Lattice structure of the random stable set in many-to-many matching market
aa r X i v : . [ ec on . T H ] J un Lattice structure of the random stable set inmany-to-many matching markets ∗ Noelia Ju ´arez † Pablo A. Neme † Jorge Oviedo † June 11, 2020
Abstract
For a many-to-many matching market, we study the lattice structure of the setof random stable matchings. We define a partial order on the random stable setand present two intuitive binary operations to compute the least upper bound andthe greatest lower bound for each side of the matching market. Then, we provethat with these binary operations the set of random stable matchings forms twodual lattices.
JEL classification:
C71, C78, D49.
Keywords:
Lattice Structure, Random Stable Matching markets, Many-to-many Match-ing Markets.
Matchings have been studied for several decades, beginning with Gale and Shapley’spioneering paper (Gale and Shapley, 1962). They introduce the notion of stable match-ings for a marriage market and provide an algorithm for finding them. Since then, aconsiderable amount of work was carried out on both theory and applications of stable ∗ We would like to thank Agustin Bonifacio, Jordi Mass ´o, Elena I ˜narra, Juan Pereyra and the GameTheory Group of IMASL for the helpful discussions and detailed comments. Our work is partiallysupported by the UNSL, through grant 319502, and from the Consejo Nacional de InvestigacionesCient´ıficas y T´ecnicas (CONICET), through grant PIP 112-201501-00505, and from Agencia Nacionalde Promoci ´on Cient´ıfica y Tecnol ´ogica, through grant PICT 2017-2355. † Instituto de Matem´atica Aplicada San Luis, Universidad Nacional de San Luis and CONICET, SanLuis, Argentina. Emails: [email protected] (N. Ju´arez), [email protected] (P.A. Neme), and [email protected] (J. Oviedo). stable if all agents have acceptable partners and there is nopair of agents, one of each side of the market, that would prefer to be matched to eachother rather than to remain with the current partner. Unfortunately, the set of many-to-one stable matchings may be empty. Substitutability is the weakest condition that hasso far been imposed on agents’ preferences under which the existence of stable match-ings is guaranteed. An agent has substitutable preference if he wants to continue beinga partner with agents from the other side of the market even if other agents becomeunavailable (see Kelso and Crawford, 1982; Roth, 1984, for more detail).One of the most important results in the matching literature is that the set of sta-ble matchings has a dual lattice structure. This is important for at least two rea-sons. First, it shows that even if agents of the same side of the market compete foragents of the other side, the conflict is attenuated since, on the set of stable match-ings, agents on the same side of the market have coincidence of interests. Second,many algorithms are based on this lattice structure. For example, algorithms thatyield stable matchings in centralized markets. In this paper, we study the lattice struc-ture of the random stable set for a general matching market, many-to-many match-ing markets with substitutable preferences and satisfiying the law of aggregated demand (L.A.D.). Random stable matchings are very useful for at least two reasons. First, therandomization allows for a much richer space of possible outcomes and may be es-sential to achieve fairness and anonymity. Second, the framework of random stablematchings admits fractional matchings that capture time-sharing arrangements, (seeRothblum, 1992; Roth et al., 1993; Teo and Sethuraman, 1998; Sethuraman et al., 2006;Ba¨ıou and Balinski, 2000; Do ˘gan and Yıldız, 2016; Neme and Oviedo, 2019a,b, amongothers).Roth et al. (1993) define binary operations to compute the least upper bound (l.u.b.) and the greatest lower bound (g.l.b.) for random stable matchings for the marriage mar-ket. To do so, they use the first-order stochastic dominance as the partial order forrandom stable matchings. This partial order, can not be applied when agents’ pref-erences are over subsets of agents of the other side of the market in a substitutablemanner. For this reason, we present a new partial order –a natural extension of thefirst-order stochastic dominance– for the random stable set of a matching market whenagents’ preferences are subsitutables and satisfies the L.A.D.. Generally, a random sta-ble matching can be represented by different lotteries. Despite this, we prove that thereis a unique way to represent a random stable matching fulfilling a special property:each stable matching involve in the lottery of this unique representation is comparablein the eyes of all firms, from now on we refer as decreasing representation . This way,our partial order is independently of the representations of the random stable match-ing. The process to construct this decreasing representation for each random stablematchings, its presented as Algorithm 1. 2o ease the definition of the binary operations and proofs, we present the splittingprocedure . Given two random stable matchings, this procedure “splits” the decreasingrepresentation of each random stable matching in a way that both lotteries have thesame numbers of terms. Moreover, both lotteries have the same scalars, term to term.The splitting procedure is formalized by Algorithm 2 presented in Appendix B.Our main contribution in this paper is that, by defining two natural binary opera-tions (pointing functions) that compute the l.u.b. and g.l.b. for random stable match-ings, the set of these matchings has a dual lattice structure. In other words, as longas the set of (deterministic) stable matchings has a lattice structure, where the binaryoperations are computed via pointing functions, the set of random stable matchingsalso has a lattice structure.Further, for the special case in which all the scalars of the lottery are rational num-bers, we show that there is a direct way to compute the l.u.b. and the g.l.b.
Other finding derived from our proofs, is a version of Rural Hospital Theoremfor random stable matchings. The Rural Hospital Theorem (for deterministic stablematchings) for a many-to-many matching market where all agents have substitutablepreference satisfying the L.A.D. is presented in Alkan (2002). The paper illustrates thesuccessive results needed to prove that the random sable set has a lattice structure withnumeric examples.
Related literature
The lattice structure of the set of stable matchings is introduced by Knuth (1976) forthe marriage market. Given two stable matchings he defines the l.u.b. for men, bymatching to each man with the best of the two partners, and the g.l.b. for men, bymatching to each man the less preferred between the two partners; these are usu-ally called the pointing functions relative to a partial order. Roth (1985) shows thatthese binary operations (pointing functions) used in Knuth (1976) do not work inthe more general many-to-many and many-to-one matching markets introduced byKelso and Crawford (1982) and Roth (1984) respectively even under substitutable pref-erences. For a specific many-to-one matching market, the so-called the college admis-sion problem, Roth and Sotomayor (1990) present a natural extension of Knuth’s resultfor q -responsive preferences. Mart´ınez et al. (2001) further extend the results proved byRoth and Sotomayor (1990). They identified a weaker condition than q -responsiveness,called q -separability, and propose two natural binary operations that give a dual latticestructure to the set of stable matchings in a many-to-one matching market with substi-tutable and q -separable preferences. Such binary operations are similar to the Knuth’sones. Pepa Risma (2015) generalizes the result of Mart´ınez et al. (2001) by showingthat their binary operations work well in many-to-one matching markets where the3references of the agents satisfy substitutability and the law of aggregate demand (a lessrestrictive than q -separability). Her paper is contextualized in many-to-one match-ing markets with contracts. Manasero (2019) extends the result in Pepa Risma (2015)to the many-to-many marching market, where one side has substitutable preferencessatisfying the law of aggregate demand, and the other side has q -responsive prefer-ences. Alkan (2002) considers a market with multiple partners on both sides. For thismarket, preferences are given by rather general path-independent choice functions thatdo not necessarily respect any ordering on individuals and satisfy the law of aggre-gated demand. He shows that the set of stable matchings in any two-sided marketwith path-independent choice functions and preferences satisfying the law of aggre-gated demand has a lattice structure under the common preferences of all agents onany side of the market. Li (2014) presents an alternative proof for Alkan’s result. Themain distinction between Li (2014) and Alkan (2002) lies in the conditions over pref-erences: Li (2014) assumes agents with complete preferences, whereas Alkan (2002)assumes agents with incomplete revealed preferences. All of these mentioned papersshare natural definitions of the binary operations via pointing functions.In other direction, there is an extensive literature that proves that the set of stablematchings has a lattice structure (see Blair, 1988; Adachi, 2000; Fleiner, 2003; Echenique and Oviedo,2004, 2006; Hatfield and Milgrom, 2005; Ostrovsky, 2008; Wu and Roth, 2018, amongothers). All of these mentioned papers define in a difficult way, by means of fixedpoints, the l.u.b and g.l.b. . That is, these papers do not compute the binary operationsvia pointing functions.Regarding to the related literature concerning to lattice structures of random sta-ble sets, Roth et al. (1993), define two binary operations for random stable matchingsin marriage markets. For these very particular markets, they proved that the set ofrandom stable matchings, endowed with a partial order (first-order stochastic domi-nance) has a lattice structure. Neme and Oviedo (2019a) proved that the strongly sta-ble fractional matching set in the marriage market, endowed with the same partialorder (first-order stochastic dominance), has a lattice structure. The binary operationsdefined in Roth et al. (1993) and also used by Neme and Oviedo (2019a), can not beextended to a more general markets, not even to the college admission problem with q -responsive preferences.The paper is organized as follows. In Section 2, we introduce the matching marketand preliminary results. In Section 3, we prove that there is a unique way to representa random stable matching with a decreasing property (Algorithm 1 and Theorem 1).Also, we present a version of the Rural Hospital Theorem for random stable matchings(Proposition 1). In section 4, we present a partial order for random matchings when Alkan (2002) calls “the law of aggregated demand” as “cardinal monotonicity”. They prove that the “stable fractional matching set” coincides with random stable matching set l.u.b. and g.l.b. for each side of the market (Proposition 5). Further, the mainresult of the paper is presented, and states that the random stable set has a dual latticestructure (Theorem 2). In subsection 5.1, we show how to compute in a direct way the l.u.b. and g.l.b. for rational random stable matchings (these are random stable match-ings where all scalars of their lotteries are rational numbers)(Corollary 2). Section 6contains concluding remarks. Finally, Appendix A contains proofs for the decreasingrepresentation and Appendix B contains proofs of the partial order, formalization ofthe splitting procedure (Algorithm 2), and the proof of the main theorem.
