A simple way to obtain the sufficient nonemptiness conditions for core of tu game Текст научной статьи по специальности «Математика»

Alexandra B. Zinchenko

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

Abstract The system of linear constraints like one that determines the core of TU game is considered. Expressing its basis solutions through characteristic function we obtain a list of sufficient conditions under which the core is nonempty. Some of them are the generalizations of known results. Keywords: cooperative TU game, core, balancedness, sufficient conditions.

1. Introduction

The core (Gillies, 1953) is the most frequently applied multivalued solution of cooperative game theory. Under the grand coalition advantageous the analysis of any TU game usually begins with verification of core existence. For concrete game this possible to do by means of linear programming problem with constraint set covers the core. The core of TU game is empty if and only if the optimal value of this problem is strictly greater than a grand coalition's weight. If we try to prove the core non-emptiness for certain class of games we need condition, expressed through characteristic function. Such is Bondareva-Shapley balancedness condition (Bondareva, 1963; Shapley, 1967). That condition is equivalent to linear system with entries corresponding to the extreme points of polytope in R2 _1. The number of extreme points and their explicit representation known only for small n.

There exist more simple but only sufficient conditions. The most known is the convexity of game. The minimal convexity test consists of 2 inequalities

(Voorneveld and Grahn, 2001). In this paper the simple procedure to generate sufficient nonemptiness conditions for core of TU game is described. For any basis matrix consisting of coalitional characteristic vectors we can obtain sufficient condition defined by a system of 2n —n—1 linear inequalities. So each condition determines a cone in linear space of all TU games.

The following table illustrates what increases the number of inequalities in conditions mentioned above when n increase.

n 3 4 5 6 7

balancedness condition 5 41 1291 200213 132422035

minimal convexity test 6 24 80 240 672

sufficient condition 4 11 26 57 120

The paper has the following contents. Next section recalls the standard facts of cooperative game theory which are useful later. The balancedness condition, minimal balanced sets and some classes of balanced games are described in third section.

The last section contains the set of sufficient nonemptiness conditions corresponding to special bases and shows that some of them are the generalizations of known results.

2. Preliminaries

A cooperative TU game is a pair (N,v) where N = {1, 2,...,n} is a player set, n > 2, v : 2n ^ R is a set function satisfying v(0) = 0. Throughout the paper we identify (N, v) and v. The class of TU games with player set N will be denoted by GN. A payoff vector for a game v e GN (ore allocation) is a vector x e R". A subset of N is called a coalition, v(S) expresses the worth of coalition S and eS is the characteristic vector of coalition S, i.e. (eS)i = 1 if i e S, (eS)i = 0 otherwise. Sometimes (for simplicity) we shall write N \ i instead of N \ {i}, v(123) instead of v({1,2, 3}) and so on. For any S e 2N and x e R" let x(S) = J2ieS x% and x(0) = 0. The cardinality of coalition S is written as \S|. The rank of matrix A is denoted as rank(A). The dual game v* of v e GN is determined by

v* (S) = v(N) - v(N \ S) for every S C N.

A game v e GN is called:

• zero-normalized if v(i) = 0 for all i e N,

• nonnegative if v(S) > 0 for all S C N,

• monotonic if v(S) < v(T) for all S C T C N,

• N-essential if ieN v(i) < v(N),

• convex (concave) if

v(S) + v(T) < (>) v(S U T) + v(S n T) for all S,T C N.

If the grand coalition is formed then players can divide the amount v(N). The order on N is a bijection n : N ^ N. The set of all orders n = {n\,..., n"} is given by n(N). The Weber set of game v e GN, denoted by W(v), is the convex hull in R" of all n-marginal vectors

W(v) = conv{mn(v) \ n e n(N)}

were

< (v) = v (Sn) - v (S-), Sn = {nh...,ni}, i e N.

