Faculty of Electrical Egieerig, Mathematics & Computer Sciece Weight Distributio i Matchig Games Frits Hof Idividual research assigmet for course program Oderzoek va wiskude i master Sciece Educatio ad Commuicatio March 016 Supervisors: dr W Ker Departmet of Aplied Mathematics Faculty of Electrical Egieerig, Mathematics ad Computer Sciece dr NC Verhoef Istitute ELAN Faculty of Behavioural, Maagemet ad Social Scieces Uiversity of Twete PO Box 17 7500 AE Eschede The Netherlads
Weight Distributio i Matchig Games Frits Hof March 7, 016 Abstract Simple games that permit a weight represetatio such that each wiig coalitio has a weight of at least 1 ad all losig coalitios have a weight of at most α, are called α-roughly weighted games For a give game the smallest such value of α is called the critical threshold of the game Freixas ad Kurz [1] improved the lower boud o α after iitial work of Gvozdeva, Hemaspaadra ad Sliko [] ad cojectured that their boud is tight I this study we give a proof of their cojecture for simple games that have miimal wiig coalitios of order 1 Itroductio A cooperative game is defied by a fiite set N of players ad a value fuctio v, assigig a certai value v(s) R to every subset S N Each subset S N is iterpreted as a coalitio of players ad the correspodig value v(s) represets the gai which the players i S ca achieve by cooperatig I the simplest case the value fuctio takes oly values 0 ad 1 I this case we simply distiguish betwee wiig (v(s) = 1) or losig coalitios (v(s) = 0) If, i additio, v is mootoe, ie, supersets of wiig coalitios are wiig, the game is referred to as a simple game Specific examples of simple games are so-called weighted votig games: Assume that each player i S has a associated weight w i 0 ad defie v(s) = 1 if w(s) := i S w i 1 ad v(s) = 0 otherwise This obviously defies a simple game Not every simple game ca be defied this way Cosider, for example, a set N = {1} of players ad defie the wiig coalitios to be the sets of the form {i, i + 1} ad supersets thereof This defies a simple game i which both sets S odd = {i N i is odd} ad S eve = {i N i is eve} are losig Assume for simplicity that is eve The, if our game were a weighted votig game, there were correspodig o-egative weights satisfyig w i + w i+1 1 for all wiig coalitios S = {i, i + 1} This implies w(n) = / ad hece either w(s odd ) or w(s eve ) must exceed / So there are losig coalitios of weight / > 1, a cotradictio Freixas ad Kurz [1] have cojectured that for every simple game there exist weights w i such that all wiig coalitios S have weight w(s) 1 ad all losig coalitios have weight at most 1 where is the umber of players I this study we will ivestigate ad prove this cojecture for some atural ad 1
iterestig subclasses of simple games ad for the special case where all miimal wiig coalitios have cardiality About simple games A ice overview of the subject of weighted simple games with refereces to early work is give by Gvozdeva ad Sliko [3] ad Taylor ad Zwicker [,5] We recommed these works to the iterested reader who wats to kow more about simple games Here we restrict ourselves to the fudametal defiitios ad ecessary otios to (partially) prove the cojecture of Freixas ad Kurz Defiitio 1 Let P = [] = {1,,, } be a set of players ad let W P be a collectio of subsets of P that satisfies the mootoicity coditio: if X W ad X Y P the Y W I such case the pair G = (P, W) is called a simple game ad the set W is called the set of wiig coalitios of G Coalitios that are ot i W are called losig A wiig coalitio is said to be miimal if every proper subset i it is a losig coalitio, so removig ay player from such coalitio will make it losig Aalogue, a losig coalitio is said to be maximal if every proper superset of it is a wiig coalitio, ie addig ay player will make it wiig The set of all losig coalitios is called L Due to the mootoicity property the set W is completely determied by the collectio W mi of all miimal wiig coalitios of G Because W = its clear that due to the mootoicity coditio P W Furthermore, a game is also fully determied by the collectio L max of maximal losig coalitios To exclude trivial games we demad / W Defiitio A