We consider many-to-many matching markets, where there are two disjoint sets ofagents, the set of firms F and the set of workers W . Each firm has an antisymmetric,transitive and complete preference relation ( > f ) over the set of all subsets of W . Inthe same way, each worker has an antisymmetric, transitive and complete preferencerelation ( > w ) over the set of all subsets of F . We denote by P the preferences profilefor all agents: firms and workers. A many-to-many matching market is denoted by ( F , W , P ) . Given a set of firms S ⊆ F , each worker w ∈ W can determine which subsetof S would most prefer to hire. We call this the w ’s choice set from S and denote it by Ch ( S , > w ) . Formally, Ch ( S , > w ) = max > w { T : T ⊆ S } .Symmetrically, given a set of workers S ⊆ W , let Ch ( S , > f ) denote firm f ’s most pre-ferred subset of S according to its preference relation > f . Formally, Ch ( S , > f ) = max > f { T : T ⊆ S } . Definition 1 A matching µ is a function from the set F ∪ W into F ∪ W such that for eachw ∈ W and for each f ∈ F:1. µ ( w ) ⊆ F;2. µ ( f ) ⊆ W;3. w ∈ µ ( f ) ⇔ f ∈ µ ( w ) We say that agent a ∈ F ∪ W is matched if µ ( a ) = ∅ , otherwise he is unmatched.A matching µ is blocked by agent a if µ ( a ) = Ch ( µ ( a ) , > a ) . We say that a matchingis individually rational if it is not blocked by any individual agent. A matching µ is5locked by a worker-firm pair ( w , f ) if w / ∈ µ ( f ) , w ∈ Ch ( µ ( f ) ∪ { w } , > f ) , and f ∈ Ch ( µ ( w ) ∪ { f } , > w ) . A matching µ is stable if it is not blocked by any individual agentor any worker-firm pair. The set of stable matchings is denoted by S ( P ) . Further, a random stable matching is a lottery over stable matchings, and denote by RS ( P ) therandom stable set for the many-to-many matching market ( F , W , P ) .Given an agent a ’s preference relation ( > a ) and two stable matchings µ and µ ′ ,let µ ( a ) ≥ a µ ′ ( a ) denote µ ( a ) = Ch ( µ ( a ) ∪ µ ′ ( a ) , > a ) . We say that µ ( a ) > a µ ′ ( a ) if µ ( a ) ≥ a µ ′ ( a ) and µ ( a ) = µ ′ ( a ) . Given a preferences profile P , and two stablematchings µ and µ ′ , let µ > F µ ′ denote the case in which all firms like µ at least as wellas µ ′ , with at least one firm preferring µ to µ ′ outright. Let µ ≥ F µ ′ denote that either µ > F µ ′ or that µ = µ ′ . Similarly, we define > W and ≥ W . Notice that, ≥ F and ≥ W arepartial orders over the set of stable matchings.An agent a ’s preferences relation satisfies substitutability if, for any subset S of theopposite set (for instance, if a ∈ F then S ⊆ W ) that contains agent b , b ∈ Ch ( S , > a ) implies b ∈ Ch ( S ′ ∪ { b } , > a ) for all S ′ ⊆ S . We say that an agent a ’s preference relation( > a ) satisfies the law of aggregated demand (L.A.D.) if for all subset S of the oppositeset and all S ′ ⊆ S , | Ch ( S ′ , > a ) | ≤ | Ch ( S , > a ) | . For a matching market ( F , W , P ) where the preference relation of each agent satis-fies substitutability and the LAD, Alkan (2002) proves that the set of stable matchingshas a lattice structure. Given two stable matchings µ and µ , l.u.b. for firms is denotedby µ ∨ F µ and g.l.b. for firms is denoted by µ ∧ F µ . Similarly, l.u.b. for workersis denoted by µ ∨ W µ and g.l.b. for workers is denoted by µ ∧ W µ . The binaryoperations are defined as follows, (see Alkan, 2002; Li, 2014, among others). µ ∨ F µ ( f ) = µ ∧ W µ ( f ) : = Ch ( µ ( f ) ∪ µ ( f ) , > f ) , for each firm f ∈ F , µ ∨ F µ ( w ) = µ ∧ W µ ( w ) : = { f : w ∈ Ch ( µ ( f ) ∪ µ ( f ) , > f ) } , for each worker w ∈ W .Similarly, µ ∨ W µ ( w ) = µ ∧ F µ ( w ) : = Ch ( µ ( w ) ∪ µ ( w ) , > w ) , for each worker w ∈ W , µ ∨ W µ ( f ) = µ ∧ F µ ( f ) : = { w : f ∈ Ch ( µ ( w ) ∪ µ ( w ) , > w ) } , for each firm f ∈ F . Remark 1
Let T ⊆ S ( P ) . We denote by _ ν ∈ TF ν ( f ) = Ch ( [ ν ∈ T ν ( f ) , > f ) | S | denotes the number of agents in S . Li (2014) present an alternative proof for Alkan’s result, Li (2014) assumes agents with completepreferences, whereas Alkan (2002) assumes agents with incomplete preferences. nd ^ ν ∈ TF ν ( f ) = { w : f ∈ Ch ( [ ν ∈ T ν ( w ) , > w ) } . By substitutability and transitivity, Li (2014) proves that that _ ν ∈ TF ν ( f ) and ^ ν ∈ TF ν ( f ) are stable matchings and coincide with the l.u.b. and g.l.b. among the stable matchings in Trespectively. In this section, we present two results that have interest in themselves and that weuse in the next section in order to prove that the set of random stable matchings hasa lattice structure. Given a random stable matching that is represented as a lotteryover stable matchings, we change its representation as a new lottery over a new set ofstable matching. To be more specific, we prove that this new set of stable matchingshave a decreasing property, namely, there is { µ , . . . , µ ˜ k } ⊆ S ( P ) with µ ℓ ≥ F µ ℓ + for ℓ =
1, . . . , ˜ k −
1. Also, we present a version for random stable matchings of the RuralHospital Theorem (Proposition (RHT)).To describe the representation of a random stable matchings, first, we need to definean incidence vector. Then, given a stable matching µ , a vector x µ ∈ {
0, 1 } | F |×| W | is an incidence vector where x µ i , j = j ∈ µ ( i ) and x µ i , j = x = ∑ ν ∈S ( P ) λ ν x ν where 0 ≤ λ ν ≤ ∑ ν ∈S ( P ) λ ν =
1, and ν ∈ S ( P ) .Notice that, each entry of a random stable matching x , fulfils that x i , j ∈ [
0, 1 ] .Given a random stable matching x , we define the support of x as follows: supp ( x ) = { ( i , j ) : x i , j > } .Given a random stable matching x , i.e. x = ∑ ν ∈S ( P ) λ ν x ν ; 0 ≤ λ ν ≤ ∑ ν ∈S ( P ) λ ν =
1, we define A to be the set of all stable matchings involve in the lottery. Formally, A = (cid:26) ν ∈ S ( P ) : x = ∑ ν ∈S ( P ) λ ν x ν ; 0 < λ ν ≤ ∑ ν ∈S ( P ) λ ν = (cid:27) .Now, in order to change the representation of the random stable matching x proceedas follows: 7 lgorithm 1:Step 0 Set B : = A [ (cid:26) _ ν ∈ TF ν : T ⊆ A (cid:27) [ (cid:26) ^ ν ∈ TF ν : T ⊆ A (cid:27) . x : = x . M : = ∅ . Λ : = ∅ . Step k ≥ Set µ k : = _ ν ∈ B k F ν . M : = M ∪ { µ k } . α k : = min { x ki , j : x µ k i , j = } . Λ : = Λ ∪ { α k } . L k : = { ( i , j ) ∈ F × W : x ki , j = α k and x µ k i , j = } . C k : = [ ( i , j ) ∈L k { ν ∈ B k : x ν i , j = } . B k + : = B k \ C k . IF B k + = ∅ , THEN , the procedure stops.
