We introduce two new monotonicity properties for core concepts: single-valued solution concepts that always select a core allocation whenever the game is balanced (has a nonempty core). We present one result of impossibility for one of the properties and we pose several open questions for the second property. The open questions arise because the most important core concepts (the nucleolus and the per capita nucleolus) do not satisfy the property even in the class of convex games.J. Arin acknowledges financial support
from Project 9/UPV00031.321-15352/2003 of the University of the Basque Country, Projects BEC2003-08182 and SEJ-2006-05455 of the Ministry of Education and Science of Spain and Project GIC07/146-IT-377-07 of the Basque Goverment

Arin Aguirre, Francisco Javier

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        MONOTONIC CORE SOLUTIONS: BEYOND YOUNG’S THEOREM  by  Javier Arin   2010  Working Paper Series: IL. 44/10              Departamento de Fundamentos del Análisis Económico I Ekonomi Analisiaren Oinarriak I Saila    University of the Basque Country Monotonic core solutions: Beyond Young´stheoremJ. ArinJuly 26, 2010AbstractWe introduce two new monotonicity properties for core concepts: single-valued solution concepts that always select a core allocation whenever thegame is balanced (has a nonempty core). We present one result of impos-sibility for one of the properties and we pose several open questions for thesecond property. The open questions arise because the most importantcore concepts (the nucleolus and the per capita nucleolus) do not satisfythe property even in the class of convex games.Keywords: Monotonicity, Core, TU games, nucleolus per capitaDpto. Ftos. A. Económico I, Basque Country University, L. Agirre 83, 48015 Bil-bao, Spain. Email: franciscojavier.arin@bs.ehu.es. J. Arin acknowledges nancial supportfrom Project 9/UPV00031.321-15352/2003 of the University of the Basque Country, ProjectsBEC2003-08182 and SEJ-2006-05455 of the Ministry of Education and Science of Spain andProject GIC07/146-IT-377-07 of the Basque Goverment. Likewise,11 IntroductionYoung (1985) formulates an impossibility result for the problem of nding coreconcepts satisfying monotonicity1 in the domain of balanced TU games. Thisresult opens up two paths of research: One is to restrict the search for monotoniccore concepts to certain classes of games2 . The other, is to dene new propertiesof monotonicity (weaker than the one formulated by Young) and to deal withthe class of all TU balanced games. This paper discusses this second path.In considering new monotonicity properties we rst analyze the compatibil-ity between monotonicity and core belonging. Only when such a compatibilityexists do we require a core concept to be monotonic. Following this approach weintroduce two new properties: strong core monotonicity and core monotonicity.For the rst property we provide a result of impossibility. For the second prop-erty we pose an open question once we have checked that core concepts such asnucleolus and per capita nucleolus do not satisfy the property.2 Preliminaries2.1 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 are calledplayers and the real-valued function v the characteristic function of the game.Any subset S of N is called a coalition. The number of players in S is denotedby jSj. Given S  N we denote by NnS the set of players of N that are not inS: Let N be a set of players and  0 a class of games. If there is no confusion, wewrite v 2  0 instead of (N; v) 2  0: A distribution of v(N) among the playersis a real-valued vector x 2 RN where xi is the payo¤ assigned by x to player i.A distribution satisfying xi  v(i) for all i 2 N is called an imputation and theset of imputations is denoted by I(v): We denotePi2Sxi by x(S). The core of agame is the set of imputations that cannot be blocked by any coalition, i.e.C(v) = fx 2 I(v) : x(S)  v(S) for all S  Ng :1See Section 2 and 3 for formal denitions of core concept and monotonicity.2This also can be seen as a weakening of the core requirements in the sense that we onlysearch in games that have a special core.2A game with a non-empty core is called a balanced game. Player i is a vetoplayer if v(S) = 0 for all S where player i is not present. A balanced game withat least one veto player is called a veto balanced game. We denote by  B theclass of balanced games and by  BV the class of veto balanced games.A solution  in a class of games  0 is a correspondence that associates aset (N; v) in RN with every 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. Thesolution is single-valued if the set contains a unique element for each game inthe class.Given x 2 RN the excess of a coalition S with respect to x in a game vis dened as e(S; x) := v(S)   x(S): Let (x) be the vector of all excesses atx arranged in non-increasing order. The lexicographic order L