Characterization Sets for the Nucleolus in Balanced Games

Transcription

1 Characterization Sets for the Nucleolus in Balanced Games Tamás Solymosi 1,2 and Balázs Sziklai 2 1 Corvinus University of Budapest, Department of Operations Research and Actuarial Sciences, H-1093 Budapest F vám tér 8. tamas.solymosi@uni-corvinus.hu 2 Institute of Economics, Research Center for Economic and Regional Studies, Hungarian Academy of Sciences, H-1112, Budapest, Budaörsi út 45. sziklai.balazs@krtk.mta.hu Abstract. We provide a new modus operandi for the computation of the nucleolus in cooperative games with transferable utility. Using the concept of dual game we extend the theory of characterization sets. Dually essential and if the game is monotonic dually saturated coalitions determine both the core and the nucleolus whenever the core is non-empty. We show how these two sets are related to the existing characterization sets. In particular we prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself. Keywords: Cooperative game theory, Nucleolus, Characterization sets 1 Introduction The nucleolus, developed by Schmeidler (1969), soon became one of the most frequently applied solution concepts of cooperative game theory. Despite its good properties it lost some popularity in the last 20 or so years. Very much like the Shapley-value it suers from computational diculties. While the former has an explicit formula and various axiomatizations, the nucleolus practically can only be computed by a sequence of linear programs and its axiomatization is less straightforward. Determining the nucleolus is a notoriously hard problem, even N P-hard for various classes of games. While N P-hardness was proven for minimum cost spanning tree games (Faigle, Kern, and Kuipers, 1998), voting games (Elkind, Goldberg, Goldberg, and Wooldridge, 2009) and ow and linear production games (Deng, Fang, and Sun, 2009), it is still unknown whether the corresponding decision problem i.e. verifying whether an allocation is the nucleolus or not belongs to N P. In recent years several polynomial time algorithms were proposed to nd the nucleolus of important families of cooperative games, like standard tree, assignment, matching and bankruptcy games (Maschler, Potters, and Reijnierse,

2 2010; Solymosi and Raghavan, 1994; Kern and Paulusma, 2003; Aumann and Maschler, 1985). In addition Kuipers (1996) and Arin and Inarra (1998) developed methods to compute the nucleolus for convex games. The main breakthrough came from another direction. In their seminal paper Maschler, Peleg, and Shapley (1979) described the geometric properties of the nucleolus and devised a computational framework in the form of a sequence of linear programs. Although these LPs consist of exponentially many inequalities they can be solved eciently if one knows which constraints are redundant. Huberman (1980); Granot, Granot, and Zhu (1998); Reijnierse and Potters (1998) provided methods to identify coalitions that correspond to non-redundant constraints. Granot, Granot, and Zhu (1998) provided the most fruitful approach. They introduced the concept of characterization set which is a collection of coalitions that determines the nucleolus by itself. They proved that if the size of the characterization set is polynomially bounded in the number of players, then the nucleolus of the game can be computed in strongly polynomial time. A collection that characterizes the nucleolus in one game need not characterize it in another one. Thus we are interested in a property of the characteristic function that describes a characterization set in every game. Huberman (1980) was the rst to show that such a property exists. He introduced the concept of essential coalitions which are coalitions that have no weakly minorizing partition. Granot, Granot, and Zhu (1998) provided another collection that characterize the nucleolus in cost games with non-empty cores. Saturated coalitions contain all the players that can join the coalition without imposing extra cost. We introduce two new characterization sets: dually essential and dually saturated coalitions. We show that each dually inessential coalition has a weakly minorizing overlapping decomposition which consists exclusively of dually essential coalitions. Thus dually essential coalitions determine the core, and if the core is non-empty they determine the nucleolus as well. If every player contributes to the value of a coalition then such coalition is called dually saturated. We show that dually saturated coalitions also determine the core, and if the core is non-empty, then also the nucleolus of a TU-game. The larger a characterization set is the easier to uncover it in a particular game class. However with smaller characterization set it comes a faster LP. Hence there is a tradeo between the diculty in identifying the members of a characterization set and its eciency. In order to exploit this technique we analyze the relationship of the four known characterizing properties. We prove that essential coalitions are a subset of dually saturated coalitions in monotonic prot games and that dually essential coalitions are a subset of saturated coalition in case of monotonic cost games. We show that in general essential and dually essential coalitions do not contain each other. In fact for additive games their intersection is empty. We prove that if the grand coalition is vital then the intersection of essential and dually essential coalitions forms a characterization set itself.