ELSE set x k + = x k − α k x µ k − α k , and continue to Step k + Theorem 1
Let x be a random stable matching and M be the output of Algorithm 1. Then, xis represented as a lottery over stable matchings that belong to M where µ ℓ > F µ ℓ + for each µ ℓ , µ ℓ + ∈ M . Moreover, the set M is unique.Proof. See proof in Appendix A. (cid:3)
The following example illustrate Algorithm 1.
Example 1
Let ( F , W , P ) be a many-to-one matching market instance where F = { f , f , f , f } ,W = { w , w , w , w } and the preference profile is given by > f = { w , w } , { w , w } , { w , w } , { w , w } , { w } , { w } , { w } , { w } . > f = { w , w } , { w , w } , { w , w } , { w , w } , { w } , { w } , { w } , { w } . > f = { w , w } , { w , w } , { w , w } , { w , w } , { w } , { w } , { w } , { w } . > f = { w , w } , { w , w } , { w , w } , { w , w } , { w } , { w } , { w } , { w } . > w = { f , f } , { f , f } , { f , f } , { f , f } , { f } , { f } , { f } , { f } . > w = { f , f } , { f , f } , { f , f } , { f , f } , { f } , { f } , { f } , { f } . > w = { f , f } , { f , f } , { f , f } , { f , f } , { f } , { f } , { f } , { f } . > w = { f , f } , { f , f } , { f , f } , { f , f } , { f } , { f } , { f } , { f } .8 t is easy to check that these preference relations are substitutable and satisfy LAD. The set ofstable matchings is represented in Table 1 and its lattice for the partial order ≥ F is representedin Figure 1. f f f f ν { w , w } { w , w } { w , w } { w , w } ν { w , w } { w , w } { w , w } { w , w } ν { w , w } { w , w } { w , w } { w , w } ν { w , w } { w , w } { w , w } { w , w } Table 1 ν ν ν ν Figure 1Let x = x ν + x ν be a random stable matching. Now, we change the representation ofx as in Theorem 1. Notice that x =
34 14 34 1414 34 14 3414 14 34 3434 34 14 14 . Then, A = { ν , ν } , and B = { ν , ν , ν , ν } . Set M : = ∅ and Λ : = ∅ . Step 1
Set µ : = ν = _ ν ∈ B F ν , M : = M ∪ { µ } , α =
14 , Λ : = Λ ∪ { α } and C = { ν , ν } . Since B = B \ C = { ν , ν } 6 = ∅ , then setx : = x − x µ − =
13 23
23 13 , and continue to Step 2. Step 2
Set µ : = ν = _ ν ∈ B F ν , M : = M ∪ { µ } , α =
23 , Λ : = Λ ∪ { α } and C = { ν } . Since B = B \ C = { ν } 6 = ∅ , then setx = x − x µ − = , and continue to Step 3. Step 3
Set µ : = ν = _ ν ∈ B F ν , M : = M ∪ { µ } , α = Λ : = Λ ∪ { α } and C = { ν } .9 ince B = B \ C = ∅ , then the procedure stops.The output of Algorithm 1 is M = { µ , µ , µ } = { ν , ν , ν } , and Λ = { , , 1 } . Therefore, x = x µ + ( − )( ) x µ + ( − )( − )( ) x µ = x µ + x µ + x µ . Since µ = ν , µ = ν and µ = ν , then x can be written as:x = x ν + x ν + x ν . As we can see in Figure 1, the stable matchings of the lottery fulfils ν ≥ F ν ≥ F ν . The following proposition it is known as
Rural Hospital Theorem . For a many-to-many matching markets where the preference relation of each agent satisfies substi-tutability and the LAD is proved in Alkan (2002).
Proposition (RHT) (Alkan (2002))
Each agent is matched with the same number of partnersin every stable matching. That is, | µ ( a ) | = | µ ′ ( a ) | for each µ , µ ′ ∈ S ( P ) and for eacha ∈ F ∪ W. Next, we present a version for random stable matchings of Proposition (RHT).
Proposition 1
Let x and x ′ be two random stable matchings, then ∑ i ∈ F x i , j = ∑ i ∈ F x ′ i , j foreach j ∈ W, and ∑ j ∈ W x i , j = ∑ j ∈ W x ′ i , j for each i ∈ F.Proof.
See proof in Appendix A. (cid:3)
From now on, by Proposition 1, we assume that each random stable matching isalready represented as a lottery over stable matchings in a decreasing way.
In this section, we define a partial order for the set of random stable set in a many-to-many matching market with substitutable preferences satisfying the L.A.D.. Thispartial order is a generalization of the first-order stochastic dominance presented inRoth et al. (1993) for the random stable set in the marriage market. Given two randomstable matching x and y for the marriage market ( M , W , P ) , Roth et al. (1993) definethe partial order as follows. They say that x weakly dominates ⋆ y for man m , (heredenoted by x (cid:23) ⋆ m y ) if ∑ j ≥ m w x m , j ≥ ∑ j ≥ m w y m , j w ∈ W . Further they say that x (cid:23) ⋆ M y if x (cid:23) ⋆ m y for each m ∈ M . The partialorder (cid:23) ⋆ W is defined analogously. Notice that the partial orders (cid:23) ⋆ M and (cid:23) ⋆ W can notorder random stable matchings when agents have preferences over subset of agents onthe other side of the market in a substitutable manner. For this reason, for the settingconsidered in this paper, we define a new partial order. Formally, Definition 2
Let x = ∑ Ii = α i x µ xi and y = ∑ Jj = β j x µ yj with µ xi ≥ F µ xi + for each i =
1, . . . , I − and µ yj ≥ F µ yj + for each j =
1, . . . , J − . We say that x weakly dominates y for the firm f , (x (cid:23) f y), if and only if for each µ yj ( f ) ∑ i : µ xi ( f ) ≥ f µ yj ( f ) α i ≥ ∑ l : µ yl ( f ) ≥ f µ yj ( f ) β l .Further, we say that x strongly dominates y for the firm f , ( x ≻ f y ), if the aboveinequalities hold with at least one strict inequality for some µ yj ( f ) . That is, x ≻ f y when x (cid:23) f y and x = y for the firm f . Further, if x (cid:23) f y for each f ∈ F we denotethat x (cid:23) F y . We define (cid:23) w , ≻ w and (cid:23) W analogously. If we interpret the x f , w as theprobability that f is matched with w , then x (cid:23) f y if x f , · stochastically dominates y f , · .Notice that, since both x and y are represented following Theorem 1, then ∑ l : µ yl ( f ) ≥ f µ yj ( f ) β l = j ∑ l = β l .Now we prove that the domination relation (cid:23) F is a partial order. The proof of (cid:23) W is analogously. Formally, Proposition 2
The domination relation (cid:23) F is a partial order.Proof. See proof in Appendix B. (cid:3)
In this subsection we explain the splitting procedure for two random stable matchings,that is formalized with an algorithm in Appendix B. Once we apply the splitting pro-cedure for two random stable matchings, we define a domination relation ( (cid:23) SF ) that isfurther used to define the binary operation in a simple way. Given two random stablematchings x = ∑ Ii = α i x µ xi and y = ∑ Jj = β j x µ yj with µ xi ≥ F µ xi + for each i =
1, . . . , I − µ yj ≥ F µ yj + for each j =
1, . . . , J −
1, the splitting procedure goes as follows: Let γ = min { α , β } . W.l.o.g. assume that γ = α . Then, x = γ µ x + I ∑ i = α i x µ xi = γ µ y + ( β − γ ) µ y + J ∑ j = β j x µ yj .Notice that the first terms of each new representation have the same scalar. Now, takethe second scalar of each representation and set γ = min { α , β − γ } . If γ = α ,then x = γ µ x + γ µ x + I ∑ i = α i x µ xi y = γ µ y + γ µ y + ( β − γ − γ ) µ y + J ∑ j = β j x µ yj .If γ = β − γ , then x = γ µ x + γ µ x + ( α − γ ) µ x + I ∑ i = α i x µ xi y = γ µ y + γ µ y + J ∑ j = β j x µ yj .Notice that the first two terms of each new representation have the same scalar. Nowtake the third scalar of each representation and set either γ = min { α , β − γ − γ } or γ = min { α − γ , β } , and so forth so on.We illustrate the splitting procedure with the following example. Example 1 (Continued)
Let x = x ν + x ν + x ν and y = x ν + x ν + x ν . Noticethat both random stable matchings are represented following Theorem 1. Let γ = min { , } = , then x = x ν + ( − ) x ν + x ν + x ν , y = x ν + x ν + x ν Notice that the first term of each new representation have the same scalar . Let γ = min { − , } = − = , then x = x ν + x ν + x ν + x ν , y = x ν + x ν + ( − ) x ν + x ν . Notice that the second term of each new representation also have the same scalar . Let γ = min { , − } = − = , thenx = x ν + x ν + x ν + ( − ) x ν + x ν , y = x ν + x ν + x ν + x ν .12 otice that the third term of each new representation also have the same scalar . Let γ = min { − , } = − = , thenx = x ν + x ν + x ν + x ν + x ν , y = x ν + x ν + x ν + x ν + ( − ) x ν . Notice that the fourth term of each new representation also have the same scalar . Let γ = min { , − } = min { , } = , thenx = x ν + x ν + x ν + x ν + x ν , y = x ν + x ν + x ν + x ν + x ν . Notice that the fifth term of each new representation also have the same scalar . Now, once thesplitting procedure is complete, both x and y have five terms in each representation. Moreover,both lotteries have the same scalar, term to term. Algorithm 2 presented in Appendix B is the formalization of the splitting procedurefor two random stable matchings. In Appendix B, using the same example, we illus-trate the splitting procedure using Algorithm 2 detailed the procedure step by step.Further, the following proposition states that the splitting procedure changes the rep-resentation of the two random stable matchings.
