Additivity Regions for Solutions in Cooperative Game Theory Branzei, M.R.; Tijs, S.H.

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1 Tilburg University Additivity Regions for Solutions in Cooperative Game Theory Branzei, M.R.; Tijs, S.H. Publication date: 200 Link to publication Citation for published version (APA): Brânzei, R., & Tijs, S. H. (200). Additivity Regions for Solutions in Cooperative Game Theory. (CentER Discussion Paper; Vol ). Tilburg: Operations research. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. - Users may download and print one copy of any publication from the public portal for the purpose of private study or research - You may not further distribute the material or use it for any profit-making activity or commercial gain - You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 24. Feb. 208

2 No ADDITIVITY REGIONS FOR SOLUTIONS IN COOPERATIVE GAME THEORY By Rodica Brânzei and Stef Tijs October 200 ISSN

3 ADDITIVITY REGIONS FOR SOLUTIONS IN COOPERATIVE GAME THEORY Rodica Br^anzei 2 and Stef Tijs 3 Abstract Superadditivity, subadditivity and additivity properties are considered for solutions and multisolutions of games in characteristic function form. Special attention is paid to the core and the fi value and to cones of games on which these solutions are additive. Introduction A breakthrough in cooperative game theory was the paper of Lloyd Shapley (953), where a solution concept for cooperative games was introduced i.e. a map assigning to each cooperative game a vector with as many coordinates as there are players. The solution which is nowadays known as the Shapley value has nice properties. One of the characterizing properties of the Shapley value in Shapley (953) is the additivity property: the Shapley value of the sum of two games equals the sum of the Shapley values of the two original games. Many other solution concepts were introduced later. Many of them have not the additivity property on the whole linear space of games. The purpose of this paper is to discover and describe subcones of the game space on which a certain solution is additive. Another important solution concept is the core (Gillies 953), which assigns to each game a set of vectors. The core turns out to be a superadditive multi solution: the core of the sum of two games is at least as large as the (Minkowski) sum of the cores of the two original games. In Dr agan, Potters This paper is dedicated to Irinel Dr agan on the occasion of his seventieth birthday 2 Faculty of Computer Science, "Alexandru Ioan Cuza" University,, Carol I Bd., 6600 Iaοsi, Romania, E mail: branzeir@infoiasi.ro 3 CentER and Department of Econometrics and Operations Research, Tilburg University, P.O. Box 9053, 5000 LE Tilburg, The Netherlands, S.H.Tijs@kub.nl

4 and Tijs (989) it is proved that on the cone of convex games the core is an additive correspondence. In this paper we will introduce other cones with this property such as the cones of simplex games, dual simplex games and general big boss games. Also other sub and superadditive correspondences get attention. For a systematic study of cones in cooperative game theory we refer to Derks (99). The outline of the paper is as follows. Section 2 recalls some notions and facts of cooperative game theory which are useful later. In Section 3 many solution concepts are investigated w.r.t. additivity properties. Special attention is given here to the fi value. Section 4 deals with simplex games and dual simplex games and Section 5 with general big boss games. There is a summary of the main results in Section 6, where also some remarks on further research are made. 2 Preliminaries In this section we recall some notions and facts of cooperative game theory which will be used in the following sections. An n-person cooperative game is a pair < N;v >; where N = f; 2; :::; ng is the the set of players and v : 2 N! IR is the characteristic function, which assigns to each element of the family 2 N of subsets of N a real number and where v(ffi) = 0. The elements in 2 N are called coalitions and N is the grand coalition. The real number v(s) istheworth or value of coalition S; it is the amount which the coalition S can obtain when working together. The value v(k) is called the individual value of player k and i(v) =(v();v(2); :::; v(n)) 4 is the individual rational vector of v. The real number v Λ (k) =v(n) v(n nfkg) is called the marginal contribution of k to the grand coalition or also the utopia payoff for player k. It is the individual value of player k in the dual game < N;v Λ >, where v Λ (S) =v(n) v(nns) for each S 2 2 N : If N is formed, the players can divide the amount v(n): Player k will not be content with a non individual rational payoff allocation, which gives less than v(k) to him, because he can obtain v(k) in staying alone. While v(k) can be seen as a lower bound for the payoff of player k, the number v Λ (k) can be seen as an upper bound for the payoff. If a player k asks an amount z, which is larger than v Λ (k) in the grand coalition, then the other players in N nfkg can exclude him and v(n nfkg) >v(n) z can be divided among the members of N nfkg if these 4 We often write v(i; j; :::) instead of v(fi; j; :::g): 2

