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 the
payoff matrix for the loyalty game:
Wife
Loyal
Loyal
Husband 6 0
3 (e)
(f)
(g) 3
0 6
Betray (a)
(b)
(c)
(d) Betray 2
2 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
better to act first or second?
What if each player literally flips a coin (with each side equally likely) to determine whether to be
loyal. 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 that
player’s relative payoff, the amount by which their original payoff exceeds the spouse’s original
payoff. 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. The
Allies 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 landing
occurred at Calais. Here is the payoff matrix for the invasion game:
Germany
Normandy
Normandy
Allies 4
2
4 (e)
(f)
(g) 0
5 0
Calais (a)
(b)
(c)
(d) Calais 4
1 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 better
to act first or second?
If Germany thinks that the Allies have a 60% chance of attacking Calais and a 40% chance of
attacking Normandy, then which action should it take?
If the Allies think that Germany has a 60% chance of defending Calais and a 40% chance of
defending 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 with
their family. The stronger candidate would not consider running as an Independent, but the weaker candidate
might consider running as an Independent. Here is the payoff matrix for the nomination game:
Strong Candidate
Run
Run 2 Run Ind. 3
6 1
Weak
Candidate Family 2
3 Family 1
5 9
2 0
0 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 better
to act first or second?
Is there a way to sequence the actions that guarantees the most efficient outcome (the highest sum
of 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 better
to act first or second?
Is there a way to sequence the actions that guarantees the most efficient outcome (the highest sum
of payoffs)?
Explain the effects of giving the weak candidate the option to run as an Independent. Who benefits
from having that option?