Consider the payoff matrix of the Battle of the Sexes we discussed in class: Wife Husband Opera Football Opera I 2, l 0, 0 Football | 0, 0 1, 2 Let...
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Consider the payoff matrix of the Battle of the Sexes we discussed in class:

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螢幕截圖 2019-10-02 下午5.05.16.png

Consider the payoff matrix of the Battle of the Sexes we discussed in class: Wife \ Husband Opera Football
Opera I 2, l 0, 0
Football | 0, 0 1, 2 Let p be the probability that the wife goes to the Opera and q be the probability that the husband goes to the
Opera. (a) Suppose that the couple still prefers going to some event together rather than alone (i.e. the payoffs
for (O, F) and (F, O) are still (0, 0)), propose ONE change to the payoff in the payoff matrix so that
the mixed strategy Nash equilibrium will change from [(2/3 0, 1/3 F), (1/3 0, 2/3 F)] to [(2/3 0, 1/3
F), (1/2 0, 1/2 F)] (i.e. q will be increased from 1/3 to 1/2). Write down the modified payoff matrix
and show that a mixed strategy Nash equilibrium is attained when p = 2/3 and q = 1/2. (b) Calculate the difference between the modified expected payoff and the original expected payoff for
the wife.

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