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MATHEMATICS
N. N. VOROBYEV
STABLE SITUATIONS IN COALITION GAMES
(Presented by Academician A. N. Kolmogorov on 27 XI 1959)
By coalition games we mean games of such a type in which certain groups (coalitions) of players possess interests inherent in these groups as such. The payoff of a group is understood as a quantitatively measured realization of the goal set before this group, and is not subject to division among the players who are members of this group. (We note that such a characterization of coalition games does not fully correspond to their description given by the author in the survey article ([1]).)
One of the simplest variants of the theory of coalition games (“coalitioners’ games”) was considered in the note ([2]), whose definitions and notation will be assumed known. Let us note at once that the model of every coalitioners’ game can be constructed within the framework of a certain noncoalitional game. Therefore, relying only on Nash’s theorem ([3]), one can prove (and moreover without using the apparatus of consistent families of measures and Markov measures ([2])) the following strengthening of Theorem 3 of the note ([2]).
Theorem 1. Every coalition game (“coalitioners’ game”) has mixed equilibrium situations in which the individual strategies of the players are independent. Moreover, one may require that the set \(K\) in the definition of an equilibrium situation be empty.
The purpose of the present note is to present a new variant of the theory of coalition games. All sets of the form \(S_i\) \((i \in I)\) will be assumed finite, and all subsets \(S_K\) \((K \subset I)\) measurable for every measure \(\mu_K\) considered below on \(S_K\).
- Let \(\mathfrak{R}\) be a complex of subsets of the set \(I\). A consistent family of measures \(\mu_{\mathfrak{R}}\) on \(S_{\mathfrak{R}} = \{S_K\}_{K \in \mathfrak{R}}\) is called internal if, for any \(K \in \mathfrak{R}\) and \(f_K \in S_K\), one has \(\mu_K(f_K) > 0\). The set of all internal consistent families of measures on \(S_K\) will be denoted by \(W(S_{\mathfrak{R}})\).
Theorem 2. There exists a topological mapping \(\varphi\) of the set \(W(S_{\mathfrak{R}})\) into a Euclidean space \(E\) of the corresponding dimension such that all conditional probabilities of the form \(\mu_{R \cup K}(f_R \mid f_K)\) turn out to be uniformly continuous, on the whole image \(\varphi W(S_{\mathfrak{R}})\), functions of the family \(\mu_{\mathfrak{R}}\) in the sense of the metric of \(E\), and the closure \(\varphi W(S_{\mathfrak{R}})\) in \(E\) is homeomorphic to a closed ball of the corresponding dimension.
- Take arbitrary \(K \subset L \subset I\) and say that a random transition \(\pi(L \mid K)\) is given on \(S_L\) if, for any \(R \subset L\), \(N \subset K\), \(x \in S_R\), \(y \in S_N\), numbers \(\pi^{L,K}_{R,N}(x \mid y)\) are defined, satisfying the following conditions:
\(1^\circ\). For any fixed \(N \subset K\) and \(y \in S_N\)
\[ \pi^{L,K}_{R,N}(x \mid y), \qquad R \subset L,\ x \in S_R, \]
constitutes a consistent family of measures on \(\{S_R\}_{R \subset L}\).
