Cooperative side payments games with restricted transferability Текст научной статьи по специальности «Математика»

Cooperative Side Payments Games with Restricted Transferability

Alexandra B. Zinchenko, Lev S. Oganyan and Gennady G. Mermelshtejn

Southern Federal University,

Melshakova s. 8”a”, Rostov-on-Don, 344090, Russia E-mail: merg@math.rsu.ru

Abstract This paper studies cooperative games in which distributed utility consists of indivisible units (discrete games). Such games are considered as a special subclass of NTU games. Relations between set-valued solutions of discrete game as well as relations between them and corresponding TU game solutions are described. Some existence conditions are obtained.

Keywords: discrete game, core, core cover, dominance core, dual core, stable set, balancedness.

1. Introduction

Transferable utility game (TU game) is defined by assumptions of transferable utility and side payments. A game with nontransferable utility (NTU game) requires neither the assumption of side payments nor the assumption of transferable utility. Intermediate position between TU games and general NTU games have the models of economic situations in which side payments are allowed, but (because of nontransferability of utility) some restrictions on them are imposed. For example, if distributed utility consists of indivisible units then the characteristic function of game and payoff vectors should be integers.

The literature on this subject started with the monograph by von Neumann and Morgenstern (1944). Von Neumann and Morgenstern (1944) studied market game with one seller, two buyers and one (indivisible) unit of utility and have demonstrated the effect of requiring indivisibility. Cooperative games with integer side payments have been called the games with restricted transferability. As it has been noticed in (von Neumann and Morgenstern, 1944) such games have economic applications, in particular, allow to construct more identical mathematical model of such phenomenon as market agreement, but generate some rather specific difficulties which represent the considerable interest. It is possible to consider the modified bankruptcy problem as the other application of such games. Let’s assume, that the bankrupt firm produced some production of indivisible type and creditors prefer to receive finished goods (which do not suffice for all) instead of monetary compensation, then the classical bankruptcy game (Curiel et al., 1987) becomes discrete.

Cooperative games with integer payoff vectors were investigated in (Morozov and Azamkhuzhaev, 1991; Azamkhuzhaev, 1991; Zinchenko, 2009; Zinchenko and Mermelshtejn, 2007) and other works by the same authors (such games are called discrete). Azamkhuzhaev (1991) obtained the necessary and sufficient non-emptiness conditions for the undominated imputations set of superadditive 3-person discrete game and for symmetric n-person discrete game. Morozov and Azamkhuzhaev (1991) have presented algorithm for computation of undominated imputations of

discrete game with 5 and 6 players. We shall notice that in (Azamkhuzhaev, 1991; Morozov and Azamkhuzhaev, 1991) the set of all undominated imputations is called the core of discrete game. It has led to the contradictions noted in (Zinchenko and Mermelshtejn, 2007).

In the current paper we consider discrete games as a special subclass of NTU games for which the generalization of some TU solution concepts, not applicable to arbitrary NTU games, is possible. The Weber set, core, core cover, dominance core, dual core and stable set of discrete game are considered. Some relations between set-valued solution concepts of discrete game, as well as relations between them and corresponding TU game solutions, are described. Existence conditions of some solutions are obtained.

2. Preliminaries

In this section we recall the facts from cooperative game theory which are useful later. A TU game is a pair (N, v) where N = {1,...,n} represents the set of players and v e Gn = {g : 2N ^ R | g(0) = 0} is the characteristic function. Often v

and (N, v) will be identified. A game v e Gn is called convex if v(Si) + v(S2) <

v(S1 U S2) + v(Si fl S2) for all S1,S2 C N .A game v e Gn is concave if (—v) e Gn is convex. A game v* e Gn is called dual to v e Gn if v* (S) = v(N) — v(N \ S) for all S C N .A game v e Gn is called simple if v (N) = 1 and v (S) e {0,1} for all S C N. Given x e RN and coalition 0 = S C N, we write x(S) = ^ieS x*.

Let n(N) be the set of all permutations n : N ^ N on N. The Weber set of game v e Gn, denoted by W(v), is the convex hull of all n-marginal vectors, i.e.

