THE NUMBER OF STABLE MATCHINGS IN MODELS OF THE GALE SHAPLEY TYPE WITH PREFERENCES GIVEN BY PARTIAL ORDERS

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1 OPERATIONS RESEARCH AND DECISIONS No DOI: /ord Ewa DRGAS-BURCHARDT 1 THE NUMBER OF STABLE MATCHINGS IN MODELS OF THE GALE SHAPLEY TYPE WITH PREFERENCES GIVEN BY PARTIAL ORDERS From the famous Gale Shapley theorem we know that each classcal marrage problem admts at least one stable matchng. Ths fact has nspred researchers to search for the maxmum number of possble stable matchngs, whch s equvalent to fndng the mnmum number of unstable matchngs among all such problems of sze n. In ths paper, we deal wth ths ssue for the Gale Shapley model wth preferences represented by arbtrary partal orders. Also, we dscuss ths model n the context of the classcal Gale Shapley model. Key words: combnatoral problems, stable matchng, Gale Shapley model 1. Prelmnares The problem of determnng the maxmum number of possble stable matchngs among all the classcal marrage problems of sze n was posed by Knuth [6] and stll remans an open queston. Knuth establshed that ths number exceeds 2 n/2 for n 2 and Gusfeld and Irvng showed [3] that when n s a power of 2 t s at least 2 n 1, whch can be mproved to (2.28) n /(1 + 3) based on the constructon gven by Irvng and Leather [4]. Some propertes of ths number analyzed as a functon of the problem sze were consdered n [10]. The frst upper bound on ths number was derved n [1]. It approxmately equals three quarters of all n! possble matchngs and s stll far from the expected number of stable matchngs, whch s asymptotc to e 1 nlnn for n [8]. In many practcal applcatons of the marrage problem, the lsts of preferences for prospectve partners allow ncomparable elements. In ths context, consder the fol- 1 Faculty of Mathematcs, Computer Scence and Econometrcs, Unversty of Zelona Góra, ul. Prof. Z. Szafrana 4a, Zelona Góra, Poland, e-mal: E.Drgas-Burchardt@wme.uz.zgora.pl

2 6 E. DRGAS-BURCHARDT lowng example. Let us assume that a company begns work on n dfferent computer programmng projects smultaneously, employng at the same tme n people as project managers. These people possess such sklls as: competence n human resource management, moblty, knowledge of spoken and programmng languages, knowledge of the topc sets connected to the projects, access to dfferent data bases. The task of the company management s to assgn managers to the projects, so that each of them s responsble for exactly one project. Such an assgnment s consdered stable f there exsts no unassocated par (manager and project j) such that manager would prefer managng project j rather than the one assgned to hm and, moreover, manager would be better for project j than the manager who s presently assgned. Note that t may be hard to decde who s a better canddate for beng the manager of a gven project, because some of the managers attrbutes are not comparable. It can happen that project needs both the knowledge of a partcular programmng language and the ablty to negotate n Fnnsh. Assume that no manager can do both thngs, but there are two, let us say m l, m k who each possesses one of these sklls. Ths fact shows the ncomparablty of m l and m k wth respect to project. Smlarly, specfc features of the projects can make two of them ncomparable for a gven manager. Usually, many attrbutes are needed to manage a gven project and people can possess some of them. Ths means that the preferences of each manager are descrbed by a partal order over the set of projects. Smlarly, the approprateness of a manager for a gven project s gven by a partal order over the set of all managers. In ths stuaton, a problem of sze n nvolves fndng a matchng between n managers and n projects wth 2n ndexed partally ordered sets, where n are partal orders descrbed over the set of projects and n over the set of managers. In ths paper, we nvestgate the mnmum number of unstable matchngs n models of the Gale Shapley type wth preferences that are not necessarly lnear. Among other thngs, we explan the dfference between the classcal model and the model consdered n ths paper. We show that f preferences are gven by arbtrary partal orders, then, unlke the classcal model, there s no upper bound on the number of stable matchngs concernng all the problems of sze n. On the other hand, f all 2n Hasse dagrams representng the preferences (for managers and for projects) have parwse the same shape for two dfferent problems of sze n (they are parwse somorphc), then the number of stable matchngs for both these problems can be bounded from above by the same expresson (Theorem 1). In partcular, we gve a smpler form of ths expresson n the case when all 4n Hasse dagrams for two dfferent problems of order n are somorphc (Corollary 3). Note that the classcal Gale Shapley model s of ths last knd, because all 4n Hasse dagrams for any two problems of sze n are somorphc (n fact they are represented by lnear orders over an n-element set).