simple game G = (P, W) is called a weighted majority game if there exist oegative weights w 1,, w ad a real umber q, called quota, such that w(s) q for all S W w(s) < q for all S L Istead of i X w i we will ofte write w(x) Not all simple games are weighted majority games Moreover, most games 1 are ot weighted [5] Games exist with w(s) q for all S W ad w(s) q for all S L These games are called roughly weighted Some games are ot eve roughly weighted I those games the lightest wiig coalitio has a weight that is less tha the most heavy losig coalitios Gvozdeva, Hemaspaadra & Sliko [] itroduced the class of α-roughly weighted games to be able to measure the distace of a game to a (roughly) weighted game I this class the quota is exteded to a iterval [1, α] for a α R 1, while w(s) 1 for all s W ad w(s) α for all S L with L = p \ W 1 Because this article is just about simple games, we will ofte omit the word simple whe we speak about simple games So each time we write game we mea simple game uless we specify otherwise
Give a α-roughly weighted game G = (P, W) we are lookig for a weight fuctio w such that α is as small as possible The idea behid this is that a game with a smaller α is earer to a (roughly) weighted game tha games with a larger α This smallest α suitable for a give α-roughly weighted game G is called the critical threshold-value α(g) of game G [1] Fidig α(g) ca be formulated as a liear program Because a simple game is fully determied by its collectio of miimal wiig coalitios W mi, this liear program is: α(g) := mi α subject to w(s) 1 for S W mi w(s) α for S L max w 1,, w 0 where L max is the collectio of maximal losig coalitios I this LP α(g) deotes the miimum weight of the maximum weighted losig coalitio of a game G(P, W) while the weight of the miimum weighted wiig coalitio is 1 This formulatio is a slight modificatio of the formulatio by Freixas ad Kurz [1] They demad α 1 as a cosequece of the defiitio of α-roughly weighted games by Gvozdeva et al However, if we omit the costrait α 1 i the LP we will fid the weight of the maximum weighted losig coalitio for ay type of weighted game, while the miimum weighted wiig coalitio has a weight 1 Notice that the LP will yield α 0 because w(s) 0 for all S P I the case of a weighted majority game we will fid 0 α < 1, i the case of a roughly weighted game we will fid α = 1 ad for α-roughly weighted games we will fid α > 1 The reaso why we allow α 0 istead of α 1 is that i the rest of this study we will cosider α(g) ot oly for games with α(g) 1 but for all games, so we eed α(g) 0 Obviously the size of the class of α-roughly weighted games varies with α A larger value of α will capture more games, ad a smaller α will capture less So a very atural questio to ask is whether a smallest α exists, such that all games are i the class of α-roughly weighted games It is clear that such a α does t exist i geeral, but for games G with the same size this α depeds o So we are lookig for a fuctio α : N R such that max G α(g ) α() for all ad α() is miimal with this property 3 Kow bouds o the critical threshold Gvozdeva, Hemaspaadra ad Sliko [] gave a lower boud for max G α(g ) for by cosiderig games with disjoit miimal wiig coalitios of two players ad proved max G α(g ) 1 Freixas ad Kurz [1] improved this boud a little for specific odd games by usig duality i liear programmig ad proved max G α(g ) 1 They showed this boud by cosiderig odd games G where all players are i 1 miimal wiig coalitios {i, i + 1} They foud a feasible solutio for the dual of the LP ad deduced α(g) 1 for this type of games For games G with a eve umber of players they followed Gvozdeva, Hemaspaadra ad Sliko [] to show α(g) = 3