5 players cooperate. The utopia point u(v) of the game < N;v > is given by u(v) =(v Λ ();v Λ (2); :::; v Λ (n)): The imputation set I(v) of <N;v>is defined by I(v) = ( x 2 IR n j i2nx i = v(n); x i v(i) for all i 2 N and consists of all vectors in IR n, satisfying the n individual rationality conditions and the efficiency condition n x i = v(n). Note that from the geometric point of view, the imputation ( set I(v) is equal ) to the intersection of the efficiency hyperplane x 2 IR n j x i = v(n) and the n orthant fx 2 IR n j x i(v)g of individual rational payoff vectors. The dual imputation set I Λ (v) of<n;v>is given by I Λ (v) = ( x 2 IR n j i2nx i = v(n); x i» v Λ (i) for all i 2 N and I Λ (v) is equal to the intersection of the efficiency hyperplane and the orthant fx 2 IR n j x» u(v)g of subutopic vectors. The set C(v)= ) ) ; ( ) x2ir n j x i =v(n); x i v(s) for each S22 N i2n i2s called the core of the game <N;v>and it consists of the efficient vectors x which are split off stable, because no coalition S can obtain more than i2sx i when splitting off. Related to the core is the core cover CC(v) =I(v) I Λ (v) (cf. Tijs and Lipperts (982)). The name is explained by the fact that for all games CC(v) ff C(v): All the introduced sets I(v);I Λ (v);c(v) and CC(v) may be empty. Necessary and sufficient conditions for the non emptiness of these sets can be obtained with the aid of balancedness conditions, where a balancedness condition is an inequality of the form v(n) S v(s) S22 N where S 0 for all S 2 2 N and S = for all i 2 N: So we have (i) I(v) 6= ; iff v(n) n S:i2S v(i): The corresponding balancedness coordinates S are given by S =ifjsj =, and S = 0 otherwise. 3 is

6 (ii) I Λ (v) 6= ; iff v(n)» n with the balancedness condition v(n) n v Λ (i): Note that this inequality is equivalent n v(n nfig): (iii) CC(v) 6= ; iff I(v) I Λ (v) 6= ;; which is equivalent to i2nv(i)» v(n)» i2nv Λ (i); and v(i)» v Λ (i) for all i 2 N: Remark that v(i)» v Λ (i) corresponds to the balancedness condition v(n) v(n nfig) + v(i). Games with CC(v) 6= ; will be called semi balanced and the cone of these games will be denoted by SB N. (iv) C(v) 6= ; iff all balancedness conditions are satisfied. This is the famous result of Bondareva (963) and Shapley (967). Given an ordering ff = (ff();ff(2); :::; ff(n)) of the players in N, the ffmarginal vector m ff (v) 2 IR n of < N;v > is the vector which has for each k 2 N as ff(k)-th coordinate the real number v(ff();ff(2); :::; ff(k)) v(ff();ff(2); :::; ff(k )): Let us denote by Π(N) the set of n! orderings of N. Then the Shapley value ffi(v) of <N;v> is equal to the average n! ff2π(n ) m ff (v) of marginal vectors (cf. Shapley (953)) and the Weber set W (v) (Weber (988)) is the convex hull in IR n of the set fm ff (v) j ff 2 Π(N)g of marginal vectors. For semi-balanced games <N;v>, the ff value is defined as the feasible compromise between u(v) and i(v): So ff(v) = ffu(v) +( ff)i(v), where the compromise factor ff is the unique real number between 0 and, such that n ff i (v) = v(n). With SB N ff we denote the cone of semi balanced games with fixed compromise factor ff. The fi value fi(v) (Tijs (98)) for a quasi balanced game is the feasible compromise between u(v) and the minimum right vector m(v); where for each i 2 N 0 m i (v) = v(s) v Λ (j) A ; S:i2S j2snfig 4

