The existence of TU \(\alpha\)-core in normal form games
The notion of an \(\alpha\)-core is a natural extension of the notion of the core from cooperative games in characteristic function form to games with transferable utility (TU). A TU game with \(n\) players is defined by its \(n\) payoff functions \(u^i:X_1\times\ldots\times X_n\to \mathbb{R}\), where \(X_i\) is the set of all possible strategies of the \(i\)-th player. For each coalition \(S\) and strategies \(x_S=\{x_i\mid i\in S\}\), the guaranteed gain \({\overline u}_S(x_S)\) is the minimum of the total gain \(\sum_{i\in S} u^i(x_S,y_{-S})\) for all possible actions of the outsiders \(y_{-S}\in X_{-S}\). The coalition's payoff \(v_\alpha(S)\) is defined as its largest possible guaranteed gain. An \(\alpha\)-core is then defined as the core of the corresponding cooperative game, i.e., as the set of all pairs \((\overline x,\sigma)\), where \(\overline x\) is a strategy that maximizes the total gain \(\sum_{i=1}^n u^i(\overline x)\), and \(\sigma=(\sigma_1,\ldots,\sigma_n)\) is an allocation in which each coalition \(S\) gets at least \(v_\alpha(S)\). The author considers weakly separable games in which, for every coalition \(S\) and for every player \(i\), the minimum of this player's gain \(u^i(x_S,y_{-S})\) is attained for exactly the same strategy \(y_{-S}\) as the minimum of the total gain \(\sum_{i\in S} u^i(x_S,y_{-S})\). He proves that for such games, if the choice sets \(X_i\) are compact and convex and the payoff functions \(u_i\) are continuous and concave, then this game has an \(\alpha\)-core. An important example of such a game is an oligopoly market in which the profit is equal to \(u^i=p(\sum x_j)\cdot x_i-c^i(x_i)\), where the price \(p\) of a unit depends on the total demand \(\sum x_j\), and the production cost \(c^i\) depends on the production volume.
scientific article; zbMATH DE number 1309800