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

were

mi (v) = v(Si ) — v(SГ-1), Si = {п1,...,пг}, i e N. (1)

For the description of other set-valued solution concepts of game v e Gn is used

the set X(v) = {x e RN I x(N) = v(N)} of payoff distributions of v(N) named the preimputation set and its subsets: the imputation set

I(v) = {x e X(v) | x* > v(i), i e N}

and the dual imputation set

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

The core, core cover and dual core of a game v e Gn are defined by

C(v) = {x e I(v) | x(S) > v(S), S C N}, CC(v) = I(v) f I*(v),

C*(v) = {x e X(v) | x(S) < v(S), S C N}.

For x,y e I(v), y is said to dominate x if there is a coalition S e 2N \ 0 such that:

x* > y* for each player i e S and x(S) < v(S). For v e Gn and T C I(v) denote

by dom(T) the set of imputations that are dominated by some element in T. The set of all undominated imputations is called D-core, or dominance core, of a game v e Gn, i.e.

DC(v) = I(v) \ dom(I(v)).

Note that the Weber set, imputation set, dual imputation set, core, core cover, dual core and D-core of a game v e Gn are the polytops. Any of these sets, except the

Weber set, may be empty. The stable set R(v) of a game v e Gn is defined by

conditions

R(v) f dom(R(v)) = 0, I(v) \ R(v) C dom(R(v)).

It is known that for any game v e Gn : W (v) = 0,

C(v) = C* (v*), C(v) C W(v), C(v) C DC(v), (2)

DC(v) C CC(v). (3)

If DC(v) = 0 then the conditions

v(S) +53 v(i) < v(N), S C N, (4)

*EN\S

are sufficient for equality of core C(v) and D-core DC(v). If C(v) = DC(v) then

C(v) = 0. DC(v) is a subset of every stable set R(v) and if DC(v) is a stable set,

then there is no other stable set. For a convex game v e Gn:

C (v) = W (v) = DC(v) = 0

and C(v) is the unique stable set. For a concave game v e Gn:

C*(v) = C (v*) = W (v*) = DC(v*) = 0.

A NTU game is described as a pair (N, V) where V is a function which maps

every set of players S C N into a subset V(S) of RS, the so-called characteristic

set, satisfying:

1. V(S) is non-empty and closed,

2. V(S) is comprehensive, i.e. if x e V(S), y e RS and x > y, then y e V(S). Sometimes it is assumed that for all S C N the sets V(S) are convex.

3. Set-valued solution concepts

Denote by Z the set of integer numbers. Discrete game can be determined as NTU game (N,VZZ), or simply Vzz, with characteristic sets

Vzz(S) = {x e ZS | x(S) < vz(S)}, S C N,

where vz e Gn = {g : 2n ^ Z | g(0) = 0} is integer-valued function. Note that

discrete game Vzz is uniquely determined by integer TU game vz. To each game vz e Gn we associate also NTU game (N,VZ) with

Vz(S) = {x e RS | x(S) < vz(S)}, S C N.

Denote by Gnz the class of all discrete games with player set N. The NTU game

(N, VZZ), or VZZ for short, with

VZz(S) = {x e ZS | x(S) < vZ(S)}, S C N,

we call the dual to game Vzz e Gnz .

The basic solution concepts of NTU game theory generalize the solutions defined on the subclass of cooperative TU games. However, at such generalization there can be a non-uniqueness and difficult problems connected, for instance, with an axiomatic characterization of some solutions and a finding of existence conditions. On the one side, standard assumptions concerning the characteristic sets of NTU game are not hold for discrete game Vzz e Gnz . But, on the other side, possibility of numerical comparison players utility in game Vzz allows to define for it the solution concepts (Weber set, core cover, dual core) which, as we know, have not been described for NTU games.