3 Stable matchngs n models of the Gale Shapley type 7 Snce people have dfferent and ncomparable attrbutes, the example mentoned above can be wrtten n the language of a marrage problem. We use ths classcal descrpton n the remanng part of the paper. 2. Man results In general, we follow the notaton and termnology of [9]. Let W be a set of n women w 1,..., w n and M be a set of n men m 1,..., m n and for a natural number n, let the notaton [n] denote the set {1,..., n}. Next, suppose that for a fxed j [n], the partal order p(m j ) represents the preferences of the man m j over the set of women. Ths means that for gven s, k, j [n] we have (w s, w k ) P(m j ) f and only f ether s = k or m j prefers w s to w k. It follows that (w s, w k ) P(m j ) f and only f k s and ether (w k, w s ) P(m j ) or w s and w k are ncomparable for m j. Smlarly, for fxed [n], a partal order P(w ) represents the preferences of the woman w over the set of men. Thus preferences for prospectve partners are represented by the set P = {P(m 1 ),..., P(m n ), P(w 1 ),..., P(w n )}. Next, a trple (M, W; P) denotes a specfc marrage market (of sze n). It should be mentoned here that, accordng to ths model, we do not allow any ndvdual to reman sngle and assume that all prospectve partners are preferred to the opton of beng sngle. Moreover, all preferences are transtve, but two alternatves can be ncomparable. Ths means that ndvduals are not necessarly ratonal. A matchng s an arbtrary bjecton of W onto M. For smplcty, we wrte () = j nstead of (w ) = m j. By ( 1, j 1, 2, j 2 ) we denote the set of all matchngs satsfyng ( 1 ) = j 1 and ( 2 ) = j 2. In ths descrpton, we mplctly assume that 1 2 and j 1 j 2. A par w, m j s sad to block a matchng (, j 1, 2, j) n(m, W; P) f (m j, m j1 ) P(w ) and(w, w 2 ) P(m j ). A matchng s unstable for (M, W; P) f there exsts at least one par blockng n (M, W; P), otherwse s stable for (M, W; P). For a gven marrage market (M, W; P) of sze n and for, j [n], let a, j = {s: (m j, m s ) P(w ) and s j} and b, j = {s: (w j, w s ) P(m ) and s j}. The number a, j gves us nformaton on how many of the men are worse than m j accordng to w, and b, j tells how many of the women are worse than w j accordng to m. Next, for fxed [n] and P(m ), by M we denote the set of all lnear orders over W, say ( w,..., w ), such that b... b. Smlarly, for fxed [n] and P(w ), by W we denote the set of all s, 1 s, n lnear orders over M, say ( m,..., m ), such that a s1 s n... a. s, 1 s, n Proposton 1. Let (M, W; P) be a marrage market of sze n and let P * be an arbtrary set {P * (m 1 ),..., P * (m n ), P * (w 1 ),..., P * (w n )} such that for all [n] the condtons s1 s n

