Symmetric core of cooperative side payments game Текст научной статьи по специальности «Математика»

Alexandra B. Zinchenko

Southern Federal University,

Faculty of Mathematics, Mechanics and Computer Science, Milchakova, 8 a, Rostov-on Don, 344090, Russia E-mail: zinch46@gmail.com

Abstract This paper concerns cooperative side payments games (with transferable utility and discrete) where at least two players are symmetric. The core and symmetric core properties are compared. The problem of symmetric core existence is considered.

Keywords: cooperative TU game, discrete game, core, symmetric core, balancedness.

1. Introduction

In many practical situations some of participants have the identical power (prestige, influence, resources, capitals). They are substitutes in associated cooperative game. Moreover, non-symmetric in the underlying problem agents may become substitutes in corresponding game. Player’s status can also changes in the zero-normalization of a game. It seems reasonable to require that symmetric players should receive the same payoff. However, almost no set-valued solution concepts (including the core, core-based solutions, von Neumann-Morgenstern stable sets, the bargaining set) that satisfy the equal treatment property. It is not difficult to provide the examples of cooperative games, where the core allocations assign to symmetric players vastly different payoffs. Even multi-solutions based on a concept of egalitarianism cannot satisfy the equal treatment property (see for instance (Dutta and Ray, 1989)).

The symmetric core is a subset of core satisfying the equal treatment property. This notion has been introduced in (Norde et al., 2002) for TU games with special structure: the airport game, generalized airport game, maintenance cost game, infrastructure cost game. The symmetric core was used to get a minimal collection of conditions that are equivalent to balancedness. In (Hougaard et al., 2001) the symmetric core was used for calculation of Lorenz-solution of a production economy with a landowner and peasants. To the best of our knowledge, the symmetric core was not yet discussed for general TU game.

Next section recalls some definitions and notations. The core and the symmetric core properties are compared in section 3. It will be shown that symmetric core satisfies the most core axioms. The last section is devoted to the problem of symmetric core existence.

2. Preliminaries

A cooperative game with transferable utility (TU game) is given as GT = (N,v), where N = {1 ,...,n}, n ^ 2, v : 2n ^ R, v(0) = 0. So-called discrete game GD differs from GT that v is integer-valued function and players payoffs must be integers (Azamkhuzhaev, 1991). In economic settings, the integer requirement reflects some forms of indivisibility. Both games summarizes the possible outcomes to a coalition

by one number, i.e. side payments are allowed. GT and GD can be also described as a games with nontransferable utility (NTU games). Let GN and GN be the sets of n-person TU and discrete games respectively, GN = GN U Gn. Denote by Q = 2n \ {N, 0} the family of proper coalitions. Given x G RN and 0 = K C N: x(K) = 5^ieK xj, x(0) = 0. The cardinality of coalition 0 = K C N is denoted by |K|. When there is no ambiguity, we write v(i), K\ i instead of v({i}), K\ {i} and so on.

Two players i,j G N are called symmetric (substitutes, interchangeable) in a game G G Gn if

v(K U i) = v(K U j) for every K G N \{i,j}. (1)

Player i G N is veto player in a game G G GN if v(K) = 0 for all K ^ i. Denote by veto(G) the set of veto players of G G GN. A game GT is called convex if v (K) + v (H) ^ v (K U H) + v(K n H) for K, H C N .A game GT is integer if v : 2n ^ Z, where Z denotes the set of integer numbers. The operator & : GN ^ GN will be used to compare TU and discrete game solutions, i.e. ^(GD) is an integer TU game corresponding to GD.

The set of feasible payoff vectors X *(GT) and pre-imputation set X (GT) of TU game GT are defined by

X*(GT) = {x G RN|x(N) < v(N)}, X(GT) = {x G RN|x(N) = v(N)}.

The related sets of discrete game GD are

X*(Gd) = X*(<f (Gd)) n ZN, X(Gd) = X(<f(Gd)) n ZN.

For any set G N C Gn a set-valued solution (or multisolution) on G N is a mapping f : G N RN which assigns to every G G G N a set of payoff vectors f (G) C X*(G). Notice that the solution set f (G) is allowed to be empty. A value of game G is a function f : G N ^ X(G). The core of TU game and core of discrete game are the sets

C(Gt) = {x G X(Gt)|x(K) > v(K), K G Q}, C(Gd) = C(<f(Gfl)) n ZN.

