International Trade

Games with Nash Equilibrium

Example - Battle of the Sexes

In this example, a man and woman are trying to choose their entertainment for the evening. Their choices are going to the football (F) or the ballet (B). The payoffs are as follows:

  • If they go to the football, the man is very happy and the woman is only happy because she is with the man. [1]
  • If they go to the ballet, the woman is very happy and the man is only happy because he is with the woman. [2]
  • If the man wants to go to the football, but the woman insists on going to the ballet, they will quarrel and end up going nowhere. A similar situation exists the other way around. [3]


    • Now consider the strategy from the man's point of view:

  • If the woman chooses F, then the man has a higher payoff with F. [4]
  • However, if she chooses B, then he has a higher payoff with B. [5]


    • In other words, there is no dominant strategy for the man. By symmetry, this is also true for the woman.

    However, although there is no dominant strategy equilibrium, there are two equilibrium points:

  • If they choose (F, F), [6] then it is an equilibrium because neither player will gain by moving away from that point unilaterally.
  • Similarly, (B, B) [7] is a similar equilibrium point.


    • These two points are known as Nash equilibrium points. [8]

    A Nash equilibrium is a point where neither player gains by moving unilaterally.

    Analysis

    Given that we have two equilibrium points, which payoff is realised? The answer is that this is a mixed strategy, based on probability and commitment. We'll see more about commitment in the next example.


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