3 2 Game theoretical framework A cooperative game with transferable utility is an ordered pair (N, v) consisting of the player set N = {1, 2,..., n} and a characteristic function v : 2 N R with v( ) = 0. The value v(s) represents the worth of coalition S. No matter how other players behave if the players of S work together they can secure themselves v(s) amount of payo. The set N when viewed as a coalition is called the grand coalition. Let P = 2 N \ {, N} denote the family of the proper coalitions. A cooperative game (N, v) is called monotonic if and superadditive if S T N v(s) v(t ), (S, T P, S T = ) v(s) + v(t ) v(s T ). In a superadditive game two disjoint coalitions can always merge without losing money. Hence we shall assume that players form the grand coalition. The main question is then how to distribute v(n) among the players in some fair way. A solution for a cooperative game Γ = (N, v) is a vector x R N that represents the payo of each player. For convenience, we introduce the following notations x(s) = i S x i for any S N, and instead of x({i}) we simply write x(i). A solution is called ecient if x(n) = v(n) and individually rational if x(i) v(i) for all i N. Ecient solutions will also be called allocations. The imputation set I(Γ ) of the game Γ consists of the ecient and individually rational solutions, formally, I(Γ ) = {x R N x(n) = v(n), x(i) v(i) for all i N}. Given a payo vector x R N, we dene the satisfaction of coalition S in game Γ as sat Γ (S, x) := x(s) v(s). The satisfaction of the grand coalition is clearly zero at any allocation. The core C(Γ ) of cooperative game Γ is the set of allocations where all the satisfaction values are non-negative. Formally, C(Γ ) = {x R N sat Γ (N, x) = 0, sat Γ (S, x) 0 for all S P}. A game is called balanced if its core in non-empty. In this paper we will consider only balanced games. A cooperative game (N, v) is called convex if the characteristic function is supermodular i.e. v(s) + v(t ) v(s T ) + v(s T ), S, T N. Deriving results for convex games is a less challenging task than in general due to their nice structure. Here we only mention a result of Shapley (1971), namely that the core of a convex game is not empty.

4 In many economic situations cooperation results in cost saving rather than prot growth. For instance such situation occurs when customers would like to gain access to some public service or public facility. The question is then, how to share the costs of the service. Cost allocation games can be modeled in a similar fashion as cooperative TU-games. A cooperative cost game is an ordered pair (N, c) consisting of the player set N = {1, 2,..., n} and a characteristic cost function c : 2 N R with c( ) = 0. The value c(s) represents the amount of cost coalition S must bear if it chooses to act separately from the rest of the players. In most cases c is monotonic and subadditive. That is, the more people use the service the more it costs, however there is also an increase of eciency. A cost game (N, c) is called subadditive if and concave if (S, T P, S T = ) c(s) + c(t ) c(s T ), c(s) + c(t ) c(s T ) + c(s T ), S, T N. It is possible to associate a prot game (N, v) with a cost game (N, c), called the savings game, which is given by v(s) = i S c(i) c(s) for all S N. Note that a cost game is subadditive (concave) if and only if the corresponding savings game is superadditive (convex). Similarly as we will see the relationship is reversed in all the previously listed concepts such as individual rationality, satisfaction, and the core. Formally, let Γ = (N, c) be a cost game and x R N an arbitrary allocation. The satisfaction of coalition S in Γ is dened as sat Γ (S, x) := c(s) x(s). As we pointed out in case of cost games the order of the characteristic function and the payo vector is reversed, hence the formula for sat Γ depends on whether Γ is a cost or prot game. However, the essence of the sat Γ (S, x) expression does not change. It still indicates the contentment of coalition S under payo vector x. Accordingly, also in a cost game Γ = (N, c) we say that x is an allocation if sat Γ (N, x) = 0, individually rational if sat Γ (i, x) 0 for all i N, and x is a core vector if it is an allocation and sat Γ (S, x) 0 for all S P. Notice that core vectors of monotonic prot and cost games are non-negative. Indeed, x(i) = x(n) x(n \ i) c(n) c(n \ i) 0 for any core allocation x and i N. For monotonic prot games this is even more trivial since x(i) v(i) v( ) = 0. We say that a vector x R m lexicographically precedes y R m (denoted by x y) if either x = y or there exists a number 1 j m such that x i = y i if i < j and x j < y j. Let Γ = (N, v) be a game and let θ P (x) R 2n 2 be the vector that contains the 2 n 2 satisfaction values sat Γ (S, x), S P, in a non-decreasing order.