which equals 1 whe is eve (see Propositio 3 o page ) So by cosiderig these games it s clear that max G α(g ) 1 Freixas ad Kurz [1] also cojectured that this boud is tight, so max G α(g ) = α() = 1 for games with four or more players Games with are (roughly) weighted [3], so max G α(g ) 1 for {1,, 3, } For it s easy to check, by cosiderig all possibilities for W ad choosig a appropriate weight distributio w that α(g 1 ) = 0 = α(1), α(g ) = 1/ = α(), α(g 3 ) = /3 = α(3) ad α(g ) = 1 = α() So the cojecture ca be relaxed to α(g ) α() = 1 1 We will get back to this after the followig prelimiaries for all G G with Prelimiaries Because α-roughly weightedess was oly defied for games with players with α 1 by Gvozdeva ad Sliko [3] we like to state the ext defiitio, which is a slight modificatio of the origial defiitio Defiitio 1 A simple game G(P, W ) is called α()-roughly weighted if there are weights w 1,, w R 0 fulfillig w(s) 1 w(s) α() for all S W for all S L with L = P \ W with = P Now we state some properties related to fractios ad to α() Propositio 1 for a 1, b 1, a, b N if a1 b 1 Proof First otice that a1 b 1 a b a b N Now a1 b 1 follows by a similar argumet = a1(b1+b) b = a1b1+a1b 1(b 1+b ) b 1(b 1+b ) the a1 b 1 Propositio for a 1, b 1, a, b N Proof a a +b a 1 b 1 a b b1 a 1 if a1 b 1 b a a b a1+a b 1+b a b is equivalet to a 1 b a b 1 for a 1, b 1, a, b a1b1+ab1 b 1(b 1+b ) the a 1 a 1+b 1 a a +b b1 a 1 + 1 b a + 1 b1+a1 a 1 Propositio 3 For eve a N a = a = a1+a b 1+b The secod iequality b+a a a1 a 1+b 1
Proof Suppose a = k with k {1,, 3, } The k = k = a Propositio For odd a N a = a 1 a = a Proof Suppose a = k + 1 with k {0, 1,, } The = k + k + 1 = k + k = k +k+1 1 = (k+1) 1 = a 1 Propositio 5 For a N a 1 a 1 a a a k = k = (k+1) = Proof First otice that a 0, so 1 a exists For odd a 3 the first iequality i the propositio follows because a+1 1 so, a 1 a a 1 a a+1 = 1 a a 1 = 1 a a ad for a = 1 by the simple substitutio 1 a a = 1 1 1 = 0 = a 1 a The secod iequality i the propositio follows by 1 a a = 1 a a 1 < 1 a a = a for odd a 1 For eve a we see that 1 a a = 1 a a = a This proves the secod iequality of the propositio for eve a The first iequality i the propositio holds for eve a because 0 (a ) = a a + = a (a 1) So (a 1) a which yields a 1 a a = 1 a a Lemma 1 Ay simple game G(N, W) with just oe miimal wiig coalitio X is α()-roughly weighted with α() = 1 Proof If G cotais dummies, we set their weights to 0 We set w(i) = 1 X for all i X, so w(x) = 1 Now the maximum weighted losig coalitio L max ca cotai at most X 1 players from X ad some or all dummies Sice X it follows that α(g) w(l max ) = X 1 X 1 Lemma Let G 1 (N 1, W 1 ) ad G (N, W ) be two disjoit simple games with N 1 N =, 1 = N 1, = N If G 1 is α( 1 )-roughly weighted ad G is α( )-roughly weighted, the the joied game G(N, W) with N = N 1 N 1 ad W = W 1 W is α()-roughly weighted with = 1 + ad α() = 1 Proof For all X W mi it is obvious that X W 1 or X W, so it clear that w(x) = 1 Because N 1 N = also the maximum weighted losig coalitios L 1 N 1 ad L N are disjoit So its clear that i the joit game G the maximum weighted losig coalitio L G = L 1 L So α(g) = w(l G ) = w(l 1 ) + w(l ) If is eve, the α() = 1 (1+ ) 1+ = (1+) = 1 + 1 1 1 + 5
1 = α( 1 ) + α( ) w(l 1 ) + w(l ) = w(l G ) = α(g) If is odd, the, wolg we may assume that 1 is eve ad is odd, so α() = 1 = 1 = 1+ 1 ( > 1+ 1+ ) 1 = 1 + 1 = 1 1 1 + 1 = α( 1 ) + α( ) w(l 1 ) + w(l ) = w(l G ) = α(g) Lemma 3 Let G(N, W) be a α()-roughly weighted game for α() = 1 Now let S N with w(s) 1 The the game G (N, W ) with W = W {S} is α()-roughly weighted Proof Note that S because w(s) 1 If S W the lemma is clear because G = G so α(g ) = α(g) α() Now let L W be a maximum weighted losig coalitio i G so X L for all X W Because W W this meas that X L for all X W, so L is a maximum weighted losig coalitio for G So α(g ) = w(l ) α(g) α() 5 Games ad graphs I the previous sectios we cosidered games from a set-theoretical poit of view Aother, very much related viewpoit is graph-theoretical A simple game ca be see as a hypergraph, where the players are the vertices ad the coalitios are the hyperedges So a weighted simple game is ow a hypergraph with weighted vertices The weights of the hyperedges are the weights of the coalitios Because simple weighted games are fully determied by their miimal wiig coalitios, a simple game ca be represeted by the hypergraph with the miimal wiig coalitios as hyperedges A coalitio is wiig if it cotais ay hyperedge as a (ot ecessarily proper) subset A special type of games are the games with a collectio of miimal wiig coalitios that all have cardiality We call