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Core many-to-one matchings by fixed-point methods
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Journal of Economic Theory 115 (2004) 358–376
Core many-to-one matchings by fixed-point methods Federico Echeniquea, and Jorge Oviedob a
Humanities and Social Sciences, California Institute of Technology, MC 228-77, Pasadena, CA 91125, USA b Instituto de Matema´tica Aplicada, Universidad Nacional de San Luis, Eje´rcito de los Andes 950, 5700 San Luis, Argentina Received 4 October 2002; final version received 21 February 2003
Abstract We characterize the core many-to-one matchings as fixed points of a map. Our characterization gives an algorithm for finding core allocations; the algorithm is efficient and simple to implement. Our characterization does not require substitutable preferences, so it is separate from the structure needed for the non-emptiness of the core. When preferences are substitutable, our characterization gives a simple proof of the lattice structure of core matchings, and it gives a method for computing the join and meet of two core matchings. r 2003 Elsevier Science (USA). All rights reserved. JEL classification: C78 Keywords: Matching; Core; Lattice; Stability; Algorithm; Complexity; Substitutability; Tarski’s fixed point theorem
1. Introduction This paper deals with the many-to-one matchings that are in the core. Given a set of workers and a set of firms, a many-to-one matching (a matching, for short) is an assignment of a group of workers to each firm. The problem is interesting because each worker has a preference relation over firms, and each firm has a preference relation over groups of workers. Given a matching, firms who are unhappy with
Corresponding author. E-mail addresses: [email protected] (F. Echenique), [email protected] (J. Oviedo).
0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-0531(03)00184-4
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their current group of workers, and workers who are unhappy with their current employers, may recontract in a mutually beneficial way, thus destroying the proposed matching. Arguably, the natural formalization of matchings that are robust to recontracting is the core (see Roth and Sotomayor [12], RS in the rest of the paper, for definitions). The standard approach to the core (RS is a classic) is to impose some structure on firms’ preferences that both simplifies the core and ensures that it is non-empty; the structure used is to assume that firms’ preferences are ‘‘substitutable’’ (see RS, or our Definition 5). When preferences are substitutable, a matching is in the core if and only if it is ‘‘stable’’—a stable matching is robust to recontracting by worker–firm pairs, but not more complicated coalitions. Further, there is an algorithm, the ‘‘deferred acceptance algorithm’’ (DAA) which finds certain distinguished core matchings: the DAA finds a core matching that is better for the firms than all other core matchings, and a core matching that is better for the workers than all other core matchings. Kelso and Crawford [7] introduced the notion of substitutability (their ‘‘gross substitutes’’ condition) and first used it to prove—via the DAA—that the core is non-empty in a class of models that, under substitutability, includes the model studied in our paper. Our approach is different. We characterize the core as the set of fixed points of a map T: We do not add structure to the model, so the core (and the fixed points of T) may be empty. Our characterization is useful because it allows us to: 1. Construct a simple algorithm that consists in iterating T; we shall call it the Talgorithm. The T-algorithm either finds a core matching or cycles in a recognizable way. Unlike the DAA, the T-algorithm does not stop at a non-core matching. We present an example where the DAA stops at a non-core matching, while the T-algorithm succeeds in finding a core matching. 2. We prove that, when preferences are substitutable, T has at least one fixed point—hence that the core is non-empty—and that the T-algorithm finds the two distinguished core matchings: the matching that is best for the firms and the matching that is best for the workers. We can bound the computational complexity of the T-algorithm. To the best of our knowledge, the computational complexity of the DAA for the many-toone case is unknown.1 3. Blair [3] proves that, when preferences are substitutable, the core matchings have a lattice structure. (a) Our characterization gives a very simple proof of Blair’s result; it follows almost immediately from Tarski’s fixed point theorem, as T is a monotone map under an appropriate order. (b) There is no formula for computing Blair’s lattice operations (join and meet). We show how the T-algorithm computes the join and meet of any two core matchings.
1
Gusfield and Irving [6] prove that the DAA is Oðn2 Þ for one-to-one matchings.
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Our results on the core build a bridge to the literature on supermodular games (see [15]); in fact, our results on the T-algorithm are adapted from similar results on computing Nash equilibria in supermodular games. We believe that this bridge is useful for understanding matching problems better, and may deliver further results in future research. Fixed-point methods have been used in the matching literature before. Tarski’s fixed-point theorem was applied by Roth and Sotomayor [11] to the assignment game, Adachi [1] to one-to-one stable matchings, and Fleiner [5] to many-toone stable matchings. Fleiner’s is the paper closest to our paper; he proves a version of Blair’s lattice result for a general graph-theoretic model. Fleiner’s model includes the many-to-one matching model with substitutable preferences as a special case. Fleiner’s paper is a paper in graph theory, though, and his results are very hard to translate into the language of matching models. For that reason, we believe that our result in 3(a) above is valuable, beyond its auxiliary role in our other results. We should clarify at this point that ‘‘core’’ is RS’s ‘‘weak core.’’ We do not have results for RS’s ‘‘strong core,’’ hence we opted, in the interest of exposition economy, for omitting all ‘‘weak’’ qualifiers from the paper. Nothing is really known about the strong core, so we feel that our omission is justified. See RS, and Sotomayor [13], for a discussion of the different core definitions. In Section 2 we define the model, and prove some preliminary results on the core many-to-one matchings. In Section 3 we introduce our fixed-point approach to the core. In Section 4 we introduce the T-algorithm. In Section 5 we study the lattice structure of the core many-to-one matchings. In Section 6 we analyze the Talgorithm for substitutable preferences.
2. Preliminary definitions and results 2.1. The model There are two disjoint sets of agents, the set of n firms F and the set of m workers W : Each firm f AF has a strict, transitive, and complete preference relation Pðf Þ over the set of all subsets of W ; and each worker has a strict, transitive, and complete preference relation PðwÞ over F ,|: Preferences profiles are ðn þ mÞ–tuples of preference relations; we denote them by P ¼ ðPðf1 Þ; y; Pðfn Þ; Pðw1 Þ; y; Pðwm ÞÞ: Given a firm’s preference relation, Pðf Þ; the sets of workers that f prefers to the empty set are called acceptable; thus we allow that firm f may prefer not hiring any worker rather than hiring unacceptable subsets of workers. Similarly, given a preference relation of a worker PðwÞ the firms preferred by w to the empty set are called acceptable; in this case we are allowing that worker w may prefer to remain unemployed rather than working for an unacceptable firm. It turns out that only acceptable partners matter, so we shall write preference relation concisely as lists of acceptable partners. For example, Pðfi Þ ¼ fw1 ; w3 g; fw2 g; fw1 g; fw3 g indicates that
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fw1 ; w3 gPðfi Þfw2 gPðfi Þfw1 gPðfi Þfw3 gPðfi Þ|: And Pðwj Þ ¼ f1 ; f3 indicates that f1 Pðwj Þf3 Pðwj Þ|: We denote by R the weak orders associated with P: So, fi RðwÞfj if fi ¼ fj or fi PðwÞfj : Similarly for Rðf Þ: We shall study assignments of workers to firms as matchings of firms with groups of workers. The assignment problem consists of matching workers with firms maintaining the bilateral nature of their relationship, and allowing for the possibility that both firms and workers may remain unmatched. Formally, Definition 1. A matching m is a mapping from the set F ,W into the set of all subsets of F ,W such that for all wAW and f AF : (a) Either jmðwÞj ¼ 1 and mðwÞDF or else mðwÞ ¼ |: (b) mðf ÞA2W : (c) mðwÞ ¼ ff g2 if and only if wAmðf Þ: Given ðF ; W ; PÞ we denote the set of matchings by M: Let P be a preference profile. Given a set SDW ; let ChðS; Pðf ÞÞ denote firm F ’s most-preferred subset of S according to its preference ordering Pðf Þ: We call ChðS; Pðf ÞÞ the choice set of S according to Pðf Þ: That is, A ¼ ChðS; Pðf ÞÞ if and only if ADS and APðf ÞB for all BDS with AaB: 2.2. The core, stable matchings, and stable matchings A matching m is stable if no worker–firm pair can benefit by deviating from m: RS prove that, if firms’ preferences are substitutable, the set of stable matchings equals the core. A matching is stable if no set of workers can, jointly with one firm, benefit from deviating from m: We prove that the set of stable matchings equals the core; the result does not require additional assumptions on preferences. We use the notion of stable matching instead of stable matching because our fixed-point approach allows us to characterize the set of stable matchings, and thus the core. It should be clear from our proofs and examples just how useful the stable definition is. Let P be a preference profile and let m be a matching. Say that m is individually rational if mðwÞRðwÞ| for all wAW ; and if mðf Þ ¼ Chðmðf Þ; Pðf ÞÞ; for all f AF : Intuitively, m is individually rational if no agent can unilaterally improve over its assignment in m—workers by choosing to remain unemployed and firms by firing some of its workers. A worker–firm pair ðw; f Þ blocks m if wemðf Þ; wAChðmðf Þ,fwg; Pðf ÞÞ; and fPðwÞmðwÞ; that is, if w and f are not matched through m; firm f wants to hire w— possibly after firing some of its current workers in mðf Þ— and worker w prefers firm f over her current match mðwÞ: Let P be a preference profile and let m be a matching. A pair ðB; f ÞA2W F ; with Ba|; blocks m if fPðwÞmðwÞ; for all wAB; and there is ADmðf Þ such that 2
We shall often abuse notation by omitting the brackets to denote a set with a unique element; here we write mðwÞ ¼ f instead of mðwÞ ¼ ff g:
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½A,B Pðf Þmðf Þ: In words, ðB; f Þ blocks m if firm f is willing to hire the workers in B—possibly after firing some of its current workers in mðf Þ—and all workers w in B prefer f over their current match mðwÞ: Definition 2. A matching m is stable if is individually rational and there is no worker– firm pair that blocks m: A matching m is stable if it is individually rational and there is no pair that blocks m: Notation. Given a preference profile P; we denote the set of stable matchings by SðPÞ; and the set of stable matchings by S ðPÞ: Lemma 1. S ðPÞDSðPÞ Proof. Let mAS ðPÞ; and assume that meSðPÞ: Since m is individually rational, and meSðPÞ; there exist ðw; f ÞAW F such that fPðwÞmðwÞ and wAChðmðf Þ, fwg; Pðf ÞÞ: This implies Chðmðf Þ,fwg; Pðf ÞÞPðf Þmðf Þ: Let B ¼ fwg; and A ¼ Chðmðf Þ,fwg; Pðf ÞÞ-mðf Þ: Then ðB; f Þ blocks m: Thus, meS ðPÞ: & We prove below that S ðPÞ equals the core, so it follows from standard results that S ðPÞ and SðPÞ may be different. We present an example as ‘‘supplementary material’’ in http://www.nyu.edu/jet/supplementary.html and in the workingpaper version ([4]) of the paper.
Definition 3. Let P be a preference profile. The core is the set of matchings m for which there is no F 0 DF ; W 0 DW with F 0 ,W 0 a|; and mAM # that satisfy (a)–(c) : (a) 0 For all wAW 0 and f AF 0 mðwÞDF # ; and mðf # ÞDW 0 : (b) For all wAW 0 and f AF 0 ; mðwÞRðwÞmðwÞ # and mðf # ÞRðf Þmðf Þ: (c) There is sAW 0 ,F 0 with mðsÞPðsÞmðsÞ: # We shall denote the core by Cw ðPÞ: Theorem 2. S ðPÞ ¼ Cw ðPÞ: Proof. First we shall prove that S ðPÞDCw ðPÞ: Let mAS ðPÞ; and suppose that me Cw ðPÞ: Let F 0 DF ; W 0 DW with F 0 ,W 0 a|; and let mAM # such that, for all wAW 0 ; 0 0 0 and for all f AF ; mðwÞDF # ; and mðf # ÞDW ; mðwÞRðwÞmðwÞ; # mðf # ÞRðf Þmðf Þ; and mðsÞPðsÞmðsÞ # for at least one sAW 0 ,F 0 : We shall need the following. Claim. There exists f AF 0 ; such that mðf # ÞPðf Þmðf Þ if and only if there is wAW 0 such that mðwÞPðwÞmðwÞ: # Proof. Let mðf # ÞPðf Þmðf Þ: Because m is individually rational, we have that mðf # ÞD / mðf Þ; so let wA # Þ\mðf Þ: By definition of m# we have that wA # ÞDW 0 ; so % mðf % mðf wemðf Þ and the definition of m# implies that mð # wÞPð wÞmð % % % wÞ: % Now suppose there exists % % Thus wAW 0 such that mðwÞPðwÞmðwÞ: # Let f% ¼ mðwÞ: # Then famðwÞ; so wemðfÞ: % % % % % mð # fÞamðfÞ; and the definition of m# implies that mð # fÞPðfÞmðfÞ: This proves the claim.
