Abstract
Full Text
UDC 518.9
MATHEMATICS
O. N. BONDAREVA
A SOLUTION FOR A CLASS OF GAMES WITH EMPTY CORE
(Presented by Academician Yu. V. Linnik on 17 V 1968)
Consider a cooperative game \(\Gamma=\langle I,v\rangle\), where \(I=\{1,\ldots,n\}\) is the set of players, and \(v(S)\) is the characteristic function of the game. The set of imputations \(A\) is defined as
\[ A=\left\{x=(x_1,\ldots,x_n): x_i \geqslant v(i),\quad \sum_{i=1}^{n}x_i=v(I_n)\right\}. \]
On \(A\) a domination relation is defined: \(x \succ y\), if there exists such an \(S \subset I\) that \(x_i>y_i,\ i\in S\), and
\[ \sum_{i\in S}x_i \leqslant v(S). \]
A solution \(V\) is a set of imputations having the following properties:
1) internal stability: it cannot be that \(x\succ y\), if \(x,y\in V\);
2) external stability: whatever \(z\notin V\), there exists such an \(x\in V\) that \(x\succ z\).
The core is the set:
\[ U=\left\{x=(x_1,\ldots,x_n)\in A:\ \sum_{i\in S}x_i \geqslant v(s)\ \text{for every } s\subset I\right\}. \]
We shall say that a game has property \(v\) if it has a solution, property \(cv\) if its solution coincides with the core, and property \(\overline{cv}\) if this solution has the following additional property:
whatever \(y\notin V\), if \(\sum_S y_i < v(S)\), then there exists such an \(x\in V\) that \(x\succ y\) by the coalition \(S\). Let us note that in all cases known to us where the solution coincides with the core, property \(\overline{cv}\) follows from \(cv\).
For any \(R\subset I\) define a game \(\Gamma_R(a_0,a_S)\) with set of players \(R\) and characteristic function \(u(S),\ S\subset R\):
\[ u(R)=a_0,\quad 0\leqslant a_0\leqslant 1; \tag{1} \]
\[ u(S)=a_S,\quad v(S)\leqslant a_S \leqslant \max_{T\subset I-R}[v(S\cup T)-v(T)] = \overline{a}_S. \]
Consider a partition of the set \(I\) into sets \(M\) and \(N\), and let \(v(M)+v(N)\geqslant 1\) (if \(v(M)+v(N)>1\), then the core is obviously empty). Construct \(\Gamma_M(a_0,a_S)\) and \(\Gamma_N(\beta_0,\beta_T)\).
Call the collections \(\{a_0,a_S\}\) and \(\{\beta_0,\beta_T\}\) conjugate if: 1) \(a_0+\beta_0=1\); 2) \(a_S+\beta_T\geqslant v(S\cup T),\quad S\subset M,\ T\subset N\).
Lemma. Whatever the collection \(\{a_0,a_S\}\) \((S\subset M)\) satisfying (1), there exists a conjugate collection \(\{\beta_0,\beta_T\}\) \((T\subset N)\).
Proof. Take \(\beta_0=1-a_0\),
\[ \beta_T=\overline{\beta}_T=\max_{S\subset N}[v(S\cup T)-v(S)]. \]
Then \(\beta_T \geqslant v(T \cup S) - v(S)\), \(S \subset M\),
\[ \alpha_S+\beta_T \geqslant v(S)+v(T\cup S)-v(S)=v(T\cup S). \]
Let \(V_M(a_0,\alpha_S)\) be a solution of \(\Gamma_M(a_0,\alpha_S)\), and \(V_N(\beta_0,\beta_T)\) a solution of \(\Gamma_N(\beta_0,\beta_T)\); then by \(V(a_0)=V_M(a_0,\alpha_S)\,\hat{\times}\,V_N(\beta_0,\beta_T)\) we denote the set of all such \(x \in A\) that
\[ x_M \in V_M(a_0,\alpha'_S), \qquad x_N \in V_N(\beta_0,\beta'_T) \]
(\(x_M\) and \(x_N\) are the projections of \(x\) onto \(M\) and \(N\), respectively), and the collections \(\{a_0,\alpha'_S\}\) and \(\{\beta_0,\beta'_T\}\) are conjugate.
Theorem 1. If in a game \(\Gamma\) \(I\) is divided into two such coalitions \(M\) and \(N\) (each with more than one player) that 1) \(v(M)+v(N)\geqslant 1\); 2) all games \(\Gamma_M(a_0,\alpha'_S)\) and \(\Gamma_N(\beta_0,\beta'_T)\), for all conjugate collections \(\{a_0,\alpha'_S\}\) and \(\{\beta_0,\beta'_T\}\), where \(1-v(N)\leqslant a_0\leqslant v(M)\), have the property \(\overline{cv}\), then
\[ V(a_0)=V_M(a_0,\alpha_S)\,\hat{\times}\,V_N(\beta_0,\beta_T) \]
is a solution of \(\Gamma\) for all \(a_0\) from the interval \(1-v(N)\leqslant a_0\leqslant v(M)\).
Proof. The internal stability of \(V(a_0)\) follows from the fact that, for \(S \supseteq M\) or \(S \supseteq N\), domination cannot occur, while for the remaining \(S\) and \(x \in V(a_0)\)
\[ \sum_S x_i \geqslant \alpha(S\cap M)+\beta(S\cap N)\geqslant v(S). \]
We now prove the external stability of \(V(a_0)\). Consider two cases:
\[ \text{1) } \sum_M y_i=a_0; \]
\[ \text{2) } \sum_M y_i<a_0 \quad \text{or} \quad \sum_N y_i<1-a_0. \]
Consider the first case. It is easy to show that, if \(y \notin V(a_0)\), then there exists a coalition \(R\) such that
\[ \sum_R y_i < v(R). \tag{2} \]
If, moreover, there exists \(R \subset M\) (or \(R \subset N\)), then by the property \(\overline{cv}\) there exists such an \(x_M \in V_M(a_0,v(S))\) that \(x_M>y_M\) (on \(R\)), and by the lemma there exists a collection \(\{1-a_0,\beta'_T\}\) conjugate with \(\{a_0,v(S)\}\); then, if \(x_N \in V_N(1-a_0,\beta'_T)\), then \(x=(x_M,x_N)>y\).