Using the equality VZZ(S) = Vz (S) f ZS we can define the Weber set W(Vzz), imputation set I (Vzz), dual imputation set I* (Vzz), core C (Vzz), core cover CC(VZZ) and dual core C*(Vzz) of discrete game Vzz as intersections of corresponding sets of NTU game Vz (or TU game vz e GN) and integer lattice ZN:

W (Vzz) = W (vz) f ZN, I (Vzz) = I (vz) f ZN, I* (Vzz) = I*(vz) f ZN, (5)

C(Vzz) = C(vz) f ZN, C* (Vzz) = C* (vz) f ZN, CC(Vzz)= CC(vz) f ZN. (6)

The D-core DC(Vzz) of discrete game Vzz we define as the set of all undominated elements in I(Vzz):

DC (Vzz) = I (Vzz) \ dom(I (Vzz)).

The stable set R(Vzz) of discrete game Vzz is defined (von Neumann and Morgen-stern, 1944) by conditions

R(Vzz) f dom(R(Vzz)) = 0, I (Vzz) \ R(Vzz) C dom(R(Vzz)).

Denote by №(Vzz) the collection of all stable sets of discrete game Vzz.

Following two propositions show that the Weber set of any discrete game Vzz always exists and the imputation set, dual imputation set, core cover of discrete game exist iff the corresponding sets of TU game vz are non-empty.

Proposition 1. Let Vzz e Gnz then:

(i) W(Vzz) = 0;

(ii) I (Vzz) = 0 iff

J2vz(i) < vz(N); (7)

*£N

(iii) I*(Vzz) = 0 iff

Y.v*z(i) > vz(N). (8)

*eN

Proof (of proposition). (i) Due to (1) it holds that mn(vz) e ZN for all n e n(N).

Because mn(vz) e W(vz), we have that mn(vz) e W(vz) f ZN. By definition of

W(Vzz) this implies W(Vzz) = 0.

(ii) It is known that (7) is a necessary and sufficient non-emptiness condition for I (vz). Hence, it is sufficient to show that I (Vzz) = 0 iff I (vz) = 0. Let’s assume, that I(Vzz) = 0. From the second equality in (5) we have that I(Vzz) C I(vz). So I (vz) = 0. To prove the converse, suppose that I (vz) = 0. Then I (vz) = conv{f%(vz) | i e N} where

fk(vz)= vz(k), k e N \ i, fl(vz) = vz(N) — ^ vz(k). (9)

keN\*

Because vz is integer-valued function, we have that f*(vz) e ZN f I(vz). Hence, P(vz) e I(Vzz), i.e. I(Vzz) = 0.

(iii) I*(Vzz) = 0 iff (8) holds. Suppose that I*(Vzz) = 0. From the last equality in (5) we have that I*(Vzz) C I*(vz). So I*(vz) = 0. Let’s assume now, that I*(vz) = 0. Then I*(vz) = conv{hl(vz) | i e N} where

hk(vz) = vz(k), k e N \ i, h\(vz) = vz(N) — £ vz(k). (10)

keN\*

Since vz(S) e Z for all S C N,we have hl(vz) e ZN fI*(vz), i.e. hl(vz) e I*(Vzz). It means that I* (Vzz) = 0. □

It is follows from the proof of proposition 1 that the imputation set I(vz) and the dual imputation set I*(vz) of integer TU game vz are integer polytopes. In general the nonempty intersection of integer polytopes can not have any integer point. However, the core cover of integer game vz e Gn defined as intersection of I (vz) and I *(vz) is also integer polytope (see the proof of proposition 2).

Proposition 2. Let Vzz e GNz. Then CC(Vzz) = 0 iff conditions (7), (8) and

vz(i) < vz(i), i e N, (11)

hold.

Proof (of proposition). Conditions (7), (8) and (11) are the necessary and sufficient for non-emptiness of CC(vz). Hence, it is sufficient to prove that CC(Vzz) = 0 iff CC(vz) = 0. Let’s assume, that CC(Vzz) = 0. From the last equality in (6) we have CC(Vzz) C CC(vz). So CC(vz) = 0. To prove the converse, suppose that CC(vz) = 0. Since CC(vz) is determined by system

x* > vz(i), x* < v*z(i)), i e N, x(N) = vz(N) (12)

then this system has a nonempty solution set. Denote by 1" n-dimentional vector with all coordinates equal to 1. Let x e RN and A be the n x (2n + 2)-matrix with columns e1,..., e", —e1,..., —e", 1", —1" where e* e R", ej = 0 for i = j and e\ = 1 for each i e N. The linear system (12) can be written down in the form xTA > bT where b = (vz(1),..., vz(n), —vz(1),..., —vz(n), vz(N), —vz(N)). Core cover CC(vz) is a polytope, hence it has at least one extreme point x. Let A be the basis corresponding to x. Obviously, det(A) = ±1. Since b is the vector with integer coordinates then x e ZN. Thus x e CC(Vzz). This implies CC(Vzz) = 0. □