5 Denition 1. The nucleolus of a cooperative game Γ with respect to the set of payo vectors X R N is the subset of those payo vectors x X that lexicographically maximize θ P (x) over X. Formally, N(Γ, X) = {x X θ P (y) θ P (x) for all y X}. It is well known (Schmeidler, 1969) that if X is nonempty and compact then N(Γ, X) and if X is convex as well then N(Γ, X) consists of a single point. Furthermore the nucleolus is a continuous function of the characteristic function. If X is chosen to be the set of allocations in Γ, we speak of the prenucleolus of Γ, if X is the set of imputations I(Γ ) then we speak of the nucleolus of Γ. Throughout the paper we will use the shorthand notation N(Γ ) for N(Γ, I(Γ )). For balanced games the prenucleolus coincides with the nucleolus, and N(Γ ) C(Γ ). 3 Characterization sets The concept of characterization sets was already used by Megiddo (1974), but somehow went unnoticed at that time. Later Granot, Granot, and Zhu (1998) and Reijnierse and Potters (1998) re-introduced the idea almost simultaneously. It is remarkable that two such closely related and important papers appeared in the same year. Here we will use the formalism of Granot, Granot, and Zhu (1998). Denition 2. Let Γ F = (N, F, v) be a cooperative game with coalition formation restrictions, where = F P consists of all coalitions deemed permissible besides the grand coalition N. Let θ F (x) R F be the restricted vector that contains the satisfaction values sat Γ (S, x), S F in a non-decreasingly order. Furthermore let N(Γ F ) be dened as the set of imputations that lexicographically maximizes θ F (x). Then F is called a characterization set for the nucleolus of the game Γ = (N, v), if N(Γ F ) = N(Γ ). Note the dierent roles of N and the coalitions in F in the above optimization. The satisfaction of the grand coalition is required to be constantly zero (partly dening the feasible set), whereas the satisfactions of the smaller permissible coalitions form the objective function. Notice that N(Γ F ) is a singleton if and only if rank({e S : S {N} F}) = n, where e S {0, 1} N denotes the membership vector of coalition S given by (e S ) i = 1 if i S and (e S ) i = 0 otherwise. The main result of Granot, Granot, and Zhu (1998) is presented in the following theorem. Theorem 1. (Granot, Granot, and Zhu, 1998) Let Γ be a cooperative (cost or prot) game, F P a non-empty collection, and x an element of the nucleolus of Γ F. Collection F is a characterisation set for the nucleolus of Γ if for every S P \ F there exists a non-empty subcollection F S of F, such that

