On existence of Nash equilibria of games with constraints on multistrategies (Q1841574)

The authors consider an \(n\) player game \(\{Y_{1},Y_{2},\ldots,Y_{n},\) \(f_{1},f_{2},\ldots,f_{n}\}\) with constraint set \(C\subset Y=\) \(\prod_{i\in G} Y_{i}\) of multistrategies where \(G=\{1,2,\ldots,n\}\) be the set of players, \(Y_{i}\) is a strategy set of player \(i\) and \(f_{i}\) is a loss function associated with a player \(i.\) The \(i\)th player has to minimize the function \(f_{i}\) with respect to the \(i\)th variable. The sufficient conditions given in this paper are less restrictive than those required by the Arrow-Debreu-Nash theorem on the existence of a Nash equilibrium of an \(n\) player game in normal form with a nonempty closed convex constraint \(C\) on the set \(Y\) of multistrategies. Two cases are considered. In the first case, \(Y\) is a real Hilbert space and the loss function class is quadratic and in the second case, \(Y\) is a Euclidean space, the loss functions are continuous and \(f_{i}\) is convex with respect to the \(i\)th variable. In both cases the existence of a Nash equilibrium is guaranteed. See also the articles by \textit{M. B. Lignola} [J. Optim. Theory Appl. 94, 137-145 (1997; Zbl 0886.90183)] and \textit{P. Dasgupta} and \textit{E. Maskin} [Rev. Econ. Stud. 53, 1-26 (1986; Zbl 0578.90098)].