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By the claim, we can assume that there exists f AF 0 such that mðf # Þamðf Þ: Let B ¼ mðf # Þ\mðf Þ and A ¼ mðf # Þ-mðf Þ then A,B ¼ mðf # ÞPðf Þmðf Þ: Now, BDmðf # ÞDW 0 ; and B-mðf Þ ¼ |: Then ðB; f Þ blocks m; this contradicts that mAS ðPÞ: Now we shall prove that Cw ðPÞDS ðPÞ: Let mACw ðPÞ; and suppose that me S ðPÞ: First we shall show that m is individually rational. Assume that there exists f AF such that mðf ÞaChðmðf Þ; Pðf ÞÞ: Let mðf # Þ ¼ Chðmðf Þ; Pðf ÞÞ; F 0 ¼ ff gDF ; W 0 ¼ 0 0 mðf # ÞDW : Thus W ; F ; and m# imply meCw ðPÞ: Thus, mðf Þ ¼ Chðmðf Þ; Pðf ÞÞ; for all f AF : Now suppose there is wAW such that |PðwÞmðwÞ: Let mðwÞ # ¼ |; W 0 ¼ fwg; F 0 ¼ |: As before, meCw ðPÞ because mðwÞPðwÞmðwÞ: # Thus, mðwÞRðwÞ|; for all wAW : Thus m is individual rational. Assume that there exist ðB; f ÞA2W F with Ba|; such that: for all wAB; fPðwÞmðwÞ; and there exists ADmðf Þ such that ½A,B Pðf Þmðf Þ: Letting mðf # Þ ¼ A,B; F 0 ¼ ff g and W 0 ¼ B; it follows that meCw ðPÞ: A contradiction. & 2.3. Lattices and chains Let X be a set, and R a partial order—a transitive, reflexive, antisymmetric binary relation—on X : For any subset A of X ; inf R A ðsupR A) is the greatest (lowest) lower (upper) bound on A; in the order R: We say that the pair ðX ; RÞ is a lattice if whenever x; yAX ; both x4R y ¼ inf R fx; yg and x3R y ¼ supR fx; yg exist in X : We say that a lattice ðX ; RÞ is complete if for every nonempty subset A of X ; inf R A and supR A exist in X : Note that any finite lattice is complete. A lattice ðX ; RÞ is a chain if, for any x; yAX ; xRy or yRx; or both, are true.
3. The core as a set of fixed points We shall construct a map T on a superset of M such that the set of fixed points of T is the core. 3.1. Pre-matchings Say that a pair n ¼ ðnF ; nW Þ; with nF : F -2W and nW : W -F ,f|g; is a prematching: Let VW ðVF Þ denote the set of all nW ðnF ) functions. Thus, V ¼ VF VW denotes the set of pre-matchings n ¼ ðnF ; nW Þ: The superset of M we need turns out to be the set of pre-matchings V (pre-matchings were first used in [1]). A pre-matching n ¼ ðnF ; nW Þ; and a matching m are equivalent if mðf Þ ¼ nF ðf Þ for all f ; and mðwÞ ¼ nW ðwÞ for all w: A pre-matching n ¼ ðnF ; nW Þ is equivalent to some matching if and only if n is such that nW ðwÞ ¼ f if and only if wAnF ðf Þ: Notation. We make two conventions. First, we shall identify matchings and prematchings when they are equivalent, so if nAV is equivalent to some matching m we shall say that n is a matching. Hence M is a subset of V: Second, if n ¼ ðnF ; nW Þ is a pre-matching, we shall some times write nðf Þ for nF ðf Þ and nðwÞ for nW ðwÞ:
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3.2. T and the core Let n be a pre-matching, and let Uðf ; nÞ ¼ fwAW : fRðwÞnðwÞg; and V ðw; nÞ ¼ ff AF : wAChðnðf Þ,fwgÞ; Pðf Þg,f|g: The set Uðf ; nÞ is the set of workers w that are willing to give up nðwÞ in exchange for f : The set V ðw; mÞ is the set of firms f that are willing to hire w; possibly after firing some of the workers it was assigned by n: Now, define T : V-V by ðTnÞðsÞ ¼ ChðUðs; nÞ; PðsÞÞ if sAF ; and ðTnÞðsÞ ¼ maxPðsÞ fV ðs; nÞg if sAW : The map T has a simple interpretation: ðTnÞðf Þ is firm f ’s optimal team of workers, among those willing to work for f ; and ðTnÞðwÞ is the firm preferred by w among the firms that are willing to hire w: We shall denote by E the set of fixed points of T; so E ¼ fnAV : n ¼ Tng: Theorem 3. E ¼ S ðPÞ: Before we prove Theorem 3, note that Theorems 2 and 3 imply: Corollary 4. E ¼ Cw ðPÞ: Proof of Theorem 3. First we shall prove that if nAE; then n is a matching and n is individually rational. Let n ¼ ðnF ; nW ÞAE: Fix wAnF ðf Þ; we shall prove that f ¼ nW ðwÞ: nAE implies that wAnF ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ: Thus wAUðf ; nÞ: The definition of Uðf ; nÞ implies fRðwÞnW ðwÞ: Now, nF ðf Þ,fwg ¼ nF ðf Þ and nAE imply that nF ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ ð%Þ: So ChðnF ðf Þ; Pðf ÞÞ
ð1Þ
¼
ð2Þ
¼
ð3Þ
¼
ChðChðUðf ; nÞ; Pðf ÞÞ; Pðf ÞÞ ChðUðf ; nÞ; Pðf ÞÞ nF ðf Þ:
Equalities (1) and (3) follow from ð%Þ: Equality (2) is a simple property of choice sets. Hence nF ðf Þ ¼ ChðnF ðf Þ; Pðf ÞÞ: Now, wAnF ðf Þ implies ChðnF ðf Þ; Pðf ÞÞ ¼ ChðnF ðf Þ,fwg; Pðf ÞÞ: So we have f AV ðw; nÞ: But nW ðwÞ ¼ ðTnÞðwÞ ¼ maxPðwÞ fV ðw; nÞg; so nW ðwÞRðwÞf : But fRðwÞnW ðwÞ; so f ¼ nW ðwÞ; as RðwÞ is antisymmetric. Let f ¼ nW ðwÞ; we shall prove that wAnF ðf Þ: First, note that f ¼ nW ðwÞ implies that wAUðf ; nÞ: Second, note that f ¼ nW ðwÞ ¼ ðTnÞðwÞ ¼ maxPðwÞ fV ðw; nÞg; because nAE: So nW ðwÞRðwÞ|; and since nF ðf Þ ¼ ChðnF ðf Þ; Pðf ÞÞ; n is individually rational. Now, f AV ðw; nÞ; so we must have wAChðnF ðf Þ,fwg; Pðf ÞÞ by the definition of V ðw; nÞ: That is, wAChðnF ðf Þ,fwg; Pðf ÞÞRðf ÞnF ðf Þ:
ð1Þ
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Now, nAE implies that nF ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ:
ð2Þ
The definition of choice set implies nF ðf ÞDUðf ; nÞ; and nF ðf ÞRðf ÞUðf ; nÞ: But wAUðf ; nÞ implies Uðf ; nÞ+ChðnF ðf Þ,fwg; Pðf ÞÞ: The definition of choice set implies nF ðf ÞRðf ÞChðnF ðf Þ,fwg; Pðf ÞÞ: Together with (1), this implies wAnF ðf Þ: We have shown that wAvF ðf Þ if and only if f ¼ vW ðwÞ: So, n is a matching. Also, n is individually rational. Let mAE: Fix f AF ; BDW such that Ba|: We assume that, for all wAB; fPðwÞmðwÞ: By the definition of Uðf ; mÞ; we have that BDUðf ; mÞ: Let ADmðf Þ: Because m is a matching, we have that for all wAA; f ¼ mðwÞ: So the definition of Uðf ; mÞ implies that ð3Þ
ADUðf ; mÞ:
Now, mAE so mðf Þ ¼ ðTmÞðf Þ ¼ ChðUðf ; mÞ; Pðf ÞÞ; BDUðf ; mÞ and statement (3) imply then mðf ÞRðf ÞChðA,B; Pðf ÞÞRðf ÞA,B:
ð4Þ
So there is no ðB; f Þ that blocks m: Thus mAS ðPÞ: Let mAS ðPÞ and assume that maTm: First assume that there exist f AF such that mðf ÞaðTmÞðf Þ ¼ ChðUðf ; mÞ; Pðf ÞÞ ¼ CDUðf ; mÞ: Let A ¼ C-mðf Þ; and B ¼ C\mðf Þ: Because m is an individually rational matching we have that mðf ÞDUðf ; mÞ and Ba|: Then ðB; f Þ blocks m; which contradicts that mAS ðPÞ: Hence, for all f AF ; mðf Þ ¼ ðTmÞðf Þ:
ð5Þ
Assume that there exists wAW such that % ðw; mÞ; mðwÞaðTmÞðwÞ ¼ max fV ðw; mÞg ¼ fAV PðwÞ
% % by the definition on V ðw; mÞ: so wAChðmðfÞ,fwg; PðfÞÞ; % and mðwÞAV ðw; mÞ: This implies that But m is a matching, so we have that wemðfÞ % fPðwÞmðwÞ; and wAChðmðf Þ,fwg; Pðf ÞÞ ¼ C: Let A ¼ C-mðf Þ ¼ C fwg; and B ¼ fwg: Then ðB; f Þ blocks m: Hence, for all wAW ; mðwÞ ¼ ðTmÞðwÞ: This and statement (5) imply m ¼ Tm: Hence mAE: &
4. The T-algorithm The T-algorithm is very simple: start at some nAV and iterate Tn until two iterations are identical. When two iterations are identical, stop. We prove that, when the T-algorithm stops, it must be at a core matching. Further, we show that once the T-algorithm hits on a matching m (not just a
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pre-matching), m must be in the core. Hence, the T-algorithm ‘‘wanders’’ around pre-matchings until it finds a matching—it will stop at the matching it finds, and it will be a core matching. It follows, of course, that if the core is empty the algorithm must cycle, and that the cycle will only involve pre-matchings that are not matchings. The algorithm is very simple, but we include a detailed description to be as clear as possible, and to introduce some notation that we shall use in the proofs below: Algorithm 1 (T-algorithm). 1. Set n0 ¼ n: Set n1 ¼ Tn0 and k ¼ 1: 2. While nk ank 1 ; do: ðaÞ set k ¼ k þ 1; ðbÞ set nk ¼ Tnk 1 : 3. Set m ¼ nk : Stop. Proposition 5. If the T-algorithm stops at mAV; then m is a matching ðmAMÞ and m is in the core. If nk is in the core, for some iteration k of the T-algorithm, then the algorithm stops at m ¼ nk : Proof. If the algorithm stops at m then nk ¼ nk 1 ¼ m: Then, m ¼ Tnk 1 ¼ Tm; so mAE: By Corollary 4, mACw ðPÞ: On the other hand, if nk is in the core, then nk is a fixed point of T by Corollary 4. Then the T-algorithm stops at m ¼ nk : & Proposition 6. If nk is a matching, for some iteration k in the T-algorithm, then the algorithm stops at m ¼ nk ; and thus nk is a core matching. We shall present the proof of Proposition 6 as two lemmata: Lemma 7. Let nAV be a pre-matching. If n1 ¼ Tn is a matching, then n1 is individually rational. Proof. For all f AF ; n1 ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ: So Chðn1 ðf Þ; Pðf ÞÞ
ð1Þ
¼
ð2Þ
ChðChðUðf ; nÞ; Pðf ÞÞ; Pðf ÞÞ
¼
ChðUðf ; nÞ; Pðf ÞÞ
ð3Þ
n1 ðf Þ:
¼
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Equalities (1) and (3) follow because n1 ¼ Tn: Equality (2) is a property of choice sets. So, for all f AF ; n1 ðf Þ ¼ Chðn1 ðf Þ; Pðf ÞÞ:
ð6Þ
For all wAW ; n1 ðwÞ ¼ ðTnÞðwÞ ¼ max fV ðw; nÞg,f|g: PðwÞ
So, n1 ðwÞRðwÞ|: Statements (6) and (7) show that n1 is individually rational.