6 i. sat Γ (T, x) sat Γ (S, x) for every T F S, ii. e S can be expressed as a linear combination of the vectors in {e T : T F S {N}}. Observe that the above conditions are sucient but not at all necessary. Take for example the (superadditive and balanced) prot game with four players N = {1, 2, 3, 4} and the following characteristic function: v(i) = 0, v(i, j) = 1, v(i, j, k) = 3 for any i, j, k N and v(n) = 4. Then the 2-player coalitions and the grand coalition are sucient to determine the nucleolus, which is given by z(i) = 1 for all i N. However, the 3-player coalitions have smaller satisfaction values at z, thus the rst condition of Theorem 1 is violated. Notice that in this game the 3-player coalitions and the grand coalition are also sucient to determine the nucleolus. In general neither the 2-player nor the 3-player coalitions (and the grand coalition) characterize the nucleolus. The fact that in this example they did was due to the particular choice (most importantly the symmetry) of the coalitional function. We would like to identify properties of coalitions that characterize the nucleolus independently of the realization of the coalitional function, at least for a large class of games. A straightforward corollary of Theorem 1 is that we can enlarge characterisation sets arbitrarily. Corollary 1. Let F P be a characterisation set that satises both conditions of Theorem 1. Then T is a characterisation set for any F T P. Now we focus on characterization sets for balanced games. As Granot, Granot, and Zhu (1998) remark, in order to apply Theorem 1 we need not nd an allocation in the nucleolus of the restricted game Γ F if we know a superset Y of N(Γ F ) such that condition (i) holds for all elements of Y. Since core allocations provide non-negative satisfaction for all coalitions, the following corollary of Theorem 1 is easily seen. Corollary 2. Let Γ be a cooperative (cost or prot) game with a non-empty core. The non-empty collection F P is a characterisation set for the nucleolus of Γ if for every S P \ F there exists a non-empty subcollection F S of F, such that i. sat Γ (T, x) sat Γ (S, x) for all x C(Γ ), whenever T F S, ii. e S can be expressed as a linear combination of the vectors in {e T : T F S {N}}. The rst characterization set for balanced games is due to Huberman (1980). Denition 3 (Essential coalitions). Let N be a set of players, (N, v) a prot, (N, c) a cost game. Coalition S is called essential in prot game Γ = (N, v) if it can not be partitioned as S = S Sk with k 2 and S j for all 1 j k such that v(s) v(s 1 ) v(s k ).

7 Similarly, coalition S is called essential in cost game Γ = (N, c) if it can not be partitioned as S = S Sk with k 2 and S j for all 1 j k such that c(s) c(s 1 ) c(s k ). The set of essential coalitions is denoted by E(Γ ), where Γ is either (N, v) or (N, c). A coalition that is not essential is called inessential. Notice that the grand coalition may or may not be essential. On the other hand, by denition, the singleton coalitions are always essential in every game. It is easily seen that in a prot / cost game each inessential coalition has a weakly majorizing / weakly minorizing partition which consists exclusively of essential coalitions. Moreover, the core is determined by the eciency equation sat Γ (N, x) = 0 and the sat Γ (S, x) 0 inequalities corresponding to the essential coalitions, all the other inequalities can be discarded from the core system. The following important result is due to Huberman (1980). Theorem 2. (Huberman, 1980) If the core of the game is non-empty then the essential coalitions form a characterization set for the nucleolus. The theorem means that in a balanced game the essential coalitions and the grand coalition are sucient to determine the nucleolus. This observation helps us to eliminate large coalitions which are redundant for the nucleolus. To detect small coalitions that are unnecessary for the nucleolus, we need the concept of dual game. Denition 4. The dual game Γ = (N, g ) of game Γ = (N, g) is dened by the coalitional function g (S) := g(n) g(n \ S) for all S N, where Γ is either (N, v) or (N, c). Notice that g ( ) = 0, g (N) = g(n), and (g ) (S) = g(s) for all S N. It will be useful to think of the dual game of a prot game as a cost game and vice versa. It can be easily checked that g is monotonic if and only if g is monotonic, and g is supermodular if and only if g is submodular. However, there is no such dualization relation between superadditivity and subadditivity. We can identify small redundant coalitions, if we apply Huberman's argument to the dual game. Denition 5 (Dually essential coalitions). Let N be a set of players, (N, v) a prot, (N, c) a cost game. Coalition S, N is called dually essential in game (N, v) if its complement can not be partitioned as N \S = (N \T 1 )..... (N \T k ) with k 2 and T j N for all 1 j k such that or equivalently, v (N \ S) v (N \ T 1 ) v (N \ T k ), v(s) v(t 1 ) v(t k ) (k 1)v(N).

