In 1972, Maschler, Peleg and Shapley proved that in the class of convex the nucleolus and the kernel coincide. The only aim of this note is to provide a shorter, alternative proof of this result.Spanish Ministerio de Ciencia e Innovación under projects SEJ2006-05455 AND ECO2009-11213, co-funded by ERDF, and by Basque Government funding for Grupo Consolidado GIC07/146-IT-377-07

Arin Aguirre, Francisco Javier

Katsev, Ilya

English

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        THE COINCIDENCE OF THE KERNEL AND NUCLEOLUS OF A CONVEX GAME:  AN ALTERNATIVE PROOF  by  J. Arín and I. Katsev  2013  Working Paper Series: IL. 67/13              Departamento de Fundamentos del Análisis Económico I Ekonomi Analisiaren Oinarriak I Saila    University of the Basque Country The coincidence of the kernel and nucleolusof a convex game: An alternative proofJ. Arinand I. KatsevyJanuary 24, 2013AbstractIn 1972, Maschler, Peleg and Shapley proved that in the class ofconvex games the nucleolus and the kernel coincide. The only aim ofthis note is to provide a shorter, alternative proof of this result.Keywords: nucleolus, kernel, convex gamesKeywords: nucleolus, kernel, convex games.JEL classication: C71, C72.Dpto. Ftos. A. Económico I, University of the Basque Country, L. Agirre 83, 48015Bilbao, Spain. Email: jeparagj@bs.ehu.es. J. Arin acknowledges the support of theSpanish Ministerio de Ciencia e Innovación under projects SEJ2006-05455 and ECO2009-11213, co-funded by ERDF, and by Basque Government funding for Grupo ConsolidadoGIC07/146-IT-377-07.ySt. Petersburg Institute for Economics and Mathematics. Email: katsev@yandex.ru11 IntroductionCoalitional game theory deals with the problem of how to divide the outcomeobtained by players that may coordinate their actions and form coalitions.Assuming transferable utility (TU games) the problem is solved by proposingreal-valued vectors where each component is the payo¤of a player. One of themost important solution concepts for TU games is the nucleolus (Schmeidler,1969). The nucleolus of a game belongs to the kernel (Davis and Maschler,1965) of the game and in some cases they coincide. Convex games weredened by Shapley (1971) and the coincidence of the nucleolus with the kernel(and the prekernel) for this class of games was established by Maschler, Pelegand Shapley in 1972. The proof of that result is quite long and complex. Inthis note we provide an alternative.2 TU gamesA cooperative n-person game in characteristic function form is a pair (N; v),where N is a nite set of n elements and v : 2N ! R is a real-valued functionin the family 2N of all subsets of N with v(;) = 0: Elements of N arecalled players and the real valued function v the characteristic function ofthe game. Any subset S of N is called a coalition. Singletons are coalitionsthat contain only one player. The number of players in S is denoted by jSj.Given S  N we denote by NnS the set of players of N that are not inS. A distribution of v(N) among the players -an allocation- is a real-valuedvector x 2 RN where xi is the payo¤ assigned by x to player i. A distributionsatisfyingPi2Nxi = v(N) is called an e¢ cient allocation. The set of e¢ cientallocations is denoted by X(v):We denotePi2Sxi by x(S). The core of a gameis the set of allocations that cannot be blocked by any coalition, i.e.C(N; v) = fx 2 X(N; v) : x(S)  v(S) for all S  Ng :Games with nonempty core are called balanced games. We say that a game(N; v) is convex if v(S)+v(T )  v(S[T )+v(S\T ) for all S; T  N: Convexgames are balanced.2A solution ' on a class of games  0 is a correspondence that associates aset '(N; v) in RNwith each game (N; v) in  0 such that x(N)  v(N) for allx 2 '(N; v). This solution is e¢ cient if this inequality holds with equality.The solution is single-valued if the set contains a single allocation for eachgame in the class.We say that the vector x weakly lexicographically dominates the vectory (denoted by x L y) if eitherx =y or there exists k such thatxi =yifor all i 2 f1; 2; :::; k   1g and xk >yk wherex andy are the vectors withthe same components as the vectorsx,y, but rearranged in a non decreasingorder (i > j ) xi xj).Given x 2 RN the satisfaction of a coalition S with respect to x in agame v is dened as e(S; x) := x(S)   v(S): Let (x) be the vector of allsatisfactions at x arranged in non decreasing order. Schmeidler (1969) intro-duced the prenucleolus of a game (N; v); denoted by PN(v); as the