Suppose now that, for any \(R\) satisfying (2), \(R\cap M\ne \Lambda\), \(R\cap N\ne \Lambda\). Denote by \(R_0\) such one of the coalitions \(R\) for which there is no coalition \(R\subset R_0\) satisfying property (2).
Put
\[ \alpha'_{R_0\cap M}=\sum_{R_0\cap M} y_i+\varepsilon_1; \qquad \alpha'_S=\sum_S y_i,\quad S\subset R_0\cap M; \qquad \alpha'_S=\bar{\alpha}_S \]
for the remaining \(S\subset M\).
\[ \beta'_{R_0\cap N}=\sum_{R_0\cap N} y_i+\varepsilon_2; \qquad \beta'_T=\sum_T y_i,\quad T\subset R_0\cap N; \qquad \beta'_T=\bar{\beta}_T \]
for the remaining \(T\subset N\),
where
\[ \varepsilon_1+\varepsilon_2=v(R_0)-\sum_{R_0} y_i,\qquad \varepsilon_1>0,\quad \varepsilon_2>0. \]
Since \(\sum_S y_i \geqslant v(S)\), \(S \underset{\ne}{\subset} R_0\), it is not hard to see that the sets \(\{\alpha_0,\alpha_S'\}\) and \(\{1-\alpha_0,\beta_T'\}\) are conjugate and, for \(R_0\),
\[ \alpha_{R_0\cap M}' + \beta_{R_0\cap N}' = v(R_0). \]
If \(|R_0\cap M|>1\) and \(|R_0\cap N|>1\), then by the property \(\overline{cv}\) there exist \(x_M\in V_M(\alpha_0,\alpha_S')\) and \(x_N\in V_N(1-\alpha_0,\beta_T')\) such that \(x_M \succ y_M\) on \(R_0\cap M\) and \(x_N \succ y_N\) on \(R_0\cap N\), and hence \(x=(x_M,x_N)\succ y\) on \(R_0\).
If, for example, \(R_0\cap M=\{i_0\}\), then, by Theorem 4.1 of \({}^{(2)}\), in the solution (core) there exists an imputation \(x_M\in V_M(\alpha_0,\alpha_S')\) such that the component \((x_M)_{i_0}\) is equal to \(\alpha(i_0)\), and the proof is preserved for this case as well.
Let now case 2 occur, and let \(\sum_M y_i<\alpha_0\); consider
\[ z_M=(z_M^{(i)}):\quad z_M^{(i)}=y_i+\varepsilon,\quad i\in M,\quad \sum_{i\in M}z_M^{(i)}=\alpha_0. \]
Either \(z_M\in V_M(\alpha_0,v(S))\). Or there exists \(x_M\in V_M(\alpha_0,v(S))\), \(x_M\succ z_M\); then in the first case \(z=(z_M,z_N)\succ y\) on \(M\), and in the second \(x=(x_M,x_N)\succ y\), where \(z_N,x_N\) are constructed as in the proof of the first case.
Consider examples.
Example 1. If for a four-person game there exists \(0\leqslant \alpha_0\leqslant 1\) such that
\[ v(1,2)\geqslant \alpha_0,\quad v(3,4)\geqslant 1-\alpha_0;\quad v(i,3)+v(i,4)\leqslant \]
\[ \leqslant 1-\alpha_0,\ i=1,2;\quad v(1,j)+v(2,j)\leqslant \alpha_0,\ j=3,4. \]
then the game has a solution lying in the plane \(x_1+x_2=\alpha_0\); note that no restrictions are imposed on the values \(v(S)\) for \(|S|=3\).
It is known that a three-person game has the property \(cv\) (\(\overline{cv}\)) if and only if \(v(i,j)+v(i,k)\leqslant v(I)\), \(\{i,j,k\}\subset \{1,2,3\}\). It can also be shown that, if a three-person game with characteristic function \(v(S)\) satisfies the condition \(cv\) (\(\overline{cv}\)), then the game with \(v'(S)=v(S)\), \(|S|>1\), \(v'(i)=a_i\), where \(a_i\leqslant v(I-\{i\})\), \(i=1,2,3\), also has the property \(cv\) (\(\overline{cv}\)). Using this, we have:
Example 2. If for a five-person game
\[ v(1,2,3)\geqslant \alpha_0;\quad v(4,5)\geqslant 1-\alpha_0;\quad \alpha_0-\frac12\leqslant v(i,j)\leqslant \alpha_0/2; \]
\[ v(i,j,l)\leqslant \alpha_0/2;\quad v(i,l)\leqslant (1-\alpha_0)/2;\quad \{i,j\}\subset \{1,2,3\};\quad l=4,5, \]
where \(0\leqslant \alpha_0\leqslant 1\); the remaining \(v(S)\) are arbitrary, then such a game has a solution lying in the hyperplane \(x_1+x_2+x_3=\alpha_0\).
Leningrad State University
named after A. A. Zhdanov
Received
7 V 1968
REFERENCES
- J. Von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton, 1947.
- O. N. Bondareva, Problems of Cybernetics, vol. 10 (1963).