8 Similarly, S, N is called dually essential in cost game (N, c) if its complement can not be partitioned as N \ S = (N \ T 1 )..... (N \ T k ) with k 2 and T j N for all 1 j k such that or equivalently, c (N \ S) c (N \ T 1 ) c (N \ T k ), c(s) c(t 1 ) c(t k ) (k 1)c(N). The set of dually essential coalitions is denoted by DE(Γ ), where Γ is either (N, v) or (N, c). A coalition that is not dually essential is called dually inessential. Notice that, by denition, the grand coalition is not dually essential. On the other hand, all (n 1)-player coalitions are dually essential in any game. If S P is dually inessential then it is contained in each of the coalitions T 1,..., T k P in the above expression, but every player in N \ S appears exactly k 1 times in this family. We call such a system of coalitions an overlapping decomposition of S. For a more general denition, where the complements of the overlapping coalitions need not form a partition of the complement coalition, see e.g. Brânzei, Solymosi, and Tijs (2005) and the references therein. It is clear that if S, T P are dually inessential coalitions and T appears in an overlapping decomposition of S, then S T so S cannot appear in an overlapping decomposition of T. Consequently, in a prot / cost game each dually inessential coalition has a weakly majorizing / weakly minorizing overlapping decomposition which consists exclusively of dually essential coalitions. Moreover, the core of Γ is also determined by the dual eciency equation sat Γ (N, x) = 0 and the sat Γ (S, x) 0 dual inequalities corresponding to the complements of the dually essential coalitions, all the other dual inequalities can be discarded from the dual core system. The main feature of dually essential coalitions for the nucleolus is that together with the grand coalition they are sucient to determine the nucleolus in balanced games. The next theorem is the dual counterpart of Theorem 2. Theorem 3. If C(Γ ), then the dually essential coalitions form a characterization set for N(Γ ). Proof. There are various ways to derive this result. A formal proof can be obtained by applying the arguments of Huberman (1980) to the dual game. Here we pursue another way and deduce it from Corollary 2. We only give the proof for prot games, for cost games it is analogous. Let S P be a dually inessential coalition in the balanced prot game Γ = (N, v). As remarked earlier, S has a weakly majorizing overlapping decomposition T 1,..., T k (k 2) which consists exclusively of dually essential coalitions. Hence ii. of Corollary 2 follows immediately. To see condition i., let

9 x C(Γ ). Then v(s) v(t 1 ) v(t k ) (k 1)v(N) v(s) x(s) v(t 1 ) v(t k ) (k 1)x(N) x(s) sat Γ (S, x) (sat Γ (T 1, x) sat Γ (T k, x)) sat Γ (S, x) sat Γ (T 1, x) sat Γ (T k, x) sat Γ (S, x) sat Γ (T j, x) 0 j = 1,..., k, where the second inequality comes from v(n) = x(n), while the third from the identity x(t 1 ) x(t k ) = (k 1)x(N) + x(s) implied by N \ S = (N \T 1 )..... (N \T k ), and the last one from the non-negativity of the satisfaction values at any core allocation. If we look for an as small characterisation set as possible for balanced games, in light of Theorems 2 and 3 it seems natural to eliminate both inessential coalitions and dually inessential coalitions at the same time. However, this might not be possible, the intersection of essential and dually essential coalitions need not yield a characterization set. For example, let Γ = (N, v) be an additive game, that is v(s) = i S v(i) for all S N. Then only the singleton coalitions are essential, and since v = v, only the (n 1)-player coalitions are dually essential. Thus E(Γ ) DE(Γ ) = that cannot characterize the nucleolus in any game. The problem lies in what we call a cycle in the decomposition 3. This occurs when coalition S can be discarded because of a collection that contains coalition T, and T can also be discarded because of a collection that contains S or another coalition R that can be eliminated because of a collection that contains S, etc.. As remarked after both denitions, such a cycle in the decomposition cannot occur if we apply only essentiality or only dually essentiality. Before we show when the two types of concepts can be mixed we need some preparation. A collection of coalitions B S P is said to be S-balanced if there exist positive weights λ T, T B S, such that T B λ T e T = e S. An N-balanced collection is simply called balanced. A coalition S is called vital if for any S- balanced collection B S and any system (λ T ) T BS of balancing weights for B S, T B S λ T v(t ) < v(s). Trivially, every vital coalition is essential, but the converse doesnot hold. The concept was introduced by Gillies (1959) and further analyzed by Shellshear and Sudhölter (2009). It is easily seen that the family of vital coalitions (just like essential coalitions) and the grand coalition are sucient to determine the core. However, Maschler, Peleg, and Shapley (1979) demonstrated that vital coalitions (unlike essential coalitions) and the grand coalition do not necessarily characterize the nucleolus in a balanced game. The next theorem provides a sucient condition for E(Γ ) DE(Γ ) to be a characterization set for the nucleolus in a balanced game. 3 Strongly essential coalitions discussed by Brânzei, Solymosi, and Tijs (2005) are not immune to this kind of failure, hence they might not form a characterization set for the nucleolus in a balanced game in which the grand coalition is not vital.