\(2^\circ.\) For any \(R \subset L,\ x \in S_R,\ N \subset K,\ y \in S_N\), as soon as \(x_{N \cap R}=y_{N \cap R}\),
\[ \pi_{R,N}^{L,K}(x \mid y)=\pi_{R\setminus N,N}^{L,K}(x_{R\setminus N}\mid y). \]
\(3^\circ.\) For any \(N_1,N_2 \subset K\) for which \(N_1\cap N_2=\Lambda\), and arbitrary \(R\subset L,\ x\in S_R,\ y\in S_N\), one has
\[ \pi_{R,N}^{L,K}(x\mid y) = \sum_{z\in S_{N_2}} \pi_{R,N_1\cup N_2}^{L,K}(x\mid y\times z)\, \pi_{N_1\cup N_2,N_1}^{N_1\cup N_2,N_1\cup N_2}(x\times y\mid x). \]
Random transitions \(\pi'(L_1\mid K_1)\) and \(\pi''(L_2\mid K_2)\) are called consistent if, for any \(R\subset L_1\cap L_2,\ N\subset K_1\cap K_2,\ x\in S_R,\ y\in S_N\),
\[ \pi_{R,N}^{\prime\,L_1,K_1}(x\mid y)= \pi_{R,N}^{\prime\prime\,L_2,K_2}(x\mid y). \]
The set of all ordered pairs of subsets \(I\) of the form \(\langle K,L\rangle\), where \(K\subset L\in\mathfrak R\), will be denoted by \(\Phi_{\mathfrak R}\), and the set of all ordered pairs \(\langle K,I\rangle\), where \(K\in\mathfrak R\), by \(\Psi_{\mathfrak R}\). By \(\pi(\Phi_{\mathfrak R})\) we shall mean a system of random transitions of the form \(\pi(L\mid K)\), where \(\langle K,L\rangle\in\Phi_{\mathfrak R}\). The system \(\pi(\Psi_{\mathfrak R})\) is defined analogously.
A consistent system \(\pi(\Psi_{\mathfrak R})\) is called an extension of the consistent system \(\pi(\Phi_{\mathfrak R})\) if every random transition from \(\pi(\Psi_{\mathfrak R})\) is consistent with every random transition from \(\pi(\Phi_{\mathfrak R})\).
Theorem 3. In order that every system of the form \(\pi(\Phi_{\mathfrak R})\) have an extension, it is necessary and sufficient that the complex \(\mathfrak R\) be regular.
In what follows, by \(\pi(\Psi_{\mathfrak R})\) we shall mean some fixed extension of \(\pi(\Phi_{\mathfrak R})\), called Markovian.
- For each \(\mu_{\mathfrak R}\in W(S_{\mathfrak R})\) (and also for each \(\varphi\mu_{\mathfrak R}\in \varphi W(S_{\mathfrak R})\)), the conditional probabilities of the form \(\mu_{R\cup K}(f_R\mid f_K)\) \((R\cup K\in\mathfrak R)\) determine a certain system of random transitions of the form \(\pi(\Phi_{\mathfrak R})\). In view of the uniform continuity of these probabilities, the correspondence between points of \(\varphi W(S_{\mathfrak R})\) and systems of random transitions can, by continuity, be extended to the closure of this set \(\overline{\varphi W}(S_{\mathfrak R})\). All systems of random transitions obtained in this way will be called realizable.
Theorem 4. The realizability of each system of random transitions is recognized by means of a finite number of elementary operations.
- Let \(I\) be the set of players; \(\mathfrak R\) a regular family of coalitions (\(\mathfrak R\) is a regular complex of subsets of \(I\)); \(S_i\), for any \(i\in I\), the set of (pure) strategies of player \(i\). The elements \(S_K\) \((K\in\mathfrak R)\) will be called pure coalition strategies, and the elements \(S_I\), pure situations. Let, further, \(\widehat{\mathfrak R}\subset\mathfrak R\), and let \(H_{\widetilde K}\) \((\widetilde K\in\widehat{\mathfrak R})\) be functions taking real values on \(S\). These values are called the payoffs of coalition \(\widetilde K\) in the corresponding situation.
The system
\[ \Gamma=\langle I,\mathfrak R,\{S_i\}_{i\in\mathfrak R},\widehat{\mathfrak R},\{H_{\widetilde K}\}_{\widetilde K\in\widehat{\mathfrak R}}\rangle \tag{1} \]
will be called a coalitional game.