these games matchig games I this type of game all miimal wiig coalitios are pairs of players These coalitios may itersect The hypergraph represetatio of this type of game is a simple graph, where the vertices are the players ad the edges are the miimal wiig coalitios With G(V, E) we deote the correspodig graph to the game G(N, W) Although the otatio of both is very similar, o cofusio will occur ad we will use both otatios For matchig games G(N, W) ay maximal weighted losig coalitio is a idepedet set i the correspodig graph G(V, E) However, a maximum weighted losig coalitio does t eed to be a maximum idepedet set Suppose, for istace, that we have a game o 3 players represeted by a star K 1, 1 Suppose that player 1 is the ceter of the star The, by givig the ceter of the star a weight of w 1 = 1 ad all other players a weight of w i = 0 (i {,, }), the ceter of the star is a maximal weighted losig coalitio L 1 = {1} with w(l 1 ) = 1 The ceter is a maximal idepedet set, but ot a maximum idepedet set I fact, we have two maximal losig coalitios L 1 with weight w(l 1 ) = 1, ad L = {,, } with weight w(l ) = 0 I order to keep the weight of the maximum weighted losig coalitio as small as possible, we ca decide to distribute the weights more equally By settig w i = 1 for i {,, } ad w 1 = 1 1 we ca create a situatio where both maximal losig coalitios 6
L 1 ad L have a weight of w(l 1 ) = w(l ) = 1 ad the wiig coalitios still have a weight 1 Notice that this weight distributio respects the boud α(g) 1 for ay star with (eve with ) by Propositio 5 6 Matchigs ad games Aother way of lookig at maximum weighted losig coalitios i matchig games is by cosiderig matchigs Ay losig coalitio ca cotai at most oe player per miimal wiig coalitio So ay maximum weighted losig coalitio ca ot have two adjacet vertices This meas we ca pick oe vertex per edge at most i a losig coalitio So ay maximum weighted losig coalitio, ca cotai at most half of the players of a maximum matchig plus all players that are o part of this maximum matchig Lemma 61 Every matchig game G(N, W) with a perfect matchig is α()- roughly weighted with α() = 1 Proof Suppose the game has = m players (m N) I the maximum matchig there are m edges, correspodig with m miimal wiig coalitios All players are matched because the matchig is perfect I ay losig coalitio there are at most m players If we would have more tha m players, there must be at least two players of the same miimal wiig coalitio i the losig coalitio, which is a cotradictio Now we ca give all edges e M a weight w(e) = 1 by givig all players i a weight w i = 1 Now the weight of the maximum weighted losig coalitio L max is α(g) = w(l max ) = m 1 = = α() Defiitio 61 A graph G is called factor-critical if deletig ay vertex from G will result i a graph with a perfect matchig Lemma 6 Every matchig game G(N, W) that ca be represeted by a factor-critical graph G(V, E) is α()-roughly weighted with α() = 1 by the weight distributio w u = 1 for all u V Proof Observe that a factor-critical graph has a odd order, because deletig ay vertex will leave a graph with a perfect matchig M, which must have a eve order M Because there is at least oe wiig coalitio of players, ad the umber of players is odd, it s clear that = N 3 Notice that ay miimal wiig coalitio W mi has a weight w(w mi ) = 1 First otice that it is impossible that all players are i a maximal losig coalitio L max So there is at least oe player u V such that u / L max Because G is factor critical, G = G u cotais a perfect matchig Now by Lemma we kow that this graph G ca have at most half of its vertices to be chose i ay maximal losig coalitio So i G there ca be at most G = 1 = vertices which are i the maximum losig coalitio Now we give all players i N the proposed weight w i = 1, so α(g) = w(lmax ) = 1 1 7
