At its core, the game theory of von Neumann is the saddlepoint result. There are two players, Red and Blue, for some positive integer n n moves, and an n x n m

At its core, the game theory of von Neumann is the saddlepoint result. There are two players, Red and Blue, for some positive integer n n moves, and an n x n matrix G = [g_ij] of real numbers of payoffs. So each of Red, Blue pick a move, that is, i, j = 1, ..., n, i for Red and j for Blue, and then at the same time they make, show, their moves. Then Red gains g_ij and Blue loses g_ij. The game theory saddlepoint result says that each of Red and Blue picks their moves probabilistically independently and randomly with some probability distribution over moves 1, 2, ..., n. Each of Red and Blue gets to pick their own distribution. So, the question for Red and Blue is, what distribution to pick? The saddlepoint result says that there is a distribution for Red and a distribution for Blue that is a saddlepoint, that is, any change by Red will result nothing better for Red and any change by Blue will result in nothing better for Blue. For the game of paper, scissors, rock, Red and Blue have the same distribution, uniform over the three moves. Von Neumann had a proof, but there is an easy proof early in linear programming theory. When I was teaching, I covered the result in a few minutes, about the same as needed just to read this post. Broad, profound consequences for civilization? Naw.