Proposition 3
Let x and y be two random stable matchings such thatx = I ∑ ℓ = α ℓ µ x ℓ and y = J ∑ ℓ = β ℓ µ y ℓ . Then, there is Ω = (cid:8)(cid:0) γ ℓ , ˜ µ x ℓ , ˜ µ y ℓ (cid:1) : ℓ =
1, . . . , ˜ k (cid:9) defined by Algorithm 2, where ˜ k is the laststep of the algorithm such thatx = ˜ k ∑ ℓ = γ ℓ ˜ µ x ℓ and y = ˜ k ∑ ℓ = γ ℓ ˜ µ y ℓ . Proof.
See proof in Appendix B. (cid:3)
Once two random stable matchings goes through the splitting procedure, we candefine the following domination relation. This domination relation and its equivalencewith the partial order defined in Section 4, are used in the next section to prove themain result.
Definition 3
Let x and y be two random stable matchings such thatx = ˜ k ∑ ℓ = γ ℓ ˜ µ x ℓ and y = ˜ k ∑ ℓ = γ ℓ ˜ µ y ℓ ,13 here for each ℓ =
1, . . . , ˜ k, < γ ℓ ≤ , ∑ ˜ k ℓ = γ ℓ = and for each ℓ =
1, . . . , ˜ k − µ x ℓ ≥ F ˜ µ x ℓ + and ˜ µ y ℓ ≥ F ˜ µ y ℓ + . We say that x splittely dominates y for all firms ( x (cid:23) SF y ) if ˜ µ x ℓ ≥ F ˜ µ y ℓ for each ℓ =
1, . . . , ˜ k . Analogously, we define x (cid:23) SW y for all workers. Remark 2
Notice that the partial order (cid:23) F ( (cid:23) W ) compares two random stable matchings in-dependently if they are or not splitted. The following proposition proves that both domination relations ( (cid:23) F and (cid:23) SF ) areequivalent. Proposition 4
The partial order (cid:23) F is equivalent to the domination relation (cid:23) SF .Proof. See proof in Appendix B. (cid:3)
Corollary 1
The domination relation (cid:23) SF is also a partial order. Given two random stable matchings represented after the splitting procedure, we de-fine binary operations for random stable matchings to compute the l.u.b. and g.l.b. foreach side of the market. Further, we state the main result of this paper: the set ofrandom stable matchings has a dual lattice structure.Recall that ∨ W , ∧ W , ∨ F and ∧ F are the binary operations relative to the partialorders ≥ W and ≥ F defined between two (deterministic) stable matchings. Now, weextend these binary operations to random stable matchings. Formally, Definition 4
Let x and y be two random stable matchings such thatx = ˜ k ∑ ℓ = γ ℓ ˜ µ x ℓ and y = ˜ k ∑ ℓ = γ ℓ ˜ µ y ℓ where for each ℓ =
1, . . . , ˜ k, < γ ℓ ≤ , ∑ ˜ k ℓ = γ ℓ = , ˜ µ x ℓ , ˜ µ y ℓ ∈ S ( P ) , and for each ℓ =
1, . . . , ˜ k − , ˜ µ x ℓ ≥ F ˜ µ x ℓ + and ˜ µ y ℓ ≥ F ˜ µ y ℓ + .Define x ⊻ F y , x ⊼ F y , x ⊻ W y and x ⊼ W y as follows: x ⊻ F y : = ˜ k ∑ ℓ = γ ℓ ( ˜ µ x ℓ ∨ F ˜ µ y ℓ ) , x ⊼ F y : = ˜ k ∑ ℓ = γ ℓ ( ˜ µ x ℓ ∧ F ˜ µ y ℓ ) , and x ⊻ W y : = ˜ k ∑ ℓ = γ ℓ ( ˜ µ x ℓ ∨ W ˜ µ y ℓ ) , x ⊼ W y : = ˜ k ∑ ℓ = γ ℓ ( ˜ µ x ℓ ∧ W ˜ µ y ℓ ) ,14 emark 3 It is straightforward by Remark 1 that x ⊻ F y , x ⊼ F y , x ⊻ W y and x ⊼ W y arerandom stable matchings. Now, we are in position to prove that these binary operations defined for randomstable matchings are actually the l.u.b. and g.l.b. for each side of the market.
Proposition 5
Let x and y be two random stable matchings. Then, for X ∈ { F , W } we havethat x ⊻ X y = l.u.b. (cid:23) X ( x , y ) and x ⊼ X y = g.l.b. (cid:23) X ( x , y ) . Also, x ⊻ F y = x ⊼ W y and x ⊻ W y = x ⊼ F y . Proof.
See proof in Appendix B. (cid:3)
Now, we are in position to state the main result as follows,
Theorem 2 ( RS ( P ) , (cid:23) F , ⊻ F , ⊼ F ) and ( RS ( P ) , (cid:23) W , ⊻ W , ⊼ W ) are dual lattices. The following example illustrate how to compute the binary operations for tworandom stable matchings.
