Coalitional Games

N. N. Vorob'ev

Teoriâ veroâtnostej i ee primeneniâ, Tome 12 (1967) no. 2, pp. 289-306

Résumé

Let $\mathfrak K$ be a regular complex with the set of vertices $I$, $\widetilde{\mathfrak K}\subset\mathfrak K$ , and let $H_{\widetilde K}$ ($\widetilde K\in\widetilde{\mathfrak K}$) be the set of real-valued functions of $\prod_{i\in I}S_i$. The vertices of $\mathfrak K$, the faces of $\mathfrak K$ and functions $H_{\widetilde K}$ are said to be players, coalitions and payoffs of coalitions correspondingly. The system $$ \Gamma=\langle I,\mathfrak K,\{S_i\}_{i\in I},\widetilde{\mathfrak K},\{H_{\widetilde K}\}_{\widetilde K\in\widetilde{\mathfrak K}}\rangle $$ is said to he a coalitional game. If the dimension of $\mathfrak K$ is 0 and $\widetilde{\mathfrak K}=\mathfrak K$ this definition coincides with that of a noncooperative game due to J. Nash. In the conditions of a coalitional game some situations can have some properties of stability. The situation $f^*_I$ is said to be stable for coalition $K\in\widetilde{\mathfrak K}$ relative to coalition $K\in\mathfrak K$ if $\widetilde K\cup K\in\mathfrak K$, $\widetilde K\cap K=\Lambda$ and for any coalitional strategies $f_K$ and $f_{\widetilde K}$ $$ H_{\widetilde K}(f^*_I\parallel(f_K,f_{\widetilde K}))\le H_{\widetilde K}(f^*_I\parallel f_K). $$ The stability of situation $f^*_I$ for $\widetilde K$ relative to $K$ essentially means that no failures of $K$ to keep its promises are to change the course of actions of $\widetilde K$. If $\varphi$ is a set of pairs $\langle\widetilde K,K\rangle$ (i.e. if $\varphi$ is a partial mapping of $\widetilde K$ into $K$) for which $\widetilde K\cup\varphi\widetilde K\in\mathfrak K$ and $\widetilde K\cap\varphi\widetilde K=\Lambda$ the situation $f_I$ is said to be $\varphi$-stable if it is stable for any $\widetilde K$ relative to $\varphi\widetilde K$. In the case of a non-copperative game when $\varphi\widetilde{\mathfrak K}=\Lambda$ the $\varphi$-stability becomes Nash's equilibrium. Naturally the existence of $\varphi$-stable situations for non-trivial sets $\varphi$ requires some mixture of strategies and situations. As formal tools for constructions of these mixtures and the proof of the corresponding theorems we use coordinated families of measures, of random transitions and their Markovian extentions. All these notions can be interpreted in terms of game $\Gamma$. For some class of sets $\varphi$ the theorem of existence of $\varphi$-stable situations is proved.