10 Theorem 4. Let Γ = (N, v) be a game with a non-empty core. If the grand coalition is vital, the collection E(Γ ) DE(Γ ) forms a characterization set of N(Γ ). Proof. Suppose coalition S is redundant, that is, either inessential or dually inessential (or both). Then there is a series of coalitions S 1,..., S k in E(Γ ) or in DE(Γ ) that have smaller satisfaction value than S and span e S. Lexicographically maximizing the satisfactions of S 1,..., S k the satisfaction of S becomes xed. If there is no cycle in the decomposition then there exists a decomposition of S i for i = 1,..., k that consist entirely of coalitions T1, i..., Tr i i that belong to E(Γ ) DE(Γ ). Lexicographically maximizing the satisfactions of T1, i..., Tr i i the satisfaction of S i becomes xed. Thus indirectly the satisfaction of S becomes xed. We conclude that, if there is no cycle in the decomposition, then by Theorem 1 E(Γ ) DE(Γ ) is a characterization set. By contradiction suppose that the grand coalition is vital, but there is a cycle T 1, T 2,..., T r in the decomposition. Note that we may assume without loss of generality that the series alternates between inessentiality and dual inessentiality. If T l and T l+1 were deemed redundant for the same reason (e.g. they are both inessential) then the inequality that shows inessentiality of T l can be rened by the inequality that shows inessentiality of T l+1. Let us assume that T 1 is inessential the proof is the same if T 1 is dually inessential. Thus using the denition of essentiality and dual essentiality k 1 v(t 1 ) v(t 2 ) + v(sj 1 ) (1) k 2 v(t 2 ) v(t 3 ) + v(sj 2 ) k 2 v(n) (2) k 3 v(t 4 ) v(t 5 ) + v(sj 3 ) (3). k r v(t r ) v(t 1 ) + v(sj r ) k r v(n) (4) In words T 1 is inessential because of the collection T 2, S1, 1..., Sk 1 1 (these are all essential coalitions, thus the inequality cannot be rened any more). Then T 2 is dually inessential because of the collection T 3, S1, 2..., Sk 2 2 compose an overlapping decomposition of T 2 (and these are all dually essential). And so on until nally T r is deemed redundant because of T 1, S1, r..., Sk r r. Note that there may be coalitions among S1, 1..., S1, 2..., S1, r..., Sk r r that coincide. Using indicator functions and the conditions of inessentiality and dual inessentiality.

11 k 1 e T1 = e T2 + e S 1 j (5) k 2 e T2 = e T3 + e S 2 j k 2 e N (6) k 3 e T4 = e T5 + e S 3 j (7). k r e Tr = e T1 + e S r j k r e N (8) Thus by summing (1)-(4) we obtain that 1 v(n) k 2 + k k r r k i v(sj) i i=1 while from (5)-(8) we gather that 1 e N = k 2 + k k r r k i i=1 e S i j i.e. the collection S1, 1..., S1, 2..., S1, r..., Sk r r fact that the grand coalition is vital. is balanced. This contradicts the 4 Monotonic balanced games The next characterization set was proposed by Granot, Granot, and Zhu (1998) for balanced monotonic cost games. Denition 6. Let (N, c) be a monotonic cost game. A coalition S N is said to be saturated if c(s) = c(s {i}) implies i S. In other words if S is a saturated coalition then every new member will impose extra cost on the coalition. Let S (Γ ) denote the set of all saturated coalitions and S(Γ ) = S (Γ ) {N \ i i N} {N}. Theorem 5. (Granot, Granot, and Zhu, 1998) Let Γ = (N, c) be a monotonic cost game with a non-empty core. Then S(Γ ) forms a characterization set for N(Γ ).