W(v) = 0 for any game v e GN. For the description of other set-valued solution concepts is used the set

X(v) = {x e Rn \ x(N) = v(N)}

of efficient payoff distributions of v(N) named the preimputation set and its subsets: the imputation set

I(v) = {x e X(v) \ xi > v(i), i e N}, the dual imputation set

I*(v) = {x e X(v) \ xi < v*(i), i e N}.

I(v) = 0 iff '%2ieN v(i) < v(N). If a game v G GN is N-essential then I(v) is an (n — l)-dimensional simplex with extreme points fi(v) G Rn, i G N, where

(>"(v»' = { v(N) — Em, j i = j foral" G N

I * (v) = 0 iff J2ieN v *(i) > v(N). In case of strict inequality I*(v) is an (n — l)-dimensional simplex with extreme points gi(v) G Rn, i G N, where

(g(v))j A v(N- EkeNM v*(k), i = j f0r all i G N

The core of a game v G GN is a subset of core cover (Branzei and Tijs, 2001) CC(v) = I(v) n I*(v) defined by

C(v) = {x G I(v) | x(S) > v(S), S C N}.

Thus v(i) < x, < v* (i) for all i G N and x G C(v). Web(v), I(v), I*(v), CC(v) and C(v) are the polytopes in Rn. The set of extreme points of polytope P will be denoted by ext(P).

Two players i,j G N are symmetric in v G GN if v(S U i) = v(S U j) for every S C N \ {i,j}. A player i G N is a veto player in v G GN if v(S) = 0 for each S $ i. The set of veto players is denoted by Veto(v).

3. Balancedness

Let

n1 = 2n \{0}, Q2 = 2n \{N, 0}, Q(ki,k2) = {S G 2N | ki <SI< k2}

are the sets of nonempty coalitions, proper coalitions and coalitions with restricted size. It is proved (Bondareva, 1963; Shapley, 1967) that the core of a game v G GN is nonempty iff

Y^ Xsv(S) < v(N) for all X G ext(An) (1)

sen1

where

A" = {X G R+ -1 | £ esXs = eN}. sen1

A game v G GN satisfying (1) is called balanced game. The condition (1) is called balancedness condition. In some game theory literature a game is balanced if it have a nonempty core.

A collection F = {F}"=1 of coalitions F, G Q2 is called minimal balanced set if there exists X G ext(An) such that Xs > 0 for S G F and Xs = 0 otherwise (this definition can be given in alternative form). The vector a(F) = (a(F,))™=1 with a(F,) = XFi is called weight vector for F. In terms of minimal balanced sets a necessary and sufficient condition for nonemptiness of the core of game v G GN can be written as

a(F,)v(F,) < v(N) for all F G Fn, (2)

FieF

where Fn denotes a family of all minimal balanced sets on N.

Definition 1. A linear inequality is called convexity-inequality, concavity-inequality, union-inequality ore balancedness-inequality if it contains in system defining corresponding property of game v.

The number of balancedness-inequalities grows very rapidly with n. Some of them define necessary and sufficient nonemptiness conditions for other sets. I(v) = 0 (I * (v) = 0 ) iff the corresponding to {{1},..., {n}} ({N \{1},...,N \ {n}}) balancedness-inequality is satisfied. CC(v) = 0 iff a game v e GN satisfies inequalities corresponding to {{1},..., {n}}, {N \ {1}, ...,N \ {n}} and {{i}, N \ {i}}, i e N.

Let us list some types of balanced games which characterization yields a sufficient conditions for core existence. A game is T-simplex (Branzei and Tijs, 2001) where T e fi1 if its core is a subsimplex of imputation set, i.e. a game is N-essential and

C(v) = conv{fi(v)\i e T}.

A game is dual-simplex (Branzei and Tijs, 2001) if its core is a subsimplex of dual imputation set, i.e. v(N) <Y1 ieN v* (i) and there is a coalition T e fi1 such that

C(v) = conv{gi(v)\i e T}.