7 and where a game is called quasi balanced if n m i (v)» v(n)» n v Λ (i);m i (v)» v Λ (i) for all i 2 IN: So fi(v) =ffu(v)+( ff)m(v); where ff is the unique real number such that n fi i (v) =v(n), which is called the fi compromise factor in the following. The cone of quasi balanced games will be denoted by QB N : 3 Superadditive, subadditive and additive solutions Denote by G N the set of all n-person games. G N is a (2 n ) dimensional vector space with the usual operations of adding of functions, and scalar multiplication with real numbers, where each game < N;v > is identified with its characteristic function v. A cone K in G N is a set of games with the property that ffv + fiw 2 K if v; w 2 K and ff; fi are non negative real numbers. A map ψ : K! IR n with as domain the cone K ρ G N is called additive on K if ψ(v + w) =ψ(v)+ψ(w) for all v; w 2 K: Examples are given in the following. Example 3.. The Shapley value is additive oneach cone K of G N. Example 3.2. We define for each i 2 N and each v 2 G N the vectors f i (v) and g i (v) by (f i (v)) k = v(i) for each k 2 N nfig and (f i (v)) i = v(n) v(k) k2n nfig (g i (v)) k = v Λ (i) for each k 2 N nfig and (g i (v)) i = v(n) k2n nfig v Λ (k): Then, obviously, f i and g i are additive maps from G N into IR n and also n n f i and n g i : n Example 3.3. Note that I N = n v 2 G N j I(v) 6= ; o is a cone of games and also I N Λ = fv 2 G N j I Λ (v) 6= ;g: Now CIS : I N! IR n is defined 5

8 n by CIS(v) = f i (v) and CIS is, in view of Example 3.2, an additive n map, called the center of the imputation set solution. The name is explained by the fact that f (v);f 2 (v); :::; f n (v) are the extreme points of I(v); which is an (n ) dimensional simplex if v(n) > v(n) = n v(i): n v(i) and a one point set if On I N Λ the map ENSR : I N Λ! IR n is additive, where ENSR(v) n = g i (v). Here ENSR stands for 'equal split of the non separable rewards'. n Example 3.4. The map ff : SB N! IR n is not additive, but for each ff 2 [0; ] the map ff : SB N ff! IR is additive which follows from the fact that for each v 2 SB N ff ff(v) =ffu(v)+( ff)i(v); and v 7! u(v) and v 7! i(v) are, obviously, additive. Example 3.5. The fi-value is not additive onqb N : Let us analyse factors, which spoil the additivity of the fi value on QB N. On one hand, although the utopia map v 7! u(v) is additive, the i-th coordinate of the minimum right map v 7! m(v) is the maximum of additive maps of the form v 7! v(s) v Λ (j), and maximum taking destroys in j2snfig general additivity. On the other hand, even if for a subcone K of QB N the minimum right map is additive, then a second additivity destroying factor can be the fact that for different games in the cone the compromise factors may bedifferent (see Example 3.6). The above insight of factors which destroy additivity gives rise to the introduction of cones where the fi-value is additive. Let ff 2 [0; ]; and for each i 2 N let B i be a subset of N containing i. Let QB N ff;b ;B 2 ;:::;B n be the cone of quasi balanced games v with ff as fixed uniform compromise factor and B i 2 arg max S:i2S 6 v(s) j2snfig v Λ (j) A :

