lie or 3 of Algorithm 2 we have b ij v j ad sice v is o-decreasig i the followig we always have u τ,v (0, i, j) = 0, so this edge will ever be proposed agai. Now assume we dispose a rigid edge i lie 6 of Algorithm. Let agai (i, j ) deote the last edge o P. Whe i was mapped to j we set v j = b ij sice v j is o-decreasig ad by defiitio of u τ,v the rigid edge will ever be proposed agai. The followig properties of the digraph of feasible proposals will be useful. Propositio 2. Cosider the situatio whe the Algorithm eters the HugariaUpdate. 9

12 . If a worker has more tha oe proposal the he has o rigid proposal. 2. D i τ,v cosists oly of flexible edges for all i P. Furthermore, τ(i) refers to a flexible edge for all i P. Proof. The first assertio follows from lie 0 4 of Algorithm 2. If Dτ,v i cotais a rigid edge or τ(i) refers to a rigid edge, the there ca be o directed path from j 0 to i for otherwise we would ot have left the ier while-loop ad hece i P. Theorem. Algorithm termiates after a fiite umber of steps. Proof. I the ier while loop (lie 5 0) we either make a ew rigid proposal, dispose a rigid proposal i lie 6, mark a ew firm as isolvet or, if P eds i a umapped worker, icrease τ(p ). If there is o more such path P, we call the routie HugariaUpdate. Note that i the begiig of that procedure o vertex i P Q is icidet with a rigid edge by Propositio 2 ad hece, > 0. Let (s, i 0, z 0 ) be such that = = u i0 u τ,v (s, i 0, z 0 ) with i 0 P, z 0 Q ad (i 0, z 0 ) Dτ,v. i0 We will show that with the dual update i lie 3 of Algorithm 3 we do ot lose a edge i the digraph of feasible proposals but add (i 0, z 0 ). Clearly, we do ot lose ay backward edge. Thus, assume (i, j) Dτ,v i was a forward edge of the digraph before the dual update. Hece, u τ,v (s, i, j) = max k= max u τ,v(s, i, k) > 0 ad τ(i) j. (4) This value is chaged i the dual update sice u τ,v (, i, j) = c ij v j (recall that all old edges i the compoet where flexible by Propositio 2). But it is chaged by the same amout for all edges coectig i to some k Q ad by defiitio of we still have u τ,v (s, i, j) = max k= max u τ,v (s, i, k) after the update. Hece, (i, j) is still a edge of the digraph after the update. Fially, sice = max k= u τ,v(s, i 0, k) max max u τ,v(s, i 0, z 0 ) we must have (i 0, z 0 ) D i0 τ,v after the update. Sice by Propositio a disposed rigid edge ever will be proposed agai, we ca add each rigid edge at most oce ad each firm is marked isolvet at most oce. Also after addig sufficietly may edges a path as required must occur. Whe = 2 a ew rigid edge eters the digraph. This ca happe at most 2 times. Whe = 3 a firm is marked isolvet. This ca happe at most times. Hece, the algorithm termiates after a fiite umber of steps. Theorem 2. Algorithm computes a stable outcome. Proof. Whe the algorithm termiates, o worker has two proposals ad all firms have a proposal, thus τ is a bijectio, or a matchig permutatio. v is a 0

13 o-egative fuctio ad for all rigid proposals τ(i) = j we have set v j = b ij. Note that, whe v is updated i lie 2 of Algorithm 3 by Propositio 2, o worker i Q is icidet with a rigid edge i the graph of feasible proposals. Before fiishig the algorithm we set u i to a iτ(i) if f(i) = 0 ad to c iτ(i) v τ(i) otherwise. Hece, the algorithm computes a outcome. I order to show that the outcome is stable we cosider the developmet of the fuctio ū : P R + defied by ū i := max k= max u τ,v(s, i, k). By defiitio the pair (ū, v) satisfies the stability coditios () ad (2). Also if τ(i) is defied we have ū i = u τ,v (f(i), i, τ(i)). Hece, whe we termiate ū = u holds. It follows that the algorithm produces a stable outcome. 5 Ruig Time Aalysis Theorem 3. The algorithm termiates i O( 4 ). Proof. Lies 4 9 i Algorithm are easily implemeted i O( 2 ) ad lies i Algorithm i liear time. PlaceProposals ca be implemeted with the same complexity as the classic Me-Propose-Wome-Dispose algorithm ad thus i O( 2 ). Hece we may focus o the ested while-loops. I the ier while-loop we either icremet τ(p ) which is possible at most times or dispose a rigid edge of which there are oly 2, itroduce a ew egde ito the graph of feasible proposals also at most 2 times or mark oe of firms isolvet. The complexity of a sigle ier while loop is domiated by the procedure PlaceProposals. Thus, the ier while loop i total accouts for O( 4 ). Fially, if HugariaUpdate itroduces ew flexible edges we may cotiue our search o the augmeted data. A stadard search procedure requires O( E ) = O( 2 ) if E deotes the set of edges i our graph. This is multiplied at worst with the umber of rigid edges which are ewly itroduced ito the digraph of feasible proposals or are disposed, the umber of augmetatios of τ(p ) ad the umber of firms that are marked isolvet. Hece, it accouts for O( 4 ) i total as well. Altogether this sums up to O( 4 ). Remark 2. Note that our algorithm i fact rus i quadratic time as the size of the iput data is O( 2 ). We thak Elisabeth Gasser for askig the right ques- Ackowledgmets. tio.

14 Refereces [] G. Demage, D. Gale, ad M. Sotomayor, Multi-item auctios, Joural of Political Ecoomy, 94 (986), pp [2] K. Eriksso ad J. Karlader, Stable matchig i a commo geeralizatio of the marriage ad assigmet models, Discrete Mathematics, 27 (2000), pp [3] D. Gale ad L. S. Shapley, College admissios ad the stability of marriage, America Mathematical Mothly, 69 (962), pp [4] D. Gusfield ad R. W. Irvig, The stable marriage problem: Structure ad algorithms, MIT Press, Cambridge, MA, USA, 989. [5] W. Hochstättler, H. Ji, ad R. Nickel, Note o a auctio procedure for a matchig game i polyomial time, i AAIM, 2006, pp [6] D. E. Kuth, Stable marriage ad its relatio to other combiatorial problems, i CRM Proceedigs ad Lecture Notes, vol. 0, America Mathematical Society, 997. [7] H. W. Kuh, The Hugaria method for the assigmet problem, Naval Research Logistics Quaterly, 2 (955), pp [8] A. E. Roth ad M. Sotomayor, Two-sided matchig: A study i gametheoretic modelig ad aalysis, Cambridge Uiversity Press, Cambridge, 99. [9] A. E. Roth ad M. Sotomayor, Stable outcomes i discrete ad cotiuous models of two-sided matchig: A uified treatmet, Revista de Ecoometria, The Brazilia Review of Ecoometrics, 6 (996). [0] D. Schiess, Mixed Matchig Markets. Bachelor s Thesis, FerUiversität i Hage, Germay, August [] L. S. Shapley ad M. Shubik, The assigmet game I: The core, Iteratioal Joural of Game Theory, (972), pp