A balanced game satisfies the CoMa-property (Hamers et al., 2002) iff the extreme points of its core are marginal vectors, i.e. ext(C(v)) C ext(Web(v)). The convex games satisfy the CoMa-property because for them C(v) = Web(v) (Shapley, 1971). The non-convex games that satisfy the CoMa-property are: information games (Kuipers, 1993), assignment games (Hamers et al., 2002), cost spanning tree games (Granot and Huberman, 1981). If a game v e GN is permutationally convex with respect to an order n e n(N) then corresponding marginal vector mn (v) is a core allocation (in other words ext(C(v)) n ext(Web(v)) = 0). But the reverse is not true in general (Velzen et al., 2005).

A game v e GN is clan game with coalition CL = 0 as clan (Potters et al., 1989) if: it satisfies the union property

v(N) - v(S) > v*(i) for all S C N with CL C S;

ieN\s

v and v*(i), i e N, are non-negative; v(S) =0 if CL C S (clan property). A game v e GN (n > 3) is called a big boss game with player 1 as big boss (Muto, et al., 1988) if: v is monotonic; v(S) = 0 for all S C N with 1 e S (boss property); v(N) - v(S) >Y1 ieN\S v*(i) for all S C N with 1 e S (union property).

4. Sufficient conditions

Consider the system

x(S) > v(S), S e fi2, —x(N) > -v(N) (3)

differs from one determines the core of game v e GN that efficiency condition

x(N) = v (N)

(4)

is replaced with inequality x(N) < v(N). If the system (3) is non-solvable than the core of game v e GN is empty. Let x is a solution to system (3). If it satisfies (4) than x e C(v). Otherwise the payoff vectors xi, i e N, where

X - -

= i = j,

j \ v(N) -EkeN\i xk, i = j,

are the core allocations. The system (3) can be presented in the form

Ax > V, (5)

where x G R", v = (v(S))Seni, V(S) = v(S), S G fi2, v(N) = -v{N), A is the (2n - 1) x n matrix with row vectors AS refering to S e fi1. The first (2n - 2) rows of matrix A are the characteristic vectors eS of coalitions S e fi2. The last row AN = -eN of A corresponds to the grand coalition. Obviously rank(A) = n.

Let B = (bij )n be a basis of A. By transposition of rows the matrix A can be represented in the form A = (B D)T and system (3) becomes

Bx>~Pb , Dx >1/£>,

where T>b = (~P(S))seB, T^d = (v(S))seD is a basis, nonbasis partition of variables in the vector v . The system Bx = vb determines the unique basis solution xB = B~1Vb ■ If xB satisfies DxB > Vd then it is the feasible solution to system (5). The following provides a simple sufficient condition. Let v e GN and B is a basis of matrix A in (5). If v satisfies

D(B-1Vb)>vd (6)

then C(v) = 0.

The next two examples illustrate the above technique for the most simple bases. Example 1. Take basis

B = {A{1} ,...,A{n} }T

with characteristic vectors of single player coalitions as rows. Thus B and B-1 are the n x n identity matrixes, xB = (v(1),... .v(n)). Condition (6) becomes

v (S) v (i), S e n1.

ieS

It determines the set of N-simplex games. For all i e N the payoff vector xi coincides with extreme point f i(v) of imputation set.

Example 2. Basis

B = {A{1},...,A{n-1},AN }T

differs from previous one that A{n} is replaced on AN = —eN. The matrixes B, B-1 = (b—1)n and basis solution xB are determined by

- b-1 = i-1- --j A (i = xB = I v (i), , i = »,

bij = bij = 1 0: «HtWise. = I v) - En.-1' v(i), i = n,

xB is a core allocation and coincide with extreme point fn(v) of imputation set. Condition (6) becomes

v(S) v(i) if n GS, v(S) < v(N) — Y v(i) if n G S, S G Q(2, n — l).

ies ieN\s

We obtain the description of subcone of balanced games containing the set of such T-simplex games that T 3 n.