9 Then, obviously, the minimum right map is additive on this cone and also the fi-value, because fi(v) =ffu(v)+( ff)m(v) for all v 2 QB N ff;b ;B 2 ;:::;B n : So we obtain Theorem 3.. Let ff; B ;B 2 ; :::; B n beasabove. Then fi:qb N ff;b ;B 2 ;:::;B n!ir n is an additive map. Example 3.6. Recall that a game v 2 G N is said to be convex (Shapley (97)) if v(s[t )+v(s T ) v(s)+v(t ) for all S; T 2 2 N : For convex games the minimum right map is additive because m(v) =(v(); v(2); :::; v(n)) for each v 2 CONV N : But fi : CONV N! IR n is not additive. An explanation of the result in Example 3.6 is that there is no uniform compromise factor for CONV N : E.g. each unanimity game u S : 2 N! IR such that u S (T )=if S ρ T, and u S (T )=0otherwise, is a convex game. But for jsj 2 the compromise factor ff(u S ) = jsj, depends on the size of S 2 2 N n f;g, which follows from u(u S )=u S (N)e S, m(u S )=0; so fi(u S ) = jsj u(u S )+( jsj )0: Here e S is the characteristic vector for S, with (e S ) i = if i 2 S, and (e S ) i = 0 otherwise. So, let us look at the subcones K r (r = ; 2; :::; n) of CONV N with uniform compromise factor r, given by K r = conefu S j S 2 2 N ; jsj = rg; the cone generated by the unanimity games based on a set S of size r: Then we have Theorem 3.2. The fi-value fi : K r! IR n is an additive map. The cones K 2 and K 3 arose in connection with telecom problems (Nouweland et al. (996)). In the mentioned paper it is proved that for all r 2 f; :::; ng the fi-value and the Shapley value coincide on K r. On K and K 2 also the nucleolus coincides with the two other values. Now we turn to multisolutions, which are correspondences D : K!! IR n from a cone K ρ G N into the family of subsets of IR n. Such a correspondence is called superadditive on K if for all v; w 2 K we have D(v + w) ff D(v)+ D(w); and subadditive if the reverse inclusion holds. A multifunction D : K!! IR n is called additive if the function is superadditive and subadditive or, equivalently, if D(v + w) = D(v) + D(w) for all v; w 2 K: 7

10 Example 3.7. C : G N!! IR n is a superadditive correspondence and W : G N!! IR n is subadditive (see e.g. Dr agan, Potters and Tijs (989)). Example 3.8. Using the results in Example 3.7 and the fact that for the cone CONV N of convex games the core and the Weber set coincide, Dr agan et al. (989) proved Theorem 3.3. C : CONV N!! IR n is additive. Example 3.9. The cone of exact games E N was introduced by Schmeidler (972) and contains the cone of convex games. Recall that a game v 2 G N is exact if for each S 2 2 N nfffig; there is an x 2 C(v) such that x(s) =v(s): The core correspondence on E N is, of course, superadditive but not additive which we learn from the symmetric 4-person exact games v; w 2 E N, given by v(s) =0; 0; 2; 3; 6ifjSj =0; ; 2; 3; 4; respectively, and w(s) =0; 0; 2; 5; 8ifjSj =0; ; 2; 3; 4; respectively: Then z = (0; 4; 4; 6) 2 C(v + w), and x = (0; 2; 2; 2) is the unique element in C(v) with as first coordinate 0, which implies that there is no y 2 C(w) with z = x + y: So C(v + w) isreally larger than C(v)+C(w). Example 3.0. The maps I : G N!! IR n, I Λ : G N!! IR n, assigning respectively to a game the imputation set and the dual imputation set are superadditive. Since the core cover CC = I I Λ is the intersection of two superadditive correspondences, it is also superadditive. The core cover is not additive. For I and I Λ we can say more as the following theorem shows. Theorem 3.4. I : I N!! IR n ; I Λ : I N Λ!! IR n are additive correspondences. Proof. We noted already that the correspondences are superadditive ong N, so certainly on I N and I N Λ, respectively. (i) To prove the additivity of I : I N!! IR n we have to show that for v; w 2 I N each z 2 I(v + w) can be written as z = x + y with x 2 I(v) and y 2 I(w): Note that z is a convex combination of the extreme points f (v + w);f 2 (v + w); :::; f n (v + w) of the simplex I(v + w), so there are non negative real numbers ff ;ff 2 ; :::; ff n with n ff i =, such that z= n ff i f i (v+w). 8

