Meni_Rosenfeld

Inspired by: Swords and Armor: A Game Theory Thought Experiment Recently, nick012000 has posted Swords and Armor: A Game Theory Thought Experiment. I was disappointed to see many confused replies to this post, even after a complete solution was given by Steve_Rayhawk. I thought someone really ought to post an explanation about mixed strategy Nash equilibria. Then I figured that that someone may as well be me. I will assume readers are familiar with the concepts of a game (a setting with several players, each having a choice of strategies to take and a payoff which depends on the strategies taken by all players) and of a Nash equilibrium (an "optimal" assignment of strategies such that, if everyone plays their assigned strategy, no player will have a reason to switch to a different strategy). Some games, like the famous prisoner's dilemma, have a Nash equilibrium in so-called "pure strategies" (as opposed to mixed strategies, to be introduced later). Consider, however, the following variant of the matching pennies game: Player 1 is a general leading an attacking army, and player 2 is the general of the defending army. The attacker can attack from the east or west, and the defender can concentrate his defenses on the east or west. By the time each side learns the strategy of its enemy, it is too late to switch strategies. Attacking where the defenses aren't concentrated gives a great advantage; additionally, due to unspecified tactical circumstances, attacking from the east gives a slight advantage. The sides have no interest in cooperating, so this is a zero-sum game (what one side wins, the other loses). This elaborate description can be summarized in the following payoff matrix (these payoffs are for the attacker; the defender's payoffs are their negatives): 2: East 2: West 1: East -1 2 1: West 1 -2 What strategy should each side play? The attacker can think, "Overall, going east is advantageous. So I'll go east." The defender, anticipating this, might sa