The formulas to obtain the CIS-value, ENSC-value, Shapey value and equal division solution of a game GT are

jeN^U)

CISi(GT) = K0 H----------------------->

n

ENSC,{GT) =

n

Sfei(GT) = Igl!(n-I,g|-1)!K^ U *) - K*0), £?A(GT) = —,

z' n! n

K^i

where i G N, v*(K) = v(N) - v(N \ K), K C N. The CIS-value is also called the equal surplus division solution. Notice that CIS-value, ENSC-value and equal division solution assign to every player some initial payoff and distribute the remainder of v(N) equally among all players. For CIS-value (the center of gravity of imputation set I(GT) = {x G X(GT)|xi > v(i),i G N}) initial payoff to player i G N is equal to its individual worth v(i). For ED-value and ENSC-value the initial payoffs are equal to zero and player’s marginal contribution v*(i) to grand coalition N, respectively. Thus, the ENSC-value assigns to any game GT the CIS-value of dual game (N, v*).

For a game G G GN denote by 9(G) the family of coalitions each of which contains only symmetric players

9(G) = {K G 2n||K| ^ 2, every i, j G K, i = j, are symmetric in G}.

Definition 1. A game G G GN is called semi-symmetric if at least two players are symmetric in G, i.e. 9(G) = 0. A game G G GN is (totally) symmetric if 9(G) = {{N}}. A game G G GN is non-symmetric if 9(G) = 0.

Let SGN = SGN U SGd be the set of semi-symmetric games G G GN.

Definition 2. The symmetric core SC(G) of a game G G GN is the set of core allocations for which the payoffs of symmetric players are equal

SC(G) = {x G C(G)|xi = xj for all i,j G K, i = j, K G 9(G)}.

Example 1. Let UH = (N, uH) be n-person (n > 3) unanimity game for a coalition H G Q: uH(K) = 1 for K D H, uH(K) = 0 otherwise. Since

f {H} if |H| = n - 1,

9(UH)= I {N \ H} if |H| = 1,

[ {H, N \ H} else,

then the game UH is semi-symmetric. Well known that any unanimity game is convex and C(UH) = {x G RN|xi =0, i G N \ H, x(H) = 1}. Therefore, the symmetric core SC(UH) consists of one point which is the Shapey value: SC(UH) = {Sh(UH)}, where Shi(UH) = for i G H, Shi(UH) = 0 otherwise.

Example 2. Consider situation with four investors having the endowments 80, 60, 50, 50 units of money (m.u. for short). Assume the following investment projects are available: a bank deposit that yields 10 interest rate whatever the outlay, two production processes that require an initial investment of 100 ore 200 m.u. and yields 15 ore 20 rate of return, respectively. The related four-person investment game (de Waegenaere et al., 2005) GT G GN is given by

N = {1, 2, 3, 4}, v(N) = 284, v(1) = 88, v(2) = 66, v(3) = v (4) = 55,

v(1, 2) = 159, v(1, 3) = v(1,4) = 148, v(2, 3) = v(2,4) = 126, v(3,4) = 115,

v(1, 2, 3) = v(1, 2, 4) = 214, v(1, 3,4) = 203, v(2, 3,4) = 181.

We obtain non-convex (v(2,4) + v(3,4) > v(4) + v(2, 3, 4)) balanced semi-symmetric game with symmetric players 3 and 4, 9(GT) = {{3,4}}. The core of game GT has 16 extreme points whereas symmetric core is the convex hull of 4 points

SC(GT) = co{xx, x2, x3, x4},

x- = (100-, 68-, 57-, 57-),

v o’ o’ o’ 2

1

3

x

2 2 2’ (98, 66, 60, 60),

c2 = (90-, 78-, 57-, 57-), v 2’ 2’ 2’ 2h

A = (88, 76, 60, 60).

Denote by GT = (N, v0), where

( 5 if |K|G{2, 3}, v0(K) = I 20 if K = N,

^ 0 else,

the zero-normalization of game GT. All players are substitutes in GT, 9(GT) = {{1, 2, 3,4}}. The symmetric core of game GT consists of one point

SC(GT) = {x0}, x0 = (5, 5,5, 5) = Sh(GT) = CIS(GT) = ENSC(GT) = ED(GT).

The payoff vector x0 corresponds to symmetric core allocation x6 = (93, 71,60, 60) of original game Gt• Notice, that x6 = x , it is equal the Shapey value SH(Gt) of original game, but does not coincide with the barycenter (94 j, 72|, 58|, 58|) of the symmetric core of game GT.

