Consider again the “Battle of the Sexes” game described in exercise 24.4. Recall that you and your partner have to decide whether to show up at the opera or a football game for your date — with both of you getting a payoff of 0 if you show up at different events and therefore aren’t together. If both of you show up at the opera, you get a payoff of 10 and your partner gets a payoff of 5, with these reversed if you both show up at the football game.
A: In this part of the exercise, you will have a chance to test your understanding of some basic building blocks of complete information games whereas in part B we introduce a new concept related to dominant strategies. Neither part requires any material from Section B of the chapter.
(a) Suppose your partner works the night shift and you work during the day — and, as a result, you miss each other in the morning as you leave for work just before your partner gets home. Neither of you are reachable at work—and you come straight from work to your date. Unable to consult one another before your date, each of you simply has to individually decide whether to show up at the opera or at the football game. Depict the resulting game in the form of a pay off matrix.
(b) In what sense is this example of a “coordination game”?
(c) What are the pure strategy Nash equilibria of the game.
(d) After missing each other on too many dates, you come up with a clever idea: Before leaving for work in the morning, you can choose to burn $5 on your partner’s nightstand — or you can decide not to. Your partner will observe whether or not you burned $5. So we now have a sequential game where you first decide whether or not to burn $5, and you and your partner then simultaneously have to decide where to show up for your date (after knowing whether or not you burned the $5). What are your four strategies in this new game?
(e) What are your partner’s four strategies in this new game (given that your partner may or may not observe the evidence of the burnt money depending on whether or not you chose to burn the money?)
(f) Illustrate the pay off matrix of the new game assuming that the original payoffs were denominated in dollars. What are the pure strategy Nash Equilibria?
B: In the text, we defined a dominant strategy as a strategy under which a player does better no matter what his opponent does than he does under any other strategy he could play. Consider now a weaker version of this: We will say that a strategy B is weakly dominated by a strategy A for a player if the player does at least as well playing A as he would playing B regardless of what the opponent does.
(a) Are there any weakly dominated strategies for you in the payoff matrix you derived in A (f)? Are there any such weakly dominated strategies for your partner?
(b) It seems reasonable that neither of you expects the other to play a weakly dominated strategy. So take your payoff matrix and strike out all weakly dominated strategies. The game you are left with is called a reduced game. Are there any strategies for either you or your partner that are weakly dominated in this reduced game? If so, strike them out and derive an even more reduced game. Keep doing this until you can do it no more — what are you left with in the end?
(c) After repeatedly eliminating weakly dominated strategies, you should have ended up with a single strategy left for each player. Are these strategies an equilibrium in the game from A (f) that you started with?
(d) Selecting among multiple Nash equilibrium to a game by repeatedly getting rid of weakly dominated strategies is known as applying the idea of iterative dominance. Consider the initial game from A (a) (before we introduced the possibility of you burning money). Would applying the same idea of iterative dominance narrow the set of Nash equilibrium in that game?
(e) True or False: By introducing an action that ends up not being used, you have made it more likely that you and your partner will end up at the opera.