Definition 2. Let F G Fn is a minimal balanced set and 3 G R. A linear inequality

£ a(F,)v(Fi) < 3 FieF

is called strengthened-balancedness-inequality if ¡ < v(N).

The next theorem provides an explicit representation the condition (6) for basis

B = (AN\1 An\2 ...An\")t (7)

consisting of characteristic vectors for all coalitions of size (n — l). The corollary 1 show that this condition consists of balancedness-inequality and strengthened-balancedness-inequalities only.

Theorem 1. Let v G GN. The following two conditions

<,(«), (8)

nl

ieN n — l

imply that C(v) = 0.

Proof. Consider the basis (7). The matrix B and inverse matrix B-1 have the form

Ti— 1 '

Therefor, xB = B is determined by

n — l

Obviously, xB (N) is equal to the left side of inequality (8). The equality

£ £ v(N \ j) = |S<| £ v(N \ i) + (|S|-1) £ v(N \ i)

ies jeN\i ieN\s ies

implies that xB (S) is equal to the right side of inequality in (9). Thus condition (6) holds. □

Corollary 1. The conditions in Theorem 1 consist of one balancedness-inequality and 2n — n — 2 strengthened-balancedness-inequalities

5 € «(!,„-2), (11)

with identical right side, where xB is determined by (10).

Proof. Known that F1 = |JieN{N \ i} is minimal balanced set with weight vector «(i11) = ..., ^rx)• Corresponding to F1 inequality in (2) coincides with (8). Therefore, (8) is balancedness-inequality. Take S G f?(1,n — 2). After transformation the right side of inequality in (9)

(\S \ + 1 — n)E v (N \ i) + |S|E i£N\S

v(N \ i)

n1

1^1 J2ieN v(N\i)-(n- 1) v(N \ i) = |g| J2ieN v(N \ i) ^

n-l n-l \%>

ies

\S\xB (N ) - £ * (N \ i),

ies

we obtain that the system (9) is equivalent with (11). The collection of coalitions F1 = {{N\S}u(Uies{i})} belongs to Fn because it is the partition of N, a(F1) = (1,..., 1). The complementation gives minimal balanced set

F2 = {S U (J {N \ i})}

ies

with weight vector a(F2) where

a(Fl)

a(F2

E^F) - 1 ST

According to (8), xB (N) < v(N). In view of definition 2 any inequality in (11) is strengthened-balanced-inequality. The number of such inequalities is equal to \D\ — 1 = 2n — n — 2. □

The following theorem show that it is possible to replace all (ore some) inequalities in system (9) by union-inequalities.

Theorem 2. Let v G GN. For any fixed r G {0,...,n — 2} the balancedness-inequality (8) together with strengthened-balancedness-inequalities

v (S ) + E ies ieN

v(N \ i)

\S\ - n - 1

and union-inequalities

< \ , Sen(l,r), (12)

/(N) - v(S) > v* (i), S e n(r + l,n- 2), (13)

ieN\s

imply than C(v) =

1

Proof. Of course every inequality in (13) is a union-inequality (for CL = S). In view of corollary 1 it is sufficient to prove that from (13) follows

ps = |S| £ v(N \ i) - (n - 1)(v(S) + £ v(N \ i)) > 0, S e n(r + 1,n- 2)

ieN ies

ore

|S| £ v(N \ i) - (n - 1)v(S) + (|S|-n + 1)£ v(N \ i) > 0, S e n(r + 1,n-2).

ieN\s ies

From (8) it follows that

(|S|-n + 1)£ v(N \ i) > (n -S ^1) £ v(N \ i) + (|S|-n +1)(n - l)v(N).

ies ieN\s

Two last inequalities implies

n - l ieN\s

v(N) - v(S) - ( £ (v(N) - v(N \ i)) = v(N) - v(S) - £ v*(i).

ieN\s ieN\s

Using (13) we have ps > 0 for all S e fi(r, n - 2). □

Let BALn be the cone of balanced games v e GN and BALn C BALn be the subcone generated by conditions in Theorem 2 . Since each inequality in (12) follows from corresponding inequality in (13) and balancedness-inequality (8) then

BALn C BALI C ... C BALnri_2.

