EC403
Game Theory and Applications
Problem Set 6
Due Tue, Feb 12
Your name: ________________________________________
Jean-Jacque Rousseau, a famous French philosopher, makes the following comment in his
Discourse on the Origin and Basis of Equality among Men:
If a group of hunters set out to take a stag,1 they are fully aware that they would
all have to remain faithfully at their posts in order to succeed; but if a hare happens
to pass near one of them, there can be no doubt that he pursued it without qualm,
and that once he had caught his prey, he cared very little whether or not he had
made his companions miss theirs.
Let us look at the game-theoretic aspect of this problem. For simplicity, we reduce the number of
hunters to just two. If both faithfully hunt stag, they get a stag, worth S units of utility, and split it
two ways, each getting S/2. If one of them chases a hare while the other one stays at his post,
then the former hunter gets a hare, worth H, while the latter gets nothing. If both go off chasing
hares, then each gets one (again, worth H).
Therefore, we have the following game:
Hunter 2
Hunt stag
Hunt hare
Hunt stag
S/2, S/2
0, H
Hunt hare
H, 0
H, H
Hunter 1
It is impossible to solve the game without knowing parameter values, neither it is our intention to
do so. On the next several pages, you are asked to analyze three versions of the game.
1
Stag, n. – an adult male deer. I am sure you know that but when I came across this word in a book on game theory
in 1997, I had to use a dictionary!
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EC403
Game Theory and Applications
Problem Set 6
Due Tue, Feb 12
Hunter 2
Hunt stag
Hunt hare
Hunt stag
S/2, S/2
0, H
Hunt hare
H, 0
H, H
Hunter 1
1. Set S = 4, H = 3 and fill the table.
Let us go over the set of questions from the last assignment one more time:
a. Is this a zero-sum, constant-sum, or variable-sum game?
b. Does Player 1 have a (weakly or strictly) dominant strategy? Does Player 2 have a (weakly or
strictly) dominant strategy? If you answered yes to any of the questions, state the dominant
strategies.
Player 1 –
Player 2 –
c. Is this game solvable by iterated dominance? If yes, list the order in which strategies are
eliminated and state the solution. If the game is not solvable but some strategies can be
eliminated, list those strategies.
d. How many Nash equilibria in pure strategies does this game have? Name all of them.
2
EC403
Game Theory and Applications
Problem Set 6
Due Tue, Feb 12
Hunter 2
Hunt stag
Hunt hare
Hunt stag
S/2, S/2
0, H
Hunt hare
H, 0
H, H
Hunter 1
2. Set S = 4, H = 2 and fill the table.
a. Is this a zero-sum, constant-sum, or variable-sum game?
b. Does Player 1 have a (weakly or strictly) dominant strategy? Does Player 2 have a (weakly or
strictly) dominant strategy? If you answered yes to any of the questions, state the dominant
strategies.
Player 1 –
Player 2 –
c. Is this game solvable by iterated dominance? If yes, list the order in which strategies are
eliminated and state the solution. If the game is not solvable but some strategies can be
eliminated, list those strategies.
d. How many Nash equilibria in pure strategies does this game have? Name all of them.
3
EC403
Game Theory and Applications
Problem Set 6
Due Tue, Feb 12
Hunter 2
Hunt stag
Hunt hare
Hunt stag
S/2, S/2
0, H
Hunt hare
H, 0
H, H
Hunter 1
3. Set S = 4, H = 1 and fill the table.
a. Is this a zero-sum, constant-sum, or variable-sum game?
b. Does Player 1 have a (weakly or strictly) dominant strategy? Does Player 2 have a (weakly or strictly)
dominant strategy? If you answered yes to any of the questions, state the dominant strategies.
Player 1 –
Player 2 –
c. Is this game solvable by iterated dominance? If yes, list the order in which strategies are eliminated
and state the solution. If the game is not solvable but some strategies can be eliminated, list those
strategies.
d. How many Nash equilibria in pure strategies does this game have? Name all of them.
(continued on the next page)
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EC403
Game Theory and Applications
Problem Set 6
Due Tue, Feb 12
e. For the last version of the Stag Hunt game, you were expected to find two N.E. in pure strategies.
Which of the two, if any, is payoff dominant?
f.
Which of the two pure-strategy N.E. above is risk dominant? Explain.
g. For the last version of the Stag Hunt game, find the mixed strategy Nash Equilibrium. (Do it at least
algebraically but provide a best response diagram if you can. Your final answer should be stated in
the following format:
At the mixed strategy NE, Player 1 plays “Stag” with probability [a numeral], and “Hare” w/prob [a
numeral]. Player 2 plays “Stag” w/prob [a numeral], and “Hare” w/prob [a numeral].
h. (extra credit) If you can, state the expected payoff of player 1 at the mixed-strategy N.E. (You can use
the logic from page 58 in the text as reference.) Use the back side of this page if needed.
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EC403
Game Theory and Applications
Problem Set 6
Due Tue, Feb 12
4. Game C from last week’s problem set, revisited.
Player 2
Left
Right
Top
3, 4
4, 3
Bottom
6, 1
2, 5
Player 1
As we previously established, this game has no equilibria in pure strategies. Every such game, however,
does have an equilibrium in mixed strategies!
Use the technique learned in class to find that equilibrium.
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