We conclude with a generalization of the minimax theorem.

His theorem is applied to a systematic treatment of interconnections between fixed point theorems, minimax theorems, variational inequalities, and monotone extension theorems.

Those applications are concerned with general equilibrium problems like as (collective) fixed point theorems, the von Neumann type intersection theorems, the von Neumann type minimax theorems, the Nash type equilibrium theorems, cyclic coincidence theorems, best approximation theorems, (quasi-) variational inequalities, and the Gale-Nikaido-Debreu theorem.

Under the constraint of fixed premium, by first identifying the special form of the optimal solution, a twist of the application of the classical minimax theorem is adopted to establish Equation (5) that characterizes the optimal values of deductibles by the marginal distribution functions of the risks.

Although the results in the previous sections conclude that stop-loss reinsurance contracts are optimal for the multivariate Problem 1, further properties of the level of the optimal deductibles can be obtained by applying the minimax theorem in view of Proposition 1 and the representation theorem in Lemma 1.

Fan, K., 1953, Minimax Theorems, Proceedings of the National Academy of Sciences of the United States of America, 39: 42-47.

where [G.sub.3](x, %) = max(f + [alpha][[sigma].sub.3], f + [alpha][[sigma].sub.3] + ([[sigma].sub.3](f - [alpha])/(1- [beta]))); interchanging of the "min" and "max" operators in the last equality is obtained by using the

minimax theorem ([22]).

The minimax theorem, the first mathematical theorem of game theory, was demonstrated by von Neumann independent of any economic considerations.

During the 1930s, von Neumann continued to show an occasional interest in the mathematics of games (46) and knew that the minimax theorem was relevant to economic theory as noted in his EEM paper.

(70) In general, these review articles expressed enthusiasm for and familiarized most economists with game theory concepts such as pure and mixed strategies, randomization, solution to a game, and the minimax theorem. Still, economists balked at accepting von Neumann and Morgenstern's work because of their aversion to mathematics and failure to read a long and technical book.

Willem,

Minimax Theorems (Progress in Nonlinear Differential Equations and Their Applications), vol.