ð7Þ &
Lemma 8. If Tn is a matching, for some nAV; then Tn is a core matching. Thus if nk ¼ Tnk 1 is a matching, then the T-algorithm must stop at nk ; and nk is a core matching. Proof. We shall prove that, if nAV is a pre-matching, and n1 ¼ Tn is a matching, then n1 AS ðPÞ: The lemma follows from Proposition 5, as S ðPÞ ¼ E ¼ Cw ðPÞ: Assume that n1 eS ðPÞ: By Lemma 7, n1 is individually rational, so there must exist a ðB; f Þ that blocks n1 : Hence, for all wAB ðBa|Þ; fPðwÞn1 ðwÞ;
ð8Þ
and there must exist ADn1 ðf Þ such that ½A,B Pðf Þn1 ðf Þ:
ð9Þ
But n1 is a matching, and n1 ¼ Tn; so we have that for all wAB; wAn1 ðn1 ðwÞÞ: But n1 ðn1 ðwÞÞ ¼ ðTnÞðn1 ðwÞÞ ¼ ChðUðn1 ðwÞ; nÞ; Pðn1 ðwÞÞÞ; so that, wAUðn1 ðwÞ; nÞ; i.e. n1 ðwÞRðwÞnðwÞ:
ð10Þ
Statements (8) and (10) imply that, for all wAB; fRðwÞnðwÞ; i.e. BDUðf ; nÞ:
ð11Þ
Now, ADn1 ðf Þ and n1 ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ; so ADUðf ; nÞ:
ð12Þ
Statements (9), (11), and (12) contradict that n1 ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ: Thus n1 AS ðPÞ: & Remark. *
Proposition 6, and Example 9 below, imply that the T-algorithm and the DAA are different algorithms; we have not simply written down the DAA using different language.
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*
Unlike the DAA, the T-algorithm only stops at a core matchings. Example 9 below shows how the T-algorithm succeeds in finding a core matching, while the DAA stops at a non-core matching. When the core is empty, the T-algorithm must cycle. We present an example as ‘‘supplementary material’’ in http://www.nyu.edu/jet/supplementary.html and in the working-paper version [4] of the paper.
Example 9. Let F ¼ ff1 ; f2 g; W ¼ fw1 ; w2 ; w3 ; w4 g; and the preference profile P be defined by: Pðf1 Þ ¼ fw1 ; w2 g; fw3 ; w4 g; fw1 ; w3 g; fw2 ; w4 g; fw1 g; fw2 g; fw3 g; fw4 g; Pðf2 Þ ¼ fw1 ; w2 g; fw1 ; w3 g; fw2 ; w4 g; fw3 ; w4 g; fw1 g; fw2 g; fw3 g; fw4 g; Pðw1 Þ ¼ f1 ; f2 ; Pðw2 Þ ¼ f2 ; f1 ; Pðw3 Þ ¼ f1 ; f2 ; and Pðw4 Þ ¼ f1 ; f2 : Note that P is not substitutable, as w1 AChðfw1 ; w2 ; w3 ; w4 g; Pðf1 ÞÞ ¼ fw1 ; w2 g while w1 eChðfw1 ; w3 ; w4 g; Pðf1 ÞÞ ¼ fw3 ; w3 g: The T-algorithm starting at f1 n0 ¼ fw1 ; w2 g
f2
w1
w2
w3
w4
fw1 ; w2 g
|
|
|
|
does: f1
f2
w1
w2
w3
w4
Uðf ; n Þ=V ðw; n Þ ¼ n1 ¼ Uðf ; n1 Þ=V ðw; n1 Þ ¼
fw1 ; w2 ; w3 ; w4 g fw1 ; w2 g fw1 ; w3 ; w4 g
fw1 ; w2 ; w3 ; w4 g fw1 ; w2 g fw2 ; w3 ; w4 g
f1 ; f2 f1 f1 ; f2
f1 ; f2 f2 f1 ; f2
| | |
| | |
n2 ¼ Uðf ; n2 Þ=V ðw; n2 Þ ¼
fw3 ; w4 g fw1 ; w3 ; w4 g
fw2 ; w4 g fw2 ; w3 ; w4 g
f1 f2
f2 f2
| f1
| f1 ; f2
n3 ¼ Uðf ; n Þ=V ðw; n3 Þ ¼ n4 ¼
fw3 ; w4 g fw1 ; w3 ; w4 g fw3 ; w4 g
fw2 ; w4 g fw1 ; w2 g fw1 ; w2 g
f2 f2 f2
f2 f2 f2
f1 f1 f1
f1 f1 ; f2 f1
0
0
3
Note that ni for i ¼ 0; 1; 2; 3 are not matchings; n4 is a matching so, by Proposition 6, n4 is a core matching. On the other hand, the DAA stops at a matching that is not in the core. For the DAA, the equivalent to starting the T-algorithm at n0 is, as we shall see in Section 5, the DAA with firms proposing. The DAA with firms proposing stops at the matching f1
f2
m ¼ fw3 ; w4 g fw2 g
w1
w2
w3
w4
|
f2
f1
f1 :
But m is not in the core, as f2 Pðw1 Þmðw1 Þ ¼ |; f2 Rðw2 Þmðw2 Þ ¼ f2 ; and fw1 ; w2 gPðf2 Þfw2 g implies that ðfw1 ; w2 g; f2 Þ blocks m:
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5. The lattice structure of the core We shall introduce a partial order on V such that, if preferences are substitutable, T is a monotone increasing map. Tarski’s fixed point theorem then delivers a lattice structure on E; and thus on Cw ðPÞ: Definition 4. Define the following partial orders on VF ; VW and V: 1. PðF Þ on VF by nF PðF Þn0F if and only if n0F anF and, for all f in F ; nF ðf Þ ¼ n0F ðf Þ or nF ðf Þ ¼ ChðnF ðf Þ,n0F ðf Þ; Pðf ÞÞ: 2. PðW Þ on VW by nW PðW Þn0W if and only if n0W anW and nW ðwÞRðwÞn0W ðwÞfor all wAW : 3. The weak partial orders associated to PðF Þ and PðW Þ are denoted RðF Þ and RðW Þ; defined as: nF RðF Þn0F if nF ¼ n0F or nF PðF Þn0F ; and nW RðW Þn0W if nW ¼ n0W or nW PðW Þn0W : 4. PðF Þ on V by nPðF Þn0 iff nF PðF Þn0F and n0W RðW ÞnW ; or nF RðF Þn0F and n0W PðW ÞnW : 5. PðW Þ on V by nPðW Þn0 iff nW PðW Þn0W and n0F RðF ÞnF ; or nW RðW Þn0W and n0F PðF ÞnF : Definition 5. A firm f ’s preference ordering Pðf Þ satisfies substitutability if for any set SDW containing workers w and w% ðwawÞ; % if wAChðS; Pðf ÞÞ then wAChðS\fwg; % Pðf ÞÞ: A preference profile P is substitutable if, for each firm f ; the preference ordering Pðf Þ satisfies substitutability. Theorem 10. Let P be substitutable. Then ðCw ðPÞ; PðF ÞÞ and ðCw ðPÞ; PðW ÞÞ are nonempty complete lattices, and inf PðF Þ Cw ðPÞ ¼ supPðW Þ Cw ðPÞ; and supPðF Þ Cw ðPÞ ¼ inf PðW Þ Cw ðPÞ: Theorem 10 says that there are two distinguished core matchings, mW and mF such that: For firms, mW ¼ inf PðF Þ Cw ðPÞ is worse, and mF ¼ supPðF Þ Cw ðPÞ is better, than any other core matchings. For workers mW ¼ supPðW Þ Cw ðPÞ is better, and mF ¼ inf PðW Þ Cw ðPÞ is worse, than any other core matching. Proof of Theorem 10. First, ðV0 ; PðF ÞÞ is a complete lattice—due to space restrictions, we refer to the working paper version of the paper [4] for the proof. We shall present the proof of the Theorem as two simple lemma. We assume that P is substitutable in the rest of the section. Notation. Let V0 ¼ fnAV : nðsÞRðsÞ|; for all sAF ,W g:
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Remark. For all nAV; TnAV0 Lemma 11. Let m and m0 be pre-matchings. If m0 RðF Þm then, for all wAW and f AF ; Uðf ; mÞDUðf ; m0 Þ; and V ðw; mÞ+V ðw; m0 Þ: Proof. Let wAUðf ; mÞ: We have that fRðwÞmðwÞ; but mðwÞRðwÞm0 ðwÞ so wAUðf ; m0 Þ: This proves Uðf ; mÞDUðf ; m0 Þ: Now we shall prove that V ðw; mÞ+V ðw; m0 Þ: First, if V ðw; m0 Þ ¼ f|g; then there is nothing to prove, as f|g ¼ V ðw; m0 ÞDV ðw; mÞ: Suppose that V ðw; m0 Þaf|g; and let f AV ðw; m0 Þ: Then, wAChðm0 ðf Þ,fwg; Pðf ÞÞ: But m0 RðF Þm; so the definition of RðF Þ implies that, for all f AF ; either 0 m ðf Þ ¼ mðf Þ so wAChðmðf Þ,fwg; Pðf ÞÞ; or m0 ðf Þ ¼ Chðm0 ðf Þ,mðf Þ; Pðf ÞÞ: Then wAChðm0 ðf Þ,fwg; Pðf ÞÞ implies that w
A ¼ ð1Þ
¼
Chðm0 ðf Þ,fwg; Pðf ÞÞ ChðChðm0 ðf Þ,mðf Þ; Pðf ÞÞ,fwg; Pðf ÞÞ Chðm0 ðf Þ,mðf Þ,fwg; Pðf ÞÞ:
Equality (1) is from [3, Proposition 2.3]. Substitutability of P implies that wAChðmðf Þ,fwg; Pðf ÞÞ: Then f AV ðw; mÞ; and thus V ðw; mÞ+V ðw; m0 Þ: & Lemma 12. E is a non-empty and complete lattice. Proof. First we show that TjV0 is monotone increasing. That is, whenever m0 RðF Þm; we have ðTm0 ÞRðF ÞðTmÞ: Let m0 RðF Þm; and fix f AF and wAW : Lemma 11 says that Uðf ; mÞDUðf ; m0 Þ: We first show that ChðUðf ; m0 Þ; Pðf ÞÞ ¼ Chð½ChðUðf ; m0 Þ; Pðf ÞÞ,ChðUðf ; mÞ; Pðf ÞÞ ; Pðf ÞÞ:
ð13Þ
To see this, let SDChðUðf ; m0 Þ; Pðf ÞÞ,ChðUðf ; mÞ; Pðf ÞÞ: Then SDUðf ; mÞ, Uðf ; m0 Þ ¼ Uðf ; m0 Þ; so ChðUðf ; m0 Þ; Pðf ÞÞRðf ÞS: But, ChðUðf ; m0 Þ; Pðf ÞÞD ChðUðf ; m0 Þ; Pðf ÞÞ,ChðUðf ; mÞ; Pðf ÞÞ; so we have established statement 13. Now, ðTm0 Þðf Þ ¼ ChðUðf ; m0 Þ; Pðf ÞÞ and ðTmÞðf Þ ¼ ChðUðf ; mÞ; Pðf ÞÞ; so statement 13 implies that ðTm0 Þðf Þ ¼ Chð½ðTm0 Þðf Þ,ðTmÞðf Þ ; Pðf ÞÞÞ:
ð14Þ
We now show that ðTmÞðwÞRðwÞðTm0 ÞðwÞ: Lemma 11 says that V ðw; m0 ÞDV ðw; mÞ: So, ðTmÞðwÞ ¼ maxfV ðw; mÞgRðwÞ max fV ðw; m0 Þg ¼ ðTm0 ÞðwÞ: Pðf Þ
Pðf Þ
ð15Þ
Statements 14 and 15 imply that ðTm0 ÞRðF ÞðTmÞ; as f AF and wAW were arbitrary. Finally, TðVÞDV0 so EDV0 ; and E equals the set of fixed points of TjV0 : TðVÞDV0 also implies that the restricted map TjV0 has range in V0 :
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Now we have that TjV0 : V0 -V0 is monotone increasing, and V0 is a complete lattice. So Tarski’s fixed point theorem implies that ðE; PðF ÞÞ is a non-empty complete lattice. & Corollary 4 and Lemma 12 finish the proof of Theorem 10. See [4] for full details.