4 8 E. DRGAS-BURCHARDT P * (m ) M and P * (w ) W are satsfed. If a matchng s stable for (M, W; P), then s stable for (M, W; P). Proof. Suppose that Proposton 1 does not hold,.e. s stable for (M, W; P * ) and unstable for (M, W; P). It follows that there exsts a par w s, m k blockng when preferences are gven by P, whch consequently means that there exsts a par (j 1, j 2 ) such that (s, j 1, 2, k) and ( mk, mj ) P(m 1 s ) and ( ws, w ) P(w 2 k ) and also k j 1 and s. From the defnton of the numbers a, j, b, j, we obtan a s, k > a s, j1 and b k, s > b k,2. Ths yelds ( mk, mj ) P * (w 1 s ) and ( ws, w ) P * (m 2 k ). Usng the defnton of a blockng par once agan, we obtan that w s, m k blocks when the preferences are gven by P *, contrary to the ntal assumpton. Let (M, W; P) denote the set of all stable matchngs for the marrage market (M, W; P). Takng nto account Proposton 1 we mmedately obtan the followng result: Corollary 1. If (M, W; P) s a marrage market of sze n, then (M, W; P * ) (M, W; P), where the unon s taken over all P * = {P * (m 1 ),..., P * (m n ), P * (w 1 ),..., P * (w n )} such that P * (m ) and P * (w ) W for each [n]. From Gale and Shapley [2] we know that f all the partal orders n P are lnear, then there exsts at least one stable matchng for (M, W; P). In the lght of the above corollary, ths fact mples the next result. Corollary 2. If (M, W; P) s a marrage market of sze n, then there exsts at least one stable matchng for (M, W; P). Unfortunately, the converson to Proposton 1 s not true. For nstance n Fg. 1 we present a set P = {P(m 1 ), P(m 2 ), P(m 3 ), P(w 1 ), P(w 2 ), P(w 3 )} of partal orders, such that the matchng defned by () = for each permssble, s stable for (M, W; P). Moreover, s unstable for any marrage market (M, W; P * ) such that P * = {P * (m 1 ), P * (m 2 ), P * (m 3 ), P * (w 1 ), P * (w 2 ), P * (w 3 )}, where P * (m ) M, P * (w ) W wth {1, 2, 3}. Indeed, for each P * (m 1 ) M 1 we have (w 2, w 1 ) P * (m 1 ) and for each P * (w 2 ) w 2 we have (m 1, m 2 ) P * (w 2 ). Thus the par w 2, m 1 blocks n the marrage market (M, W; P * ). Fg. 1. An llustraton of the fact that the contanment n Proposton 1 can be strct

5 Stable matchngs n models of the Gale Shapley type 9 Of course there are many examples for whch the ncluson n Corollary 1 can be substtuted by an equalty. Analyss of partcular marrage markets havng preferences wth tes gves us nfntely many such examples. Markets of ths type have been consdered n many papers, especally from the aspect of the exstence of stable matchngs [5, 7]. A partal order over [n] s called smple f there exsts a partton of [n] nto k parts X 1,..., X k, such that (a, b) f and only f a = b or a X, b X j and < j. Proposton 2. Let (M, W; P) be a marrage market of order n such that all the partal orders from P are smple. If s a stable matchng for (M, W; P), then there exsts a set P * consstng of lnear orders P * (m 1 ),..., P * (m n ), P * (w 1 ),..., P * (w n ) such that P * (m ) M and P * (w ) W for [n] and s stable for (M, W; P * ). Proof. Let, p [n] satsfy () = p. Because P(m p ), P(w ) are smple, there exst parttons X 1,..., X k and Y 1,..., Y q of W and M, respectvely, and ndces s [k], j [q] such that w X s and m p Y j. In addton, we know that all the elements of X s are ncomparable based on P(m p ) and all the elements of Y j are ncomparable based on P(w ). We construct P * (m p ) M p and P * (w ) W n an arbtrary way that satsfes (w, w l ) P * (m p ) for each w l X s and (m p, m l ) P * (w ) for each m l Y q. For every other par of ndces 1, p 1 [n] satsfyng the equalty ( 1 ) = p 1, we use a smlar argument to produce a set P * = {P * (m 1 ),..., P * (m n ),..., P * (w 1 ),..., P * (w n )} that satsfes the theorem. It should be mentoned that marrage markets wth all preferences gven by smple orders are sometmes consdered n the lterature as markets wth lnear preferences and tes. Indvduals n such markets are ndfferent between some elements. Above we ndcated a connecton between any marrage market wth preferences gven by partal orders and some set of markets wth preferences gven by lnear orders. It was shown n [1] that the lnearty of preferences guarantees the exstence of a non-trval lower bound on the number of unstable matchngs. The rato of ths bound to the number n! of all possble matchngs tends to 1/4 when n tends to nfnty. We have no such concluson f for at least one man or at least one woman the preferences for potental spouses are gven by a non-lnear order. One specfc stuaton occurs when for each woman any two men are ncomparable and for each man any two women are ncomparable. Namely, n ths case, any matchng s stable. Thus we cannot guarantee the exstence of unstable matchngs at all but f they exst, then ther number can be bounded from below by parameters defned here. The correspondng result wll be preceded by some helpful defntons. For a marrage market (M, W; P) of sze n and for, j [n], let A, j (M, W; P) denote the set of all tuples (, j 1, 1, j) such that w, m j s a blockng par for each matchng (, j 1, 1, j). Evdently, A, j (M, W; P) = {(, j 1, 1, j): (m j, m j1 ) P(w ) and (w, w 1 )