11 Because f i (v + w) =f i (v)+f i (w) for each i 2 N; we obtain the result z = x + y with x = n I : I N!! IR n is additive. ff i f i (v) 2 I(v) and y = n ff i f i (w) 2 I(w): Hence, (ii) The proof of the additivity of I Λ : I N Λ!! IR n runs in a similar way since each z 2 I Λ (v + w) is a convex combination of the extreme points g (v + w);g 2 (v + w); :::; g n (v + w) ofi Λ (v + w), and g i (v + w) =g i (v)+g i (w) for all i 2 N: 2 4 Simplex games and dual simplex games Let us call a game <N;v> a simplex game if the core C(v) of the game v is non empty and equals the simplex I(v). A game <N;v>is called a dual simpex game if ;6= C(v) =I Λ (v): Let us denote the set of simplex games by SI N and the set of dual simplex games by SI N Λ. It will turn out that these sets are cones on which the core is an additive correspondence. Theorem 4.. (i) SI N = ( v 2 G N j v(s)» i2sv(i) for all S 6= N; i2n ) v(i)» v(n) (ii) SI N is a cone (iii) C : SI N!! IR n is additive. Proof. (i.a) Suppose v 2 SI N : Then I(v) 6= ;; so v(n) Let i2nv(i): S 2 2 N, S 6= N. Take k 2 N n S: Since f k (v) 2 I(v) = C(v); we obtain = (f i2sv(i) k (v)) i v(s): So we have proved that SI N is a subset of the i2s set of the right side of the equality sign in (i). (i.b) For the converse inclusion, suppose that v 2 G N is such that v(s)» v(i) for all S 6= N and» v(n): Then the last inequality implies that I(v) 6= ;: Of course, C(v) ρ I(v); but the i2s i2nv(i) reverse also holds because for each x 2 I(v) we have n x i = v(n) and for all 9

12 S 6= N : x i v(s). So ;6= I(v) =C(v); v 2 SI i2s i2sv(i) N. This finishes the proof of (i). (ii) From the characterization of SI N (i) it follows directly that SI N is a cone. in terms of linear inequalities in (iii) The additivity of C : SI N!! IR n follows directly from (ii) and Theorem Note that (i) of Theorem 4. tells that in simplex games it does not pay to form coalitions unequal to the grand coalition. Let us look at the subcone SUSI N of SI N of superadditive simplex games. Recall that an n person game is called a superaddtive game if v(s [ T ) v(s)+v(t ) for all S; T 2 2 N. Theorem 4.2. (i) For each v 2 SUSI N : (ii) fi : SUSI N! IR n is additive CIS(v) =ENSR(v) =fi(v); CC(v) =C(v) (iii) CC : SUSI N!! IR n is additive. Proof. (i) For v 2 SUSI N and i 2 N : v Λ (i) = v(n) v(n n fig) = v(n) So ENSR i (v) = v k2n nfigv(k). Λ (i) ψ n! v Λ (k) v(n) = n k= v(n) v Λ (i) n v(k) A = v(i)+ n n k2n Further, m i (v) =v(i) for each i 2 N: Then v(n) k2n v(n) v(k) A = CISi (v): fi i (v) = n vλ (i)+ n n m i(v) =v(i)+ v(k) A = CIS(v) n k2n ( ) n CC(v) = x 2 IR n j x i = v(n); v(i)» x i» v Λ (i) for each i 2 N = 8 < : x 2 IRn j n = I(v) =C(v): x i = v(n); v(i)» x i» v(n) 0 k2n nfig 9 = v(k) ; =

