A matrix for this game would show that both players can choose from four strategies. Alice can always pass, always bet, pass with red and bet with black, or bet with red and pass with black. Bob can always fold, always call, fold with red and call with black, or fold with black and call with red. If you calculate the payoffs, you will see that Alice should bet three-fifths of the time no matter what card she has; the other two-fifths of the time she should bet only if she has black. Bob, on the other hand, should call Alice’s bet two-fifths of the time no matter what card he has; three-fifths of the time he should fold if he has red and call if he has black.29 (By the way, another thing game theory can show you is that this game is stacked in favor of Alice, if she always goes first. Playing the mixed strategies dictated by the game matrix assures her an average of 30 cents per hand.)
The notion of a mixed strategy, using some random method to choose from among the various pure strategies, is the essence of von Neumann’s proof of the minimax theorem. By choosing the correct mixed strategy, you can guarantee the best possible outcome you can get—if your opponent plays as well as possible. If your opponent doesn’t know game theory, you might do even better.
Game theory was not supposed to be just about playing poker or chess, or even just about economics. It was about making strategic decisions—whether in the economy or in any other realm of real life. Whenever people compete or interact in pursuit of some goal, game theory describes the outcomes to be expected by the use of different strategies. If you know what outcome you want, game theory dictates the proper strategy for achieving it. If you believe that people interacting with other people are all trying to find the best possible strategy for achieving their desires, it makes sense that game theory might potentially be relevant to the modern idea of a Code of Nature, the guide to human behavior.
In their book, von Neumann and Morgenstern did not speak