6. The T-algorithm and substitutability When preferences are substitutable, the T-algorithm 1. finds the worker-optimal core matching ðinf PðF Þ Cw ðPÞÞ and the firm-optimal core matching ðsupPðF Þ Cw ðPÞÞ; 2. is computationally efficient—we can give a bound on its computational complexity and show that the bound is likely to be quite small, 3. calculates the lattice operations ‘‘join’’ and ‘‘meet,’’ on the Cw ðPÞ lattice. Result 1 is important because it shows that the T-algorithm does the job of the DAA. Further, by result 2 we have an idea of the T-algorithm’s computational complexity, which is an open problem for the DAA in the many-to-one case. In a very simple example, we show that, while the ‘‘size’’ of the problem is 30; 000 iterations, the T-algorithm needs at most nine iterations to find a core matching. Our result on the complexity of the T-algorithm is similar to Topkis’s [14] (see also [15]) results on algorithms for finding equilibria in supermodular games. Result 3: A number of papers have sought to improve on Blair’s lattice operations [2,8] by restricting firms’ preferences. We show that, with only substitutability, the Talgorithm can compute the lattice operations—thus allowing for a practical solution to the problem of Blair’s lattice operations. 6.1. The firm- and worker-optimal core matchings Proposition 13. If P is substitutable, the T-algorithm starting at inf PðF Þ V0 stops at the worker-optimal core matching inf PðF Þ Cw ðPÞ; the T-algorithm starting at supPðF Þ V0 stops at the firm-optimal core matching supPðF Þ Cw ðPÞ: Proof. When preferences are substitutable, T is a monotone map on the complete lattice V0 : Proposition 13 follows then from standard results, see [14,15]. & Remark. 1. The pre-matching inf PðF Þ V0 is nAV with nðf Þ ¼ f|g and nðwÞ ¼ maxPðwÞ F : The pre-matching supPðF Þ V0 is n with nðf Þ ¼ ChðW ; Pðf ÞÞ and nðwÞ ¼ f|g: 2. The T-algorithm starting at points that are not inf PðF Þ V0 or supPðF Þ V0 may not converge. For example, one can show that the T-algorithm starting at the prematching n ¼ ððsupPðF Þ V0 ÞF ; ðinf PðF Þ V0 ÞW Þ cycles.
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6.2. Complexity of the T-algorithm We shall now give a bound on the computational complexity of the T-algorithm, but we need some auxiliary definitions first. For each f ; let Xf ¼ fADW : ARðf Þ|g; and let Xf be the partial order on Xf defined by AXB iff A ¼ ChðA,B; Pðf ÞÞ or A ¼ B: Let qf be the size of the longest Xf -chain in Xf : For each w; let Xw ¼ ff AF : fRðwÞ|g; and let qw be the number of elements in Xw : We present an example as ‘‘supplementary material’’ in http:// www.nyu.edu/jet/supplementary.html and in the working-paper version [4] of the paper. Proposition 14. If P is substitutable, the T-algorithm starting at inf PðF Þ V0 ðsupPðF Þ V0 Þ stops at the worker-optimal (firm-optimal) core matching in less than X ðqs 1Þ sAF ,W
iterations. Proof. Let n0 ; n1 ; ynk be the sequence generated by the T-algorithm before it stops k at a core matching nk : The P number of iterations before the algorithm stops at n is k; we shall prove that kp sAF ,W ðqs 1Þ: Assume—without loss of generality—that we started the T-algorithm at n0 ¼ inf PðF Þ V0 : Since n1 ¼ ðTn0 ÞPðF Þn0 and T is monotone increasing (see the proof of Theorem 10), nk PðF Þnk 1 PðF Þ?n1 PðF Þn0 : For each l; 1plpk; it must be that nl ðf Þ ¼ nl 1 ðf Þ or that nl ðf Þ ¼ Chðnl ðf Þ,nl 1 ðf Þ; Pðf ÞÞ for all f ; and nl 1 ðwÞRðwÞnl ðwÞ for all w: Also, for each l; 1plpk; either nl ðf Þanl 1 ðf Þ and nl ðf Þ ¼ Chðnl ðf Þ,nl 1 ðf Þ; Pðf ÞÞ for some for f or nl 1 ðwÞPðwÞnl ðwÞ for some w—recall that PðF Þ is a strict order, see Definition 4. Say that l is an f -step if nl ðf Þanl 1 ðf Þ and nl ðf Þ ¼ Chðnl ðf Þ,nl 1 ðf Þ; Pðf ÞÞ for firm f : Say that l is a w-step if nl 1 ðwÞPðwÞnl ðwÞ for worker w: There cannot be more than qs 1 s-steps, as a number of, say r steps involves a chain of r þ 1 elements in Xs : Now k is smaller than the sum over all sAW ,F of all s-steps, so X ðqs 1Þ: kp sAF ,W
But k is the number of iterations of the T-algorithm before it finds nk :
&
Remark. Recall that jW j ¼ m and jF j ¼ n; so the number of pre-matchings is 2nm ðn þ 1Þm ; a number which is normally orders of magnitude larger than the bound in Proposition 14: In an example we develop as ‘‘supplementary material’’ in http:// www.nyu.edu/jet/supplementary.html (see also [4]) Proposition 14 implies that
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the T-algorithm converges before X ðqs 1Þ ¼ 9 sAF ,W
iterations. Compare with the number of pre-matchings: 32768. The number of matchings is smaller than the number of pre-matchings, but still substantially larger than 9. It may be unfair to compare the complexity of the T-algorithm to the number of pre-matchings, after all the T-algorithm will not involve pre-matchings where agents are matched to unacceptable partners—but still, the number of pre-matchings that only involve acceptable partners is 9261. 6.3. Computing the lattice operations Let m1 ; m2 ACw ðPÞ: We shall look at the problem of finding m1 3PðF Þ m2 ; the join (least upper bound) of m1 and m2 in the ðCw ðPÞ; PðF ÞÞ-lattice. One can solve the problem of finding m1 4PðF Þ m2 in the ðCw ðPÞ; PðF ÞÞ-lattice in a similar way. Martı´ nez et al. [8,9] prove that, when preferences satisfy substitutability and an additional assumption (which they call q-separability) there is a simple formula for m1 3PðF Þ m2 (see [2] for a similar result). Martı´ nez et al. prove that m1 3PðF Þ m2 is 8 < Chðm1 ðsÞ,m2 ðsÞ; PðsÞÞ if sAF ; ð16Þ n% ðsÞ ¼ min fm1 ðsÞ; m2 ðsÞg if sAW : : PðsÞ In general, though, when preferences are only substitutable, there is no formula for m1 3PðF Þ m2 : In fact, n% defined in (16) is a pre-matching but not necessarily a matching. We prove that, if one starts the T-algorithm at n% ; it will stop at m1 3PðF Þ m2 : Hence we give a practical solution to Blair’s problem of calculating m1 3PðF Þ m2 :3 Theorem 15. Let P be substitutable, let m1 and m2 be core matchings, and n% be defined as in (16). The T-algorithm started at n% stops at m1 3PðF Þ m2 : # ¼ fnAV : nRðF Þn% g: First we prove (1) that T VD # V: # Then we prove Proof. Let V 1 2 (2) that T-algorithm starting at n% stops at m 3PðF Þ m : (1) First we shall prove that n% is the least upper bound on fm1 ; m2 g in V: By the definition of n% ; n% RðF Þm1 and n% RðF Þm2 : 3 Blair’s proof of the existence of m1 3PðFÞ m2 can also be expressed as an algorithm for calculating m1 3PðFÞ m2 :
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Let n be an upper bound on fm1 ; m2 g in V; we shall prove that nRðF Þn% : For any f AF : ð1Þ
nðf ÞRðf Þm1 ðf Þ
3 nðf Þ ¼ Chðnðf Þ,m1 ðf Þ; Pðf ÞÞ
nðf ÞRðf Þm2 ðf Þ
3 nðf Þ ¼ Chðnðf Þ,m2 ðf Þ; Pðf ÞÞ:
ð2Þ
We have that Chðnðf Þ,n% ðf Þ; Pðf ÞÞ
ð3Þ
¼
Chðnðf Þ,Chðm1 ðf Þ,m2 ðf Þ; Pðf ÞÞ; Pðf ÞÞ
ð4Þ
¼
Chðnðf Þ,m1 ðf Þ,m2 ðf Þ; Pðf ÞÞ
ð5Þ
¼
ChðChðnðf Þ,m1 ðf Þ; Pðf ÞÞ,m2 ðf Þ; Pðf ÞÞ
ð6Þ
Chðnðf Þ,m2 ðf Þ; Pðf ÞÞ
¼
ð7Þ
¼
nðf Þ:
Where equality (3) follows from the definition of n% : Equalities (4) and (5) follow from substitutability and [3, Proposition 2.3]. Equality (6) follows from equality (1). Equality (7) follows from equality (2). Thus, nF RðF Þn% F : Now, nRðF Þm1 and nRðF Þm2 imply that, for any wAW ; m1 ðwÞRðwÞnðwÞ; and m2 ðwÞRðwÞnðwÞ: But then, n% ðwÞ ¼ minfm1 ðwÞ; m2 ðwÞgRðwÞnðwÞ: PðwÞ
This and nF RðF Þn% F implies that nRðF Þn% : Hence n% is the least upper bound on fm1 ; m2 g in V: # V: # Let nAV; # then nRðF Þn% RðF Þm1 and Now we shall prove that T VD 2 1 nRðF Þn% RðF Þm : But then TnRðF ÞTm and TnRðF ÞTm2 ; as T is monotone. Now, m1 and m2 are core matchings, so m1 ¼ Tm1 and m2 ¼ Tm2 ; then Tn is an upper bound on # fm1 ; m2 g in V: But n% is the least upper bound, so TnRðF Þn% and thus TnAV: # V; # so Tj # : V# V: # T is monotone and n% is the smallest (2) We know that T VD V # pre-matching in V: By Topkis’s [14] results, the T-algorithm stops at the RðF Þsmallest fixed point of TjV# ; but this is the RðF Þ-smallest fixed point of T; of those fixed points that are RðF Þ-larger than both m1 and m2 : Then the T-algorithm stops at m1 3PðF Þ m2 : & Remark. 1. A key part of the proof of Theorem 15 is that n% is the join of m1 and m2 in V: By a similar argument, the T-algorithm calculates the meet, m1 4PðF Þ m2 ; of any two core matchings m1 and m2 : One only needs to start the algorithm at the meet of m1 and m2 in V instead of the join.4 4
See [2] on how to calculate the meet of m1 and m2 in V:
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2. The bound presented in Proposition 14 also bounds the number of T-iterations necessary to calculate m1 3PðF Þ m2 : We now illustrate how the algorithm finds the join of two core matchings: Example 16. (Roth [10]). Let F ¼ ff1 ; f2 ; f3 ; f4 ; f5 g be the set of firms and W ¼ fw1 ; w2 ; w3 ; w4 ; w5 ; w6 g be the set of workers. As in [10], we specify only preferences among the choices we shall need. Pðf1 Þ ¼ fw4 g; fw1 g; fw2 ; w3 ; w5 ; w6 g; y; fw5 g; Pðf2 Þ ¼ fw2 g; fw1 ; w3 g; Pðf3 Þ ¼ fw3 g; fw2 g; Pðf4 Þ ¼ fw5 g; fw4 ; w6 g; Pðf5 Þ ¼ fw6 g; fw5 g; Pðw1 Þ ¼ f2 ; f1 ; Pðw2 Þ ¼ f1 ; f3 ; f2 ; Pðw3 Þ ¼ f1 ; f2 ; f3 ; Pðw4 Þ ¼ f4 ; f1 ; Pðw5 Þ ¼ f1 ; f5 ; f4 ; Pðw6 Þ ¼ f1 ; f4 ; f5 : Consider the following two core matchings f1 m1 ¼ fw1 g f1 m2 ¼ fw4 g
f2 fw2 g
f3 fw3 g
f4 fw4 ; w6 g
f5 and fw5 g
f2
f3
f4
f5
fw1 ; w3 g
fw2 g
fw5 g
fw6 g
n1 ¼ Uðf ; n1 Þ=V ðw; n1 Þ ¼ n2 ¼
:
f1 w4
f2 w2
f3 w3
f4 w5
f5 w6
w1 f1
w2 f2
w3 f3
w4 f1
w5 f4
w6 f5
w1 ; w4 w4
w2 w2
w3 w3
w5 w5
w6 w6
| |
f2 f2
f3 f3
f1 f1
f4 f4
f5 f5 :
n2 is a matching and n2 ¼ m1 3PðF Þ m2 : Acknowledgments We are grateful to Jinpeng Ma, Ruth Martı´ nez, and Alejandro Neme for their helpful comments.
References [1] H. Adachi, On a characterization of stable matchings, Econ. Letters 68 (2000) 43–49. [2] A. Alkan, A class of multipartner matching markets with a strong lattice structure, Econ. Theory 19 (2002) 737–746. [3] C. Blair, The lattice structure of the set of stable matchings with multiple partners, Math. Operations Research 13 (1988) 619–628. [4] F. Echenique, J. Oviedo, Core many-to-one matchings by fixed-point methods, Caltech Social Science Working Paper, No. 1140, 2002. [5] T. Fleiner, A fixed-point approach to stable matchings and some applications, Technical Report, Egervary Research Group on Combinatorial Optimization, 2001. [6] D. Gusfield, R. Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989. [7] A. Kelso, V. Crawford, Job matching, coalition formation, and gross substitutes, Econometrica 50 (1982) 1483–1504. [8] R. Martı´ nez, J. Masso´, A. Neme, J. Oviedo, Single agents and the set of many-to-one stable matchings, J. Econ. Theory 91 (2000) 91–105.