13 (ii) Since CIS : I N!! IR n is additive also CIS : SUSI N! IR n is additive and fi = CIS on SUSI N by (i). (iii) CC : SUSI N!! IR n is additive since CC = C on SUSI N and C is additive onsusi N by Theorem The next theorem can be seen as a dual of Theorem 4.. Theorem 4.3. (i) SI N Λ = ( v 2 G N j v Λ (S) i2sv Λ (i) for all S 6= N; i2n ) v Λ (i) v Λ (N) (ii) SI N Λ is a cone (iii) C : SI N Λ!! IR n is an additive correspondence. Proof. (i.a) First we prove that SI N Λ is a subset of the set on the right side of the equality in (i). In (i.b) we prove the converse inclusion. Let v 2 SI N Λ (v): Then ;6= C(v) =I Λ (v) implies that i2nv Λ (i) v(n) =v Λ (N): Let S 2 2 N n fng: Take i 2 N n S: Then g i (v) 2 I Λ (v)) = C(v) implies that v(n n S)» i2n ns g i (v) = k2n ns[fig v Λ (k)+ v(n) = v(n) v Λ (k): So v Λ (S) =v(n) v(n n S) k2s k2sv Λ (k): k2n nfig v Λ (k) (i.b) Suppose that v Λ (S) v Λ (i) for all S 6= N and i2s i2nv Λ (i) v Λ (N): Then the last inequality implies that I Λ (v) 6= ;: Further C(v) ρ I Λ (v): For the converse we prove that g i (v) 2 C(v) for each i 2 N. Clearly, n k= A (g i (v)) k = v(n): Since g i (v) 2 I Λ (v)wehave for each S 2 2 N, S 6= N : (g i (v)) k» k2n ns (g i (v)) k = v(n) (g i (v)) k v(n) k2n nsv Λ (k)» v Λ (N n S): So k2s v Λ (N n S) =v(s): Hence g i (v) 2 C(v): k2n ns (ii) follows from (i), and (iii) follows from the additivity ofi Λ on SI N Λ.2 In Driessen and Tijs (983) and Driessen (988) the cone CONV N of -convex games was studied. CONV N was defined as fv 2 G N j 0»

14 g v (N)» g v (S) for each S 2 2 N g, where g v (S) := i2sv Λ (i) v(s) is the gap of coalition S in the game v, or the gap between the sum of the utopia payoffs of S and the reality that only v(s) can be obtained by S in cooperation. For these games v 2 CONV N it was proved that fi(v) = ENSR(v). Note that g v (N) 0 is equivalent with i2nv Λ (i) v(n) and that for S 6= N: g v (N n S) g v (N) is equivalent to v Λ (i) v(n) i2n nsv Λ (i) v(n n S) i2n and so also to i2sv Λ (i)» v(n) v(n n S) = v Λ (S). This implies that CONV N = SI N Λ. So we have in view of the above Theorem 4.4. (i) fi(v) =ENSR(v) for all v 2 SI N Λ (ii) fi : SI N Λ! IR n is additive. 5 General big boss games In this section we look at n-person games < N;v >; where one player namely player n has a special role. To be precise, we consider the class of games GBB N = fv 2 G N j f n (v) 2 C(v);g n (v) 2 C(v)g and call its elements general big boss games for reasons which become clear soon. Theorem 5.. Let v 2 G N : The following assertions are equivalent (i) v 2 GBB N (ii) v satisfies the following three properties: v(i)»v Λ (i) for all i2n; i2n v(i)»v(n)» i2nv Λ (i) [core cover non-emptiness] v(s)» i2sv(i) for all S 2 2 N with n =2 S [general big boss property] v Λ (N n S) i2n ns v Λ (i) for all S 2 2 N with n2s [union property] (iii) ffi 6= ( C(v) = x 2 IR n j v(i)» x i» v Λ (i) for all i 2 N nfng; 2 n x i = v(n) ) :