12 In fact Granot, Granot, and Zhu (1998) proved a slightly stronger statement, namely, that the intersection of saturated and essential coalitions forms a characterization set in monotonic cost games with a non-empty core. Similarly to the other characterization sets, S(Γ ) also induces a representation of the core C(Γ ) as well. Let us mention here that just because a collection of coalitions determines the core it does not necessarily characterize the nucleolus of the game. Maschler, Peleg, and Shapley (1979) presented two games with the same core, but with dierent nucleoli. We now convert the concept of saturatedness to monotonic prot games based on the dualization correspondence between prot and cost games. Let (N, v) be a monotonic prot game and S N be an arbitrary coalition. We say that S is dually saturated if v(s \ i) < v(s) for any i S. In other words every member contributes to the worth of coalition S. Notice that S is saturated in the monotonic cost game c if and only if N \ S is dually saturated in the monotonic prot game c. Let DS (Γ ) denote the set of all dually saturated coalitions and DS(Γ ) = DS (Γ ) {i i N} {N}. The following denition is needed for our next theorem. Let (N, v) be a monotonic game and S N a dually non-saturated coalition, then we say that S is a lower closure of S if S S, v(s) = v(s) and S is a dually saturated coalition. Note that if S has no lower closure, then no member contributes to the worth of S or to any subset of S. Hence v(s) = v(i) = v( ) = 0 for any i S. Theorem 6. Let Γ = (N, v) be a monotonic game with a non-empty core, then DS(Γ ) forms a characterization set for N(Γ ). Proof. Again we will use Theorem 1. Let S be a dually non-saturated coalition. If S has no lower closure then v(s) = v(i) = 0 for any i S. From this observation it also follows that sat Γ ({i}, x) sat Γ (S, x) for any i S and for any allocation x. Since all the singleton coalitions are included in DS(Γ ) by Theorem 1, S can be discarded. Finally let S be a lower closure of S and let S \ S = T, then sat Γ (S, x) + x(t ) = x(s) + x(t ) v(s) = x(s) v(s) = sat Γ (S, x). Since core vectors are non-negative this also means sat Γ (S, x) sat Γ (S, x) for any x C(Γ ). Now we show that sat Γ ({i}, x) sat Γ (S, x) for any i T. sat Γ ({i}, x) = x(i) v(i) x(i) = x(s) x(s \ i) + v(s \ i) v(s) = sat Γ (S, x) sat Γ (S \ i, x) sat Γ (S, x) We have shown that for any S 2 N \ DS(Γ ) there exist a subcollection F of DS(Γ ), such that F fullls both conditions of Theorem 1. Hence DS(Γ ) is a characterization set for N(Γ ).

13 Next we show a relationship between dually essential and saturated coalitions. Lemma 1. Let Γ = (N, c) be a monotonic cost game, then DE(Γ ) S(Γ ) Proof. The grand coalition and the n 1 player coalitions are all members of both S(Γ ) and DE(Γ ). Let S be a non-saturated coalition with at most n 2 players. We will show that S is dually inessential. As S is not saturated there exists i N \ S such that c(s) = c(s i). Let S 1 := S i and S 2 := N \ i. Then S 1 S 2 = N and S 1 S 2 = S therefore we can use Denition 5 since c(n) c(n \ i), c(s) c(s) + c(n \ i) c(n), c(s) c(s 1 ) + c(s 2 ) c(n). In other words S appears in an overlapping decomposition of S 1 and S 2, therefore it can not be dually essential. The above lemma suggests that dually essential coalitions are more useful since they dene a smaller characterization set, which in turn implies a smaller LP. However usually it is also harder to determine whether a coalition is dually essential or not. Saturatedness on the other hand can be checked easily. For instance for airport games 4 there exist at most n saturated coalitions, which can be easily determined from the characteristic function. In fact it is enough to know the value of the singleton coalitions to identify the saturated coalitions, which gives us an alternative way to derive an ecient algorithm for the nucleolus. There is a symmetrical result for essential and dually saturated coalitions. Lemma 2. Let Γ = (N, v) be a monotonic prot game, then E(Γ ) DS(Γ ). Proof. Observe that the singleton coalitions are all members of both E(Γ ) and DS(Γ ). Let S be a dually non-saturated coalition such that S > 1. Then there exists i S such that v(s) = v(s \ i). By monotonicity v(i) 0, hence v(s) v(s \ i) + v(i). Thus S is inessential. 5 Verifying the nucleolus Faigle, Kern, and Kuipers (1998) conjecture the decision problem: "given x R N, is x the nucleolus?" to be N P-hard in general. The most resourceful tool regarding this problem is the criterion developed by Kohlberg (1971). Theorem 7. (Kohlberg, 1971) Let Γ = (N, v) be a game with non-empty core and let x I(Γ ). Then x = N(Γ ) if and only if for all y R the collection { S N sat Γ (S, x) y} is balanced or empty. 4 This class of games was introduced by Littlechild and Owen (1973).