Example 1 (Continued)
Given x and y represented as in Proposition 3, we compute x ⊻ F yand x ⊼ F y as follows: (The other two cases are similar)x = x ν + x ν + x ν + x ν + x ν , y = x ν + x ν + x ν + x ν + x ν . x ⊻ F y = x ν ∨ F ν + x ν ∨ F ν + x ν ∨ F ν + x ν ∨ F ν + x ν ∨ F ν = x ν + x ν + x ν + x ν + x ν , = x ν + x ν + x ν . x ⊼ F y = x ν ∧ F ν + x ν ∧ F ν + x ν ∧ F ν + x ν ∧ F ν + x ν ∧ F ν = x ν + x ν + x ν + x ν + x ν , = x ν + x ν + x ν .15 .1 Binary operations for rational random stable matchings In this subsection, we compute the g.l.b. and l.u.b. for two random stable matchingswhere each scalar of the lottery is a rational number . These random stable matchingsare called rational random stable matchings . In this case, the splitting procedure isdirectly and different to the procedure described by Algorithm 2.Let x and y be two rational random stable matchings, represented as follows: x = I ∑ i = α i µ xi , (1)such that, 0 < α i ≤ i =
1, . . . , I ; ∑ Ii = α i = µ xi ∈ S ( P ) , α i is a rationalnumber and µ xi > F µ xi + ; where i =
1, ..., I − y = J ∑ j = β j µ yj , (2)such that, 0 < β j ≤ j =
1, . . . , J ; ∑ j ∈ J β j = µ yj ∈ S ( P ) , β j is a rationalnumber and µ yj > F µ yj + ; where j =
1, ..., J − α i and β j are positive rational numbers, we have that for each α i there arenatural numbers a i , b i such that α i = a i b i . Similarly, for each β j there are natural numbers c j , d j such that β j = c j d j .Denote by e the least common multiple ( lcm ) of all denominators b i , d j for each i =
1, . . . , I and for each j =
1, . . . , J . That is, e = lcm ( b , . . . , b I , d , . . . , d J ) .Then, we can write α i = a i b i = a i ebi e and β i = c j d j = c j edj e for each i =
1, . . . , I and foreach j =
1, . . . , J . Hence, we can write all the scalars α and β with the same denomina-tor.Denote by γ k = e and define ˜ µ xk : = µ x for k =
1, . . . , a b e µ x for k = a b e +
1, . . . , (cid:16) a b + a b (cid:17) e ... ... µ xI for k = I − ∑ n = a n b n ! e +
1, . . . , I ∑ n = a n b n ! e µ yk : = µ y for k =
1, . . . , c d e µ y for k = c d e +
1, . . . , (cid:16) c d + c d (cid:17) e ... ... µ yJ for k = J − ∑ m = c m d m ! e +
1, . . . , J ∑ m = c m d m ! e Then, we have that x = I ∑ i = α i µ xi = I ∑ i = a i b i µ xi = I ∑ i = a i eb i e µ xi = e ∑ k = e ˜ µ xk . (3)Analogously, we have that y = J ∑ j = β j µ yj = J ∑ j = c j d j µ yj = J ∑ j = c j ed j e µ yj = e ∑ k = e ˜ µ yk . (4)Given two rational random stable matchings x and y , represented as in (3) and (4),to compute x ⊻ F y , x ⊼ F y , x ⊻ W y and x ⊼ W y we state the following corollary of Theorem5. Corollary 2
Let x and y be two rational random stable matchings (i.e. each α and each β in(1) and (2) are rational numbers). Then, for X ∈ { F , W } we have thatx ⊻ X y = e ∑ k = e ( ˜ µ xk ∨ X ˜ µ yk ) and x ⊼ X y = e ∑ k = e ( ˜ µ xk ∧ X ˜ µ yk ) . Example 1 (Continued)
Let x and y be two random stable matchings represented as in Propo-sition 1, x = x ν + x ν + x ν , y = x ν + x ν + x ν . Let e = lcm (
2, 3, 4, 6 ) = . Then, the random stable matchings x and y can be representedas: x = x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν , y = x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν .17 hen, x ⊻ F y = x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν + x ν = x ν + x ν + x ν . Analogously for x ⊼ F y , x ⊻ W y and x ⊼ W y . In this paper, we prove an important result that involves two very much studied topicsin the matching literature: random stable matchings and lattice structure. The many-to-many matching markets with substitutable preferences satisfying the L.A.D. are themost general matching markets in which it is known that the binary operations be-tween two stable matchings ( l.u.b. and g.l.b. ) are computed via pointing functions.For these markets, we prove that the set of random stable matchings endowed with apartial order has a dual lattice structure. Moreover, we present natural binary opera-tions to compute l.u.b. and g.l.b. between two random stable matching for each sideof the markets. The partial order defined in this paper is a generalization of the first-order stochastic dominance for the case in which agents have substitutable preferencessatisfying the L.A.D.. For more general matching markets, for instance markets thatonly satisfy substitutability (not L.A.D.), the binary operations between (determinis-tic) stable matchings are computed as fixed points. Then, the lattice structure of the setof random stable matchings for these markets is still an open problem, left for futureresearch.
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The following technical results are used in the proof of Theorem 1.
Lemma 1 µ k ∈ B k for each k =
1, . . . , ˜ k.Proof.
Let µ = _ ν ∈ B F ν . Then, by definition of B , we have that µ ∈ B . Let µ = W ν ∈ B ν and assume that µ / ∈ B , then µ ∈ C . Since µ is computed via pointingfunctions, there is ν ′ ∈ B such that ν ′ ∈ C , which is a contradiction, then µ ∈ B .Similar arguments proves that µ k ∈ B k for each k =
1, . . . , ˜ k , where ˜ k is the last step ofAlgorithm 1. (cid:3) Lemma 2
If B k = ∅ , then B k + ⊂ B k .Proof. By definition of µ k and C k , µ k ∈ B k ∩ C k . Then, B k + = B k \ C k ⊂ B k . (cid:3) Lemma 3
Let ˜ ν = ^ ν ∈ B F ν , and ˜ k the step of Algorithm 1 in which B ˜ k = ∅ and B ˜ k + = ∅ . Then, ˜ ν ∈ B ˜ k . Proof.
Let ˜ ν = ^ ν ∈ B F ν and ˜ k the step of Algorithm 1 in which B ˜ k = ∅ and B ˜ k + = ∅ . Bydefinition of B , ˜ ν ∈ B . Assume that ˜ ν / ∈ B ˜ k , then there is a k ′ < ˜ k such that ˜ ν ∈ B k ′ and ˜ ν / ∈ B k ′ + . Then, ˜ ν ∈ C k ′ . Hence, by definition of C k ′ , there is a pair ( i ′ , j ′ ) suchthat x k ′ i ′ , j ′ = α k ′ and x ˜ ν i ′ , j ′ = x µ k ′ i ′ , j ′ =
1. Notice that, by definition of µ k ′ , we have that j ′ ∈ µ k ′ ( i ′ ) = Ch ( ∪ ν ∈ B k ′ ν ( i ′ ) , > i ′ ) . Since the preferences relation > i ′ is substitutableand ˜ ν ∈ B k ′ , we have that j ′ ∈ Ch ( ˜ ν ( i ′ ) ∪ { j ′ } , > i ′ ) . (5)By Lemma 2 and k ′ < ˜ k , we have that B k ′ + = ∅ . Then, there is ν ′ ∈ B k ′ such that ν ′ / ∈ C k ′ . We claim that j ′ / ∈ ν ′ ( i ′ ) . If j ′ ∈ ν ′ ( i ′ ) , for ( i ′ , j ′ ) we have x k ′ i ′ , j ′ = α k ′ and x ν ˜ k i ′ , j ′ = x µ k ′ i ′ , j ′ =
1, then ν ′ ∈ C k ′ , which is a contradiction. Thus, j ′ / ∈ ν ′ ( i ′ ) . Since ν ′ ∈ B k ′ ⊆ B , then ν ′ ≥ F ˜ ν . That is, ν ′ ( i ′ ) = Ch ( ˜ ν ( i ′ ) ∪ ν ′ ( i ′ ) , > i ′ ) . Now, given that j ′ ∈ ˜ ν ( i ′ ) \ ν ′ ( i ′ ) , we have that j ′ / ∈ Ch ( ˜ ν ( i ′ ) ∪ { j ′ } , > i ′ ) , which is a contradiction with(5). Therefore, ˜ ν ∈ B ˜ k . (cid:3) Lemma 4
Let µ ∈ S ( P ) , x a random stable matching, and x k = x k − − α k − x µ k − − α k − be the matrixconstruct by Algorithm 1 in Step k. Then, for each k, we have that ∑ i ∈ F x ki , j = | µ ( j ) | for eachj ∈ W, and ∑ j ∈ W x ki , j = | µ ( i ) | for each i ∈ F. roof. Let µ ∈ S ( P ) and let k = B = ∅ , then B = C .That is, ˜ ν ∈ C . Hence, there is ( i , j ) ∈ L such that x µ i , j = x ˜ ν i , j = x i , j = α .Then, for each ν ∈ B such that µ ≥ F ν ≥ F ˜ ν we have that x ν i , j =
1. Hence, α = supp ( x µ ) ⊆ supp ( x ) and α = min { x i , j : x µ i , j = } , then x = x µ . Thus, byProposition (RHT) and definition of incidence vector, we have that ∑ i ∈ F x µ i , j = | µ ( j ) | for each j ∈ W , and ∑ j ∈ W x µ i , j = | µ ( i ) | for each i ∈ F .Assume that B = ∅ and ∑ i ∈ F x k − i , j = | µ ( j ) | for each j ∈ W , then by Proposition(RHT) and definition of x k we have that ∑ i ∈ F x ki , j = ∑ i ∈ F x k − i , j − α k − ∑ i ∈ F x µ k − i , j − α k − = | µ ( j ) | − α k − | µ ( j ) | − α k − = | µ ( j ) | .Therefore, ∑ i ∈ F x ki , j = | µ ( j ) | for each j ∈ W and for each k =
1, . . . , ˜ k . Similarly, we canprove that ∑ j ∈ W x ki , j = | µ ( i ) | for each i ∈ F and for each k =
1, . . . , ˜ k . (cid:3) Lemma 5 B k + = ∅ if and only if α k < .Proof. = ⇒ ) Let B k + = ∅ , then B k = C k . Hence, | B k | >
1. By Lemma 3 we have that˜ ν ∈ B k . Also, by definition of µ k , we have that ˜ ν = µ k . Then, by Proposition (RHT),there are at least three agents i ′ ∈ F and ˜ j , j ′ ∈ W such that: x ki ′ , j ′ > x ki ′ ,˜ j > x µ k i ′ , j ′ = x µ k i ′ ,˜ j = x ˜ ν i ′ , j ′ =
0, and x ˜ ν i ′ ,˜ j = ∑ j ∈ W x ki ′ , j = | µ k ( i ′ ) | = | ˜ ν ( i ′ ) | . Since supp ( x µ k ) ⊂ supp ( x k ) ,and supp ( x ˜ ν ) ⊂ supp ( x k ) , we have that |{ j ∈ W : x ki ′ , j > }| > | µ k ( i ′ ) | . Then there isan agent ˆ j ∈ W such that x µ k i ′ ,ˆ j = < x ki ′ ,ˆ j <
1. Thus, α k = min { x ki , j : x µ k i , j = } ≤ x ki ′ ,ˆ j < ⇐ =) Let α k <
1. Then there is a pair ( i ′ , j ′ ) such that x µ k i ′ , j ′ = x ki ′ , j ′ = α k < ( i ′ , ˜ j ) such that x µ k i ′ ,˜ j = x ki ′ ,˜ j > x ˜ ν i ′ ,˜ j =
1. Hence,for each pair ( i , j ) such that x µ k i , j = x ˜ ν i , j =
1, by Proposition (RHT) we have that x ν i , j = ν ∈ B k . Thus, x ki , j ≥ x ki ′ ,˜ j + x ki ′ , j ′ = x ki ′ ,˜ j + α k > α k , Then, ˜ ν / ∈ C k .Therefore, B k + = ∅ . (cid:3) Corollary 3 If α k = , then x k = x µ k . Proof.