In game theory literature there exist two (equivalent) versions of TU game balancedness: a game GT G GN is called balanced if it has a nonempty core ore if it satisfies the Bondareva-Shapley condition

AKv(K) < v(N), A : Q ^ R+, ^ AK = 1, i G N, (2)

KEfi KEfi, K3i

see (Bondareva, 1963) and (Shapley, 1967). Since (2) is necessary but not sufficient condition for the nonemptiness of core of discrete game, the unified definition is required.

Definition 3. A game G G GN with nonempty core is called balanced.

We need the following axiom to be satisfied by solution y>.

Axiom 3.1 (equal treatment). For all G G G N, all x G y>(G) and every symmetric players i, j in G: xi = xj.

Known that Sh(GT), CIS(GT), ENSC(GT) and ED(GT) satisfy equal treatment. From above definitions it straightforwardly follows that:

• the symmetric core of a game G G GN may be empty;

• the symmetric core of TU game GT is a convex subset of its core;

• the symmetric core of non-symmetric game G G GN coincides with its core, therefore, apart from their different definitions the real difference is exposed for semi-symmetric balanced games;

• the symmetric core of balanced symmetric TU game consists of one point which is the equal division solution SC(GT) = {ED(GT)};

• the symmetric core of balanced semi-symmetric TU game contains all core selectors satisfying equal treatment, in particular, the nucleolus that realizes a fairness principle based on lexicographic minimization of maximum excess for all coalitions;

• if the Shapley value of semi-symmetric TU game satisfies the core inequalities then it belongs to symmetric core, the Shapley value is always symmetric core allocation on the domain of convex TU games;

• the CIS-value, the ENSC-value, the equal division solution which "have some egalitarian flavour" (Brink and Funaki, 2009) and any convex combination of these

solutions cannot belong to symmetric core of balanced semi-symmetric TU game.

A nonempty core of NTU game (even 3-person) may contains no equal treatment outcomes (Aumann, 1987). The following two propositions show that balancedness of TU game is the necessary and sufficient condition for nonemptiness of its symmetric core, but the same is not true for balanced discrete game.

Proposition 1. Let GT G SGN. Then SC(GT) = 0 iff C(GT) = 0.

Proof. If SC(GT) = 0 then C(GT) = 0 by inclusion SC(GT) C C(GT). Assume now that C(GT) = 0 and take x1 G C(GT). If x1 G SC(GT) then SC(GT) = 0. Otherwise, there exist a coalition K G 9(GT) and players i,j G K such that x1 < xj. Construct x2 G RN as follows: x2 = xj, x2 = x1, x^ = x1 for l G N\{i, j}.

Using (1) we see that x2 € C(GT)- By core convexity, x3 = xl+G C(GT)- So, we get the core allocation x3 satisfying x3 = x3, xf = x^ for l G N \ {i, j}. If x3 G SC(GT) then by repeated application of above procedure one obtains the payoff vector belonging to SC(GT). □

Proposition 2. There exist discrete games GD G SGd such that C(GD) = 0 but SC (Gd) = 0.

Proof. Consider discrete games GD, defined by set function vs on N: vs(K) G {0,1} for K c N and vs(N) = 1. The associated TU game ^(GD) = (N, vs) is simple. Assume |veto(^(GD))| > 2. Then C(^(GD)) = co{ei G ZN|i G veto(^(GD))} and C(GD) = {ei G ZN|i G veto(^(GD)), where ej =0 for i = j, ei = 1. Obviously, veto players are substitutes in games ^(GD) and GD. However xi = xj for all x G C(GD) and every (i, j) G veto(GD). Thus SC (GD) = 0. □

The core of TU game has been intensely studied and axiomatized. We shall formulate some convenient properties of a solution concept on G N C GN which has been employed in the well-known core axiomatizations. The axiomatic characterizations of discrete game solutions are not yet provided.

Axiom 3.2 (efficiency). x(N) = v(N) for all x G y>(G) and all G G GN.

Axiom 3.3 (symmetry). For all G G G N and every symmetric players i, j in G: if x G y>(G) then there exists y G y>(G) such that xi = yj, xj = yi and xp = for p G N \ {i, j}.

Axiom 3.4 (modularity). For any modular game G G GN generated by the vector x G RN: y>(G) = {x} .