It is follows from (5) - (6) that for any discrete game Vzz e GNz, as for related TU game vz e GN, the core coincides with the dual core of dual game Vzz and the Weber set contains the core

C(Vzz) = C*(Vzz), C(Vzz) C W (Vzz).

For convex game vz the core of associated discrete game Vzz is nonempty and coincides with Weber set

C (Vzz) = W (Vzz) = 0.

For concave game vz the dual core of associated discrete game Vzz is nonempty and coincides with Weber set of dual game Vzz, the dual core of game Vzz also coincides with the core of dual game Vzz

C *(Vzz ) = C (Vzz )= W (Vzz ) = 0.

In propositions 3, 4 (below) other relations between set-valued solution concepts of discrete game, analogous to the relations (2) between TU game solutions, are proved. Thus, the core of any discrete game is a subset of its D-core and all stable sets of any discrete game contain the D-core. The proofs of these propositions is similar to proof of theorem 2.11 from (Branzei et al., 2005)

Proposition 3. Let Vzz e GNz. Then C(Vzz) C DC(Vzz).

Proof (of proposition). Suppose that C(Vzz) = 0 and take x e C(Vzz). If x e DC(Vzz) then x e dom(I (Vzz)), i.e. there exists an ye I (Vzz) which dominate x. Hence, there exists a coalition 0 = S C N for which y(S) < vz(S) and y* > x* for each player i e S. This yields that x(S) < y(S) < v(S), i.e. x e C(vz). It is follows from the definition of C(Vzz) that C(Vzz) C C(vz). Thus, x e C(Vzz). Hence, the assumption x e DC(Vzz) is false. □

Proposition 4. Let Vzz e Gnz and №(Vzz) = 0. Then

DC (Vzz) C R(Vzz) for all R(Vzz) e K(Vzz).

Proof (of proposition). Suppose DC(Vzz) = 0 and take x e DC(Vzz). Assume that R(Vzz) e №(Vzz) and x e R(Vzz). From the definition of stable set this implies that there exists payoff vector y e R(Vzz) such that y dominate x. Since R(Vzz) C I(Vzz) we have y e I(Vzz). This yields a contradiction x e DC(Vzz).

□

Unlike previous statements, propositions 5 and 6 show that some relations between set-valued solutions of TU game vz e Gn do not hold for corresponding sets of discrete game Vzz. The convex function vz satisfy the condition (4) but even convexity of function vz is not sufficient for the equality of core and D-core of game Vzz. The core of discrete game Vzz defined by convex function vz can be not equal to any stable set R(Vzz) e =K(Vzz). Unlike (3) the core cover of discrete game can be own subset of its D-core.

Proposition 5. Let vz be a convex game. Then there are discrete games Vzz e GNz for which:

(i) C(Vzz) = DC (Vzz) and C (Vzz) = 0,

(ii) C(Vzz) = R(Vzz) for every R(Vzz) e ^(Vzz).

Proof (of proposition). (i) Consider the convex TU game (N, vz) with N = {1, 2, 3} and

vz(i) =0, ieN, vz(1, 2) = 0, vz(1, 3) = vz(2, 3) = 1, vz(N) = 2. (13)

This game defines discrete game Vzz e Gnz with characteristic sets

Vzz(i) = {xe Z« | x* < 0}, i e N,

Vzz(1, 2) = {x e Z{1,2} | x1 + x2 < 0}, Vzz(1, 3) = {x e Z{1,3} | x1 + x3 < 1}, Vzz(2, 3) = {x e Z{2,3} | x2 + x3 < 1}, Vzz(N) = {x e ZN| x1 + x2 + x3 < 2} Discrete game Vzz has non-empty core C(Vzz) = {(0, 0, 2), (1,1,0), (1,0,1), (0,1,1)} which is not equal to its D-core DC(Vzz) = C(Vzz) U {(2,0, 0), (0, 2,0)}.