Lemma 63 Ay game represeted by a biregular graph G(A, B; E) is α()- roughly weighted with α() = 1 where = A + B Proof Wlog we assume A B Let 0 < λ = A 1 be the fractio of the umber of players that are i A, so A = λ ad B = (1 λ) Ay maximal losig coalitio L will cotai a umber l A = L A players i A ad a umber l B = L B players i B Those l A vertices i A are icidet to deg(a) l A edges that are icidet to deg(a) deg(b) l A vertices i B Its clear that i the biregular graph deg(a) A = deg(b) B, so l A vertices i A cover deg(a) deg(b) l A = B A l A = 1 λ λ l A vertices i B Assume L cotais a fractio ρ of the players i A, so l A = ρ A (0 ρ 1 such that ρ A N {0}) The there are 1 λ λ l A vertices i B that are coected with these vertices i A Because o losig coalitio ca cotai two players from a miimal wiig coalitio, these vertices ca ot be preset i the maximal losig coalitio So its clear that l B B 1 λ λ l A = (1 λ) 1 λ λ ρλ = (1 λ) (1 λ)ρ = (1 ρ)(1 λ) If we chose for each player a A a weight w a = 1 λ ad for each player b B a weight w b = λ the the weights of all edges, which are the miimal wiig coalitios, are 1 Ay maximal losig coalitio L will have a weight w(l) w a l A +w b l B (1 λ)ρλ+λ(1 λ)(1 ρ) = λ(1 λ)(ρ+1 ρ) = λ(1 λ) By simple calculus we kow that P (λ) = λ(1 λ) has a maximum value of for λ = 1 A But sice λ = the value λ = 1 ca oly occur whe is eve The maximum weighted losig coalitio L max will have a weight w(l max ) = whe λ = 1, so whe A = B Whe λ < 1, so whe A < B the weight of the maximum weighted losig coalitio w(l max ) < This proves α(g) = w(l max ) < = 1 whe is eve If is odd, the maximum weight is reached for A =, so λ = / This yields w(l max ) = 1 (1 1 1 1 ) = (1 1 ) = 1 +1 = 1 1 = 1 So for every game G represeted by a biregular graph choosig the proposed λ-weight distributio, will guaratee that α(g) α() Lemma 6 Ay game represeted by a bipartite graph G(A, B; E) which cotais a Hamilto path, is α()-roughly weighted with α() = 1 where = A + B Proof Wlog we may assume A B If A = B, its clear that G has a perfect matchig, so the lemma follows from Lemma 61 Suppose A < B The the Hamiltopath starts i B, ad the vertices i the path are alterately i A ad B, with the last vertex i B, so B = A + 1 Now we kow that is odd ad A = 1 ad B = +1 Its clear that if we take λ A vertices from A i ay maximal losig coalitio, this coalitio ca cotai at most B (λ A + 1) = (1 λ) A vertices i B Now we give weights w a = B for a A ad w b = A for b B The weight of ay miimal wiig coalitio {a, b} with a A ad b B is w a + w b = B + A = 1 The weight of a maximal losig coalitio L max is bouded by α(g) = w(l max ) λ A w a + (1 λ) A w b < λ A w a + (1 λ) A w a = A w a = 1 +1 = 8
1 1 = 1 = α() where the secod last equality holds because of Propostio sice is odd Lemma 65 Ay game represeted by a bipartite graph G(A,B;E) with a matchig of A ito B is α()-roughly weighted with α() = where = A + B Proof Sice a matchig of A ito B exists its clear that A B ad, by Halls coditio, S N(S) for S A Let σ S be the ratio of the umber of vertices i S ad the umber of eighbors of S i B, so σ S = S N(S) Now we decompose G i the followig way: Let A 1 A be a largest subset of A amog the subsets S A with the largest σ S If A 1 A the remove A 1 from A to get A ad remove N(A 1 ) from B to get B ad iterate this procedure o the remaiig subgraph G (A, B ; E ) to fid A, A 3,, A k I this way we partitio A ito {A 1,, A k } such that A i A j = for i j ad i A i = A with i, j {1,, k} Next let B 1 = N(A 1 ) ad B i = N(A i ) \ i 1 j=1 B j for i {,, k} So {B 1,, B k } is a partitio of B 1 We have to be sure that at ay time i the partitioig N(A i ) to be sure B i exists For A 1 this is obvious For i > 1 suppose that N(A i ) = B i = ad B i 1 The σ Ai 1 A i = Ai 1 Ai B = Ai 1 + Ai i 1 B i B i 1 > Ai 1 B = σ i 1 S i 1 which is a cotradictio to the costructio of the partitioig of G because S i 1 S i should have bee chose i the partitio istead of S i 1 Also otice that σ Ai > σ Ai+1 Suppose for the cotrary, that σ Ai σ Ai+1, so Ai B i Ai+1 B But the, by Propositio 1, σ i+1 A i = Ai B i Ai + Ai+1 B i + B i+1 = A i A i+1 B = σ i B i+1 A i A i+1 This agai is a cotradictio to the costructio of the partitioig of G because ow A i A i+1 should have bee chose i the partitio istead of A i So ow we have a partitio of the graph G ito subgraphs G i (A i, B i ; E i ) ad σ i = Ai B where σ i i > σ j for i < j Let i = A i + B i ad let λ i be the fractio of the vertices of G i that are i A i So A i = λ i i ad B i = (1 λ i ) i Now suppose there is a fractio