Lecture 8 Practice Question Answers
From Browning and Zupan:
14.1. A dominant-strategy equilibrium is a set of strategies (one for each player) in
which every player is playing a dominant strategy – that is, a strategy that is best
regardless of what the other players do. A Nash equilibrium is a set of strategies (one for
each player) in which every player is playing a strategy that is a best response to what the
other players’ equilibrium strategies. It is possible to have a Nash equilibrium that is not
a dominant-strategy equilibrium; for example, Battle of the Sexes has two Nash
equilibria, but neither one is a dominant-strategy equilibrium.
14.3. A prisoners’ dilemma is a game in which every player has a dominant strategy
(see above), and the resulting dominant-strategy equilibrium is Pareto-inferior to the
outcome that would have occurred if everyone had played a different strategy. This is
relevant to analyzing cheating in a cartel, because the members of a cartel usually have a
dominant strategy of producing more output (or setting a lower price). As a result, they
have a strong incentive to cheat on the cartel’s agreement to produce less output (set
higher prices).
Additional questions:
1. (a) Here is the strategy-and-payoff matrix for Rock-Paper-Scissors. Player 1 chooses
the row, and player 2 chooses the column.
Rock
Paper
Scissors
Rock
0, 0
1, -1
-1, 1
Paper
-1, 1
0, 0
1, -1
Scissors
1, -1
-1, 1
0, 0
(b) There is no Nash Equilibrium of this game. Why? Because for any pair of
strategies, at least one player would wish to change strategy in response to the other
person’s strategy. For instance, if player 1 is playing Rock and player 2 is playing
Paper (the top middle cell), then player 1 wishes to switch to Scissors.
(Also, there is actually a Nash Equilibrium in “mixed strategies,” which means the
players randomize over their strategies. Specifically, each player plays each strategy
with 1/3 probability. But you don’t have to know this.)
2. (a) Here is the strategy-and-payoff matrix for “Chicken.” Spike picks the row and
Biff picks the column. (“K” means 1000.)
Continue
Stop
Continue
-10K, -10K
-1K, 1K
Stop
1K, -1K
0, 0
(b) There is no dominant strategy, because Continue is the best response to Stop, but
Stop is the best response to Continue. There is no dominant strategy equilibrium
(DSE), because both players must have dominant strategies to have a DSE.
(c) There are two Nash Equilibria: (Continue, Stop) and (Stop, Continue). Why?
Given that the other guy is going to Stop, you want to Continue. Given that the
other guy is going to Continue, you want to Stop.
(d) There can be a first-mover advantage if one player can commit in advance to
playing Continue. For example, Spike could throw his brake pedal out the
window so that he’s incapable of stopping.
3. (a) If firm 2 picks quantity q2, then firm 1's demand curve is P = [1500 - q2] - q1, and
his marginal revenue is MR1 = [1500 - q2] - 2q1. Set this equal to MC1 = 300 and
solve for q1 like so:
[1500 - q2] - 2q1 = 300
1200 - q2 = 2q1
q1 = 600 - ½q2
Doing the same thing for firm 2 (with the quantities reversed and using MC2 =
600 instead), we get q2 = 450 - ½q1. Now solve the system of equations by
plugging the equation for q2 into the equation for q1, like so:
q1 = 600 - ½[450 - ½q1]
q1 = 375 + (1/4)q1
(3/4)q1 = 375
q*1 = 500
and plug this back into the equation for q2 to get q*2:
q*2 = 450 - ½[500] = 200
So the Nash Equilibrium is q*1 = 500, q*2 = 200. The total quantity is 700, so the
Nash Equilibrium price is P* = 1500 - 700 = 800.
(b) Firm 1, which has the lower marginal cost, will set his price just below firm 2's
marginal cost. Firm 2 doesn't produce, so firm 1 gets the whole market at P1 =
599. The corresponding quantity is found by plugging the price into the demand
curve: 599 = 1500 - Q, and thus Q = 901.
4. (a) Every DSE is a NE, but not every NE is a DSE. (b) In a DSE, each player’s
strategy must be a best response to any possible strategies chosen by the other
players. But in a NE, a player’s strategy only needs to be a best response to the
equilibrium strategies of the other players.
5. Battle of the Sexes, Market Segmentation, Sides of the Road. (There are many other
possible examples.)
6. Here is a diagram of Battles of the Sexes converted into a sequential game with Terry
choosing first:
Terry
Opera
Fight
Pat
Opera
Pat
Fight
Opera
Fight
I have used heavier lines to show the predicted choices of the players (we did this in
class using arrows). When Terry has already chosen Opera, Pat compares 1 and 0
and therefore chooses Opera. When Terry has already chosen Fight, Pat compares 0
and 3 and therefore chooses Fight. Terry can predict all this, which means choosing
Opera will lead to a payoff of 3 and choosing Fight will lead to a payoff of 1, and
therefore Terry will choose Opera. Notice that Terry has a first-mover advantage in
this game.
7. (a) In Cournot, firms choose quantities, and the market sets the resulting price. In
Bertrand, firms choose prices, and the market sets the resulting quantities sold. (b)
Cournot leads to a price and total quantity that are between those that would occur
under monopoly and perfectly competition. Bertrand leads to a price and total
quantity identical to those that occur under perfect competition.
8. (a) If firm 2 picks quantity q2, then firm 1's demand curve is P = [100 - q2] - q1, and
his marginal revenue is MR1 = [100 - q2] - 2q1. Setting this equal to MC1 = 20 (firm
1's MC) and solving for q1, we get q1 = 40 - ½q2. Doing the same thing for firm 2
(with the quantities reversed and using MC2 = 40 instead), we get q2 = 30 - ½q1.
Solving the system of equations, we get q*1 = 100/3 and q*2 = 40/3. The total
quantity is Q* = 140/3, and the corresponding price is P* = 100 - 140/3 = 160/3.
(b) Firm 1, which has the lower marginal cost, will set his price just below firm 2's
marginal cost. Firm 2 doesn't produce, so firm 1 gets the whole market at P1 = 39.99.
The corresponding quantity is found by plugging the price into the demand curve:
39.99 = 100 - Q, and thus Q = 60.01.