Let L k = { ( i , j ) ∈ F × W : x ki , j = α k and x µ k i , j = } and recall that by definition of µ k , we have that supp ( x µ k ) ⊆ supp ( x k ) . If α k =
1, then L k = supp ( x µ k ) . By Lemma 4,we have that ∑ i ∈ F x ki , j = | µ k ( j ) | for each i ∈ F , then supp ( x k ) = supp ( x µ k ) . Therefore, x k = x µ k . (cid:3) roof of Theorem 1. Let x be a random stable matching. The output of Algorithm 1are the sets M = { µ , . . . , µ ˜ k } and Λ = { α , . . . , α ˜ k } .By Lemma 2, we have that B k + ⊂ B k . By the finiteness of the set of stable match-ings, we have that there is a step of Algorithm 1, say Step ˜ k , such that B ˜ k + = ∅ . Then,the algorithm stops. Hence, by Lemma 5 we have that α ˜ k =
1. Therefore, by Corollary3, we have that x ˜ k = x µ ˜ k .Recall that µ k = _ ν ∈ B k F ν , and µ k + = _ ν ∈ B k + F ν . By Lemma 2 we have that B k + ⊂ B k ,by Lemma 1 we have that µ k ∈ B k , and by definition of C k we have that µ k ∈ C k . Hence, µ k / ∈ B k + . Then, µ k > F µ k + .Let β = α , β = ( − α ) α , β = ( − α )( − α ) α , . . . , and β ˜ k = ∏ ˜ k − k = ( − α k ) .Now, we prove that ∑ ˜ kk = β k = ˜ k ∑ k = β k = ˜ k − ∑ k = β k + β ˜ k = ˜ k − ∑ k = k − ∏ ℓ = ( − α ℓ ) + ˜ k − ∏ ℓ = ( − α ℓ ) .Note that, β ˜ k − + β ˜ k = ˜ k − ∏ ℓ = ( − α ℓ ) α ˜ k − + ˜ k − ∏ ℓ = ( − α ℓ ) = ˜ k − ∏ ℓ = ( − α ℓ )( α ˜ k − + ( − α ˜ k − )) = ˜ k − ∏ ℓ = ( − α ℓ ) .Also, we have that β ˜ k − + β ˜ k − + β ˜ k = ˜ k − ∏ ℓ = ( − α ℓ ) α ˜ k − + ˜ k − ∏ ℓ = ( − α ℓ )( − α ˜ k − ) = ˜ k − ∏ ℓ = ( − α ℓ ) .Continuing this inductive process, β + · · · + β ˜ k = ( − α ) . Then, ˜ k ∑ k = β k = β + ˜ k ∑ k = β k = α + ( − α ) = x = ˜ k ∑ k = β k x µ k where 0 < β k ≤ ∑ ˜ kk = β k =
1, and µ k > F µ k + for each k =
1, . . . , ˜ k − Uniqueness:
Assume that x has two different representations: x = ∑ ν ∈ A λ ν x ν = ∑ ν ′ ∈ A ′ λ ′ ν ′ x ν ′ where 0 < λ ν ≤
1, 0 < λ ′ ν ′ ≤ ∑ ν ∈ A λ ν = ∑ ν ′ ∈ A ′ λ ′ ν ′ =
1, and ν , ν ′ ∈ S ( P ) .Since, S ν ∈ A ν ( i ) = { j : x i , j > } = S ν ′ ∈ A ′ ν ′ ( i ) then, µ ( i ) = Ch ( S ν ∈ B ν ( i ) , > i ) = Ch ( S ν ′ ∈ B ′ ν ′ ( i ) , > i ) = µ ′ ( i ) for each i ∈ F . Therefore, µ = µ ′ .23et k > µ = µ ′ , . . . , µ k − = µ ′ k − . Then, x k = x k − − α k − x µ k − − α k − = x k − − α k − x µ ′ k − − α k − .We claim that { ( i , j ) : x ki , j > } = { ( i , j ) : S ν ∈ B k x ν i , j = } (and { ( i , j ) : x ki , j > } = { ( i , j ) : S ν ′ ∈ B ′ k x ν ′ i , j = } ). If not, there is a pair ( i , j ) such that x ki , j > x ν i , j = ν ∈ B k . Then, x ˜ ki , j > ν ∈ B ˜ k such that x ν i , j =
0. This contradictsthat x ˜ ki , j = x µ ˜ k i , j . Hence, { ( i , j ) : S ν ∈ B k x ν i , j = } ⊇ { ( i , j ) : x ki , j > } .Assume that there is ν ∈ B k such that x ν i , j = x ki , j =
0. Since x ν i , j =
1, then x i , j >
0. Hence, there is k ′ < k such that x k ′ i , j > x k ′ + i , j =
0. Since x k ′ + i , j = x k ′ i , j − α k ′ x µ k ′ i , j − α k ′ = x k ′ i , j = α k ′ x µ k ′ i , j . Hence, ( i , j ) ∈ L k ′ and ν ∈ C k ′ because we assume that x ν i , j = ν / ∈ B k ′ + ⊇ B k and this implies that x ki , j >
0, which is a contradiction. Therefore, { ( i , j ) : S ν ∈ B k x ν i , j = } ⊆ { ( i , j ) : x ki , j > } .Similar arguments prove that { ( i , j ) : x ki , j > } = { ( i , j ) : S ν ′ ∈ B ′ k x ν ′ i , j = } .Since S ν ∈ B k ν ( i ) = { j : x ki , j > } = S ν ′ ∈ B ′ k ν ′ ( i ) , then µ k ( i ) = Ch ( S ν ∈ B k ν ( i ) , ≥ i ) = Ch ( { j : x ki , j > } , > i ) = Ch ( S ν ′ ∈ B ′ k ν ′ ( i ) , > i ) = µ ′ k ( i ) for each i ∈ F . Therefore, µ k = µ ′ k . (cid:3) Proof of Proposition 1.