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[9] R. Martı´ nez, J. Masso´, A. Neme, J. Oviedo, On the lattice structure of the set of stable matchings for a many-to-one model, Optimization 50 (2001) 439–457. [10] A. Roth, Conflict and coincidence of interest in job matching: some new results and open questions, Math. Operations Research 10 (1985) 379–389. [11] A. Roth, M. Sotomayor, Interior points in the core of two-sided matching markets, J. Econ. Theory 45 (1988) 85–101. [12] A. Roth, M. Sotomayor, Two-sided Matching: A Study in Game-Theoretic Modelling and Analysis, Econometric Society Monographs No. 18, Cambridge University Press, Cambridge, England, 1990. [13] M. Sotomayor, Three remarks on the many-to-many stable matching problem, Math. Soc. Sci. 38 (1999) 55–70. [14] D.M. Topkis, Equilibrium points in non-zero sum n-person submodular games, SIAM J. Control Optim. 17 (1979) 773–787. [15] D.M. Topkis, Supermodularity and Complementarity, Princeton University Press, Princeton, NJ, 1998.
Journal of Economic Theory 115 (2004) 358–376
Core many-to-one matchings by fixed-point methods Federico Echeniquea, and Jorge Oviedob a
Humanities and Social Sciences, California Institute of Technology, MC 228-77, Pasadena, CA 91125, USA b Instituto de Matema´tica Aplicada, Universidad Nacional de San Luis, Eje´rcito de los Andes 950, 5700 San Luis, Argentina Received 4 October 2002; final version received 21 February 2003
Abstract We characterize the core many-to-one matchings as fixed points of a map. Our characterization gives an algorithm for finding core allocations; the algorithm is efficient and simple to implement. Our characterization does not require substitutable preferences, so it is separate from the structure needed for the non-emptiness of the core. When preferences are substitutable, our characterization gives a simple proof of the lattice structure of core matchings, and it gives a method for computing the join and meet of two core matchings. r 2003 Elsevier Science (USA). All rights reserved. JEL classification: C78 Keywords: Matching; Core; Lattice; Stability; Algorithm; Complexity; Substitutability; Tarski’s fixed point theorem
1. Introduction This paper deals with the many-to-one matchings that are in the core. Given a set of workers and a set of firms, a many-to-one matching (a matching, for short) is an assignment of a group of workers to each firm. The problem is interesting because each worker has a preference relation over firms, and each firm has a preference relation over groups of workers. Given a matching, firms who are unhappy with
Corresponding author. E-mail addresses: [email protected] (F. Echenique), [email protected] (J. Oviedo).
0022-0531/03/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0022-0531(03)00184-4
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their current group of workers, and workers who are unhappy with their current employers, may recontract in a mutually beneficial way, thus destroying the proposed matching. Arguably, the natural formalization of matchings that are robust to recontracting is the core (see Roth and Sotomayor [12], RS in the rest of the paper, for definitions). The standard approach to the core (RS is a classic) is to impose some structure on firms’ preferences that both simplifies the core and ensures that it is non-empty; the structure used is to assume that firms’ preferences are ‘‘substitutable’’ (see RS, or our Definition 5). When preferences are substitutable, a matching is in the core if and only if it is ‘‘stable’’—a stable matching is robust to recontracting by worker–firm pairs, but not more complicated coalitions. Further, there is an algorithm, the ‘‘deferred acceptance algorithm’’ (DAA) which finds certain distinguished core matchings: the DAA finds a core matching that is better for the firms than all other core matchings, and a core matching that is better for the workers than all other core matchings. Kelso and Crawford [7] introduced the notion of substitutability (their ‘‘gross substitutes’’ condition) and first used it to prove—via the DAA—that the core is non-empty in a class of models that, under substitutability, includes the model studied in our paper. Our approach is different. We characterize the core as the set of fixed points of a map T: We do not add structure to the model, so the core (and the fixed points of T) may be empty. Our characterization is useful because it allows us to: 1. Construct a simple algorithm that consists in iterating T; we shall call it the Talgorithm. The T-algorithm either finds a core matching or cycles in a recognizable way. Unlike the DAA, the T-algorithm does not stop at a non-core matching. We present an example where the DAA stops at a non-core matching, while the T-algorithm succeeds in finding a core matching. 2. We prove that, when preferences are substitutable, T has at least one fixed point—hence that the core is non-empty—and that the T-algorithm finds the two distinguished core matchings: the matching that is best for the firms and the matching that is best for the workers. We can bound the computational complexity of the T-algorithm. To the best of our knowledge, the computational complexity of the DAA for the many-toone case is unknown.1 3. Blair [3] proves that, when preferences are substitutable, the core matchings have a lattice structure. (a) Our characterization gives a very simple proof of Blair’s result; it follows almost immediately from Tarski’s fixed point theorem, as T is a monotone map under an appropriate order. (b) There is no formula for computing Blair’s lattice operations (join and meet). We show how the T-algorithm computes the join and meet of any two core matchings.
1
Gusfield and Irving [6] prove that the DAA is Oðn2 Þ for one-to-one matchings.
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Our results on the core build a bridge to the literature on supermodular games (see [15]); in fact, our results on the T-algorithm are adapted from similar results on computing Nash equilibria in supermodular games. We believe that this bridge is useful for understanding matching problems better, and may deliver further results in future research. Fixed-point methods have been used in the matching literature before. Tarski’s fixed-point theorem was applied by Roth and Sotomayor [11] to the assignment game, Adachi [1] to one-to-one stable matchings, and Fleiner [5] to many-toone stable matchings. Fleiner’s is the paper closest to our paper; he proves a version of Blair’s lattice result for a general graph-theoretic model. Fleiner’s model includes the many-to-one matching model with substitutable preferences as a special case. Fleiner’s paper is a paper in graph theory, though, and his results are very hard to translate into the language of matching models. For that reason, we believe that our result in 3(a) above is valuable, beyond its auxiliary role in our other results. We should clarify at this point that ‘‘core’’ is RS’s ‘‘weak core.’’ We do not have results for RS’s ‘‘strong core,’’ hence we opted, in the interest of exposition economy, for omitting all ‘‘weak’’ qualifiers from the paper. Nothing is really known about the strong core, so we feel that our omission is justified. See RS, and Sotomayor [13], for a discussion of the different core definitions. In Section 2 we define the model, and prove some preliminary results on the core many-to-one matchings. In Section 3 we introduce our fixed-point approach to the core. In Section 4 we introduce the T-algorithm. In Section 5 we study the lattice structure of the core many-to-one matchings. In Section 6 we analyze the Talgorithm for substitutable preferences.
2. Preliminary definitions and results 2.1. The model There are two disjoint sets of agents, the set of n firms F and the set of m workers W : Each firm f AF has a strict, transitive, and complete preference relation Pðf Þ over the set of all subsets of W ; and each worker has a strict, transitive, and complete preference relation PðwÞ over F ,|: Preferences profiles are ðn þ mÞ–tuples of preference relations; we denote them by P ¼ ðPðf1 Þ; y; Pðfn Þ; Pðw1 Þ; y; Pðwm ÞÞ: Given a firm’s preference relation, Pðf Þ; the sets of workers that f prefers to the empty set are called acceptable; thus we allow that firm f may prefer not hiring any worker rather than hiring unacceptable subsets of workers. Similarly, given a preference relation of a worker PðwÞ the firms preferred by w to the empty set are called acceptable; in this case we are allowing that worker w may prefer to remain unemployed rather than working for an unacceptable firm. It turns out that only acceptable partners matter, so we shall write preference relation concisely as lists of acceptable partners. For example, Pðfi Þ ¼ fw1 ; w3 g; fw2 g; fw1 g; fw3 g indicates that
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fw1 ; w3 gPðfi Þfw2 gPðfi Þfw1 gPðfi Þfw3 gPðfi Þ|: And Pðwj Þ ¼ f1 ; f3 indicates that f1 Pðwj Þf3 Pðwj Þ|: We denote by R the weak orders associated with P: So, fi RðwÞfj if fi ¼ fj or fi PðwÞfj : Similarly for Rðf Þ: We shall study assignments of workers to firms as matchings of firms with groups of workers. The assignment problem consists of matching workers with firms maintaining the bilateral nature of their relationship, and allowing for the possibility that both firms and workers may remain unmatched. Formally, Definition 1. A matching m is a mapping from the set F ,W into the set of all subsets of F ,W such that for all wAW and f AF : (a) Either jmðwÞj ¼ 1 and mðwÞDF or else mðwÞ ¼ |: (b) mðf ÞA2W : (c) mðwÞ ¼ ff g2 if and only if wAmðf Þ: Given ðF ; W ; PÞ we denote the set of matchings by M: Let P be a preference profile. Given a set SDW ; let ChðS; Pðf ÞÞ denote firm F ’s most-preferred subset of S according to its preference ordering Pðf Þ: We call ChðS; Pðf ÞÞ the choice set of S according to Pðf Þ: That is, A ¼ ChðS; Pðf ÞÞ if and only if ADS and APðf ÞB for all BDS with AaB: 2.2. The core, stable matchings, and stable matchings A matching m is stable if no worker–firm pair can benefit by deviating from m: RS prove that, if firms’ preferences are substitutable, the set of stable matchings equals the core. A matching is stable if no set of workers can, jointly with one firm, benefit from deviating from m: We prove that the set of stable matchings equals the core; the result does not require additional assumptions on preferences. We use the notion of stable matching instead of stable matching because our fixed-point approach allows us to characterize the set of stable matchings, and thus the core. It should be clear from our proofs and examples just how useful the stable definition is. Let P be a preference profile and let m be a matching. Say that m is individually rational if mðwÞRðwÞ| for all wAW ; and if mðf Þ ¼ Chðmðf Þ; Pðf ÞÞ; for all f AF : Intuitively, m is individually rational if no agent can unilaterally improve over its assignment in m—workers by choosing to remain unemployed and firms by firing some of its workers. A worker–firm pair ðw; f Þ blocks m if wemðf Þ; wAChðmðf Þ,fwg; Pðf ÞÞ; and fPðwÞmðwÞ; that is, if w and f are not matched through m; firm f wants to hire w— possibly after firing some of its current workers in mðf Þ— and worker w prefers firm f over her current match mðwÞ: Let P be a preference profile and let m be a matching. A pair ðB; f ÞA2W F ; with Ba|; blocks m if fPðwÞmðwÞ; for all wAB; and there is ADmðf Þ such that 2
We shall often abuse notation by omitting the brackets to denote a set with a unique element; here we write mðwÞ ¼ f instead of mðwÞ ¼ ff g:
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½A,B Pðf Þmðf Þ: In words, ðB; f Þ blocks m if firm f is willing to hire the workers in B—possibly after firing some of its current workers in mðf Þ—and all workers w in B prefer f over their current match mðwÞ: Definition 2. A matching m is stable if is individually rational and there is no worker– firm pair that blocks m: A matching m is stable if it is individually rational and there is no pair that blocks m: Notation. Given a preference profile P; we denote the set of stable matchings by SðPÞ; and the set of stable matchings by S ðPÞ: Lemma 1. S ðPÞDSðPÞ Proof. Let mAS ðPÞ; and assume that meSðPÞ: Since m is individually rational, and meSðPÞ; there exist ðw; f ÞAW F such that fPðwÞmðwÞ and wAChðmðf Þ, fwg; Pðf ÞÞ: This implies Chðmðf Þ,fwg; Pðf ÞÞPðf Þmðf Þ: Let B ¼ fwg; and A ¼ Chðmðf Þ,fwg; Pðf ÞÞ-mðf Þ: Then ðB; f Þ blocks m: Thus, meS ðPÞ: & We prove below that S ðPÞ equals the core, so it follows from standard results that S ðPÞ and SðPÞ may be different. We present an example as ‘‘supplementary material’’ in http://www.nyu.edu/jet/supplementary.html and in the workingpaper version ([4]) of the paper.
Definition 3. Let P be a preference profile. The core is the set of matchings m for which there is no F 0 DF ; W 0 DW with F 0 ,W 0 a|; and mAM # that satisfy (a)–(c) : (a) 0 For all wAW 0 and f AF 0 mðwÞDF # ; and mðf # ÞDW 0 : (b) For all wAW 0 and f AF 0 ; mðwÞRðwÞmðwÞ # and mðf # ÞRðf Þmðf Þ: (c) There is sAW 0 ,F 0 with mðsÞPðsÞmðsÞ: # We shall denote the core by Cw ðPÞ: Theorem 2. S ðPÞ ¼ Cw ðPÞ: Proof. First we shall prove that S ðPÞDCw ðPÞ: Let mAS ðPÞ; and suppose that me Cw ðPÞ: Let F 0 DF ; W 0 DW with F 0 ,W 0 a|; and let mAM # such that, for all wAW 0 ; 0 0 0 and for all f AF ; mðwÞDF # ; and mðf # ÞDW ; mðwÞRðwÞmðwÞ; # mðf # ÞRðf Þmðf Þ; and mðsÞPðsÞmðsÞ # for at least one sAW 0 ,F 0 : We shall need the following. Claim. There exists f AF 0 ; such that mðf # ÞPðf Þmðf Þ if and only if there is wAW 0 such that mðwÞPðwÞmðwÞ: # Proof. Let mðf # ÞPðf Þmðf Þ: Because m is individually rational, we have that mðf # ÞD / mðf Þ; so let wA # Þ\mðf Þ: By definition of m# we have that wA # ÞDW 0 ; so % mðf % mðf wemðf Þ and the definition of m# implies that mð # wÞPð wÞmð % % % wÞ: % Now suppose there exists % % Thus wAW 0 such that mðwÞPðwÞmðwÞ: # Let f% ¼ mðwÞ: # Then famðwÞ; so wemðfÞ: % % % % % mð # fÞamðfÞ; and the definition of m# implies that mð # fÞPðfÞmðfÞ: This proves the claim.