(ii) Discrete game defined by integer convex 3-person game (13) has a unique stable set. №(Vzz) = {R(Vzz)}, where R(Vzz) = C(Vzz) U {(2,0,0), (0, 2,0)}. Hence, C(Vzz) = R(Vzz). □

Proposition 6. There are the discrete games Vzz e GNz for which

CC(Vzz) C DC (Vzz).

Proof (of proposition). Discrete game defined by integer convex 3-person game (13) have CC(Vzz) = C(Vzz) = {(0, 0, 2), (1,1, 0), (1, 0,1), (0,1,1)} and DC(Vzz) = C(Vzz) U{(2,0,0), (0, 2,0)}. Hence, CC(Vzz) C DC(Vzz). □

In last proposition of this section we show that the intersection of D-core of game vz e GN and integer lattice ZN can be unequals the D-core of associated discrete game Vzz e GNz and the intersection of every stable set of game vz e GN and integer lattice ZN can not belongs to family of stable sets of game Vzz e Gnz .

Proposition 7. Let Vzz e Gnz then

(i) DC(vz) f ZN C DC(Vzz) and it is possible strict inclusion,

(ii) it is possible that R(vz) f ZN / №(Vzz) for each R(vz) e =K(vz).

Proof (of proposition). (i) The proved inclusion follows from DC(vz )fZN = I (vz )\ dom(I (vz)) f ZN C (I (vz) f ZN) \ dom(I (vz) f ZN) = I (Vzz) \ dom(I (Vzz)) = DC(Vzz). Consider discrete game Vzz defined by symmetric 3-person TU game vz with

vz (i) =0, ieN = {1, 2, 3}, vz (1, 2) = vz (1, 3) = vz (2, 3) = vz (N) = 1. (14)

Since DC(vz) = 0 and DC(Vzz) = {(1, 0,0), (0,1, 0), (0, 0,1)} we have DC(vz) f ZN C DC (Vzz).

(ii) TU game (14) has symmetric stable set

«°(^) = . (|.°.|) ■ (°'|-|)1

and three families of stable sets

R1,c(vz) = {(c, x2, 1 — c — x2) | 0 < x2 < 1 — c},

R2,c(vz) = {(1 — c — x3, c, x3) | 0 < x3 < 1 — c},

R3,c(vz) = {(x1, 1 — c — x1, c) | 0 < x1 < 1 — c},

were c G [0, i). Let’s find the intersections of stable sets of game (14) and integer lattice ZN:

W{vz) = R°(vz) n ZN = 0,

R^{vz) = Rl’c{vz) n ZN = {(0,0,1), (0,1,0)},

RM{vz) = R2’c{vz) n ZN = {(0,0,1), (1,0,0)},

W^{vz) = R3’c(iyz) n ZN = {(0,1,0), (1,0,0)}.

Discrete game Vzz has the unique stable set: №(Vzz) = {R(Vzz)} were R(Vzz) =

{(0, 0,1), (0,1, 0), (1, 0,0)}. Hence, RP(vz) £ 3t(Vzz) and Rl’c(vz) <£ 3t(Vzz) for

each i = 1, 2, 3. □

4. Balancedness

It is known that integer game vz e GN (as TU game v e Gn) has a non-empty core iff it is balanced, i.e.

^2 Asvz (S) < vz (N) (15)

Sen

where H = (2n \ 0) \ N, sen S3* As = 1 for each i e N and XS > 0. It is follows from inclusion C(Vzz) C C(vz) that (15) is a necessary non-emptiness condition for the core C(Vzz) of discrete game Vzz. Obviously, condition (15) is hot sufficient for the existence of C(Vzz). Next proposition gives the necessary non-emptiness condition for the D-core of discrete game.

Proposition 8. Let Vzz e GNz then

(i) condition (15) is not necessary for the existence of D-core of game Vzz,

(ii) a necessary non-emptiness condition for the D-core of game Vzz is balancedness of TU-game vz G Gz were

VZ{N) = vz(N), VZ{S) = vz{S) - IS1 + 1 for all Sett.