Let µ ∈ S ( P ) . By Lemma 4 of Appendix A, for the case inwhich k = x = x k , we have that ∑ i ∈ F x i , j = | µ ( j ) | for each j ∈ W . Analogously, wehave that ∑ i ∈ F x ′ i , j = | µ ( j ) | for each j ∈ W . Hence, ∑ i ∈ F x i , j = ∑ i ∈ F x ′ i , j .The proof of ∑ j ∈ W x i , j = ∑ j ∈ W x ′ i , j for each i ∈ F is analogously. (cid:3) Partial order and the splitting procedure
Proof of partial order
Proof of Proposition 2.
Let x , y and z be random stable matchings represented as x = I ∑ i = α i x µ xi , y = J ∑ j = β j x µ yj and z = K ∑ k = γ k x µ zk . Reflexivity: x (cid:23) F x . By the uniqueness of the representation of x following Theorem 1, we have that foreach µ xk ∑ i : µ xi ≥ F µ xk α l ≥ ∑ i : µ xi ≥ F µ xk α l . Transitivity: If x (cid:23) F y and y (cid:23) F z , then x (cid:23) F z . Since y (cid:23) F z , then ∑ l : µ yl ≥ F µ zk β l ≥ ∑ n : µ zn ≥ F µ zk γ n for each µ zk . Since x (cid:23) F y , then ∑ i : µ xi ≥ F µ yj α i ≥ ∑ l : µ yl ≥ F µ yj β l for each µ yj . Recall that x , y and z are represented followingTheorem 1. Then, for each µ zk there is an unique µ yj = min ≥ F { µ yl : µ yl ≥ F µ zk } such that ∑ m : µ xm ≥ F µ zk α m = ∑ i : µ xi ≥ F µ yj α i by { µ yl : µ yl ≥ F µ yj } = { µ yl : µ yl ≥ F µ zk } ∑ i : µ xi ≥ F µ yj α i ≥ ∑ l : µ yl ≥ F µ yj β l by x (cid:23) F y ∑ l : µ yl ≥ F µ yj β l = ∑ l : µ yl ≥ F µ zk β l by { µ xm : µ xm ≥ F µ zk } = { µ xm : µ xm ≥ F µ yj } ∑ l : µ yl ≥ F µ zk β l ≥ ∑ n : µ zn ≥ F µ zk γ n by y (cid:23) F z Hence, for each µ zk we have that ∑ m : µ xm ≥ F µ zk α m ≥ ∑ n : µ zn ≥ F µ zk γ n .Therefore, x (cid:23) F z . Antisymmetry: If x (cid:23) F y and y (cid:23) F x , then x = y . Assume that x (cid:23) F y and x = y , then we prove that y (cid:15) F x . By definition of x (cid:23) F y we have that x (cid:23) f y for each f ∈ F . Since x = y , then there is at least one f ′ ∈ F suchthat x ≻ f ′ y . Hence, by definition of x ≻ f ′ y , there is µ yj ( f ′ ) such that ∑ i : µ xi ( f ′ ) ≥ f ′ µ yj ( f ′ ) α i > ∑ l : µ yl ( f ′ ) ≥ f ′ µ yj ( f ′ ) β l .Then, y (cid:15) f ′ x , which in turns implies that y (cid:15) F x .Therefore, the domination relation (cid:23) F is a partial order. (cid:3) lgorithm 2 Let x and y be two random stable matchings such that, x = I ∑ i = α i µ xi and y = J ∑ j = β j µ yj .where 0 < α i ≤ I for i =
1, . . . , I , 0 < β j ≤ J for j =
1, . . . , J , ∑ Ii = α i = ∑ Jj = β j =
1. Let I = {
1, . . . , I } and J = {
1, . . . , J } . Set Ω = ∅ . Algorithm 2:Step k ≥ IF | I k − | = | J k − | = THEN , the procedure stops.Set, γ k = α k − = β k − , ˜ µ xk = µ xI , ˜ µ yk = µ yJ .Set Ω = Ω ∪ { ( γ k , ˜ µ xk , ˜ µ yk ) } . ELSE ( | I k − | > | J k − | > γ k = min { α k − , β k − } . IF γ k = α k − , THEN , set I k : = I k − and α k ℓ : = ( α k − − γ k if ℓ = α k − ℓ if ℓ > ℓ ∈ I k − . ELSE ( γ k = α k − ), set I k : = I k − \ max ℓ { ℓ ∈ I k − } and α k ℓ − = α k − ℓ for each ℓ ∈ I k − . IF γ k = β k − , THEN , J k : = J k − and β k ℓ : = ( β k − − γ k if ℓ = β k − ℓ if ℓ > ℓ ∈ J k − . ELSE ( γ k = β k − ), set J k : = J k − \ max ℓ { ℓ ∈ J k − } and β k ℓ − = β k − ℓ for each ℓ ∈ J k − .Set p = | I | − | I k − | and r = | J | − | J k − | .Set ˜ µ xk = µ xp + and ˜ µ yk = µ yr + .Set Ω = Ω ∪ { ( γ k , ˜ µ xk , ˜ µ yk ) } , and continue to Step k+1. Lemma 6
Algorithm 2 stops in a finite number of steps. That is, there is a ˜ k such that | I ˜ k − | = | J ˜ k − | = and α ˜ k = β ˜ k . Proof.
Note that in each step of Algorithm 2, we have that | I k | = | I k − | − | J k | = | J k − | −
1. We also have that in each Step k of the algorithm, ∑ ℓ ∈ I k α k ℓ = ∑ ℓ ∈ I k − α k − ℓ − γ k and ∑ ℓ ∈ J k β k ℓ = ∑ ℓ ∈ J k − β k − ℓ − γ k .26ence, ∑ ℓ ∈ I k α k ℓ = ∑ ℓ ∈ I α ℓ − k ∑ t = γ t = − k ∑ t = γ t .Similarly, ∑ ℓ ∈ J k β k ℓ = ∑ ℓ ∈ J β ℓ − k ∑ t = γ t = − k ∑ t = γ t .That is, for each k we have that ∑ ℓ ∈ I k α k ℓ = ∑ ℓ ∈ J k β k ℓ = − k ∑ t = γ t . (6)By the finiteness of the sets I and J , and given that in each step of Algorithm 2we have that | I k | = | I k − | − | J k | = | J k − | −
1, we claim that there is a ˜ k suchthat | I ˜ k − | = | J ˜ k − | =
1. Assume that there is a Step k − | I k − | = | J k − | >
1. By equality (6), we have that α k − = ∑ ℓ ∈ J k − β k − ℓ . Hence. α k − > β k − ℓ for each ℓ ∈ J k − and | I k | = | I k − | . Thus, α k = α k − − γ k = α k − − β k − and J k = J k − \ max ℓ { ℓ ∈ J k − } , and β k ℓ = β k − ℓ + for each ℓ ∈ J k . Then, | I k − | = | I k | = | J k | = | J k − | − ≥
1. If | J k | >
1, then proceed with Algorithm 2 untilthere is a step ˜ k such that | I ˜ k − | = | J ˜ k − | = α ˜ k = β ˜ k = γ ˜ k . (cid:3) Proof of Proposition 3.
First we prove that there is k such that α = ∑ k t = γ t . Since γ = min { α , β } , we analyze two cases. Case 1: γ = α . In this case k = Case 2: γ < α . In this case we have that | I | = | I | and α = α − γ . Then, in thenext step, γ ≤ α .If γ = α , then α = γ + γ .If γ < α , then repeat this procedure until k is found such that γ k = α k − . Then α = ∑ k t = γ t . Note that | I | = | I | = . . . = | I k | . Then, wehave that ˜ µ xt = µ x for t =
1, . . . , k and k ∑ t = γ t ˜ µ xt = k ∑ t = γ t µ x = α µ x .Notice that | I k | = | I k − | −
1. That is, 1 = p = | I | − | I k | and ˜ µ xk + = µ x .Then, for each k ≥ k + µ xk = µ x .Once we find k , we have to repeat this procedure with each α ℓ for ℓ ≥ β is similar. (cid:3) We illustrate Algorithm 2 with two random matchings of Example 1.