14 The Kohlberg-criterion is often used to verify the nucleolus in practical computation when the size of the player set is not too large. As the following LP shows it is easy to tell whether a given collection of coalitions is balanced or not. Let S 1,..., S m be the collection which balancedness is in question and let q [0, 1] m. For k = 1,..., m let p k = max q k m q i e Si = e N (9) i=1 q 1,..., q m 0. Lemma 3. The collection S 1,..., S m is balanced if and only if p k > 0 for each k = 1,..., m. Proof. Trivially p k > 0 is a necessary condition. Let a 1,..., a m [0, 1] be arbitrary reals such that a a m = 1. Furthermore let q k be an optimal solution of the k th LP. We claim that λ = m i=1 a iq i is a vector of balancing weights. Note that λ j = m i=1 a iqj i > 0 as qj j = p j > 0 and m qi j e S j = e N for all i = 1,..., m by (9). Then m m m m λ j e Sj = a i qje i Sj = i=1 m a i i=1 q i je Sj = (a a m )e N = e N. Sobolev (1975) extended Theorem 7 to the prenucleolus (where instead of x I(Γ ) we only require x to be an allocation). A direct consequence of the Sobolev-criterion is that the prenucleolus of monotonic games is non-negative. Theorem 8. Let Γ = (N, v) be a monotonic game and let z denote its prenucleolus, then z(i) 0 for all i N. Proof. By contradiction suppose that z(i) < 0 for some i N. Let B 0 contain the coalitions with the smallest satisfaction values under z. By the Sobolev-criterion B 0 is a balanced collection. For every S B 0, i S otherwise S {i} would have an even smaller satisfaction due to the monotonicity of the characteristic function and the fact that z(i) < 0. By balancedness of B 0, S B 0 λ S e S = e N. As i S for all S B 0 this also means that S B 0 λ S = 1. Then for all j i and for all S B 0, S must contain j. Thus the only coalition in B 0 is the grand coalition. However the grand coalition has zero satisfaction under any allocation, while coalition {i} has negative satisfaction under z, which contradicts that B 0 contains the coalitions with the smallest satisfaction values. Reijnierse and Potters (1998) proved that for every game (N, v) there exists a collection with at most 2(n 1) coalitions that determine the nucleolus. However, they remarked that nding these coalitions is as hard as computing the

15 nucleolus itself. Unfortunately this result does not make the verication of the nucleolus belong to N P. Even if somehow we could eortlessly put the satisfaction values of the 2 n coalition in increasing order. It can happen that these 2(n 1) coalitions are scattered among the dierent balanced coalition arrays and we have to evaluate exponential many of them before we could conrm that the given allocation is indeed the nucleolus. The Kohlberg-criterion applied to games with coalition formation restrictions yields the following theorem. Theorem 9. (Maschler, Potters, and Tijs, 1992) Let F be a characterization set and x be an imputation of the game Γ with C(Γ ). Then x = N(Γ ) if and only if for all y R the collection {S F sat Γ (S, x) y} is balanced or empty. A similar criterion appears in (Groote Schaarsberg, Borm, Hamers, and Reijnierse, 2013). With the help of the Kohlberg-criterion the problem of nding the nucleolus is reduced to nding the right characterization set. Acknowledgements The authors would like to thank Prof. Tamás Kis of Institute for Computer Science and Control (MTA SZTAKI) for his valuable comments. Research was funded by OTKA grants K and K and by the Hungarian Academy of Sciences under its Momentum Programme (LD-004/2010).

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