between twovectors x and y is dened by x L y if there exists an index k such that xl = ylfor all l < k and xk < yk and the weak lexicographic order Lby x L y ifx L y or x = y:Schmeidler (1969) introduced the nucleolus of a game v; denoted by (v); asthe unique imputation that lexicographically minimizes the vector of non in-creasingly ordered excesses over the set of imputations. In formula:f(N; v)g = fx 2 I(N; v) j(x) L (y) for all y 2 I(N; v)g :For any game v with a non-empty imputation set, the nucleolus is a single-valued solution, is contained in the kernel and lies in the core provided that thecore is non-empty.The per capita nucleolus is dened analogously by using the concept of percapita excess instead of excess. Given S and x the per capita excess of S at x isepc(S; x) :=v(S)  x(S)jSjIn this paper, we study solution concepts that select precisely one core allo-cation for each balanced game. We call such concepts core concepts.2.2 On monotonicity propertiesWe present some monotonicity properties for single-valued solution concepts.3Let v; w 2  0, such that for all T containing player i; v(T )  w(T ); and forall S  Nn fig v(S) = w(S) :We say that game w is monotonic with respect togame v and player i.Let  be a single-valued solution in a class of games  0.We say that solution  satises monotonicity if for all v; w 2  0, suchthat w is monotonic with respect to game v and player i then i(w)  i(v):We say that solution  satises N-monotonicity (Meggido, 1974) if for allv; w 2  0, such that for all S 6= N; v(S) = w(S) and v(N) < w(N); then for alli 2 N; i(v)  i(w):We say that solution  satises strong N -monotonicity if for all v; w 2  0,such that for all S 6= N; v(S) = w(S) and v(N) < w(N); then for all i; j 2 N;i(w)  i(v) = j(w)  j(v)  0:Clearly, monotonicity implies N-monotonicity.Young (1985) shows that in the class of balanced games there is no coreconcept that satises monotonicity. The result is presented by means of twobalanced games with only one core allocation each. Therefore in these games acore concept must choose the allocation contained in the core. To some extent,we can say that in this case the core restrictions do not allow a core conceptto be monotonic. This observation motivates the following note. We investi-gate new monotonicity properties that require for some compatibility betweenmonotonicity and core belonging.In the class of convex games the Shapley value is a core concept that satisesmonotonicity. Arin and Feltkamp (2007) show that in the class of veto balancedgames there are several core concepts that satisfy monotonicity. It is also shownthat the nucleolus and the per capita nucleolus do not satisfy the property.3 On core monotonicityWe introduce new core monotonicity properties3 . The idea of the new proper-ties is that a core concept should be monotonic whenever core restrictions arecompatible with this monotonicity.Denition 1 Let v; w 2  0, such that game w is monotonic with respect togame v and player i. We say that game w has a monotonic core with respect to3We dene the properties for core concepts.4game v and player i if there exist x 2 C(w) and z 2 C(v) such that xi  zi:The existence of a monotonic core is certainly a necessary condition forthe monotonicity of a core concept. If this requirement is not satised anycore concept must violate monotonicity. In the example given by Young therequirement is not satised.Following this observation we dene strong core monotonicity.Denition 2 Strong core monotonicity: A core concept  in  0 is stronglycore monotonic if for all v; w 2  0 such that:1.- game w is monotonic with respect to game v and player i2.- game w has a monotonic core with respect to game v and player iit holds that; i(w)  i(v):Note that if a core concept is monotonic then is strongly core monotonic.Theorem 3 In the class of balanced games there is no core concept that satisesstrong core monotonicity .Proof. Let N = f1; 2; 3; 4g a set of players and consider the following 4-person balanced games:v(S) =8>><>>:12 if S 2f1; 3g ; f1; 4g ; f2; 3g ; f2; 4gf1; 2; 3g ; f1; 2; 4g f1; 3:4g ; f2; 3; 4g24 if S = N0 otherwiseandwi(S) =v(S) if S 6= Nn fig24 if S = Nn figwhere i 2 N: The games wi are monotonic with respect to to game v and playerl; l 6= i: They also have a monotonic core with respect to game v and player l;l 6= i: For the games w1 and w2 we need to consider the only core allocation ofthe two games, z1 = (0; 0; 12; 12) and for games w3 and w4 we need to considerthe allocation z3 = (12; 12; 0; 0):Let  a core concept satisfying strong core monotonicity. By applying strongcore monotonicity between games w1 and v we conclude that (v) = z1: Byapplying strong core monotonicity between games w3 and v we conclude that(v) = z3:In this example, core restrictions do not allow a core concept to behavemonotonically from game v to games w1 and w3: We try to nd a denition of5core