ρ i of the vertices of A i i a maximal losig coalitio Let A i be the set of these vertices, so A i = ρ iλ i i By defiitio of the partitio of A its clear A that i N(A i ) Ai N(A i) = Ai B Ideed, A i i N(A i ) > Ai N(A i) would cotradict the choice of A i (sice the maximality of σ i ) So N(A i ) A i A i B i = ρ i B i This meas that ρ i λ i i vertices i A i cover at least ρ i of the vertices i B i So, a maximal losig coalitio that cotais ρ i λ i i vertices i A i ca cotai at most (1 ρ i ) B i of the vertices of B i Now chose w a = 1 λ i for a A i ad w b = λ i for b B i The the weight of a maximum weighted losig coalitio L max will be bouded by w(l max ) w a ρ i λ i i + w b (1 ρ i )(1 λ i ) i = λ i (1 λ i ) i This yields, as show already 9
i the biregular case i Lemma 63, that α(g i ) 1 i i The rest of the proof follows from Lemma ad Lemma 3 Because all G i are α( i )-roughly weighted, the uio i G i is α()-roughly weighted by Lemma Now we have to add edges e(a, b) E(G) with a A i ad b B j (i j) to add the coalitios that are ot i ay G i Observe that for i < j o edges e(a, b) exist with a A i ad b B j due to the defiitio of the partitio of G I the graph G(A, B; E) edges e(a, b) ca exist with a A j ad b B i for i < j By defiitio of the costructio of the partitio of G we kow that for i < j it holds that Ai So by Propositio we kow B i > Aj B j λ i = Ai A > Aj i + B i A = λ j + B j j ad thus w(e) = w a +w b = (1 λ j )+λ i > 1 So the wiig coalitio represeted by e suffices the miimum weight demaded by the defiitio of α()-roughly weighted games ad ca be added to the game i G i by Lemma 3 without violatig the boud This ca be doe for all miimal wiig coalitios e(u, v) with u A j ad v B i for i < j 7 Provig the cojecture Now we come to the prove of the cojecture of Freixas ad Kurz for games with two-player miimal wiig coalitios Assume the game G(N, W) is such a matchig game Note that all players that are ot part of a miimal wiig coalitio are dummies So we assume all players are i at least oe miimal wiig coalitio We costruct a graph G, where the vertices are the players, ad edges are the miimal wiig pairs As see before the weighted losig coalitio ca cotai at most half of the players of a maximum matchig plus all players that are o part of this maximum matchig To ivestigate the structure of the matchig we use the followig decompositio of graphs Defiitio 71 The Gallai-Edmods decompositio of a graph G(V, E) is the partitio D A C of V (G) give by D = {v V (G) some maximum matchig i G fails to match v} A = {u V (G) D u is adjacet to a vertex i D} C = V (G) D A Whe a graph is decomposed accordig to the Gallai-Edmods decompositio we kow some special properties of the sets i the decompositio: Theorem 71 (Gallai-Edmods Structure Theorem) Let A, C, D be the sets i the Gallai-Edmods Decompositio of a graph G Let D 1,, D k be the compoets of G[D] If M is a maximum matchig i G, the the followig properties hold: a M covers C ad matches A ito distict compoets of G[D] b Each D i is factor critical ad has a ear-perfect matchig i M c If S A, the N(S) itersects at least S + 1 of D 1,, D k 10
C A D 1 D D 3 D D Figure 1: Gallai-Edmods Decompositio of graph represetig a arbitrary matchig game Thick lies idicate a maximum matchig Proof See [6] So ow let the graph G(V, E) represet the matchig game G(N, W) with = N The miimal wiig coalitios may itersect, but wolg we may assume that all players are i at least oe miimal wiig coalitio We will show that we ca chose a weight fuctio w : P R 0 such that α(g) 1 I order to get a upper boud o the weight of a maximum weighted losig coalitio with respect to a weight fuctio, we fix a maximum matchig M ad decompose G accordig to the Gallai-Edmods structure All compoets i C i C cotai a perfect matchig, so by Lemma 61 they are α()-roughly weighted for = C i by givig all players a weight of 1/ By Lemma it s clear that C is α()-roughly weighted for = C I order to prove that A D is α()-roughly weighted, we will decompose this game ito three parts The first is a bipartite game, cosistig of the players i A, the sigletos i D ad the players of D that are matched ito A by M The secod part is the game cosistig of