uniqueallocation that lexicographically maximizes the vector of non decreasinglyordered satisfactions over the set of allocations. In formula:PN(N; v) = fx 2 X(N; v) j(x) L (y) for all y 2 X(N; v)g :Analogously, the nucleolus (Schmeidler, 1969) of a game (N; v), denoted byN(v), is the unique allocation that lexicographically maximizes the vectorof non decreasingly ordered satisfactions over the set of imputations. Animputation is an allocation that satises individual rationality, i.e each playeri receives at least his/her individual worth v(fig):Given a TU game (N; v) and an allocation x we denote by B(x) the setof coalitions with minimal satisfaction according to x:Given a TU game (N; v) and an allocation x 2 X(N; v) the complaint ofplayer i against player j is dened as follows:sij(x) = minS:i2S;j 62SF (S; x):The prekernel of a TU game (N; v) is:PK(N; v) = fx 2 X(N; v) : sij(x) = sji(x) for all i 6= jg3For any game (N; v) the prenucleolus is single-valued, is contained in theprekernel and lies in the core whenever the core is nonempty. In general, theprekernel and the prenucleolus do not coincide1.The Davis-Maschler reduced game property is dened as follows:Let (N; v) be a game, T  N , T 6= N; ; and x 2 RT . Then the Davis-Maschler reduced game with respect toNnT and x is the game (NnT; vx) wherevNnTx (S) :=8>>><>>>:0 if S = ;v(N) Pi2Txi if S = NnTmaxRTv(S [R) Pi2Rxifor all S  NnTThe prekernel satises the Davis-Maschler reduced game property: Let (N; v)be a game and let S be a nonempty coalition. If x 2 PK(N; v) then xS 2PK(S; vx):3 Antipartitions and partitionsThe notion of antipartitions (Arin and Inarra, 1998) plays a central role inthe main result of this note.A collection of sets C =n_S : S  Nois called a partition if [S2CS = Nand for any S; T 2 C it holds that S \ T = ?:A collection of sets C =n_S : S  Nois called an antipartition if thecollection of sets fNnS : S 2 Cg is a partition of N . For any game (N; v) thesatisfaction of an antipartition C is dened byF (C) :=v(N) PS2C1jCj 1v(S)jCj :1Consider a 4-person game (N; v) dened as follows:v(S) =8><>:4 if S = N0 if jSj = 1 or S 2 ff1; 2g ; f3; 4gg2 otherwise.It is immediately apparent that (2; 2; 0; 0) is an element of the prekernel (kernel) of thegame and that the prenucleolus (nucleolus) of (N; v) is (1; 1; 1; 1):4For any game (N; v) the satisfaction of a partition C is dened byF (C) :=v(N) PS2C v(S)jCj :Lemma 1 (Arin and Inarra, 1998) Let (N; v ) be a game and x be anallocation:a) if the collection of sets with minimal satisfaction B(x) contains anantipartition C then F (S; x) = F (C) for all S belonging to B(x).b) if the collection of sets with minimal satisfaction B(x) contains a par-tition C then F (S; x) = F (C) for all S belonging to B(x).Note that if the set of coalitions with minimal satisfaction with respectto an allocation contains an antipartition or a partition then the satisfactionof those coalitions depends only on the characteristic function of the game.Let (N; v) be a TU game and x be an allocation. Denote by B(x) the setof coalitions with minimal satisfaction at x:In our paper we also use Theorem 5.2. and Lemma 5.7 in Maschler et al.(1972), which we now restate without the proof2.Theorem 2 For any convex game (N; v), coalition S and core allocation x,the reduced game (S; vx) is convex.Lemma 3 Let (N; v) be a convex game, and consider a core allocation x,a coalition S in B(x) and the reduced game (S; vx): Then for all R  S itfollows thatvx(R) = maxQ2NnS(v(R [Q)  x(Q)) = max fv(R); v(R [ (NnS))  x(NnS)g :4 Convex games: The kernel and the nucle-olus coincideIn what follows we prove that in the class of convex games the prekernel andthe prenucleolus coincide.2The proofs of the statements are not long and complex.5A collection C of subsets of a set N is called near-ring ifA;B 2 C then either A [ B = N; or A \ B = ; or both A [ B 2C and A \B 2 C:It is immediately apparent that if (N; v) is convex and x is an allocationthen B(x) is a near-ring3.Theorem 4 Let (N; v) be a convex game and let x 2 PK(N; v). Then,either B(x) contains a partition or B(x) contains an antipartitionProof. Take any maximal coalition S in B(x)4 and dene K as thesubset of coalitions in B(x) which are contained in NnS and are maximalunder these conditions. Two cases may occur:a) K 6= ;Then S together with coalitions in K form a partition. By maximalitydi¤erent coalitions in K do not intersect. It su¢ ces to show that for everyi 2 NnS there is R 2 K