One of the most famous games treated in early game theory courses is known as the “Battle of the Sexes” — and it bears close resemblance to the game in which you and I choose sides of the street when you are British and I am American. In the “Battle of the Sexes” game, two partners in a newly blossoming romance have different preferences for what to do on a date, but neither can stand the thought of not being with the other. Suppose we are talking about you and your partner. You love opera and your partner loves football.2 Both you and your partner can choose to go to the opera and today’s football game, with each of you getting 0 payoff if you aren’t at the same activity as the other, 10 if you are at your favorite activity with your partner, and 5 if you are at your partner’s favorite activity with him/her.
A: In this exercise, we will focus on mixed strategies.
(a) Begin by depicting the game in the form of a pay off matrix.
(b) Let ρ be the probability you place on going to the opera, and let δ be the probability your partner places on going to the opera. For what value of δ are you indifferent between showing up at the opera or showing up at the football game?
(c) For what values of ρ is your partner indifferent between these two actions?
(d) What is the mixed strategy equilibrium to this game?
(e) What are the expected payoffs for you and your partner in this game?
B: In the text, we indicated that mixed strategy equilibria in complete information games can be interpreted as pure strategy equilibria in a related incomplete information game. We will illustrate this here. Suppose that you and your partner know each other’s ordinal preferences over opera and football—but you are not quite sure just how much the other values the most preferred outcome. In particular, your partner knows your payoff from both showing up at the football game is 5, but he thinks your payoff from both showing up at the opera is (10+α) with some uncertainty about what 2Since this game dates back quite a few decades, you can imagine which of the two players was referred to as the “husband” and which as the “wife” in early incarnation. I will attempt to write this problem without any such gender (or other) bias and apologize to the reader if he/she is not a fan of opera. Exactly is Similarly, you know your partner gets a payoff of 5 if both of you show up at the opera, but you think his/her payoff from both showing up at the football game is (10 + β), with you unsure of what exact value β takes. We will assume that both α and β are equally likely to take any value in the interval from 0 to x; i.e. α and β are drawn randomly from a uniform distribution on [0, x]. We have thus turned the initial complete information game into a related incomplete information game in which your type is defined by the randomly drawn value of α and your partner’s type is defined by the randomly drawn value of β, with [0,x] defining the set of possible types for both of you.
(a) Suppose that your strategy in this game is to go to the opera if α > a (and to go to the football game otherwise), with a falling in the interval [0, x]. Explain why the probability (evaluated in the absence of knowing α) that you will go to the opera is (x−a)/x. What is the probability you will go to the football game?
(b) Suppose your partner plays a similar strategy: go to the football game if β > b and otherwise go to the opera. What is the probability that your partner will go to the football game? What is the probability that he/she will go to the opera?
(c) Given you know the answer to (b), what is your expected payoff from going to the opera for a given α? What is your expected payoff from going to the football game?
(d) Given your partner knows the answer to (a), what is your partner’s expected payoff from going to the opera? What about the expected payoff from going to the football game?
(e) Given your answer to (c), for what value of α (in terms of b and x) are you indifferent between going to the opera and going to the football game?
(f) Given your answer to (d), for what value of β (in terms of a and x) is your partner indifferent between going to the opera and going to the football game?
(g) Let a be equal to the value of α you calculated in (e), and let b be equal to the value of β you calculated in (f). Then solve the resulting system of two equations for a and b (using the quadratic formula).
(h) Why do these values for a and b make the strategies defined in (a) and (b) pure (Bayesian Nash) equilibrium strategies?
(i) How likely is it in this equilibrium that you will go to the opera? How likely is it that your partner will go to the football game? How do your answers change as x approaches zero— and how does this compare to the probabilities you derived for the mixed strategy equilibrium in part A of the exercise?
(j) True or False: The mixed strategy equilibriumto the complete information Battle of the Sexes game can be interpreted as a pure strategy Bayesian equilibrium in an incomplete information game that is almost identical to the original complete information game — allowing us to interpret the mixed strategies in the complete information game as arising from uncertainty that players have about the other player.