monotonicity that requires monotonic behavior of a core concept whenevercore restrictions allow such behavior.Let N be a set of players and  0 a class of games. By a pair (w; i(w)) werefer to a game w 2  0 and a player i 2 N associated to game w:Denition 4 Let N be a set of players and v 2  0. We say that game v hasan extendable monotonic core if there exists z 2 C(v) such that for any pair(w; i(w)) where w is a monotonic game with respect to v and i(w) we can ndx 2 C(w) such that xi(w)  zi(w):In the class of convex games any game has an extendable monotonic coresince, in this class, the Shapley value is a monotonic core concept.In the above proof it is immediate that game v does not have an extendablemonotonic core in the class of all balanced games. The following propertyincorporates this new aspect of the compatibility between monotonicity andcore belonging.Denition 5 Extendable core monotonicity: A core concept  on  0 isextendable core monotonic if for all v; w 2  0 such that:1.- game w is monotonic with respect to game v and player i2.- game w has a monotonic core with respect to game v and player i3.- i(w) < i(v)then game v does not have an extendable monotonic core.This denition is not completely satisfactory as the following example illus-trates4 .Let N = f1; 2; 3:4g a set of players and consider the following 4-personbalanced games:v(S) =8>><>>:12 if S 2f1; 3g ; f1; 4g ; f2; 3g ; f2; 4gf1; 2; 3g ; f1; 2; 4g f1; 3:4g ; f2; 3; 4g24 if S = N0 otherwiseandq(S) =v(S) if S 6= N28 if S = N.It seems clear that a core concept  such that i(v) > i(q) for a player icannot be considered as a monotonic core concept. But since game v has no4The property does not imply N -monotonicity. That is, a solution can be monotonic inthe sense of Denition 5 but violates N -monotonicity.6extendable monotonic core according to the denition above, we cannot claimthat this solution  is not extendable core monotonic using the above facts.In this example, it does not seem reasonable to argue that game v hasno extendable monotonic core in order to justify non-monotonic behavior of acore concept while moving from game v to game q: For such an argument, itseems necessary to show that game q is necessary to imply that game v has noextendable monotonic core. This idea is formalized in the property we call coremonotonicity.Denition 6 Let v 2  0. We say that game v has an extendable monotoniccore with respect to the list of pairs (wi; j(wi))i=1;:::;k if:1.- for any pair (wi; j(wi)) it holds that wi 2  0 and wi is a monotonic gamewith respect to v and j(wi):2.- there exists z 2 C(v) such that for any pair (wi; j(wi)) we can ndx 2 C(wi) such that xj(wi)  zj(wi):Note that, in the proof of Theorem 3, game v has no extendable monotoniccore with respect to the list of pairs: (w1; 2); (w2; 1); (w4; 3) and (w3; 4):Denition 7 Core monotonicity: A core concept  in  0 is core monotonicif for all v; w 2  0 such that:1.- game w is monotonic with respect to game v and player i2.- game w has a monotonic core with respect to game v and player i3.- i(w) < i(v)then there exists a list of pairs (wl; j(wl))l=1;:::;k, such that1.- game v has an extendable monotonic core for the list of pairs (wl; j(wl))l=1;:::;k.2,.- game v has no extendable monotonic core for the list of pairs (w; i);(wl; j(wl))l=1;:::;k.The property means that any violation of the requirement of monotonicityshould be justied. If such a violation occurs between two games, from gamev to game w, then game w should be a member of a minimal list of games forwhich game v has no extendable monotonic core. Minimality implies that forany other subset of the list of games, game v has an extendable monotonic core.Remark 8 If a core concept is strongly core monotonic then is core monotonic.7In the class of balanced games there exist core concepts that satisfy N-monotonicity such as the per capita nucleolus. Therefore core restrictions and N-monotonicity are compatible. The following proposition relates core monotonic-ity and N-monotonicity.Proposition 9 Core monotonicity implies N-monotonicity.Proof. Assume that there exists a core concept  that is core monotonicbut not N-monotonic. Then there exist two balanced games (N; v) and (N;w)such that:1.- w(N) > v(N) and w(S) = v(S) for any other coalition S:2.- game w has a monotonic core with respect to game v and player i3.- i(w) < i(v)4.- there exists a list of pairs ((N; ql); j(ql))l=1;:::;k, such that:a.- game v has an extendable monotonic core for the list of pairs ((N; ql); j(ql))l=1;:::;k.b.