satisfying i 2 R. Since x 2 PK(v) there is acoalition in B(x) containing i. Not every T of this type intersects S since wewould contradict separability5 observing that T  NnS by maximality of S.b) K = ;In this case every coalition in B(x) which intersectsNnS also intersects S,hence it contains NnS by maximality of S. Let F be the subset of maximalcoalitions of B(x) which contain NnS. By maximality the union of twodi¤erent coalitions in F coincides with N .To show that S together with all coalitions of F form an antipartition itremains to be demonstrated that every player i of S is a member of preciselyj F j  1 coalitions of F . By separability there is a coalition in F notcontaining i and the union of two di¤erent coalitions in F coincides with N .The alternative proof is based in the following facts:3The result arises by applying the denition of convexity to the sets in B(x):4There is no coalition T in B(x) such that S  T:5Separability implies that if there exists S 2 B(x) such that i 2 S and j =2 S then thereexists T 2 B(x) such that j 2 T and i =2 T: Clearly, if x is an allocation of the prekernelthis condition of separability must hold in B(x):61.- The prekernel and the prenucleolus satisfy the DM-RGP and theycoincide if the game has three or fewer players.2.- Let x be a prekernel allocation of a convex game (N; v). The set ofcoalitions with minimal satisfaction according to x contains a partition or anantipartition.3.- Let (N; v) be a convex game and let x; y 2 PK(N; v): There existsa partition or antipartition, say Q; that must be included in B(x) and inB(y). The satisfaction of coalitions in Q depends only on the characteristicfunction of the game. Therefore, for any S 2 Q it must be the case thatx(S) = y(S):4.- Let (N; v) be a convex game, S a nonempty coalition of N and x aprekernel allocation. In order to compute the DM-reduced game (S; vx) onlythe total payo¤ of coalition NnS; x(NnS); is needed.5.- Combining facts 3 and 4 it must be concluded that the reduced gameswith respect to a coalition in Q and allocations x and y are identical. Thesereduced games are convex and therefore the arguments can be repeated untilgames of three or fewer players are obtained.Theorem 5 Let (N; v) be a convex game. Then PK(N; v) = fPN(N; v)g :Proof. Let x = PN(N; v) and assume that there exists y; y 6= x; suchthat y 2 PK(N; v): First of all note that B(y) and B(x) contain either anantipartition or a partition, denoted by Q. Since the satisfaction of an an-tipartition or a partition only depends on the characteristic function B(x)and B(y) should contain the same antipartition or partition. Therefore, forany coalition S in the antipartition (partition) it must hold that y(S) = x(S)and consequently y(NnS) = x(NnS): Finally, DM-reduced games (S; vy) and(S; vx) are convex and identical (by applying Lemma 3). The prekernel satis-es DM-reduced game property. If (xi)i2S and (yi)i2S are di¤erent we repeatthe argument with the games (S; vy) and (S; vx): At some point the reducedgames should have three or fewer players and in this cases the prekernel con-tains only one allocation. Therefore;it must be concluded that for all S 2Qand for all i 2 S it holds that xi = yi: Since Q is either a partition or anantipartition [S2Q= N and therefore for all i 2 N it holds that xi = yi:7In the class of balanced games the prenucleolus and the nucleolus coincide.The nucleolus of a game belongs to the kernel of the game Therefore,Corollary 6 Let (N; v) be a convex game. Then PK(N; v) = K(N; v) =fN(N; v)g :References[1] Arin J and Inarra E (1998): A characterization of the nucleolus for convexgames. Games and Econ Behavior 23:12-24.[2] Davis M and Maschler M (1965). The kernel of a cooperative game. NavalResearch Logistics Quart. 12, 223-259.[3] Maschler M, Peleg, B and Shapley LS (1972) The kernel and the bargain-ing set for convex games. Int J of Game Theory 15: 73-93.[4] Peleg B (1986) On the reduced game property and its converse. Int J ofGame Theory 15:187-200[5] Schmeidler D (1969) The nucleolus of a characteristic function game.SIAM J on Applied Mathematics 17:1163-1170[6] Shapley LS (1971). Cores and convex games. Int. J. of Game Theory1, 11-268

The coincidence of the kernel and nucleolus of a convex game: an alternative proof

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The coincidence of the kernel and nucleolus of a convex game: an alternative proof

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