QUESTION 1 (40 Points)
Yuppietown has two food stores, La Boulangerie, which sells bread, and La Fromagerie,
which sells cheese. It costs $1 to make a loaf of bread and $2 to make a pound of cheese.
If La Boulangerie’s price is P1 dollars per loaf of bread and La Fromagerie’s price is P2
dollars per pound of cheese, their respective weekly sales, Q1 thousand loaves of bread and
Q2 thousand pounds of cheese, are given by the following equations:
Q1 = 10 − P1 − 0.5P2, Q2 = 12 − 0.5P1 − P2.
(a) Find the two stores’ best-response rules, illustrate the best-response curves, and find
the Nash equilibrium prices in this game. (15 Points)
(b) Suppose that the two stores collude and set prices jointly to maximize the sum of
their profits. Find the joint profit-maximizing prices for the stores. (15 points)
(c) Provide a short intuitive explanation for the differences between the Nash equilibrium
prices and those that maximize joint profit. Why is joint profit maximization not a
Nash equilibrium? (10 Points)
QUESTION 2 (60 Points)
Imagine a match between the two all-time best women players-Martina Navratilova and
Chris Evert. Navratilova at the net has just volleyed a ball to Evert on the baseline, and
Evert is about to attempt a passing shot. She can try to send the ball either down the line
(DL; a hard, straight shot) or crosscourt (CC; a softer, diagonal shot). Navratilova must
likewise prepare to cover one side or the other. Each player is aware that she must not give
any indication of her planned action to her opponent, knowing that such information will
be used against her. Navratilova would move to cover the side to which Evert is planning
to hit or Evert would hit to the side that Navratilova is not planning to cover. Both must
act in a fraction of a second, and both are equally good at concealing their intentions until
the last possible moment; therefore their actions are effectively simultaneous, and we can
analyze the point as a two-player simultaneous-move game.
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Payoffs in this tennis-point game are given by the fraction of times a player wins the
point in any particular combination of passing shot and covering play. Given that a downthe-line passing shot is stronger than a crosscourt shot and that Evert is more likely to win
the point when Navratilova moves to cover the Wrong side of the court, we can work out
a reasonable set of payoffs. Suppose Evert is successful with a down-the-line passing shot
80% of the time if Navratilova covers crosscourt; she is successful with the down-the-line
shot only 50% of the time if Navratilova covers down the line. Similarly, Evert is successful
with her crosscourt passing shot 90% of the time if Navratilova covers down the line. This
success rate is higher than when Navratilova covers crosscourt, in which case Evert wins
only 20% of the time. Clearly, the fraction of times that Navratilova wins this tennis point
is just the difference between 100% and the fraction of time that Evert wins.
(a) Represent this game in a payoff-matrix form. (20 Points)
(b) Show that this game does not have a Nash Equilibrium in pure strategies. (20 Points)
(c) Find the mixed strategy Nash Equilibrium of this game. (20 Points)
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Sample Solution