6 10 E. DRGAS-BURCHARDT P(m j ) wth j j 1 and 1 }, whch means that A, j (M, W; P) = a, j b j,. Snce each tuple (, j 1, 1, j) belongng to A, j (M, W; P) corresponds to (n 2)! unstable matchngs from (, j 1, 1, j) and because for any two tuples from A, j (M, W; P) these sets of unstable matchngs are dsjont (each of cardnalty (n 2)!) we obtan the followng fact. Remark 1. Let (M, W; P) be a marrage market of sze n. If, j [n], then (M, W; P) n! (n 2)!a, j b j,. Recall that two partal orders 1, 2 over the sets X, Y, respectvely, are somorphc f there exsts a bjecton : X Y such that for any, w X we have (v, w) 1 f and only f ((), (w)) 2. In the next theorem, we use Remark 1 to obtan an upper bound on the number max(m, W; P), where the maxmum s taken over all marrage markets (M, W; P) of sze n, n whch the partal orders that represent preferences for correspondng women and men are parwse somorphc. Frst, we focus on the general case, next we turn our attenton to a specal case, thus mprovng the upper bound obtaned usng the frst approach. Theorem 1. (M, W; P) be a marrage market of sze n wth n 2 and let for [n] the orderngs ( s1,..., s ), ( k1,..., k ) of [n] satsfy a... a and b... b. If n n s, 1 s, n s, n/21 k, n/2 1, d1 mn : [ ] when s odd a n n s, k, n1/2 n1/2 c k, 1 k, n mn a : [ n] when n s even mn b : n when [ n] s even mn b : n when [ n] s odd and c a n, n s k, n/2 n mn : [ ] when s even mn b : [ n] when n s even, /2 1 d c when n s odd d when n s odd then (M, W; P) n! (n 2)! max{c 1 d 1, c 2 d 2 }. Moreover, n! (n 2)! max{c 1 d 1, c 2 d 2 } s an upper bound on (M, W; P ) for any other marrage market (M, W; P ) of sze n such that for each [n] the partal orders P(m ) and P (m ) are somorphc and the partal orders P(w ) and P (w ) are somorphc. Proof. Frst we shall show that the theorem holds for (M, W; P). It s suffcent to prove that there exst ndces 1, j 1, 2, j 2 [n] such that a 1, j 1 c 1 and b j1, 1 d 1 whle

