CYBERNETICS AND THE THEORY OF REGULATION

O. N. BONDAREVA

SOME THEOREMS OF THE THEORY OF \(\Psi\)-STABILITY IN COOPERATIVE GAMES

(Presented by Academician P. S. Novikov, 3 VI 1963)

Consider a cooperative game given by a characteristic function in \(0\)-\(1\)-reduced form (see \((^{1})\)). Consider the set of imputations

\[ A=\left\{a=(a_1,\ldots,a_n):\quad a_i \geqslant 0,\quad \sum_{i=1}^{n} a_i = 1\right\}. \]

To each coalition \(S \subset I_n\) assign the vector \(S=\{s_1,\ldots,s_n\}\), where \(s_i=1\) if \(i \in S\), and \(s_i=0\) if \(i \notin S\).

We shall call a system of numbers \((\lambda_1,\ldots,\lambda_m)\), \(\lambda_j \geqslant 0\), a \(q\)-\(\theta\)-covering of \(I_n\) if

\[ \sum_{j=1}^{m} \lambda_j S_j = I_n,\quad S_j \subset I_n, \]

where \(q\) is the number of positive \(\lambda_j\), and \(\theta\) is the set of corresponding coalitions \(S_j\) (see \((^{3})\)). The extreme points of the set of coverings are called reduced coverings; their number is finite. A subset \(U\) of the set \(A\) is called a core if, for any \(\alpha \in U\), the condition

\[ \sum_{i \in S} \alpha_i = S \cdot \alpha \geqslant v(S) \qquad \text{for all } S \subset I_n \]

is satisfied.

Any partition \(\tau\) of the set \(I_n\) into disjoint coalitions is called a coalition structure.

Suppose that for each \(\tau\) a mapping \(\Psi(\tau)\) into the set of all coalitions \(S \subset I_n\) is given, with \(\tau \subset \Psi(\tau)\). A pair \([\alpha,\tau]\) \((\alpha \in A)\) is called \(\Psi\)-stable if the following conditions are satisfied: 1) \(S \cdot \alpha \geqslant v(S)\) for all \(S \in \Psi(\tau)\); 2) if \(\alpha_i=0\) \((=v(\{i\}))\), then \(\{i\}\in \tau\).

The concept of \(\Psi\)-stability was introduced by Luce (see, for example, \((^{1,4})\)), as was the concept of \(k\)-stability. The known theorems on \(k\)-stability concerning classes of symmetric games and games with a quota also belong to him.

In the present paper \(\Psi\)-stability is studied by means of methods of linear programming.

Lemma. In order that a system of inequalities of the form

\[ S \cdot \alpha \geqslant v(S),\quad S \in \Xi, \]

\[ I_n \cdot \alpha = 1 \]

(\(\Xi\) is some set of coalitions) have a solution, it is necessary and sufficient that, for every reduced \(q\)-\(\theta\)-covering \((\lambda_1,\ldots,\lambda_m)\) such that \(\theta \subset \Xi\), the inequality

\[ \sum_{j=1}^{m} \lambda_j v(S_j) \leqslant 1, \]

be satisfied.

moreover, in order that at least one of the inequalities be strict, it is necessary that

\[ \sum_{j=1}^{m} \lambda_j v(S_j) < 1 \]

for coverings in which the \(\lambda_j\) corresponding to this inequality is \(>0\).

The proof is based on the theorems on the solvability of systems of linear inequalities from \((2)\).

Theorem 1. In order that, for some \(\tau\), there exist a \(\Psi\)-stable pair \([a,\tau]\), it is necessary and sufficient that, for every reduced \(q\)-\(\theta\)-covering \((\lambda_1,\ldots,\lambda_m)\), for which \(\theta \subset \{\Psi(\tau),\{1\},\ldots,\{n\}\}\), the inequality

\[ \sum_{j=1}^{m} \lambda_j v(S_j) \leqslant 1 \]

hold, and the inequality must be strict for coverings containing one-element sets not belonging to \(\tau\).

Proof. In order that the pair \([a,\tau]\) be \(\Psi\)-stable, it is necessary and sufficient that \(a\) be a solution of the system of inequalities

\[ \begin{aligned} S\cdot a &\geqslant v(S), \qquad S \in \Psi(\tau) && \text{(condition 1)},\\ a_i &> 0, \qquad i \in S \in \tau \ \text{and}\ |S| \geqslant 2 && \text{(condition 2)},\\ a_i &\geqslant 0 \quad \text{for the remaining } i,\\ \mathbf{I}_n\cdot a &= 1 \end{aligned} \qquad \left\{ \begin{array}{l} a \in A, \end{array} \right. \]

and it remains for us only to refer to the lemma just formulated, which completes the proof.

Denote by \(\bar{\Gamma}\) a game in which there is no such \(q\)-\(\theta\)-covering \((\lambda_1,\ldots,\lambda_m)\) that

\[ \sum_{j=1}^{m} \lambda_j v(S_j) \leqslant 1. \]

Corollary 1. Whatever the mapping \(\Psi(\tau)\) may be, in the game \(\bar{\Gamma}\) there exist no \(\Psi\)-stable pairs.