The system
\[ \Gamma^*=\langle I,\mathfrak R,\{S_i^*\}_{i\in\mathfrak R},\widehat{\mathfrak R},\{H_{\widetilde K}^*\}_{\widetilde K\in\widehat{\mathfrak R}}\rangle \]
is called a mixed extension of the game (1) if \(S_K^*\) \((K\in\mathfrak R)\) is the set of all realizable systems of random transitions of the form
\(\pi(\Phi_{I_K})\) (\(I_K\) denotes the complex of all subsets of the set \(K\)). All realizable Markov continuations \(\pi(\Psi_{\mathfrak K})\) of consistent systems of random transitions of the form \(\pi(\Phi_{\mathfrak K})\), consisting of systems of the form \(\pi(\Phi_{I_K})\) \((K \in \mathfrak K)\), shall be called refined situations \(\Gamma^*\), and the measures \(\pi(\cdot \mid f_\Lambda)\) on \(S_I\) included in them shall be called situations. The payoff function \(H^*_{\widetilde K}\) in \(\Gamma^*\) is defined as the mathematical expectation
\[ H^*_{\widetilde K}(\pi)=\sum_{f_I\in S_I} H_{\widetilde K}(f_I)\pi(f_I\mid f_\Lambda). \]
If \(\pi\) is some refined situation \(\Gamma^*\), then for \(K\in\mathfrak K\) and \(f_K\in S_K\), by \(\pi\|f_K\) we shall mean the situation (i.e., the measure on \(S_I\)) for which, for any \(f'_I\in S_I\),
\[ (\pi\|f_K)(f'_I)=\pi(f'_I\mid f_K). \]
Consider some mapping \(\varphi\) of the complex \(\widetilde{\mathfrak K}\) into \(\mathfrak K\), under which \(\widetilde K\cup\varphi\widetilde K\in\mathfrak K\) and \(\widetilde K\cap\varphi\widetilde K=\Lambda\). A situation \(\pi\) is called \(\varphi\)-stable if, for every \(K\in\widetilde{\mathfrak K}\) and arbitrary \(f_{\widetilde K}\in S_{\widetilde K}\) and \(f_{\varphi\widetilde K}\in S_{\varphi\widetilde K}\),
\[ H^*_{\widetilde K}(\pi\|(f_{\widetilde K},f_{\varphi\widetilde K}))\leqslant H^*_{\widetilde K}(\pi\|f_{\varphi\widetilde K}). \]
If the complex \(\mathfrak K\) is zero-dimensional, \(\widetilde{\mathfrak K}=\mathfrak K\setminus\{\Lambda\}\), and \(\varphi\widetilde K=\Lambda\) for every \(\widetilde K\in\widetilde{\mathfrak K}\), then the game \(\Gamma\) becomes noncoalitional, and the concept of \(\varphi\)-stability of situations turns into the concept of their equilibrium in the sense of Nash.
Let
\[ \mathfrak K=\mathfrak K_n\supset \mathfrak K_{n-1}\supset\cdots\supset \mathfrak K_1\supset \mathfrak K_0=\Lambda \]
be a sequence of subcomplexes of \(\mathfrak K\), in which \(\mathfrak K_{i-1}\) \((i=1,\ldots,n)\) is a normal subcomplex of \(\mathfrak K_i\) and is obtained from \(\mathfrak K_i\) by deleting the stars of all vertices belonging to the set \(Q_i\) of all proper vertices of an outer skeleton \(T_i\) in \(\mathfrak K_i\). Partition each of the sets \(Q_i\) into pairwise disjoint sets of players
\[ Q_i=K_{i1}\cup K_{i2}\cup\cdots\cup K_{ir_i} \]
and take as \(\widetilde{\mathfrak K}\) the family of all sets of the form \(K_{ij}\) \((i=1,\ldots,n;\ j=1,\ldots,r_i)\). Put \(\varphi^*K_{11}=\Lambda\), and, for \(i\ne1\) or \(j\ne1\), \(\varphi^*K_{ij}\subset K_{kl}\), where \(k<i\) or \(k=i\) and \(l<j\). At the same time we shall require that, for any \(i\) and \(j\),
\[ K_{ij}\cup\varphi^*K_{ij}\in\mathfrak K. \]
Theorem 5. Every coalitional game \(\Gamma\), for any mapping of the form \(\varphi^*\), has \(\varphi^*\)-stable situations in its mixed extension.
Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR
Received
26 XI 1959
REFERENCES
- N. N. Vorob’ev, Uspekhi Mat. Nauk, 14, 4(88), 21 (1959).
- N. N. Vorob’ev, DAN, 124, No. 2, 253 (1959).
- J. Nash, Ann. of Math., 54, 286 (1951).