7 Stable matchngs n models of the Gale Shapley type 11 a 2, j 2 c 2 and b j2, 2 d 2, before usng Remark 1. By symmetry, t s enough to show the exstence of ndces 1, j 1. To obtan a contradcton, assume that such ndces 1, j 1 do not exst. Ths means that for any par of ndces (, j) ether a, j c 1 1 or b j, d 1 1. We defne two sets A = {(, j, a,j ):, j [n]} and B = {(, j, b, j ):, j [n]} and a mappng that assgns the element (j,, b j, ) from B to the element (, j, a, j ) from A. Evdently, A = B = n 2 and s a bjecton. In the remanng part of the proof we shall construct two sets E A and F B such that maps E to a subset of F and the cardnalty of F s less than the cardnalty of E. Obvously, ths contradcts the fact that s a bjecton. Frst consder the case of even n. From the defnton of c 1 t follows that for each [n], the nequalty a c holds. Moreover, a... a. Hence for each [n], the numbers b,..., b s1, j sn/2 1, j b s, ( n/2) 1,..., b 1 s1, j sn/2 1, j s, 1 s, n/21 do not exceed d 1 1. Thus the values j n l satsfy j D, where D { k p : p[ n], l{ n/2 1,..., n}}. Now the m- age under of the set E {(, j, a, j) : { n], j { s1,..., s( n/2) 1}} of cardnalty (n 2 /2) + n s a subset of F {(, j, b, j) : [ n], j D}. Note that the cardnalty of F s equal to n 2 /2, whch proves the theorem when n s even. We use the same reasonng n the case of an odd n by constructng the same mappng, sets A, B and ther subsets E {(, j, a, j) : [ n], j{ s1,..., s( n 1)/2}} and F {(, j, b, j) : [ n], jd}, where D { k p s : p[ n], s{( n3) / 2,..., n}}. By the defntons of c 1, d 1, we know that (E) F. Smlarly to the prevous case, the cardnalty of E equals n(n + 1)/2 and the cardnalty of F equals n(n 1)/2, whch contradcts the fact that s a bjecton. Thus n! (n 2)!max{c 1 d 1, c 2 d 2 } s an upper bound on (M, W; P). The last statement of the theorem follows snce all the numbers c 1, c 2, d 1, d 2 depend only on the shape of the Hasse dagrams and are ndependent of the assgnments of women and men to them (somorphc partal orders produce the same numbers c 1, c 2, d 1, d 2 ). We llustrate Theorem 1 usng a marrage market (M, W; P) of sze 5, where P = {P(m 1 ),..., P(m 5 ), P(w 1 ),..., P(w 5 )} (Fg. 2). In ths case, the orderngs whose exstence s assumed n Theorem 1 could be ( s1,..., s = (1, 2, 3, 4,, ( s1,..., s = (5, , 1, 2, 3), ( s1,..., s = (2, 3, 4, 5, 1), ( s1,..., s = (1, 2, 3, 4,, ( s1,..., s = (4, 3, , 5, 1), ( k1,..., k = (3, 1, 5, 2, 4), ( k1,..., k = (1, 5, 4, 3, 2), ( k1,..., k = (5, 1, 4, 2, 3), ( k 4,..., k 4 ) = (2, 4, 3, 5, 1), ( k 5,..., k 5 ) = (1, 5, 2, 3, 4). Next, we have , 4, s a 1 a 2 a 3 a 5 2 and a whch gves c 1, s3 2, s3 3, s3 5, s3 4 1 = 1 (5 s an odd number). 3 Moreover, b 1 b 2 b 3 b 4 b 5 whch mples that d 1 = 2. Thus, from 2, 1, k3 2, k3 3, k3 4, k3 5, k3

8 12 E. DRGAS-BURCHARDT Theorem 1, we can fnd at least 12 unstable matchngs for (m, W; P) and t does not depend on the assgnment of the elements from the sets W, M to the vertces of Hasse dagrams presented n Fg. 2. Fg. 2. The Hasse dagrams that represent preferences For example, the same number of 12 unstable matchngs s guaranteed n the marrage market (M, W; P) of sze 5 whose set of preferences P = {P (m 1 ),..., P (m n ), P (w 1 ),..., P (w n )} s gven n Fg. 3. Corollary 3. Let (M, W; P) be a marrage market of sze n such that P = {P(m 1 ),..., P(m n ), P(w 1 ),..., P(w n )} and let all the partal orders from P be somorphc to. Next, let be defned over [n] such that for e = {j [n]: j and (, j) }. It follows that e 1... e n. If e,when n s even n/2 1 en /2,when n s even c and d e 1/2, when n s odd e 1/2, when n s odd n n then (M, W; P) n! cd(n 2)!.

9 Stable matchngs n models of the Gale Shapley type 13 Fg. 3. The Hasse dagrams that represent preferences Fg. 4. Hasse dagrams