The next example show that BALn = 0.

Example 3. Consider 5-person game :

v(i) =0 for all i e N = {1,..., 5}, v(N) = 10, v (12) = v (13) = v (14) = v (l5) = 2, v (23) = v (24) = v (25) = v (34) = v (35) = 1, v (123) = v (124) = v (125) = v (134) = v (135) = 5, v(234) = v(235) = v(345) = 4,

v(1234) = v(1235) = v(1245) = v(1345) = 8, v(2345) = 7. The core have five extreme points

ext(C(v)) = {(2, 2, 2, 2, 2), (3,1, 2, 2, 2), (3, 2,1, 2, 2), (3, 2, 2,1, 2), (3, 2, 2, 2,1)}. This is monotonic, superadditive, but non-convex even for the grand coalition (v(1234) + v(1235) > v(N) + v(123)) game. Players marginal contributions to the grand coalition are:

v*(1) = 3, v*(2) = v*(3) = v*(4) = v* (5) = 2. Take r = 0. Since the players 2-5 are symmetric it is sufficient to verify:

S| = 1=^ v(N) - v(1) > 4v*(2), v(N) - v(2) > v*(1) + 3v*(2), S| = 2=^ v(N) - v(12) > 3v*(3), v(N) - v(23) > v*(1) + 2v*(4), S| = 3=^ v(N) - v(123) > 2v*(2), v(N) - v(234) > v*(1) + v* (2). All inequalities hold. From (10) we obtain

xB = (2.75,1.75,1.75,1.75,1.75).

As v(N) — xB (N) = 0.25 then xB ^ C(v). But we have five core allocations associated with basis B:

x1 = (3,1.75,1.75,1.75,1.75), x2 = (2.75, 2,1.75,1.75,1.75), x3 = (2.75,1.75, 2,1.75,1.75), x4 = (2.75,1.75,1.75, 2,1.75), x5 = (2.75,1.75,1.75,1.75, 2).

Next theorem provides the sufficient condition corresponding to basis

ie(N\H)\i

B = ((A{i})iEH, (AN\i)ie{N\H)V< ,An)t

where H e (2N \{N})

Theorem 3. Let v e GN. Let also i* e N, H e (2N \ {N}) \ {i*} are fixed and n2H = {S e n2\ S = {i} for i e H, S = {N \ i} for i e (N \ H) \ i*}. The following two conditions

v(S) < E v(i)+ E v*(i), S e n2H, i* e S, (14)

iesnH ies\H

v(S) < v(N) - E v(i) - £ v*(i), S e n2H, i* e S, (15)

ieH\s ie(N\H)\s

imply that C(v) =

Proof. Consider the vector xH determined by

( v(i), i e H,

(xH)t = v* (i) i e (N \ H) \ *

( H I v(N) -£ v j) - £ v(N \ j), i = i*. K jeH je(N\H)\i*

If i* e S then xH(S) coincides with right side of inequality in (14). If i* e S then

x H (S) = E v(i)+ E v* (i)+ v(N) - E v(i) - E v(N \ i)

iesnH ies\H\i* ieH ie(N\H)\i*

is equals to the right side of of inequality in (15). Since xH(S) = v(S) for all S e n1 \ Q2h we have xH e C(v). □

Corollary 2. The system (14)-(15) characterizes such class of TU games that at least one extreme point of imputation set I (v) or dual imputation set I * (v) or core cover CC(v) belongs to core.

Proof. Let H and i* be the same as in Theorem 3. If H = N\ i* then xH is defined by

b (v(i), i e N \ i*,

(xH)i =\ v(N) - £ v(j), i = i*.