Example 1 (Continued)
Let x = x ν + x ν + x ν and y = x ν + x ν + x ν . Noticethat both random stable matchings are represented as in Theorem 1. We use Algorithm 2 tochange their representation. Let I = {
1, 2, 3 } and J = {
1, 2, 3 } . Set Ω = ∅ . Step 1
Since I = {
1, 2, 3 } and J = {
1, 2, 3 } , set γ = min { , } = , α = − = α = α = β = β = Then, I = {
1, 2, 3 } , J = {
1, 2 } , ˜ µ x = ν and ˜ µ y = ν . Set Ω = Ω ∪ { ( ν , ν , ) } and continue to Step 2. Step 2
Since I = {
1, 2, 3 } , J = {
1, 2 } , set γ = min { , } = , α = α = β = − = β = Then, I = {
1, 2 } , J = {
1, 2 } , ˜ µ x = ν and ˜ µ y = ν . Set Ω = Ω ∪ { ( ν , ν , ) } andcontinue to Step 3. Step 3
Since I = {
1, 2 } , J = {
1, 2 } , set γ = min { , } = , α = − = α = β = Then, I = {
1, 2 } , J = { } , ˜ µ x = ν and ˜ µ y = ν . Set Ω = Ω ∪ { ( ν , ν , ) } andcontinue to Step 4. Step 4
Since I = {
1, 2 } , J = { } , ˜ µ x = ν , set γ = min { , } = , α = β = − = Then, I = { } , J = { } , ˜ µ x = ν and ˜ µ y = ν . Set Ω = Ω ∪ { ( ν , ν , ) } andcontinue to Step 5. Step 5
Since I = { } , J = { } , then the procedure stops. Set γ = min { , } = , ˜ µ x = ν and ˜ µ y = ν . Set Ω = Ω ∪ { ( ν , ν , ) } Therefore, we can represent the random stable matchings x and y as follows:x = x ν + x ν + x ν + x ν + x ν , y = x ν + x ν + x ν + x ν + x ν .28bserve that x and y have five terms in each representation. Moreover, both lotter-ies have the same scalar, term to term. Proof of Proposition 4. (= ⇒ ) Let x and y be two random stable matchings represented after the splittingprocedure. Assume that x (cid:23) F y . Fix f ∈ F . We prove that x (cid:23) Sf y . That is, ˜ µ x ℓ ( f ) ≥ f ˜ µ y ℓ ( f ) for each ℓ =
1, . . . , ˜ k .If ˜ µ y ( f ) > f ˜ µ x ( f ) , we have that0 = ∑ ℓ : ˜ µ x ℓ ( f ) ≥ f ˜ µ y ( f ) γ ℓ ≥ ∑ ℓ : ˜ µ y ℓ ( f ) ≥ f ˜ µ y ( f ) γ ℓ = γ > µ x ( f ) ≥ f ˜ µ y ( f ) . Assume that there is k ≤ ˜ k such thatfor each ℓ < k we have that ˜ µ x ℓ ( f ) ≥ f ˜ µ y ℓ ( f ) , and ˜ µ xk ( f ) < f ˜ µ yk ( f ) .Note that ˜ µ y ℓ ( f ) ≥ f ˜ µ y ℓ + ( f ) for each ℓ =
1, . . . , ˜ k − k ∑ ℓ = γ ℓ = ∑ ℓ : ˜ µ y ℓ ( f ) ≥ f ˜ µ yk ( f ) γ ℓ . (7)By hypothesis ( x (cid:23) F y ), in particular for w = ˜ µ yk ( m ) we have that ∑ ℓ : ˜ µ y ℓ ( f ) ≥ f ˜ µ yk ( f ) γ ℓ ≤ ∑ ℓ : ˜ µ x ℓ ( f ) ≥ f ˜ µ yk ( f ) γ ℓ .Notice that for k , we have that ˜ µ xk − ( f ) ≥ f ˜ µ yk − ( f ) and ˜ µ xk ( f ) < f ˜ µ yk ( f ) . Then,˜ µ xk − ( f ) ≥ f ˜ µ yk − ( f ) ≥ f ˜ µ yk ( f ) > f ˜ µ xk ( f ) . Hence, ∑ ℓ : ˜ µ x ℓ ( f ) ≥ f ˜ µ yk ( f ) γ ℓ = ∑ ℓ : ˜ µ x ℓ ( f ) ≥ f ˜ µ xk − ( f ) γ ℓ = k − ∑ ℓ = γ ℓ . (8)Thus, by equalities (7) and (8), we have that ∑ k ℓ = γ ℓ ≤ ∑ k − ℓ = γ ℓ , and this is a con-tradiction since γ k >
0. Then, there is no k such that for each ℓ < k we have that˜ µ x ℓ ( f ) ≥ f ˜ µ y ℓ ( f ) , and ˜ µ xk ( f ) < f ˜ µ yk ( f ) . Thus, ˜ µ x ℓ ( f ) ≥ f ˜ µ y ℓ ( f ) for each ℓ =
1, . . . , ˜ k ,which in turns implies that x (cid:23) SF y . ( ⇐ =) Recall that both x and y are represented by the splitting procedure. That is,both representations have the same numbers of terms and the same scalar term toterm. Moreover, ˜ µ x ℓ ≥ F ˜ µ x ℓ + and ˜ µ y ℓ ≥ F ˜ µ y ℓ + for each ℓ =
1, . . . , ˜ k −
1. Also, since x (cid:23) SF y , then ˜ µ x ℓ ≥ F ˜ µ y ℓ for each ℓ =
1, . . . , ˜ k . Fix ℓ ′ , then { γ ℓ : ˜ µ x ℓ ≥ F ˜ µ x ℓ ′ } = { γ ℓ : ˜ µ y ℓ ≥ F ˜ µ y ℓ ′ } ⊆ { γ ℓ : ˜ µ x ℓ ≥ F ˜ µ y ℓ ′ } .Hence, ∑ ℓ : ˜ µ y ℓ ( f ) ≥ f ˜ µ y ℓ ′ ( f ) γ ℓ ≤ ∑ ℓ : ˜ µ x ℓ ( f ) ≥ f ˜ µ y ℓ ′ ( f ) γ ℓ f ∈ F and for each ℓ ′ =
1, . . . , ˜ k . Then, x (cid:23) F y .Therefore, the partial order (cid:23) F is equivalent to the domination relation (cid:23) SF . (cid:3) Proof of Proposition 5.
We prove that x ⊻ X y = l.u.b. (cid:23) X ( x , y ) . Recall that by Propo-sition 4, x (cid:23) F y if and only if x (cid:23) SF y (analogously for (cid:23) W and (cid:23) SW ).(i) x ⊻ X y (cid:23) X x : Since ˜ µ x ℓ ∨ X ˜ µ y ℓ ≥ X ˜ µ x ℓ for each ℓ =
1, . . . , ˜ k , then x ⊻ X y (cid:23) SX x . Hence, x ⊻ X y (cid:23) X x .(ii) x ⊻ X y (cid:23) X y : Since ˜ µ x ℓ ∨ X ˜ µ y ℓ ≥ X ˜ µ y ℓ for each ℓ =
1, . . . , ˜ k , then x ⊻ X y (cid:23) SX y . Hence, x ⊻ X y (cid:23) X y .(iii) If z (cid:23) X x and z (cid:23) X y , then z (cid:23) X x ⊻ y : We have that ˜ µ z ℓ ≥ X ˜ µ x ℓ and ˜ µ z ℓ ≥ X ˜ µ y ℓ for each ℓ =
1, . . . , ˜ k . Since, ˜ µ x ℓ ∨ X ˜ µ y ℓ is the l.u.b. ≥ X ( ˜ µ x ℓ , ˜ µ y ℓ ), then ˜ µ z ℓ ≥ X ˜ µ x ℓ ∨ X ˜ µ y ℓ for each ℓ =
1, . . . , ˜ k . Hence, z (cid:23) SX x ⊻ X y .Therefore, z (cid:23) X x ⊻ X y .The proof for x ⊼ X y = g.l.b. (cid:23) X ( x , y ) is analogous.To prove that x ⊻ F y = x ⊼ W y , recall that the lattices of stable matchings are dual,that is, given µ , µ ∈ S ( P ) µ ∨ F µ = µ ∧ W µ . By Definition 4, we have that if0 < γ ℓ ≤ ∑ ˜ k ℓ = γ ℓ =
1, ˜ µ x ℓ ∈ S ( P ) , ˜ µ x ℓ ≥ F ˜ µ x ℓ + and ˜ µ y ℓ ≥ F ˜ µ y ℓ + , then x ⊻ F y = ˜ k ∑ ℓ = γ ℓ ( ˜ µ x ℓ ∨ F ˜ µ y ℓ ) = ˜ k ∑ ℓ = γ ℓ ( ˜ µ x ℓ ∧ W ˜ µ y ℓ ) = x ⊼ W y .The proof for x ⊻ W y = x ⊼ F y is analogous. (cid:3)(cid:3)