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Playing for Pennies
Not all game theory games have a saddle point. Consider a simple game that requires a more complex playing strategy: Matching Pennies, another classic game theory problem. Here is how the game works: two players each have a penny. Hiding their penny from view, both players pick a side, heads up or heads down, and then simultaneously reveal their pennies. If they match, Player 1 gets both pennies. If they don't match, Player 2 gets them. We can graph this game on a payoff grid : ing to a certain probability ratio. For Matching Pennies, the mixed strategy requires rational players to randomly pick heads or tails, with a 50/50 chance of selecting either one.
Remember that rational players will attempt to maximize their own gains in utility while minimizing the gains of their opponents. If rational players play many, many games of Matching Pennies, they will end up with an average utility of zero. This means that neither player will ever come out ahead, but that is the best that they can hope for in this "game." 