the odd compoets D i without the matched players ad the third part are the odd compoets that are ot coected to A via the matchig M Notice that due to the Gallai-Edmods structure theorem we kow that i the bipartite graph the umber of players i A is less tha or equal to the umber of matched players i D plus the umber of sigletos This will give a weight distributio such that the players i A have a weight greater tha or equal to 1/ ad the players i D a weight less tha or equal to 1/ A weight less tha 1/ is good for the sigletos, but for the matched players i the odd compoets we ca ot allow this Because these compoets are factor critical, we would like a weight of 1/ for all players i those compoets We will show that this is possible What remais is to add coalitios / edges that are ot i ay of the compoets This are coalitios i A, coalitios that coect players i C with a player i A ad coalitios that coect a player i A with a umatched player i D But these coalitios ca be formed, because the weight of those players is at least 11
1/ So, after this outlie of the proof of the cojecture of Freixas ad Kurz, we state the cojecture formally ad proof it Theorem 7 For ay simple game G(N, W) with = N ad X = for all X W mi, we have α(g) 1 Proof Let M be a maximum matchig i the graph G Ay maximal losig coalitio i G ca cotai at most half of the vertices i M plus the umatched vertices Now decompose the graph represetatio of the game accordig to the Gallai-Edmods decompositio (see Figure 1 for a example of such a game) We set w c = 1/ for all c C, so the games represeted by these compoets respect the boud due to Lemma Also set w d = 1/ for all d D i with D i 3 Now chose i each odd compoet D i with D i 3 the vertex d i M D i D is the uio of all these vertices plus the sigletos i D Note that due to properties 1 ad 3 of the Gallai-Edmods structure theorem there ca be compoets D i for which such a vertex d i does ot exist We costruct the bigraph G (A, D ; E ), represetig the game G We oly add the edges / coalitios betwee vertices i A ad D to get a bipartite graph Notice that i this bipartite graph A D ad that A is matched by M We apply Lemma 65 to show that G is α()-roughly weighted with = V (G ) ad to achieve a weight distributio w a = D A + D 1 for a A ad w d = A A + D 1 for d D Now the miimal wiig coalitios represeted by a edge e(a 1, a ) with a 1, a A will have a weight w(e) 1, so by Lemma 3 we ca add the desired coalitios i A to the game Also coalitios represeted by e(a, c) with a A, c C ca be added to the game without violatig the boud because w(a, c) = w a + w c 1 However, the weight distributio i the bipartite graph G chaged the weights of the vertices d i D ito a weight less tha 1 while we would like to have all the players i the odd compoets D i with D i 3 to have a weight of at least 1 to fulfill the demad o the miimal weight of wiig coalitios (i D i) We claim that we ca give those players that have a weight less tha 1 a weight 1 without makig the maximum weighted losig coalitio L max i the total game too heavy We will prove this claim iductively by icreasig the weights of all those players d i D oe by oe Cosider a arbitrary D i for which D i M Let Di = D i\{d i } ad cosider the game G = G Di Notice that d i G ad d i / Di The maximum umber of vertices i Di that ca be chose at the same time i ay maximal losig coalitio is mi with m i = V (D i ) 1 = V (Di ) Now let w d i = 1 ad cosider two cases I the first case, whe d i / L max, we have w(l max G ) α(g ) 1 with = V (G ) ad w(l max D ) = 1 i mi So both games G ad Di respect the boud Sice G Di = we kow by Lemma that the game represeted by G Di is α()-roughly weighted with = V (G Di ) I the secod case, whe d i L max, we see that w(l max G ) 1 + 1 with = V (G ) because we icreased the weight of d i from w di 1 to w d i = 1 1