10 14 E. DRGAS-BURCHARDT Note that Corollary 3 can be appled f all the partal orders from P are somorphc. One such possblty occurs when the preferences for all women and all men are lnear. In ths case, Corollary 3 mples one of the man results gven n [1]. To llustrate ths corollary usng another example, consder an arbtrary marrage market (M, W; P) of sze n for whch all P(m ) and all P(w ) are somorphc to the one gven n Fg. 4a. Note that the numbers e defned n Corollary 3 (the ndexes correspond to the vertces of Hasse dagram from Fg. 4a satsfy e 1 = 14, e 2 = 10, e 3 = e 4 = 9, e 5 = e 6 = e 7 = 8, e 8 = 7, e 9 = 6, e 10 = 5, e 11 = 4, e 12 = 3, e 13 = 2, e 14 = 1, e 15 = 0. Ths means that c = e (15+1)/2 = e 8 = 7. Hence, there exst at least 13! 49 unstable matchngs for the market (M, W; P) and ths does not depend on the correspondence between the ndvduals and the elements from the set [15]. Concludng remarks It s worth notng that the upper bound on the number of stable matchngs derved n ths paper depends on the number of ndvduals that are worse than gven partcpants of the marrage market. A detaled analyss of the numbers c 1, c 2, d 1, d 2 or c, d, respectvely, leads to the concluson that the value of ths bound s very dependent on the partcpants who occupy central postons n the preference orderngs. For a better understandng of ths fact, let us consder four classes of marrage markets of sze 15. The frst, where all of the preferences are represented by lnear orders, the second, where all the partal orders descrbng preferences are somorphc to the one from Fg. 4a, the thrd and the fourth where all the partal orders representng preferences are somorphc to those from Fgs. 4b and 4c, respectvely. In the frst and second cases, Corollary 3 demonstrates that a randomly selected matchng s unstable wth the probablty of at least 49/(14 1 = 7/30, n the thrd one wth the probablty of at least 25/(14 1 = 5/42, and n the last one wth the probablty of at least 4/(14 1 = 2/105. Thus, the best possble upper bounds correspond to marrage markets n whch the worst elements are ordered lnearly and any ncomparable elements are relatvely good. In [1] the well-known prncple of ncluson and excluson was used to derve an exact expresson for the number of stable matchngs n a marrage market wth preferences gven by lnear orders. A smlar expresson can be derved for marrage markets wth preferences gven by arbtrary partal orders. Indeed, n both cases, we count all the matchngs whch do not le n any of the sets A, j. Unfortunately, n the case of arbtrary partal orders, t s harder to determne the sets A, j, whose cardnaltes determne the number of stable matchngs. For completeness, we gve the desred ex-

11 Stable matchngs n models of the Gale Shapley type 15 presson, together wth the requred notons but we omt the proof whch s exactly analogous to the one presented for the smpler (lnear) case. Let (M, W; P) be a marrage market of sze n and A, jnn A, j (M, W; P). Next, let G A be a bpartte graph (X, Y; E A ) such that X = {x 1,..., x n }, Y = {y 1,..., y n } and E A ={x p y q : ( p, q, 2, j 2 ) A or ( 1, j 1, p, q) A}. By F(M, W; P) we denote a set { A A ( M, W; P): AA ( M, W; P) 1 and (G A ) = 1}, where, jnn j, jnn,, j k (G) s the maxmum degree of a vertex n the graph G. Fnally, let Fs ( M, W, P ) = {A F(M, W; P): A = s and E A = k}. Theorem 2. Let n 2. If (M, W; P) s a marrage market of sze n, then nn 1 s k, W; P) = n! 1 nk! F, W; P ) n s1 k2 s Acknowledgement The author would lke to thank the referees for many valuable comments. Research fnancally supported by the grant DEC-2011/01/B/HS4/ References [1] DRGAS-BURCHARDT E., ŚWITALSKI Z., A number of stable matchngs n models of the Gale Shapley type, Dscrete Appled Mathematcs, 2013, 162 (18), [2] GALE D., SHAPLEY L.S., College Admssons and the Stablty of Marrage, Amercan Mathematcal Monthly, 1962, 69, [3] GUSFIELD D., IRVING R.W., The Stable Marrage Problem: Structure and Algorthms, MIT Press, Cambrdge, MA, [4] IRVING R.W., LEATHER P., The complexty of countng stable marrages, SIAM Journal on Computng, 1986, 15, [5] IRVING R.W., Stable marrage and ndfference, Dscrete Appled Mathematcs, 1994, 48, [6] KNUTH D.E., Marages Stables, Les Presses de l Unversté de Montréal, Montréal [7] MANLOVE D.F., IRVING R.W., IWAMA K., MIYAZAKI S., MORITA Y., Hard varants of stable marrage, Theoretcal Computer Scence, 2002, 276, [8] PITTEL B., The Average Number of Stable Matchngs, SIAM Journal on Dscrete Mathematcs, 1989, 2 4, [9] ROTH A.E., SOTOMAYOR M.A.O., Two-sded matchngs, A study n game-theoretc modelng and analyss, Econometrc Socety Monographs, 18, [10] THURBER E.G., Concernng the maxmum number of stable matchngs n the stable marrage problem, Dscrete Mathematcs, 2002, 248, Receved 12 July 2014 Accepted 3 February 2015