The Battle of the Sexes is a canonical game in game theory where two players want to coordinate on an action,

but have different preferences over the actions. The story is that for date night, Alice and Bob have tickets to

both a boxing match and the opera. Alice would prefer to go to the boxing match, and Bob would prefer to go to

the opera, but they always prefer to go to the same event as opposed to different events. They both must choose

one of the events to go to$,$ and they cannot observe each other$$s choice or communicate about their choices until

they arrive at the event. If the players arrive at the same event, the player who prefers that event gets a payoff

of $4,$ and the player who does not prefer that event gets a payoff of $2.$ If the players miscoordinate and arrive at

different events, they both get payoffs of $0.$

$($a$)$ Who are the players and what are their strategy sets? Write out their payoffs in a payoff matrix. $(3$ points$)$

$($b$)$ What are the pure strategy Nash equilibria of the game? $(2$ points$)$

$($c$)$ What is the mixed strategy Nash equilibria of the game? What are the players equilibrium expected payoffs?

$(4$ points$)$

$($d$)$ Suppose the Alice$$s payoff from attending the boxing match with Bob increases to $7.$ Her payoff from

attending the opera with Bob decreases to $1.$ Her payoffs from miscoordinating $($attending a different

event from Bob$)$ remain $0.$ Bob$$s payoffs also remain unchanged. What is the new mixed strategy Nash

equilibrium? $(3$ points$)$

$($e$)$ Provide some intuition for how the equilibrium probabilities change in part $($d$).$ Are the results surprising

at all? $(2$ points$)$

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