1 3 5 6 7 8 9 Figure : Bipartite graph G(A, B; E) represetig a two player miimal wiig coalitio game with A = {1,, 3} ad B = {, 5, 6, 7, 8, 9} ad a collectio of miimal wiig coalitios W = {{1, }, {1, 5}, {, 5}, {, 6}, {3, 6}, {3, 7}, {3, 8}, {3, 9}} The umber of players i D i that ca be chose i L max is mi So because d i is chose i L max ad d i D i = Di {d i} we ca chose from Di mi at most 1 players i L max Now w(l max ) w(l max G )+w(lmax D ) 1 i + 1 + ( 1 mi 1) = 1 + 1 m i for = V (G ) Sice m i is eve, this yields w(l max ) 1 + mi So G respects the boud 1 (+mi) +m i Now we add all the miimal wiig coalitios {d i, d i } with d i D i to costruct the odd compoet D i This ca be doe because w(d i ) = w(d i ) = 1/ so we ca apply Lemma 3 We repeat the iductive step util all matched D i are i G We also add the compoets D i for which D i M = to the game This ca be doe due to Lemma 6 ad Lemma i order Now we apply Lemma 3 agai to add the remaiig miimal wiig coalitios {a, d} with a A ad d D \ D, which ca be doe because w a 1/ ad w d = 1/ so w a +w d 1 Fially, agai by Lemma 3, we ca add the miimal wiig coalitios i A because w a 1/ for all players a A 8 Discussio ad coclusio The decompositio of a matchig game that we used i the proof of the cojecture, guaratees that the maximum weighted losig coalitio has a weight below α() = 1 However, the weight of this maximum weighted losig coalitio is ot the miimum possible weight Let G(N, W) be a simple game with N = [9] ad the collectio of miimal wiig coalitios W = {{1, }, {1, 5}, {, 5}, {, 6}, {3, 6}, {3, 7}, {3, 8}, {3, 9}} This game ca be represeted by a bipartite graph G(A, B; E) with A = {1,, 3} ad B = {, 5, 6, 7, 8, 9}, see Figure Notice that A ca be matched ito B, so we ca apply Lemma 65 The costructio of a partitio accordig to Lemma 65 will yield A 1 = {1, }, B 1 = {, 5, 6} with w 1 = w = 3/5 ad w = w 5 = w 6 = /5 ad A = {3} ad B = {7, 8, 9} with w 3 = 3/ ad w 7 = w 8 = w 9 = 1/ Now A ad B are maximum weighted 13
losig coalitios with weight w(a) = w(b) = 39/0 This weight respects the boud α(8) = However, if we choose weights w 1 = w = w = w 5 = w 6 = 1/, w 3 = 7/8 ad w 7 = w 8 = w 9 = 1/8 still A ad B are maximum weighted losig coalitios, but w(a) = w(b) = 15/8 which is slightly better tha the total weight for a maximum weighted losig coalitio of 39/0 that we got as the result of the partitioig accordig to Lemma 65 So a decompositio of the game like i the proof of the cojecture of Freixas ad Kurz gives a maximum weighted losig coalitio that respects the boud but does t yield a weight distributio that makes the maximum weighted losig coalitio as light as possible I this study we preseted a way of proofig the cojecture of Freixas ad Kurz for games G(N, W) that have miimal wiig coalitio which all have order A proof for the case where all miimal wiig coalitios have a order of at least is easy Givig all players i a game a weight of 1 will make w(s) 1 for all S W Now ay maximum weighted losig coalitio L L ca cotai at most 1 players from the grad coalitio N, so for all S L it sclear that w(s) 1 ( 1) For eve this yields 1 1 ( 1) = < = 1 For odd we see that 1 < ( 1) < 1 = 1 A atural follow-up would be to ivestigate games with miimal wiig coalitios which all have order 3 ad to ivestigate games with miimal wiig coalitios of various orders Moreover, we suppose that the critical threshold o games with players is bad for games with small miimal wiig coalitios ad the boud is oly reached i games with miimal coalitios of order So aother iterestig questio is how the critical threshold depeds o the size of the smallest miimal wiig coalitio i relatio to the largest losig coalitio This would tell us more about the cost of stability i coalitio games tha just the value of α() Refereces [1] J Freixas ad S Kurz O α-roughly weighted games Iteratioal Joural of Game Theory, 3(3):659 69, 01 [] T Gvozdeva, L A Hemaspaadra, ad A Sliko Three hierarchies of simple games parameterized by resource parameters Iteratioal Joural of Game Theory, (1):1 17, 013 [3] T Gvozdeva ad A Sliko Weighted ad roughly weighted simple games Mathematical Social Scieces, 61(1):0 30, 011 [] A Taylor ad W Zwicker A characterizatio of weighted votig Proceedigs of the America Mathematical Society, 115():1089 109, 199 [5] A D Taylor ad W S Zwicker Simple Games: Desirability Relatios, Tradig, Pseudoweightigs Priceto Uiversity Press, 1999 [6] D B West A short proof of the Berge Tutte formula ad the Gallai Edmods structure theorem Europea Joural of Combiatorics, 3(5):67 676, 011 1