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ECON 2112. SOLVING NASH EQUILIBRIA IN PURE AND MIXED STRATEGIES

Exercise 1. Find every Nash equilibria in pure and mixed strategies in the following version of the

“Battle of the Sexes” .

Solution. (Recall that the symbol “∈” reads “belongs to” and that “[0, 1]” represents the set of numbers between 0 and 1, including 0 and 1.)  Player 1 can compute his best reply to player 2’s strategy in the following way. Suppose that player 2 plays L with probability y and R with probability 1 一 y. Player 1’s

expected payof from each of his strategies is

Therefore, player 1 strictly prefers T to B if and only if 3y > 1 一 y, which is equivalent to y > . Also, player 1 strictly prefers B to T if and only if 3y < 1 一 y, i.e., if y < . Finally, Player 1 is indiferent between playing T or B if and only if y = .  If x represents the probability that player 1 plays T, we have that

We can represent R1(y) in the xy-plane:

Player 2 striclty prefers L to R if and only if x > 3(1 一 x), which holds when x > . Player 2 striclty prefers R to L if and only if x < 3(1 一 x), that is, if x < . Finally, Player 2 is indiferent between R and L if x = . Thus, the best response of player 2 to player 1’s strategies is:

If we represent R2(x) in the same plane as R1(y), we obtain that every point where R1(y) and R2(x) intersect is a Nash equilibrium point. That is, (T, L) correspond to the point x = y = 1, ( T + B, L+R) corresponds to the point (x, y) = ( , ) and (B, R) corresponds to the point x = y = 0.

So there are 3 Nash equilibria (T, L), ( T + B, L + R) and (B, R).