Corollary 2. For a game with a superadditive payoff function there exist \(\Psi\)-stable pairs for a certain choice of \(\Psi\).

The assertion is true if, for example, the mapping \(\Psi(\tau)\) is a “partition” of the coalitions from \(\tau\).

Corollary 3. Suppose that in a game
\[ (\lambda_1^{(1)},\ldots,\lambda_m^{(1)}),\ldots,(\lambda_1^{(t)},\ldots,\lambda_m^{(t)}) \]
are all the reduced \(q_i\)-\(\theta_i\)-coverings for which
\[ \sum_{j=1}^{m} \lambda_j^{(i)} v(S_j) \leqslant 1; \]
then, in order that a \(\Psi\)-stable pair \([a,\tau]\) exist, it is necessary that

\[ \Psi(\tau) \subset \bigcup_{i=1}^{t} \theta_i. \]

Corollary 4 (Theorem 1 from \((3)\)). In order that the game \(\Gamma\) have a core, it is necessary and sufficient that, for every reduced \(q\)-\(\theta\)-covering \((\lambda_1,\ldots,\lambda_m)\), one have

\[ \sum_{j=1}^{m} \lambda_j v(S_j) \leqslant 1. \]

The proof follows from the fact that if \(\Psi(\tau)\), for every \(\tau\), is a mapping onto the set of all coalitions, then every pair \([a,\tau]\), where \(a\in U\), is \(\Psi\)-stable, and conversely, for every \(\Psi\)-stable pair \(a\in U\).

Let us now consider a function \(\Psi(\tau)\) of a certain special form (see \((1)\), pp. 289–290): we assume that \(\Psi(\tau)\) consists of all coalitions \(T\) for which there exists an \(S\in\tau\) such that
\[ |T\setminus S|+|S\setminus T|\leqslant k \]
(the modulus sign for a set denotes the number of its elements), where \(k\) is a given integer. \(\Psi\)-stability in this case is called \(K\)-stability.

Recall that a game is called symmetric if \(v(S)=v(|S|)\).

Theorem 2 (Theorem 1 from \((^4)\)). In order for a symmetric game to be \(k\)-stable, it is necessary and sufficient that the condition

\[ v(S)\leqslant \frac{|S|}{n}\quad \text{for}\quad |S|=2,\ldots,k+1 \]

be satisfied.

Proof. Sufficiency. If \(v(S)\) satisfies the condition of the theorem, then the pair

\[ \left[\left(\frac{1}{n},\ldots,\frac{1}{n}\right),(\{1\},\ldots,\{n\})\right] \]

is \(k\)-stable (see the definition).

Necessity. Let \([a,\tau]\) be \(k\)-stable, and let \(\tau=(S_1,\ldots,S_l)\). Fix some \(r\): \(2\leqslant r\leqslant k+1\). Consider

\[ rI_n=\{1,\ldots,n,\;1,\ldots,n,\ldots,1,\ldots,n\}. \]

Redistribute the players in \(rI_n\) so that a partition of \(rI_n\) into sets \(S'_1,\ldots,S'_m\) is obtained such that \(|S'_j|=a_jr\) (\(a_j\) an integer). This operation can be carried out so that to each set from \(\tau\) there will be added (or subtracted) the least negative (positive) remainder upon division by \(r\). Since \(r\leqslant k+1\), the remainders will not exceed \(k\), and this means that the sets \(S'_1,\ldots,S'_m\in\Psi(\tau)\). They form an \(m\)-\(\theta\)-covering, since

\[ \sum_{j=1}^{m} S'_j=rI_n \quad\text{or}\quad \sum_{j=1}^{m}\frac{1}{r}S'_j=I_n . \]

By Theorem 1, it is necessary that

\[ \sum_{j=1}^{m}\frac{1}{r}v(S'_j)\leqslant 1. \]

Since \(|S'_j|=a_jr\), considering \(v(S)\) superadditive, we obtain

\[ v(S'_j)\geqslant a_j v(r),\quad j=1,\ldots,m. \]

Thus,

\[ 1\geqslant \sum_{j=1}^{m}\frac{1}{r}v(S'_j)\geqslant \frac{v(r)}{r}\sum_{j=1}^{m}a_j = v(r)\frac{n}{r}, \]

i.e. \(v(r)\leqslant r/n\) for any \(r=2,\ldots,k+1\), as was required to prove.

Corollary. In order for a symmetric game to have a core, it is necessary and sufficient that \(v(S)\leqslant \dfrac{|S|}{n}\) for every \(S\subset I_n\).

Received
30 V 1963

REFERENCES

\({}^1\) R. D. Luce, H. Raiffa, Games and Decisions, IL, 1961. \({}^2\) Fan Tzu, Systems of linear inequalities, Linear Inequalities and Related Systems, IL, 1959. \({}^3\) O. N. Bondareva, Vestn. Leningr. Univ., No. 13, 141 (1962). \({}^4\) R. D. Luce, Ann. Math., 62, No. 3, 517 (1955).