15 Proof. We prove respectively that (i) implies (ii), (ii) implies (iii), and (iii) implies (i). (a) Suppose v 2 GBB N : Then C(v) is non empty, which implies that CC(v) is non empty since C(v) ρ CC(v): The fact that f n (v) 2 C(v) implies for S not containing n that v(s)» (f n (v)) i = So the general big i2s i2sv(i): boss property holds. The property g n (v) 2 C(v) implies the union property, since for S with n 2 S we have v(s)» i2s(g n (v)) i = v(n) i2n nfng or v Λ (N n S) =v(n) v(s) v Λ (i) A + i2n ns i2snfng v Λ (i) v Λ (i); (b) Suppose v satisfies the three properties in (ii). We have to show that (iii) holds. It is sufficient to show that ffi 6= P (v) =C(v) where ( ) n P (v) = x 2 IR n j v(i)» x i» v Λ (i) for all i 2 N nfng; x i = v(n) : Note that C(v) ρ CC(v) ρ P (v): Because of the property in (ii) that the core cover is not empty we obtain that P (v) 6= 0: Further C(v) ρ P (v): We have finished the proof if we show that x 2 P (v) satisfies the split off stability conditions: i2sx i v(s) for each S 2 N: Take first an S 2 2 N with n =2 S. Then x i v(s) by the general big boss property. Take i2s i2sv(i) finally an S 2 2 N with n 2 S. Then x i» v Λ (i)» v Λ (N n S) by the union property. So i2sx i = n i2n ns x i i2n ns i2n ns x i v(n) v Λ (N n S) =v(s): (c) That (iii) implies (i) is obvious, because (f n (v)) i = v(i) and (g n (v)) i = v Λ (i) for each i 2 N nfng; further n (f n (v)) i = v(n) = n (g n (v)) i : 2 Remark 5.. For v 2 GBB N ; player n is called the big boss, because of the general big boss property, which tells that a coalition not containing n cannot profit from cooperation. Such a coalition is called a union. The union property tells that for a union N n S (so n 2 S) its marginal contribution 3

16 v(n) v(s) to the grand coalition is at least as large as the sum of the individual marginal contributions v Λ (i) of its members. Remark 5.2. For a general big boss game with three players, the core is a parallelogram in the efficiency plane. For v 2 GBB n the core is the intersection ( of n strips ) S ;S 2 ; :::; S n and the efficiency hyperplane n x 2 IR n j x i = v(n). Here S i is the region fx 2 IR n j v(i)» x i» v Λ (i)g between the two parallel hyperplanes fx 2 IR n j x i = v(i)g and fx 2 IR n j x i = v Λ (i)g: We will denote the average 2 (f n (v)+g n (v)) by fi(v), because this point is the barycenter of the core. Remark 5.3. In Muto et al. (989) the cone BB N of big boss games was introduced and many economic situations giving rise to such games were discussed. Here we call a game v a big boss game with player n as big boss if (i) v 2 GBB N, (ii) v(s) = for all S 2 2 i2sv(i) N with n 2 S: In Muto et al. (989) only 0 normalized big boss games were considered, i.e. games with v(i) = 0 for each i 2 N: For big boss games it was proved that the fi value is in the center of the core and coincides with the nucleolus. We collect some interesting facts for general big boss games in the next theorem. Theorem 5.2. (i) GBB N is a cone, (ii) CC(v) =C(v) for all v 2 GBB N, (iii) C : GBB N!! IR n is an additive correspondence, (iv) fi : GBB N! IR n is an additive map, (v) fi(v) =fi(v) for v 2 GGB N iff v(n nfng) = n v(i); (vi) For v 2 BB N : fi(v) =fi(v) and fi is additive on BB N. 4

17 Proof. (i) That GBB N is a cone follows from characterization (ii) of general big boss games in Theorem 5.. (ii) That CC(v) = C(v) for v 2 GBB N follows from part (b) of the proof of Theorem 5.. (iii) Let v; w 2 GBB N and let z 2 C(v+w): We have to prove that there are x 2 C(v); y 2 C(w) such that z = x + y. In view of characterization (iii) of general big boss games, we can find ff ;ff 2 ; :::; ff n 2 [0; ] such that z i = ff i (v+w)(i)+( ff i )(v+w) Λ (i) for each i 2 N nfng: Take x 2 C(v) and y 2 C(w) such that x i = ff i v(i)+( ff i )v Λ (i) and y i = ff i w(i)+( ff i )w Λ (i) for all i 2 N nfng: Then z = x + y: (iv) That fi is additive follows from the additivity off n and g n : (v) Note that in view of (ii) of Theorem 5., for v 2 GBB N ψ! n m(v) = v();v(2); :::; v(n );v(n) v Λ (i) u(v) =(v Λ (); :::; v Λ (n );v(n) v(n nfng)) : So fi(v) = ffu(v) +( ff)m(v) for some ff 2 [0; ]: Then fi(v) = fi(v) iff ff = 2 and ff = n 2 iff v(n nfng) = v(i) because n u i (v) v(n) = v(n) n n m i (v) = v Λ (i) v(n nfng); n v Λ (i) n v(i): (vi) For v 2 BB N we have v(s) = i2sv(i) for each S with n =2 S. So by (v), fi(v) =fi(v), and by (iv) fi : BB N! IR n is an additive map. 2 6 Summary and concluding remarks Starting points of the paper were the observations that the Shapley value is additive on all cones of games and that the core is additive on the cone of convex games. Although most of the other interesting solutions do not have 5