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By the claim, we can assume that there exists f AF 0 such that mðf # Þamðf Þ: Let B ¼ mðf # Þ\mðf Þ and A ¼ mðf # Þ-mðf Þ then A,B ¼ mðf # ÞPðf Þmðf Þ: Now, BDmðf # ÞDW 0 ; and B-mðf Þ ¼ |: Then ðB; f Þ blocks m; this contradicts that mAS ðPÞ: Now we shall prove that Cw ðPÞDS ðPÞ: Let mACw ðPÞ; and suppose that me S ðPÞ: First we shall show that m is individually rational. Assume that there exists f AF such that mðf ÞaChðmðf Þ; Pðf ÞÞ: Let mðf # Þ ¼ Chðmðf Þ; Pðf ÞÞ; F 0 ¼ ff gDF ; W 0 ¼ 0 0 mðf # ÞDW : Thus W ; F ; and m# imply meCw ðPÞ: Thus, mðf Þ ¼ Chðmðf Þ; Pðf ÞÞ; for all f AF : Now suppose there is wAW such that |PðwÞmðwÞ: Let mðwÞ # ¼ |; W 0 ¼ fwg; F 0 ¼ |: As before, meCw ðPÞ because mðwÞPðwÞmðwÞ: # Thus, mðwÞRðwÞ|; for all wAW : Thus m is individual rational. Assume that there exist ðB; f ÞA2W F with Ba|; such that: for all wAB; fPðwÞmðwÞ; and there exists ADmðf Þ such that ½A,B Pðf Þmðf Þ: Letting mðf # Þ ¼ A,B; F 0 ¼ ff g and W 0 ¼ B; it follows that meCw ðPÞ: A contradiction. & 2.3. Lattices and chains Let X be a set, and R a partial order—a transitive, reflexive, antisymmetric binary relation—on X : For any subset A of X ; inf R A ðsupR A) is the greatest (lowest) lower (upper) bound on A; in the order R: We say that the pair ðX ; RÞ is a lattice if whenever x; yAX ; both x4R y ¼ inf R fx; yg and x3R y ¼ supR fx; yg exist in X : We say that a lattice ðX ; RÞ is complete if for every nonempty subset A of X ; inf R A and supR A exist in X : Note that any finite lattice is complete. A lattice ðX ; RÞ is a chain if, for any x; yAX ; xRy or yRx; or both, are true.
3. The core as a set of fixed points We shall construct a map T on a superset of M such that the set of fixed points of T is the core. 3.1. Pre-matchings Say that a pair n ¼ ðnF ; nW Þ; with nF : F -2W and nW : W -F ,f|g; is a prematching: Let VW ðVF Þ denote the set of all nW ðnF ) functions. Thus, V ¼ VF VW denotes the set of pre-matchings n ¼ ðnF ; nW Þ: The superset of M we need turns out to be the set of pre-matchings V (pre-matchings were first used in [1]). A pre-matching n ¼ ðnF ; nW Þ; and a matching m are equivalent if mðf Þ ¼ nF ðf Þ for all f ; and mðwÞ ¼ nW ðwÞ for all w: A pre-matching n ¼ ðnF ; nW Þ is equivalent to some matching if and only if n is such that nW ðwÞ ¼ f if and only if wAnF ðf Þ: Notation. We make two conventions. First, we shall identify matchings and prematchings when they are equivalent, so if nAV is equivalent to some matching m we shall say that n is a matching. Hence M is a subset of V: Second, if n ¼ ðnF ; nW Þ is a pre-matching, we shall some times write nðf Þ for nF ðf Þ and nðwÞ for nW ðwÞ:
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3.2. T and the core Let n be a pre-matching, and let Uðf ; nÞ ¼ fwAW : fRðwÞnðwÞg; and V ðw; nÞ ¼ ff AF : wAChðnðf Þ,fwgÞ; Pðf Þg,f|g: The set Uðf ; nÞ is the set of workers w that are willing to give up nðwÞ in exchange for f : The set V ðw; mÞ is the set of firms f that are willing to hire w; possibly after firing some of the workers it was assigned by n: Now, define T : V-V by ðTnÞðsÞ ¼ ChðUðs; nÞ; PðsÞÞ if sAF ; and ðTnÞðsÞ ¼ maxPðsÞ fV ðs; nÞg if sAW : The map T has a simple interpretation: ðTnÞðf Þ is firm f ’s optimal team of workers, among those willing to work for f ; and ðTnÞðwÞ is the firm preferred by w among the firms that are willing to hire w: We shall denote by E the set of fixed points of T; so E ¼ fnAV : n ¼ Tng: Theorem 3. E ¼ S ðPÞ: Before we prove Theorem 3, note that Theorems 2 and 3 imply: Corollary 4. E ¼ Cw ðPÞ: Proof of Theorem 3. First we shall prove that if nAE; then n is a matching and n is individually rational. Let n ¼ ðnF ; nW ÞAE: Fix wAnF ðf Þ; we shall prove that f ¼ nW ðwÞ: nAE implies that wAnF ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ: Thus wAUðf ; nÞ: The definition of Uðf ; nÞ implies fRðwÞnW ðwÞ: Now, nF ðf Þ,fwg ¼ nF ðf Þ and nAE imply that nF ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ ð%Þ: So ChðnF ðf Þ; Pðf ÞÞ
ð1Þ
¼
ð2Þ
¼
ð3Þ
¼
ChðChðUðf ; nÞ; Pðf ÞÞ; Pðf ÞÞ ChðUðf ; nÞ; Pðf ÞÞ nF ðf Þ:
Equalities (1) and (3) follow from ð%Þ: Equality (2) is a simple property of choice sets. Hence nF ðf Þ ¼ ChðnF ðf Þ; Pðf ÞÞ: Now, wAnF ðf Þ implies ChðnF ðf Þ; Pðf ÞÞ ¼ ChðnF ðf Þ,fwg; Pðf ÞÞ: So we have f AV ðw; nÞ: But nW ðwÞ ¼ ðTnÞðwÞ ¼ maxPðwÞ fV ðw; nÞg; so nW ðwÞRðwÞf : But fRðwÞnW ðwÞ; so f ¼ nW ðwÞ; as RðwÞ is antisymmetric. Let f ¼ nW ðwÞ; we shall prove that wAnF ðf Þ: First, note that f ¼ nW ðwÞ implies that wAUðf ; nÞ: Second, note that f ¼ nW ðwÞ ¼ ðTnÞðwÞ ¼ maxPðwÞ fV ðw; nÞg; because nAE: So nW ðwÞRðwÞ|; and since nF ðf Þ ¼ ChðnF ðf Þ; Pðf ÞÞ; n is individually rational. Now, f AV ðw; nÞ; so we must have wAChðnF ðf Þ,fwg; Pðf ÞÞ by the definition of V ðw; nÞ: That is, wAChðnF ðf Þ,fwg; Pðf ÞÞRðf ÞnF ðf Þ:
ð1Þ
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Now, nAE implies that nF ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ:
ð2Þ
The definition of choice set implies nF ðf ÞDUðf ; nÞ; and nF ðf ÞRðf ÞUðf ; nÞ: But wAUðf ; nÞ implies Uðf ; nÞ+ChðnF ðf Þ,fwg; Pðf ÞÞ: The definition of choice set implies nF ðf ÞRðf ÞChðnF ðf Þ,fwg; Pðf ÞÞ: Together with (1), this implies wAnF ðf Þ: We have shown that wAvF ðf Þ if and only if f ¼ vW ðwÞ: So, n is a matching. Also, n is individually rational. Let mAE: Fix f AF ; BDW such that Ba|: We assume that, for all wAB; fPðwÞmðwÞ: By the definition of Uðf ; mÞ; we have that BDUðf ; mÞ: Let ADmðf Þ: Because m is a matching, we have that for all wAA; f ¼ mðwÞ: So the definition of Uðf ; mÞ implies that ð3Þ
ADUðf ; mÞ:
Now, mAE so mðf Þ ¼ ðTmÞðf Þ ¼ ChðUðf ; mÞ; Pðf ÞÞ; BDUðf ; mÞ and statement (3) imply then mðf ÞRðf ÞChðA,B; Pðf ÞÞRðf ÞA,B:
ð4Þ
So there is no ðB; f Þ that blocks m: Thus mAS ðPÞ: Let mAS ðPÞ and assume that maTm: First assume that there exist f AF such that mðf ÞaðTmÞðf Þ ¼ ChðUðf ; mÞ; Pðf ÞÞ ¼ CDUðf ; mÞ: Let A ¼ C-mðf Þ; and B ¼ C\mðf Þ: Because m is an individually rational matching we have that mðf ÞDUðf ; mÞ and Ba|: Then ðB; f Þ blocks m; which contradicts that mAS ðPÞ: Hence, for all f AF ; mðf Þ ¼ ðTmÞðf Þ:
ð5Þ
Assume that there exists wAW such that % ðw; mÞ; mðwÞaðTmÞðwÞ ¼ max fV ðw; mÞg ¼ fAV PðwÞ
% % by the definition on V ðw; mÞ: so wAChðmðfÞ,fwg; PðfÞÞ; % and mðwÞAV ðw; mÞ: This implies that But m is a matching, so we have that wemðfÞ % fPðwÞmðwÞ; and wAChðmðf Þ,fwg; Pðf ÞÞ ¼ C: Let A ¼ C-mðf Þ ¼ C fwg; and B ¼ fwg: Then ðB; f Þ blocks m: Hence, for all wAW ; mðwÞ ¼ ðTmÞðwÞ: This and statement (5) imply m ¼ Tm: Hence mAE: &
4. The T-algorithm The T-algorithm is very simple: start at some nAV and iterate Tn until two iterations are identical. When two iterations are identical, stop. We prove that, when the T-algorithm stops, it must be at a core matching. Further, we show that once the T-algorithm hits on a matching m (not just a
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pre-matching), m must be in the core. Hence, the T-algorithm ‘‘wanders’’ around pre-matchings until it finds a matching—it will stop at the matching it finds, and it will be a core matching. It follows, of course, that if the core is empty the algorithm must cycle, and that the cycle will only involve pre-matchings that are not matchings. The algorithm is very simple, but we include a detailed description to be as clear as possible, and to introduce some notation that we shall use in the proofs below: Algorithm 1 (T-algorithm). 1. Set n0 ¼ n: Set n1 ¼ Tn0 and k ¼ 1: 2. While nk ank 1 ; do: ðaÞ set k ¼ k þ 1; ðbÞ set nk ¼ Tnk 1 : 3. Set m ¼ nk : Stop. Proposition 5. If the T-algorithm stops at mAV; then m is a matching ðmAMÞ and m is in the core. If nk is in the core, for some iteration k of the T-algorithm, then the algorithm stops at m ¼ nk : Proof. If the algorithm stops at m then nk ¼ nk 1 ¼ m: Then, m ¼ Tnk 1 ¼ Tm; so mAE: By Corollary 4, mACw ðPÞ: On the other hand, if nk is in the core, then nk is a fixed point of T by Corollary 4. Then the T-algorithm stops at m ¼ nk : & Proposition 6. If nk is a matching, for some iteration k in the T-algorithm, then the algorithm stops at m ¼ nk ; and thus nk is a core matching. We shall present the proof of Proposition 6 as two lemmata: Lemma 7. Let nAV be a pre-matching. If n1 ¼ Tn is a matching, then n1 is individually rational. Proof. For all f AF ; n1 ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ: So Chðn1 ðf Þ; Pðf ÞÞ
ð1Þ
¼
ð2Þ
ChðChðUðf ; nÞ; Pðf ÞÞ; Pðf ÞÞ
¼
ChðUðf ; nÞ; Pðf ÞÞ
ð3Þ
n1 ðf Þ:
¼
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Equalities (1) and (3) follow because n1 ¼ Tn: Equality (2) is a property of choice sets. So, for all f AF ; n1 ðf Þ ¼ Chðn1 ðf Þ; Pðf ÞÞ:
ð6Þ
For all wAW ; n1 ðwÞ ¼ ðTnÞðwÞ ¼ max fV ðw; nÞg,f|g: PðwÞ
So, n1 ðwÞRðwÞ|: Statements (6) and (7) show that n1 is individually rational.