I jeN\i*

Therefore xB (H) = fi* (v) e ext(I(v)). If H = 0 then

b f v*(i), i e N \ i*,

(xH)i = i v(N) - £ v*(j), i = i*,

jeN\i*

and xH = gi' (v) e ext(I*(v)). Let mow H e f2 \{N \ i*}. The core cover CC(v) is defined by the system

xi > v(i), Xi < v* (i), i e N, x(N) = v(N).

The vector xH satisfies as equality n — 1 linearly independent inequalities in this system and xH(N) = v(N). So xH e ext(CC(v)). □

Now we give example to illustrate the conditions in Theorem 3.

Example 4. Let v e and (without loss of generality) i* = 3. The conditions

in Theorem 3, corresponding to all possible choices for coalition H e (2N \ {N}) \ i* are given in the following table. For each H the inequalities listed in the last

H S en2H inequalities in (14), (15) transformed inequalities

{3} {12} {12} {13} {23} v (12) < v(1) + v (2) v (3) < v(N) - v (1) -v(13) < v(N) - v (2) ^(23) < v{n) - v{ 1) v(2) v(1)+ v (2) + v(3) < v (N) v(2) + v (13) < v (N) iy[l) +^(23) < v{n)

0 {1} {2} {3} {12} Kl) <"•(!) v (2) < v* (2) v(12) < v*(1) + v*(2) 1/(3) < v(N) - - similar H = {12}, S = {23} similar H = {12}, S = {13} v(12) + v(13) + v(23) < 2v(N) ^(13) + i/(23) > v(N) + 7/(3)

{1} {2} {3} {12} {23} v{2) < v*{2) v(12) < v(1) + v*(2) v(3) < v(N) - v(1) -similar H = {12}, S - v * (2) = {23} similar H = {12}, S = {13} v(12) + v (13) < v (N) + v (1) v(1) + v(3) < v(13)

{2} similar H = {1}

column of table are balancedness-inequalities, convexity-inequalities ore concavity-inequalities only. Consider the game:

v(1) = 1, v(2) = —1, v(3) = 2, v(12) = 0, v(13) = 6. v(23) = 4. v(N) = 6. It is monotonic, superadditive, but non-convex even for the grand coalition (v(13) + v(23) > v(N) + v(3)) game. Players marginal contributions to the grand coalition are: v*(1) = 2, v*(2) = 0, v* (3) = 6. We have

ext(Web(v)) = {(2, —1, 5), (2, 2, 2), (4, 0, 2), (1, —1, 6), (1,0, 5)}, ext(I(v)) = {(5, —1, 2), (1, 3, 2), (1, —1, 6)}, ext(I*(v)) = {(0,0, 6), (2, —2, 6), (2,0, 4)},

ext(C(v)) = ext(CC(v)) = {(2, —1, 5), (1, 0, 5), (1, —1, 6), (2,0, 4)}. For every coalition T e f1 the given game is not T-simplex and dual simplex because ext(C(v)) % ext(I(v)), ext(C(v)) % ext(I*(v)). It do not satisfy the CoMa-property because ext(C(v)) % ext(Web(v)). The conditions (14), (15) are satisfied for all H e (2N \ {N}) \ {3}. We have

xf12} = (1, —1, 6) e ext(I(v)), xf0} = (2, 0, 4) e ext(I*(v)), xf!} = (1, 0, 5) e ext(CC(v)), xf2} = (2, —1, 5) e ext(CC(v)).

Remark 1. The set of games satisfying (14)-(15) contains T-simplex games, dual simplex games with T 3 i*, zero-normalized monotonic games with Veto(v) B i*, clan games with CL 3 i*, big boss games with player i* as big boss.

Remark 2. For basis containing the rows AN, A{ni}, A{nin2},...,A{nin2'"'nfc}, where 2 < k < n - 1, the conditions (6) define TU games with some convexity behaviour. For k = n -1 we obtain condition that determines such class of games that at least one extreme point mn (v) of Weber set belongs to core, i.e. permutationally convex games (in particular, convex games and games satisfy the CoMa-property).

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