- game v has not an extendable monotonic core for the list of pairs ((N;w):i);((N; ql); j(ql))l=1;:::;k.Let z be a core allocation in game (N; v) that allows game v to have anextendable core with respect to the list of pairs ((N; ql); j(ql))l=1;:::;k: Addingthe pair ((N;w); i(w)) to the list does not change this fact since the allocationx = z + (w(N) v(N)jN j ; :::;w(N) v(N)jN j ) is a core allocation in game w and satisesxi  zi for all i 2 N:It is well-known that in the class of balanced games, convex games and vetobalanced games the nucleolus does not satisfy N-monotonicity. (See Hokari(2000) for the case of convex games and Arin and Feltkamp (2005) for the caseof veto balanced games.)Corollary 10 In the class of balanced games, convex games and veto balancedgames the nucleolus is not core monotonic.The per capita nucleolus is a core concept that satises N-monotonicity inthe class of all balanced games, which is why some authors consider it a goodcandidate when seeking to select monotonic core allocations. The following re-sult shows that the per capita nucleolus violates core monotonicity and therefore,even if it does satisfy N-monotonicity, it can hardly be seen as a monotoniccore concept (at least in the class of all balanced games and in the subclass ofveto balanced games).8Proposition 11 In the class of balanced games the per capita nucleolus is notcore monotonic.Proof. Let N = f1; 2; 3:4:5; 6g a set of players and consider the following6-person veto balanced games:v(S) =8<: 4 if 1 2 S and jSj = 510 if S = N0 otherwiseandw(S) =8<: 8 if S = Nn f6g12 if S = Nv(S) otherwiseThe per capita nucleolus selects the allocation (5; 1; 1; 1; 1; 1) in the rst gameand in the second game selects ( 8621 ;3821 ;3821 ;3821 ;3821 ;1421 ): Therefore, the per capitanucleolus is not monotonic since player 1 receives a lower payo¤ in the secondgame. The pair (w; 1) is not necessary in a list of pairs in order to make gamev a game without an extendable monotonic core with respect to these pairs. Ifgiven a list of pairs we have that game v has an extendable monotonic core withrespect to that list, adding the pair (w; 1) to the list does not change this factsince the allocation x = (12; 0; 0; 0; 0; 0) is a core allocation in the game w andsatises x1  z1 for all z 2 C(v):Note that the proof is constructed using veto balanced convex games.Corollary 12 In the class of veto balanced games and in the class of convexgames the per capita nucleolus is not core monotonic.4 Results and open questionsThe foregoing analysis suggests the following general question:Problem 13 Are there any core monotonic core concept in the class of balancedgames?It also suggests several other questions concerning classes of balanced games.In the class of convex games the Shapley value is a core concept and thereforea core monotonic core concept. But, in general, the Shapley value is not a coreconcept. This motivates the following question: if there are core monotonic coreconcepts in the class of balanced games do those core concepts coincide with9the Shapley value in the class of convex games? The answer is given by thefollowing exampleLet N = f1; 2; 3:4g a set of players and consider the following 4-personbalanced games:v(S) =4 if S 2 ff1; 2; 3g ; f1; 2; 4g f1; 3:4g ; Ng0 otherwiseandw(S) =v(S) if S 6= N8 if S = NThe game w is convex and therefore its Shapley value, (3; 53 ;53 ;53 ) is a coreallocation. The only core allocation of the game (N; v) is (4; 0; 0; 0): Note thatgame w is monotonic with respect to game v and any player in N . Therefore amonotonic core concept does not select an allocation where player 1 receives apayo¤ lower than 4 and therefore such a core concept will not coincide with theShapley value of the game.In the class of veto balanced games there are core monotonic core concepts(see Arin and Feltkamp (2007)). The solution concepts analyzed in this paperare not dened in the class of balanced games and therefore a similar questioncan be asked. That is, given a core monotonic concept dened on the class ofveto balanced games, say ; are there any core monotonic core concepts in theclass of balanced games coinciding with  in the class of veto balanced games?References[1] Arin J and Feltkamp V (2005) Monotonicity properties of the nucleolus onthe domain of veto balanced games. TOP 13, 2:331-342[2] Arin J and V. Feltkamp V (2007) On monotonic core allocations for coali-tional games with veto players. Working paper IL2807[3] Hokari T, (2000), The nucleolus is not aggregate-monotonic on the domainof convex games. Int J of Game Theory 29:133-137[4] Meggido N (1974) On the monotonicity of the bargaining set, the kernel andthe nucleolus of a game. SIAM J of Applied Mathematics 27:355-358[5] Schmeidler D (1969) The nucleolus of a characteristic function game. SIAMJ of Applied Mathematics 17:1163-117010[6] Shapley LS (1953) A Value for n-Person Games in Contributions to theGame Theory, Vol. II Kuhn and Tucker editors[7] Young HP (1985) Monotonic solutions of cooperative games. Int J of GameTheory 14:65-7211

Monotonic core solutions: Beyond Young's theorem

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Monotonic core solutions: Beyond Young's theorem

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