ð7Þ &
Lemma 8. If Tn is a matching, for some nAV; then Tn is a core matching. Thus if nk ¼ Tnk 1 is a matching, then the T-algorithm must stop at nk ; and nk is a core matching. Proof. We shall prove that, if nAV is a pre-matching, and n1 ¼ Tn is a matching, then n1 AS ðPÞ: The lemma follows from Proposition 5, as S ðPÞ ¼ E ¼ Cw ðPÞ: Assume that n1 eS ðPÞ: By Lemma 7, n1 is individually rational, so there must exist a ðB; f Þ that blocks n1 : Hence, for all wAB ðBa|Þ; fPðwÞn1 ðwÞ;
ð8Þ
and there must exist ADn1 ðf Þ such that ½A,B Pðf Þn1 ðf Þ:
ð9Þ
But n1 is a matching, and n1 ¼ Tn; so we have that for all wAB; wAn1 ðn1 ðwÞÞ: But n1 ðn1 ðwÞÞ ¼ ðTnÞðn1 ðwÞÞ ¼ ChðUðn1 ðwÞ; nÞ; Pðn1 ðwÞÞÞ; so that, wAUðn1 ðwÞ; nÞ; i.e. n1 ðwÞRðwÞnðwÞ:
ð10Þ
Statements (8) and (10) imply that, for all wAB; fRðwÞnðwÞ; i.e. BDUðf ; nÞ:
ð11Þ
Now, ADn1 ðf Þ and n1 ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ; so ADUðf ; nÞ:
ð12Þ
Statements (9), (11), and (12) contradict that n1 ðf Þ ¼ ðTnÞðf Þ ¼ ChðUðf ; nÞ; Pðf ÞÞ: Thus n1 AS ðPÞ: & Remark. *
Proposition 6, and Example 9 below, imply that the T-algorithm and the DAA are different algorithms; we have not simply written down the DAA using different language.
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*
Unlike the DAA, the T-algorithm only stops at a core matchings. Example 9 below shows how the T-algorithm succeeds in finding a core matching, while the DAA stops at a non-core matching. When the core is empty, the T-algorithm must cycle. We present an example as ‘‘supplementary material’’ in http://www.nyu.edu/jet/supplementary.html and in the working-paper version [4] of the paper.
Example 9. Let F ¼ ff1 ; f2 g; W ¼ fw1 ; w2 ; w3 ; w4 g; and the preference profile P be defined by: Pðf1 Þ ¼ fw1 ; w2 g; fw3 ; w4 g; fw1 ; w3 g; fw2 ; w4 g; fw1 g; fw2 g; fw3 g; fw4 g; Pðf2 Þ ¼ fw1 ; w2 g; fw1 ; w3 g; fw2 ; w4 g; fw3 ; w4 g; fw1 g; fw2 g; fw3 g; fw4 g; Pðw1 Þ ¼ f1 ; f2 ; Pðw2 Þ ¼ f2 ; f1 ; Pðw3 Þ ¼ f1 ; f2 ; and Pðw4 Þ ¼ f1 ; f2 : Note that P is not substitutable, as w1 AChðfw1 ; w2 ; w3 ; w4 g; Pðf1 ÞÞ ¼ fw1 ; w2 g while w1 eChðfw1 ; w3 ; w4 g; Pðf1 ÞÞ ¼ fw3 ; w3 g: The T-algorithm starting at f1 n0 ¼ fw1 ; w2 g
f2
w1
w2
w3
w4
fw1 ; w2 g
|
|
|
|
does: f1
f2
w1
w2
w3
w4
Uðf ; n Þ=V ðw; n Þ ¼ n1 ¼ Uðf ; n1 Þ=V ðw; n1 Þ ¼
fw1 ; w2 ; w3 ; w4 g fw1 ; w2 g fw1 ; w3 ; w4 g
fw1 ; w2 ; w3 ; w4 g fw1 ; w2 g fw2 ; w3 ; w4 g
f1 ; f2 f1 f1 ; f2
f1 ; f2 f2 f1 ; f2
| | |
| | |
n2 ¼ Uðf ; n2 Þ=V ðw; n2 Þ ¼
fw3 ; w4 g fw1 ; w3 ; w4 g
fw2 ; w4 g fw2 ; w3 ; w4 g
f1 f2
f2 f2
| f1
| f1 ; f2
n3 ¼ Uðf ; n Þ=V ðw; n3 Þ ¼ n4 ¼
fw3 ; w4 g fw1 ; w3 ; w4 g fw3 ; w4 g
fw2 ; w4 g fw1 ; w2 g fw1 ; w2 g
f2 f2 f2
f2 f2 f2
f1 f1 f1
f1 f1 ; f2 f1
0
0
3
Note that ni for i ¼ 0; 1; 2; 3 are not matchings; n4 is a matching so, by Proposition 6, n4 is a core matching. On the other hand, the DAA stops at a matching that is not in the core. For the DAA, the equivalent to starting the T-algorithm at n0 is, as we shall see in Section 5, the DAA with firms proposing. The DAA with firms proposing stops at the matching f1
f2
m ¼ fw3 ; w4 g fw2 g
w1
w2
w3
w4
|
f2
f1
f1 :
But m is not in the core, as f2 Pðw1 Þmðw1 Þ ¼ |; f2 Rðw2 Þmðw2 Þ ¼ f2 ; and fw1 ; w2 gPðf2 Þfw2 g implies that ðfw1 ; w2 g; f2 Þ blocks m:
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5. The lattice structure of the core We shall introduce a partial order on V such that, if preferences are substitutable, T is a monotone increasing map. Tarski’s fixed point theorem then delivers a lattice structure on E; and thus on Cw ðPÞ: Definition 4. Define the following partial orders on VF ; VW and V: 1. PðF Þ on VF by nF PðF Þn0F if and only if n0F anF and, for all f in F ; nF ðf Þ ¼ n0F ðf Þ or nF ðf Þ ¼ ChðnF ðf Þ,n0F ðf Þ; Pðf ÞÞ: 2. PðW Þ on VW by nW PðW Þn0W if and only if n0W anW and nW ðwÞRðwÞn0W ðwÞfor all wAW : 3. The weak partial orders associated to PðF Þ and PðW Þ are denoted RðF Þ and RðW Þ; defined as: nF RðF Þn0F if nF ¼ n0F or nF PðF Þn0F ; and nW RðW Þn0W if nW ¼ n0W or nW PðW Þn0W : 4. PðF Þ on V by nPðF Þn0 iff nF PðF Þn0F and n0W RðW ÞnW ; or nF RðF Þn0F and n0W PðW ÞnW : 5. PðW Þ on V by nPðW Þn0 iff nW PðW Þn0W and n0F RðF ÞnF ; or nW RðW Þn0W and n0F PðF ÞnF : Definition 5. A firm f ’s preference ordering Pðf Þ satisfies substitutability if for any set SDW containing workers w and w% ðwawÞ; % if wAChðS; Pðf ÞÞ then wAChðS\fwg; % Pðf ÞÞ: A preference profile P is substitutable if, for each firm f ; the preference ordering Pðf Þ satisfies substitutability. Theorem 10. Let P be substitutable. Then ðCw ðPÞ; PðF ÞÞ and ðCw ðPÞ; PðW ÞÞ are nonempty complete lattices, and inf PðF Þ Cw ðPÞ ¼ supPðW Þ Cw ðPÞ; and supPðF Þ Cw ðPÞ ¼ inf PðW Þ Cw ðPÞ: Theorem 10 says that there are two distinguished core matchings, mW and mF such that: For firms, mW ¼ inf PðF Þ Cw ðPÞ is worse, and mF ¼ supPðF Þ Cw ðPÞ is better, than any other core matchings. For workers mW ¼ supPðW Þ Cw ðPÞ is better, and mF ¼ inf PðW Þ Cw ðPÞ is worse, than any other core matching. Proof of Theorem 10. First, ðV0 ; PðF ÞÞ is a complete lattice—due to space restrictions, we refer to the working paper version of the paper [4] for the proof. We shall present the proof of the Theorem as two simple lemma. We assume that P is substitutable in the rest of the section. Notation. Let V0 ¼ fnAV : nðsÞRðsÞ|; for all sAF ,W g:
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Remark. For all nAV; TnAV0 Lemma 11. Let m and m0 be pre-matchings. If m0 RðF Þm then, for all wAW and f AF ; Uðf ; mÞDUðf ; m0 Þ; and V ðw; mÞ+V ðw; m0 Þ: Proof. Let wAUðf ; mÞ: We have that fRðwÞmðwÞ; but mðwÞRðwÞm0 ðwÞ so wAUðf ; m0 Þ: This proves Uðf ; mÞDUðf ; m0 Þ: Now we shall prove that V ðw; mÞ+V ðw; m0 Þ: First, if V ðw; m0 Þ ¼ f|g; then there is nothing to prove, as f|g ¼ V ðw; m0 ÞDV ðw; mÞ: Suppose that V ðw; m0 Þaf|g; and let f AV ðw; m0 Þ: Then, wAChðm0 ðf Þ,fwg; Pðf ÞÞ: But m0 RðF Þm; so the definition of RðF Þ implies that, for all f AF ; either 0 m ðf Þ ¼ mðf Þ so wAChðmðf Þ,fwg; Pðf ÞÞ; or m0 ðf Þ ¼ Chðm0 ðf Þ,mðf Þ; Pðf ÞÞ: Then wAChðm0 ðf Þ,fwg; Pðf ÞÞ implies that w
A ¼ ð1Þ
¼
Chðm0 ðf Þ,fwg; Pðf ÞÞ ChðChðm0 ðf Þ,mðf Þ; Pðf ÞÞ,fwg; Pðf ÞÞ Chðm0 ðf Þ,mðf Þ,fwg; Pðf ÞÞ:
Equality (1) is from [3, Proposition 2.3]. Substitutability of P implies that wAChðmðf Þ,fwg; Pðf ÞÞ: Then f AV ðw; mÞ; and thus V ðw; mÞ+V ðw; m0 Þ: & Lemma 12. E is a non-empty and complete lattice. Proof. First we show that TjV0 is monotone increasing. That is, whenever m0 RðF Þm; we have ðTm0 ÞRðF ÞðTmÞ: Let m0 RðF Þm; and fix f AF and wAW : Lemma 11 says that Uðf ; mÞDUðf ; m0 Þ: We first show that ChðUðf ; m0 Þ; Pðf ÞÞ ¼ Chð½ChðUðf ; m0 Þ; Pðf ÞÞ,ChðUðf ; mÞ; Pðf ÞÞ ; Pðf ÞÞ:
ð13Þ
To see this, let SDChðUðf ; m0 Þ; Pðf ÞÞ,ChðUðf ; mÞ; Pðf ÞÞ: Then SDUðf ; mÞ, Uðf ; m0 Þ ¼ Uðf ; m0 Þ; so ChðUðf ; m0 Þ; Pðf ÞÞRðf ÞS: But, ChðUðf ; m0 Þ; Pðf ÞÞD ChðUðf ; m0 Þ; Pðf ÞÞ,ChðUðf ; mÞ; Pðf ÞÞ; so we have established statement 13. Now, ðTm0 Þðf Þ ¼ ChðUðf ; m0 Þ; Pðf ÞÞ and ðTmÞðf Þ ¼ ChðUðf ; mÞ; Pðf ÞÞ; so statement 13 implies that ðTm0 Þðf Þ ¼ Chð½ðTm0 Þðf Þ,ðTmÞðf Þ ; Pðf ÞÞÞ:
ð14Þ
We now show that ðTmÞðwÞRðwÞðTm0 ÞðwÞ: Lemma 11 says that V ðw; m0 ÞDV ðw; mÞ: So, ðTmÞðwÞ ¼ maxfV ðw; mÞgRðwÞ max fV ðw; m0 Þg ¼ ðTm0 ÞðwÞ: Pðf Þ
Pðf Þ
ð15Þ
Statements 14 and 15 imply that ðTm0 ÞRðF ÞðTmÞ; as f AF and wAW were arbitrary. Finally, TðVÞDV0 so EDV0 ; and E equals the set of fixed points of TjV0 : TðVÞDV0 also implies that the restricted map TjV0 has range in V0 :
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Now we have that TjV0 : V0 -V0 is monotone increasing, and V0 is a complete lattice. So Tarski’s fixed point theorem implies that ðE; PðF ÞÞ is a non-empty complete lattice. & Corollary 4 and Lemma 12 finish the proof of Theorem 10. See [4] for full details.