Proof (of proposition). (i) Let N = {1, 2, 3}, A' = {AS}Sen were A^ 2} = A^ 3} = A{2 3} = \ and ^{*} = 0 for i £ N. For vector A' and 3-person game (14) condition (15) is not holds, i.e. this game is non-balanced. However discrete game Vzz defined by integer TU game (14) have non-empty D-core: D(Vzz) = {(0, 0,1), (0,1,0), (1, 0,0)}.

(ii) Let x = y be two imputations x,y e I(Vzz) of discrete game Vzz. If y dominate x then there exists coalition S such that y(S) < vz(S) and y* > x* for each i S. It easily be verified that coalition S satisfies the condition 1 < | S| < n. Since x,y e ZN then for each i e S we have y* > x* + 1. We obtain that x(S) + S| < y(S) < vz(S). Equivalently, x(S) < vz(S) — |S|. For integer-valued

function vz we have that the inequality x(S) < vz(S) — |S| + 1 holds. Hence, each undominated imputation x e DC(Vzz) must satisfy conditions: x(N) = vz(N), x(S) > vz(S) — |S| + 1, 1 < S11 < n. Thus, x must be the core payoff vector of TU game vz. This implies that C(VZ) ^ 0, i.e. the game vz is balanced. □

The balancedness conditions for NTU-game: standart balancedness, balancedness for convex games, b-balancedness, (b, <)-balancedness, n-balancedness, balancedness with respect to the transfer rate mapping (their description can be found in (Bonnisseau and Iehle, 2007)) are not applicable to games with discrete characteristic sets Vzz(S). The problem of obtaining the necessary and sufficient nonemptiness conditions for the core of discrete game Vzz is reduced to problem of existence integer point in polytope C(vz). This problem, as we know, is not solved yet in general, however, for simple games balancedness conditions are found simply.

Proposition 9. Let vz be a simple game. Then C(Vzz) = 0 iff

Veto(vz) = f{S | vz(S) = 1} = 0. (16)

Proof (of proposition). The core of simple game vz has the following representation C(vz) = {x e RN | J2i.eveto(vZ) x* = 1 and x* =0 for i e N \ Veto(vz)}. It is known that C(vz) = 0 iff (16) holds. Since C(vz) is the convex hull of vectors e1, i e Veto(vz), then C(Vzz) = 0 iff (16) holds. □

Next proposition gives the sufficient non-emptiness conditions for the core of discrete game, which are satisfied for some models of real situations.

Proposition 10. If one of the following conditions holds

(i) there exists i e N such that vz(S) + 5^keN\s vz (k) < vz(N) for each S C N with i e S, and vz(S) <^2kES vz(k) otherwise;

(ii) there exists i e N such that vz(S) < (|S| + 1— n)vz(N) + 5^keN\s vz(N\k) for each S C N with i e S, and vz(S) < |S^z(N) —^2kes vz(N \ k) otherwise;

(iii) there is such n-marginal vector mn (vz) that vz (S) < ^s ml (vz) for

S = {n1, .. .,n}, i e N;

then the core of discrete game Vzz e GNz is non-empty.

Proof (of proposition). It is easy to check that under maded assumptions vector f *(vz) defined by (9) belongs to the core of TU game vz e GN. Since f *(vz) e ZN then P(vz) e C(Vzz). This means that C(Vzz) = 0.

(ii) Under maded assumptions hl(vz) e C(vz) were hl(vz) defined by (10). Since hl(vz) e ZN then h1 (vz) e C(Vzz). Hence C(Vzz) = 0.

(iii) From the definition of n-marginal vector mn(vz) we have ^£s ml. (vz) =

vz (S) for S = Sf, i e N. From maded assumptions follows mn (vz) e C(vz). Since mn(vz) e ZN tlien mn(vz) e C(Vzz). Thus C(Vzz) = 0. □

From properties of discrete games follows that a nonempty core have the discrete analogues of following subclasses of the class of balanced TU games: bankruptcy games (Curiel et al., 1987), big boss games (Muto et al., 1988) (in particular, holding games (Tijs et al., 2005)), clan games (Potters et al., 1989), T-simplex (0 = T C N) and dual simplex games (Branzei and Tijs, 2001), games satisfying the CoMa-property (Hamers et al., 2002), k-convex games (k e N).

