I’m a bit interested in game theory, although i never studied it at school. I’m satisfied with reading books about it, and without going in depth too far I try to get the most important concepts.
Now I’ve read The Compleat Strategyst - Being a Primer on the theory of Games of Strategy by J.D. Williams, published in the 50s. It’s all about zero-sum games with the row player being the maximizing and the column player being the minimizing guy.
Williams comes up with something he calls “saddle-point”. You have a saddle-point if the greatest of the minima of each row equals the smallest of the maxima of each column (i.e. if each player chooses the strategy where their damage in the worst case is the smallest). For Williams, a saddle-point (if there is one) is the solution and, finally, the outcome of the game.
Other game theory books I’ve read explain the notion of the “Nash equilibrium”, named after Russell Crowe. In a NE, neither of the two players has reason to change their strategy because it wouldn’t improve his payoff. The saddle-point thing is not mentioned (is this what game theorists refer to when they speak about the “minmax theorem”?).
I’m inclined to suppose that “saddle-point” and Nash equilibrium are two different definitions of the same thing. Are they? Is every saddle-point (after Williams’ definition) an equilibrium after Nash’s definition, and vice versa?