Axiom 3.5 (antimonotonicity). For any pair of games G1, G2 G G N defined by set functions v1, v2 on N such that v1(N) = v2(N) and v1(K) < v2(K) for all K c N, it holds that ^(G2) C ^(G1).

Axiom 3.6 (reasonableness (from above)). For all G G G N, all x G ^(G) and every

i G N: xi ^ max {v(K U i) — v(K)}.

KCN\i

Axiom 3.7 (covariance). For any pair of games G1,G2 G G N defined by set functions v1, v2 such that v2 = av1 + ^ for some a >0 and some ft G RN it holds that ^(G2) = a^(G1) + ft.

Axiom 3.8 (projection consistency (or reduced game property)). Let G G G N, 0 = H C N and x G ¥>(G), then = (H, ) G GH and xH G ^(fiH), where xH =

is the projected reduced game with respect to H and x.

Known (Llerena and Carles, 2005) that the core is the only solution on GN satisfying projection consistency, reasonableness (from above), antimonotonicity and modularity. Notice that projection consistency is one of the fundamental principle used in this field. By summarizing the statements formulated above we can say that the symmetric core of balanced semi-symmetric TU and discrete games satisfies equal treatment, efficiency, symmetry, modularity, reasonableness (from above) and many other core axioms based on only the original game. Theorem 1 (below) shows that for the class of balanced semi-symmetric games the symmetric core is in conflict with antimonotonicity, covariance and projection consistency. All these properties involve the pairs of games.

Lemma 1. Let G G SGN is a balanced game and G0 is its zero-normalization.

Then G0 G SGN, SC(G0) C SC(G) and there exist games G G SGN such that SC(G0) = SC (G).

Proof. The zero-normalization G0 of any game G G GN is uniquely determined by set function v0 on N, where

(1) and (3) imply that v0(KUi) = v0(KUj) for all K C N\{i, j}. Thus, symmetric players in G remain symmetric in G0. Example 2 shows that non-symmetric in G players can become symmetric in G0. If G = GT then a linear system defining SC(GT) contains the one for SC(GT) and, perhaps, additional equality constraints. So SC (GT) C SC(GT). In view of Example 2 this inclusion can be strict. For discrete game G = GD the final part of lemma is proved analogously. □□

Theorem 1. Let G G SGN is a balanced game. Then SC(G) does not satisfy

(i) Axiom 3.5;

(ii) Axiom 3.7 even for a =1 and ft = (v(1),..., v(n));

(iii) Axiom 3.8.

Proof, (i) Consider two balanced four-person TU games GT, GT defined by set functions v1, v2 such that

(xi)ieH G RH and

0 if K = 0,

v(K) if 0 = K c H,

v(N) — x(N \ H) if K = H,

0 = K C N.

(3)

ieK

Obviously, G0 G SGN. Let i, j G N, i = j, are symmetric players in G. The formulas

v 1(K)

2 |K | = 2, 4 |K | = 3, 6 K = N,

0 else,

The games G^ and G^ are symmetric and semi-symmetric, respectively. Q(GT) = {{1, 2,3,4}}, Q(GT) = {{3,4}}, vX(N) = v2(N) and v^K) < v2(K) for all K C N. It holds that

SC(G2t) = co{(2,1, li, 1±), (2, 0, 2, 2), (1,1, 2, 2)} £ SC(G^) = {(l|, l|, l|, 1^)}. Consider now discrete games GD, GD corresponding to given TU games. We have

SC(GD) = {(2,0, 2, 2), (1,1, 2, 2)} C SC(G^) = 0.

Thus, antimonotonicity is violated by SC(G).

(ii) This statement follows from lemma 1.

(iii) In four-person TU game GT defined by

N = {1, 2, 3, 4}, v(N) = 8, v(i) = 0, i G N, Ï

v(1, 2) = v(1, 3) = v(1,4) = v(2, 3) = v(2, 4) = 2, v(3,4) = 3, v(1, 2, 3) = v(1, 2,4) = 6, v(1, 3, 4) = 5, v(2, 3,4) = 4

players 3 and 4 are symmetric, Q(GT) = {{3,4}}. The symmetric core is the convex hull of four points SC(GT) = co{x1, x2, x3, x4}, where x1 = (4,0, 2, 2), x2 = (4,1,1^, 1^), x3 = (1,3, 2,2) and x4 = (2, 3,1^, 1^). The projected reduced game RH* = (H, ) relative to H = {1,2, 3} at x2 is defined by:

rf2( 1,2,3) = 6^, rf2(*) = 0, *6iî, rf2(l,2) = rf2(l,3)=rf2(2,3) = 2.

