Homework 12 – Each member of a couple must decide

Homework #12(due November 22) You should solve all problems analytically, not using a calculator. 1.Each member of a couple must decide whether to remain loyal or to betray the other. Here is thepayoff matrix for the loyalty game:WifeLoyalLoyalHusband 6 03 (e)(f)(g) 30 6Betray (a)(b)(c)(d) Betray 22 What are the dominant strategies in this game (if any)?What are the pure Nash equilibria in this game (if any)?What is the maxmin outcome of this game?If the actions were sequential in this game, then does it matter who acts first? If so, then is itbetter to act first or second?What if each player literally flips a coin (with each side equally likely) to determine whether to beloyal. Is that a mixed equilibrium?Calculate and compare the players’ average total payoffs (sum of wife’s and husband’s payoffs)in each of the pure and mixed equilibria that you have found in parts (a)-(e).Extra credit: Suppose that in each cell, each player’s payoff is changed so that it now equals thatplayer’s relative payoff, the amount by which their original payoff exceeds the spouse’s originalpayoff. What is the new payoff matrix? Repeat parts (a)-(d) for this new game. 2.In 1944 the Allies are planning the invasion of France and Germany is preparing its defenses. TheAllies can choose to land at Normandy or Calais, and Germany can choose to defend Normandy or Calais.Normandy is a more difficult landing for the Allies but provides a more valuable beachhead, if successful.If the Allies land successfully, then Germany has better opportunities for inland defense if the landingoccurred at Calais. Here is the payoff matrix for the invasion game:GermanyNormandyNormandyAllies 424 (e)(f)(g) 05 0Calais (a)(b)(c)(d) Calais 41 What are the dominant strategies in this game (if any)?What are the pure Nash equilibria in this game (if any)?What is the maxmin outcome of this game?If the actions were sequential in this game, then does it matter who acts first? If so, then is it betterto act first or second?If Germany thinks that the Allies have a 60% chance of attacking Calais and a 40% chance ofattacking Normandy, then which action should it take?If the Allies think that Germany has a 60% chance of defending Calais and a 40% chance ofdefending Normandy, then which action should they take?Extra credit: Find all of the mixed Nash equilibria of this game. 3.Two candidates are deciding whether to run for their party’s nomination or spend more time withtheir family. The stronger candidate would not consider running as an Independent, but the weaker candidatemight consider running as an Independent. Here is the payoff matrix for the nomination game:Strong CandidateRunRun 2 Run Ind. 36 1WeakCandidate Family 23 Family 15 92 00 For parts (a) through (e), suppose that running as an Independent is not an option (eliminate that action).(a)(b)(c)(d)(e) What are the dominant strategies in this game (if any)?What are the pure Nash equilibria in this game (if any)?What is the maxmin outcome of this game?If the actions were sequential in this game, then does it matter who acts first? If so, then is it betterto act first or second?Is there a way to sequence the actions that guarantees the most efficient outcome (the highest sumof payoffs)? For the remaining parts, suppose that running as an Independent becomes an option (include that action).(f)(g)(h)(i)(j)(k) What are the dominant strategies in this game (if any)?What are the dominated strategies in this game (if any)?What are the pure Nash equilibria in this game (if any)?If the actions were sequential in this game, then does it matter who acts first? If so, then is it betterto act first or second?Is there a way to sequence the actions that guarantees the most efficient outcome (the highest sumof payoffs)?Explain the effects of giving the weak candidate the option to run as an Independent. Who benefitsfrom having that option?