(Defending territory) General A is defending territory accessible by two mountain passes against an attack by general B. General A has three divisions at her disposal, and general B has two divisions. Each general allocates her divisions between the two passes. General A wins the battle at a pass if and only if she assigns at least as many divisions to the pass as does general B; she successfully defends her territory if and only if she wins the battle at both passes. Formulate this situation as a strategic game and find all its mixed strategy equilibria. (First argue that in every equilibrium B assigns probability zero to the action of allocating one division to each pass. Then argue that in any equilibrium she assigns probability ½ to each of her other actions. Finally, find A’s equilibrium strategies.) In an equilibrium do the generals concentrate all their forces at one pass, or spread them out? An implication of Proposition 113.2 is that a nondegenerate mixed strategy equilibrium (a mixed strategy equilibrium that is not also a pure strategy equilibrium) is never a strict Nash equilibrium: every player whose mixed strategy assigns positive probability to more than one action is indifferent between her equilibrium mixed strategy and every action to which this mixed strategy assigns positive probability. Any equilibrium that is not strict, whether in mixed strategies or not, has less appeal than a strict equilibrium because some (or all) of the players lack a positive incentive to choose their equilibrium strategies, given the other players’ behavior. There is no reason for them not to choose their equilibrium strategies, but at the same time there is no reason for them not to choose another strategy that is equally good. Many pure strategy equilibria—especially in complex games—are also not strict, but among mixed strategy equilibria the problem is pervasive. Given that in a mixed strategy equilibrium no player has a positive incentive to choose her equilibrium strategy, what determines how she randomizes in equilibrium? From the examples above we see that a player’s equilibrium mixed strategy in a two-player game keeps the other player indifferent between a set of her actions, so that she is willing to randomize. In the mixed strategy equilibrium of BoS, for example, player 1 chooses B with probability 2 3 so that player 2 is indifferent between B and S, and hence is willing to choose each with positive probability. Note, however, that the theory is not that the players consciously choose their strategies with this goal in mind! Rather, the conditions for equilibrium are designed to ensure that it is consistent with a steady state. In BoS, for example, if player 1 chooses B with probability 2/3 and player 2 chooses B with probability 1 3 then neither player has any reason to change her action. We have not yet studied how a steady state might come about, but have rather simply looked for strategy profiles consistent with steady states. In Section 4.9 I briefly discuss some theories of how a steady state might be reached.