6. The T-algorithm and substitutability When preferences are substitutable, the T-algorithm 1. finds the worker-optimal core matching ðinf PðF Þ Cw ðPÞÞ and the firm-optimal core matching ðsupPðF Þ Cw ðPÞÞ; 2. is computationally efficient—we can give a bound on its computational complexity and show that the bound is likely to be quite small, 3. calculates the lattice operations ‘‘join’’ and ‘‘meet,’’ on the Cw ðPÞ lattice. Result 1 is important because it shows that the T-algorithm does the job of the DAA. Further, by result 2 we have an idea of the T-algorithm’s computational complexity, which is an open problem for the DAA in the many-to-one case. In a very simple example, we show that, while the ‘‘size’’ of the problem is 30; 000 iterations, the T-algorithm needs at most nine iterations to find a core matching. Our result on the complexity of the T-algorithm is similar to Topkis’s [14] (see also [15]) results on algorithms for finding equilibria in supermodular games. Result 3: A number of papers have sought to improve on Blair’s lattice operations [2,8] by restricting firms’ preferences. We show that, with only substitutability, the Talgorithm can compute the lattice operations—thus allowing for a practical solution to the problem of Blair’s lattice operations. 6.1. The firm- and worker-optimal core matchings Proposition 13. If P is substitutable, the T-algorithm starting at inf PðF Þ V0 stops at the worker-optimal core matching inf PðF Þ Cw ðPÞ; the T-algorithm starting at supPðF Þ V0 stops at the firm-optimal core matching supPðF Þ Cw ðPÞ: Proof. When preferences are substitutable, T is a monotone map on the complete lattice V0 : Proposition 13 follows then from standard results, see [14,15]. & Remark. 1. The pre-matching inf PðF Þ V0 is nAV with nðf Þ ¼ f|g and nðwÞ ¼ maxPðwÞ F : The pre-matching supPðF Þ V0 is n with nðf Þ ¼ ChðW ; Pðf ÞÞ and nðwÞ ¼ f|g: 2. The T-algorithm starting at points that are not inf PðF Þ V0 or supPðF Þ V0 may not converge. For example, one can show that the T-algorithm starting at the prematching n ¼ ððsupPðF Þ V0 ÞF ; ðinf PðF Þ V0 ÞW Þ cycles.
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6.2. Complexity of the T-algorithm We shall now give a bound on the computational complexity of the T-algorithm, but we need some auxiliary definitions first. For each f ; let Xf ¼ fADW : ARðf Þ|g; and let Xf be the partial order on Xf defined by AXB iff A ¼ ChðA,B; Pðf ÞÞ or A ¼ B: Let qf be the size of the longest Xf -chain in Xf : For each w; let Xw ¼ ff AF : fRðwÞ|g; and let qw be the number of elements in Xw : We present an example as ‘‘supplementary material’’ in http:// www.nyu.edu/jet/supplementary.html and in the working-paper version [4] of the paper. Proposition 14. If P is substitutable, the T-algorithm starting at inf PðF Þ V0 ðsupPðF Þ V0 Þ stops at the worker-optimal (firm-optimal) core matching in less than X ðqs 1Þ sAF ,W
iterations. Proof. Let n0 ; n1 ; ynk be the sequence generated by the T-algorithm before it stops k at a core matching nk : The P number of iterations before the algorithm stops at n is k; we shall prove that kp sAF ,W ðqs 1Þ: Assume—without loss of generality—that we started the T-algorithm at n0 ¼ inf PðF Þ V0 : Since n1 ¼ ðTn0 ÞPðF Þn0 and T is monotone increasing (see the proof of Theorem 10), nk PðF Þnk 1 PðF Þ?n1 PðF Þn0 : For each l; 1plpk; it must be that nl ðf Þ ¼ nl 1 ðf Þ or that nl ðf Þ ¼ Chðnl ðf Þ,nl 1 ðf Þ; Pðf ÞÞ for all f ; and nl 1 ðwÞRðwÞnl ðwÞ for all w: Also, for each l; 1plpk; either nl ðf Þanl 1 ðf Þ and nl ðf Þ ¼ Chðnl ðf Þ,nl 1 ðf Þ; Pðf ÞÞ for some for f or nl 1 ðwÞPðwÞnl ðwÞ for some w—recall that PðF Þ is a strict order, see Definition 4. Say that l is an f -step if nl ðf Þanl 1 ðf Þ and nl ðf Þ ¼ Chðnl ðf Þ,nl 1 ðf Þ; Pðf ÞÞ for firm f : Say that l is a w-step if nl 1 ðwÞPðwÞnl ðwÞ for worker w: There cannot be more than qs 1 s-steps, as a number of, say r steps involves a chain of r þ 1 elements in Xs : Now k is smaller than the sum over all sAW ,F of all s-steps, so X ðqs 1Þ: kp sAF ,W
But k is the number of iterations of the T-algorithm before it finds nk :
&
Remark. Recall that jW j ¼ m and jF j ¼ n; so the number of pre-matchings is 2nm ðn þ 1Þm ; a number which is normally orders of magnitude larger than the bound in Proposition 14: In an example we develop as ‘‘supplementary material’’ in http:// www.nyu.edu/jet/supplementary.html (see also [4]) Proposition 14 implies that
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the T-algorithm converges before X ðqs 1Þ ¼ 9 sAF ,W
iterations. Compare with the number of pre-matchings: 32768. The number of matchings is smaller than the number of pre-matchings, but still substantially larger than 9. It may be unfair to compare the complexity of the T-algorithm to the number of pre-matchings, after all the T-algorithm will not involve pre-matchings where agents are matched to unacceptable partners—but still, the number of pre-matchings that only involve acceptable partners is 9261. 6.3. Computing the lattice operations Let m1 ; m2 ACw ðPÞ: We shall look at the problem of finding m1 3PðF Þ m2 ; the join (least upper bound) of m1 and m2 in the ðCw ðPÞ; PðF ÞÞ-lattice. One can solve the problem of finding m1 4PðF Þ m2 in the ðCw ðPÞ; PðF ÞÞ-lattice in a similar way. Martı´ nez et al. [8,9] prove that, when preferences satisfy substitutability and an additional assumption (which they call q-separability) there is a simple formula for m1 3PðF Þ m2 (see [2] for a similar result). Martı´ nez et al. prove that m1 3PðF Þ m2 is 8 < Chðm1 ðsÞ,m2 ðsÞ; PðsÞÞ if sAF ; ð16Þ n% ðsÞ ¼ min fm1 ðsÞ; m2 ðsÞg if sAW : : PðsÞ In general, though, when preferences are only substitutable, there is no formula for m1 3PðF Þ m2 : In fact, n% defined in (16) is a pre-matching but not necessarily a matching. We prove that, if one starts the T-algorithm at n% ; it will stop at m1 3PðF Þ m2 : Hence we give a practical solution to Blair’s problem of calculating m1 3PðF Þ m2 :3 Theorem 15. Let P be substitutable, let m1 and m2 be core matchings, and n% be defined as in (16). The T-algorithm started at n% stops at m1 3PðF Þ m2 : # ¼ fnAV : nRðF Þn% g: First we prove (1) that T VD # V: # Then we prove Proof. Let V 1 2 (2) that T-algorithm starting at n% stops at m 3PðF Þ m : (1) First we shall prove that n% is the least upper bound on fm1 ; m2 g in V: By the definition of n% ; n% RðF Þm1 and n% RðF Þm2 : 3 Blair’s proof of the existence of m1 3PðFÞ m2 can also be expressed as an algorithm for calculating m1 3PðFÞ m2 :
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Let n be an upper bound on fm1 ; m2 g in V; we shall prove that nRðF Þn% : For any f AF : ð1Þ
nðf ÞRðf Þm1 ðf Þ
3 nðf Þ ¼ Chðnðf Þ,m1 ðf Þ; Pðf ÞÞ
nðf ÞRðf Þm2 ðf Þ
3 nðf Þ ¼ Chðnðf Þ,m2 ðf Þ; Pðf ÞÞ:
ð2Þ
We have that Chðnðf Þ,n% ðf Þ; Pðf ÞÞ
ð3Þ
¼
Chðnðf Þ,Chðm1 ðf Þ,m2 ðf Þ; Pðf ÞÞ; Pðf ÞÞ
ð4Þ
¼
Chðnðf Þ,m1 ðf Þ,m2 ðf Þ; Pðf ÞÞ
ð5Þ
¼
ChðChðnðf Þ,m1 ðf Þ; Pðf ÞÞ,m2 ðf Þ; Pðf ÞÞ
ð6Þ
Chðnðf Þ,m2 ðf Þ; Pðf ÞÞ
¼
ð7Þ
¼
nðf Þ:
Where equality (3) follows from the definition of n% : Equalities (4) and (5) follow from substitutability and [3, Proposition 2.3]. Equality (6) follows from equality (1). Equality (7) follows from equality (2). Thus, nF RðF Þn% F : Now, nRðF Þm1 and nRðF Þm2 imply that, for any wAW ; m1 ðwÞRðwÞnðwÞ; and m2 ðwÞRðwÞnðwÞ: But then, n% ðwÞ ¼ minfm1 ðwÞ; m2 ðwÞgRðwÞnðwÞ: PðwÞ
This and nF RðF Þn% F implies that nRðF Þn% : Hence n% is the least upper bound on fm1 ; m2 g in V: # V: # Let nAV; # then nRðF Þn% RðF Þm1 and Now we shall prove that T VD 2 1 nRðF Þn% RðF Þm : But then TnRðF ÞTm and TnRðF ÞTm2 ; as T is monotone. Now, m1 and m2 are core matchings, so m1 ¼ Tm1 and m2 ¼ Tm2 ; then Tn is an upper bound on # fm1 ; m2 g in V: But n% is the least upper bound, so TnRðF Þn% and thus TnAV: # V; # so Tj # : V# V: # T is monotone and n% is the smallest (2) We know that T VD V # pre-matching in V: By Topkis’s [14] results, the T-algorithm stops at the RðF Þsmallest fixed point of TjV# ; but this is the RðF Þ-smallest fixed point of T; of those fixed points that are RðF Þ-larger than both m1 and m2 : Then the T-algorithm stops at m1 3PðF Þ m2 : & Remark. 1. A key part of the proof of Theorem 15 is that n% is the join of m1 and m2 in V: By a similar argument, the T-algorithm calculates the meet, m1 4PðF Þ m2 ; of any two core matchings m1 and m2 : One only needs to start the algorithm at the meet of m1 and m2 in V instead of the join.4 4
See [2] on how to calculate the meet of m1 and m2 in V:
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2. The bound presented in Proposition 14 also bounds the number of T-iterations necessary to calculate m1 3PðF Þ m2 : We now illustrate how the algorithm finds the join of two core matchings: Example 16. (Roth [10]). Let F ¼ ff1 ; f2 ; f3 ; f4 ; f5 g be the set of firms and W ¼ fw1 ; w2 ; w3 ; w4 ; w5 ; w6 g be the set of workers. As in [10], we specify only preferences among the choices we shall need. Pðf1 Þ ¼ fw4 g; fw1 g; fw2 ; w3 ; w5 ; w6 g; y; fw5 g; Pðf2 Þ ¼ fw2 g; fw1 ; w3 g; Pðf3 Þ ¼ fw3 g; fw2 g; Pðf4 Þ ¼ fw5 g; fw4 ; w6 g; Pðf5 Þ ¼ fw6 g; fw5 g; Pðw1 Þ ¼ f2 ; f1 ; Pðw2 Þ ¼ f1 ; f3 ; f2 ; Pðw3 Þ ¼ f1 ; f2 ; f3 ; Pðw4 Þ ¼ f4 ; f1 ; Pðw5 Þ ¼ f1 ; f5 ; f4 ; Pðw6 Þ ¼ f1 ; f4 ; f5 : Consider the following two core matchings f1 m1 ¼ fw1 g f1 m2 ¼ fw4 g
f2 fw2 g
f3 fw3 g
f4 fw4 ; w6 g
f5 and fw5 g
f2
f3
f4
f5
fw1 ; w3 g
fw2 g
fw5 g
fw6 g
n1 ¼ Uðf ; n1 Þ=V ðw; n1 Þ ¼ n2 ¼
:
f1 w4
f2 w2
f3 w3
f4 w5
f5 w6
w1 f1
w2 f2
w3 f3
w4 f1
w5 f4
w6 f5
w1 ; w4 w4
w2 w2
w3 w3
w5 w5
w6 w6
| |
f2 f2
f3 f3
f1 f1
f4 f4
f5 f5 :
n2 is a matching and n2 ¼ m1 3PðF Þ m2 : Acknowledgments We are grateful to Jinpeng Ma, Ruth Martı´ nez, and Alejandro Neme for their helpful comments.
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F. Echenique, J. Oviedo / Journal of Economic Theory 115 (2004) 358–376
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