The reduced game is symmetric. Its symmetric core consists of one point (2|, 2|, 2|). The restriction of x2 to H, x2H = (4,1,1^-), does not belong to the symmetric core of reduced game. For discrete game GD corresponding to last TU game GT we have SC(Gd) = {x1,x3,x5,x6}, where x5 = (3,1, 2, 2), x6 = (2, 2, 2, 2). The projected reduced game RH relative to H = {1, 2,3} at x1 is defined by:

rfx (1, 2, 3) = 6, rfx (i)=0, i G H, rfx(1, 2)= rfx (1, 3) = rfx (2, 3) = 2.

Since the reduced game is symmetric SC(RX ) = {(2, 2, 2)}. The restriction of x1 to H, xH = (4,0,2), does not belong to SC (Ri ). So SC (G) does not provide projection consistency. □

It has been interesting to study the interrelation between the symmetric core of a game G G SGN and strongly egalitarian core allocations.

Definition 4. Let G G Gw, x G G(G) and x G Rw is obtained from x by permuting its coordinates in a non-decreasing order: xi < X2 < ... < x„. A core allocation x is Lorenz allocation (Lorenz maximal, strongly egalitarian ) iff it is undominated in the sense of Lorenz, i.e. there does not exist y G C(G) such that Ylï= 1 Vi ^ Sf=i for all p G {1,..., n — 1} with at least one strict inequality.

For a game G G GN denote by LA(G) the set of its Lorenz allocations.

Example 3. Consider balanced four-player TU game GT defined by

( 7 if (K = {1, 2}) V (K = {1, 3}),

v(K) = I 12 if K = N,

I 0 else.

In was proved (Arin et al., 2008) that the set of Lorenz allocations is of the form LA(Gt) = {xG C(Gt)\ x = (7 — fi, fi, fi, 5 — fi), 2- ^ jj, ^ 3-^-}.

Taking ¡jl = 3^, ¡jl = 2^ and ¡jl = 3 yield the lexmax solution Lmax^Gx) = (3^, 3^, 3 lj), the lexmin solution Lm*n(G-r) = (4^, 2^, 2^-, 2^-) and least squares solution LS(GT) = (4, 3, 3, 2), respectively ((Arin et al., 2008, p.571)). By the formulas in section 2 one obtains SIi(Gt) = (4^,3, 3,l|) G LA(Gt), CIS(Gt) = ENSC(GT) = ED(GT) = (3,3, 3,3) G LA(GT).

The next theorem states that the symmetric core of balanced semi-symmetric TU game contains all Lorenz allocations. Besides, SC(GT) is externally stabile with respect to Lorenz domination, but internal stability does not hold.

Theorem 2. Let GT G SGN is a balanced game. Then

(i) LA(GT) Ç SC(GT) and the inclusion can be strict;

(ii) SC(GT) Lorenz dominates every other core allocation.

Proof, (i) LA(GT) satisfies equal treatment and LA(GT) Ç C(GT). Therefore, LA(GT) Ç SC(GT). The four-person TU game in Example 3 is semi-symmetric Q(GT) = {{2, 3}},

i^>=~«4444>’(4444>

C SC(GT) = co{(2, 5, 5,0), (7,0, 0, 5), (12,0, 0,0)}.

(ii) If C(GT) = SC(GT) then the statement is straightforward. Let C(GT) = SC(GT) and take x0 G C(GT)\SC(GT). Then there exists K G Q(GT) and i,j G K such that x° > x0. By symmetry there is y G C(GT) with y* = x0, yj = x0, y; = x0

for l G N \ Consider x1 = æ £y ■ By core convexity x1 G C(Gt)■ Vector x1

Lorenz dominates x0 (x1 >-L x0) since xj = x1 = x° — â = x0 + â, x1 = xj0 for l G N\{i,j}, â > 0. Repetition of this procedure gets the sequence x0, x1,..., xp core allocations, where xk L xk-1 for all k G {1,...,p}, x0 G SC(GT), xp G SC(GT). The transitive property of Lorenz domination completes the proof. □

4. Existence conditions

The balancedness condition (2) is derived by means of dual linear programming problems associated with a game GT G GN

f(x) = x* ^ min, x* ^ v(K), K G i?, (4)

iGN iGK

g(A) = ^ v(K)AK ^ max, ^ AK = 1, i G N, A G R+-2. (5)

Kefi,ieK

The condition (2) can be as well written as

^ AKv(K) < v(N), A G ext(Mn),

where ext(Mn) is the set of extreme points of problem (5) constraint set Mn. The number of extreme points and their explicit representation known only for small n

|ext(M 3)| = 5, |ext(M 4)| = 41, |ext(M5)| = 1291, |ext(M 6)| = 200213.