5. Conclusion

The propositions given before show that for special situations modelled by discrete games sometimes it is more convenient to work with related integer TU game instead of describing discrete game as a NTU game. The intersections of integer lattice ZN and the TU game solution sets determined by linear constraints (Weber set, core, dual core, core cover) are identical to corresponding sets of discrete game. The relations between these sets are saved (see propositions 2-4). However, the setvalued solutions determined by dominance relation (D-core and stable sets) do not satisfy to similar properties (see proposition 5-7). As it is illustrated before, there are infinite many stable sets of integer game (14), but associated discrete game Vzz has a unique stable set containing all integer payoff vectors which belong to stable sets of TU game vz.

Von Neumann and Morgenstern (1944) introduced the notion of a kernel of directed graph as an generalization of their concept of stable set. Let r = (U, E) be the directed graph with vertex set U corresponding to imputations set I(Vzz) of discrete game Vzz e GNz and let (i,j) e E iff xj e I(Vzz) dominates x1 e I(Vzz). In contrast to TU case, graph r is finite. Any kernel of such graph, obviously, defines stable sets of discrete game Vzz. Known existence conditions of kernels of directed graph give existence conditions of stable sets of discrete game.

For computation of payoff vectors belonging to core, dual core or D-core of discrete game it is possible to use the integer programming algorithms. For computation of payoff vectors which belong to stable sets we can use the algorithms

of graph theory. Thus, the unique kernel in an acyclic directed graph can be constructed efficiently by "backward induction” procedure. Note that even in simple cases it is difficult to calculate all (or only one of) the stable sets TU game and non-discrete NTU game.

References

Azamkhuzhaev, M. Kh. (1991). Nonemptiness conditions for cores of discrete cooperative game. Computational Mathematics and Modeling. , 2(4), 406-411.

Bonnisseau, J.-M. and V. Iehle (2007). Payoff-dependent balancedness and cores. Games and Economic Behavior, 61(1), 1-26.

Branzei, R., D. Dimitrov and S. Tijs (2005). Models in cooperative game theory: crisp, fuzzy and multichoice games. Lecture notes in economics and mathematical systems, 556, Springer-Verlag, Berlin.

Branzei, R. and S.H. Tijs (2001). Cooperative games with a simplical core. Balkan Journal of Geometry and Application. 6, 7-15.

Curiel, I. J., M. Maschler and S. H. Tijs (1987). Bankruptcy games. Mathematical Methodes and Operations Reseach, 31(5), 143-159.

Hamers, H. J.M., F. Klijn, T. Solymosi, S.H. Tijs and J. Pere Villar (2002). Assignment games satisfy the CoMa-property. Games and Economic Behavior, 38(2), 231-239. Morozov, V. V. and M.,Kh. Azamkhuzhaev (1991). About search of imputations of discrete cooperative game. In: Application of computing means in scientific researches and educational process, pp. 49-62. MSU: Moscow, (in Russian).

Muto, S., M. Nakayama, J. Potters and S. Tijs (1988). On big boss games. The Economic Studies Quarterly, 39, 303-321.

Potters, J., R. Poos, S. Tijs and S. Muto (1989). Clan games. Games and Economic Behavior, 1, 275-293.

Tijs, S.H., A. Meca and M. A. Lopez (2005). Benefit sharing in holding situations. European Journal of Operational Research, 162(1), 251-269. von Neumann, J. and O. Morgenstern (1944). Theory of Games and Economic Behavior.

Princeton University Press: Princeton, NJ.

Zinchenko, A. B. (2009). Core properties of discrete cooperative game. News of high schools.

The North Caucasian region. Natural sciences , 2, 4-7 (in Russian).

Zinchenko, A. B. and G. G. Mermelshtejn (2007). Balanced and dual balanced discrete cooperative games. Economic herald of Rostov state university, 5(2), 109-115 (in Russian).