U3. In the small town of Embarrass, Minnesota, famous for its funny name, the Miss Embarrass contest is no laughing matter. Each August, several young ladies from around the Embarrass River Valley vie for the honor of being crowned Queen at the Embarrass Region Fair. (Everything so far is true; the rest is made up.) The Miss Embarrass contest has traditionally used the plurality method to determine the winner. However, one of the organizers recently read about a FairVote.org study that found that instant runoff voting (IRV) promotes more civil campaigning, and that format is now being considered. All 1,000 Embarrass area residents who attend the fair are eligible to vote. This year, Ashley (A) and Becky (B) are the front-runners, trailed by Olivia (O). At the start of fair season, 420 people are "Ashley fans," with the preference ordering A B O; 400 people are "Becky fans," with the preference ordering B A O; and 180 are "Olivia fans," with the preference ordering O A = B. However, fairgoers' preferences may change before the vote, depending on whether Ashley and/or Becky engages in "negative campaigning" by spreading a nasty rumor during the fair about her opponent. (For simplicity, we assume that Olivia is sure to run a positive campaign.) As you consider the questions below, assume (i) that Ashley and Becky each prefer to remain civil and each prefer not to have a nasty rumor spread about them, but most of all want to win the election; (ii) that negative campaigning suppresses turnout, causing 40 fans of the targeted candidate to forgo voting in the election; and (iii) that if only one candidate stays civil, all Olivia fans will prefer the civil candidate as their second favorite. (If Ashley and Becky are both civil or both negative, Olivia fans will flip a coin if forced to decide between them.)

(a) Suppose that the plurality rule continues to be used for the election. Draw the ordinal payoff matrix for the "civility game" played by Ashley and Becky when they choose between Civil and Negative. Show that this game does not have a pure-strategy Nash equilibrium. Explain in words why, in the unique mixed-strategy Nash equilibrium, Ashley and Becky are each sometimes crowned Miss Embarrass. (b) Suppose that the organizers decide to switch to instant runoff voting for the election. Show that being civil is now a superdominant strategy for both Ashley and Becky. Hint: Draw the ordinal payoff matrix for the civility game with Ashley as the row player. Civil is superdominant for Ashley if the best two payoffs (4 and 3) appear in the Civil row. Similarly, Civil is superdominant for Becky if the two best payoffs appear in the Civil column.