We concentrate now on n-person non-negative semi-symmetric TU games in zeronormal form (SGN)+. The following example illustrates how the problem (4) is modified by replacing the core by symmetric core.

Example 4- Consider two four-person games (Gy)1, (Gy)2 G (SGN)+ with two and three symmetric players, 9((Gy)1) = {{3,4}}, 9((Gy)2) = {{2, 3,4}}. The explicit representations of (4) and modified problems given in table 1. It is remarkable that the number of extreme points of modified dual problems constraint sets M;4, where s is the number of symmetric players, decreases as s increases: |ext(M24)| = 21, |eXt(M;4)| = 6.

Table 1.

Original problem Modified problem 1, 9((G^)1) = {{3,4}} Modified problem 2, Q((G°T)2) = {{2,3,4}}

fix) = Xl + X2 + xs + X4 —s- min Xi 0, i £ {1, 2, 3, 4} Xl + X2 > u(l, 2) xi +®3 >^(1,3) Xl + X4 ^ 1/(1, 4) X2+X3 > v(2, 3) X2 + X4 > i/(2,4) X3 + X4 i/(3, 4) X1 + X2 + X3 >^(1,2,3) X1 + X2 + X4 v(l, 2, 4) xi + X3 + X4 v(l, 3,4) X2 + X3 + X4 > i/(2, 3,4) fix) = Xl + X2 + 2X3 —> rnin Xi 0, i £ {1,2, 3} Xl + X2 > u(l, 2) Xl + £3 > v(l, 3) X2 + X3 > 1/(2, 3) 2x3 ^ i/(3,4) *1 + *2 + *3 > i/(l, 2, 3) æi + 2®3 i/(l, 3,4) + 2x3 > v(2, 3,4) fix) = Xl + 3x2 —>■ rnin Xi^Q,i£ {1,2} Xl + X2 > ¡/(1, 2) 2æ2 ^ v(2, 3) æi + 2æ2 ^ v( 1, 2, 3) 3®2 ^ i/(2, 3,4)

The symmetry of all players makes a game especially easy to handle. The criterion for existence of its core (and, by Proposition 1, for symmetric core too) contains (n — 1) inequalities only

v(K) v(N)

i for all Ken.

|K | n

It is then natural to focus the attention on games with (n — 1) symmetric players. Notice that any such game is determined by 2(n — 2) numbers v(K), K G ^1 U «2, where

«1 = {{2, 3}, {2, 3,4},..., {2,..., n}}, ^2 = {{1, 2}, {1, 2, 3},..., {1,...,n — 1}}.

A few of their applications:

• market with one seller and symmetric buyers;

• games with a landlord and landless workers;

• weighted majority game with one large party and (n — 1) equal sized smaller parties;

• patent licensing game with the firms each producing an identical commodity and a licensor of a patented technology (Watanabe and Muto, 2008);

• subclass of games related information collecting situations under uncertainty (Branzei et al., 2000) where an action taker can obtain more information from other agents;

• big boss games (Muto et al.,1988) with symmetric powerless players.

The characterization of such games and the sufficient conditions under which the symmetric core is a singleton have been provided in (Zinchenko, 2012). Let

GT G (SGN)+, Q(GT) = {{2, ...,n}} and n > 3. The symmetric core of game G^ is nonempty iff the system

v0(T) + < v°(N), < v°(N), H€nuT€Ü2

|HI |H I

is consistent. Notice that system consists of (n — 1)(n — 2) inequalities. If G^ G (SGn)+ is a balanced game, Q(GT) = {{2, ...,n}}, n > 4 and v0 satisfies at least one of three equalities

\ {1, n}) = 1,0(N), \ 1) + V°{1, 2) =

n — 2 n — 1

1/0 + v°{N \n) = v°{N)

n — 1

then SC(GT) consists of a unique allocation.

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