18 this global additivity property it turned out that many of them have the additivity property on interesting cones of games such as the CIS solution and the ENSR solution. By analysing two additivity destroying factors for the fi-value, we obtained a collection of cones (Theorem 3.), where the fivalue is additive. Similarly, cones based on uniform compromise factors were found for the ff-value (Example 3.4). For multi solutions such as the core and the core cover also new regions of additivity were found. The core is additive not only on the cone of convex games but also e.g. on the cones of simplex games, dual simplex games and general big boss games and so is the fi-value on subcones of these cones. A negative result was found for the cone of exact games when the core is not additive (Example 3.9). We conclude with some remarks on further research. A systematic approach in looking for additivity regions of the core will yield new interesting cones of games. Few attention is paid to additivity regions of the nucleolus (Schmeidler (969)) although we know that the nucleolus coincides with the fi-value for the cones of big boss games and of dual simplex games, so on these cones the nucleolus is also additive. Exploration of other cones of additivity for the nucleolus and other solution concepts may also enrich cooperative game theory. References. Bondareva, O.N. (963). Some applications of linear programming methods to the theory of cooperative games (in Russian). Problemy Kibernet 0, Derks, J. (99). On Polyhedral Cones of Cooperative Games. Ph.D. Dissertation, University of Limburg, Maastricht, The Netherlands. 3. Dr agan, I., Potters, J. and S.H. Tijs (989). Superadditivity for solutions of coalitional games, Libertas Mathematica 9, Driessen, T.S.H. (988). Cooperative Games, Solutions and Applications. Theory and Decision Library, Series C: Game Theory, Mathematical Programming and Operations Research, Boston, Kluwer Academic Publishers. 5. Driessen, T.S.H. and S.H. Tijs (983). The fi-value, the nucleolus and 6

19 the core for a subclass of games. Methods of Operational Research 46, Gillies, D.B. (953). Some theorems on n-person games, Ph.D. Dissertation, Princeton, New Jersey. 7. Muto, S., Nakayama, M., Potters, J., and S.H. Tijs (988). On big boss games, The Economic Studies Quarterly 39, Nouweland, A. van den, Borm, P., Golstein Brouwers, W. van, Groot Bruinderink, R., and S.H. Tijs (996). Management Science 42, Schmeidler, D. (969). The nucleolus of a characteristic function game, SIAM Journal of Applied Mathematics 7, Schmeidler, D. (972). Cores of exact games, Journal of Mathematical Analysis and Applications 40, Shapley, L.S. (953). A value for n-person games, Annals of Mathematical Studies, 28, Shapley, L.S. (967). On balanced sets and cores, Naval Research Logistics Quarterly, 4, Shapley, L.S. (97). Cores of convex games, International Journal of Game Theory, Tijs, S.H. (98). Bounds for the core and the fi-value, in: O. Moeschlin and D. Pallaschke (eds.), Game Theory and Mathematical Economics, 23 32, Amsterdam, North Holland. 5. Tijs, S.H. and F.A.S. Lipperts (982). The hypercube and the core cover of n-person cooperative games, Cahiers du Centre d'etudes de Recherche Opérationelle, 24, Weber, R.J. (988). Probabilistic values for games, in: A.E. Roth (ed.), The Shapley Value: Essays in Honor of L.S. Shapley, 0 9. Cambridge, Cambridge University Press. 7