1. Consider the following strategic interaction which is in a class of games referred to as Cournot duopolies.
Assume there are two competing firms (A and B) who must simultaneously choose their output levels qa
and qb, respectively, such that qi ∈ [0, ∞). Assume the marginal cost c = 12. Let demand be a function
of the total quantity sold Q, such that Q = qa+qb. The price can then be written as p(Q) = 120−qa−qb.
Each firms payoffs are their profits πi
, which are given by the price times its quantity minus its costs:
πa = qa ∗ (120 − qa − qb) − cqa
πb = qb ∗ (120 − qa − qb) − cqb
(a) Identify each players’ best response function.
(b) Solve for the Nash equilibrium (q
∗
a
, q∗
b
).
(c) What is the price in equilibrium?
Next, allow firm A to choose their quantity of production qa ∈ [0, ∞) first, followed by firm B choosing
their quantity of production qb ∈ [0, ∞) after observing A’s choice. As before, assume marginal costs
are constant at c = 12 and define Q = qa + qb, p(Q) = 120 − qa − qb, and the payoff functions (profits)
the same as before.
(d) Solve for the Nash equilibrium of this new game that incorporates sequential moves.
(e) How has the market priced changed by moving from a simultaneous to sequential move game? Are
either, both, or neither players made better off? Explain your answer.
2. Consider the following static game of complete information that involves the simultaneous contribution
of two players to some joint endeavor. In this case, we will assume the task can only be accomplished if
both players exert the costly effort. They are better off when they both exert effort and perform the task
than they are if neither exerts effort (and nothing is accomplished). However, the worst case scenario
for each person is that they exert effort and the other player does not – in which case they have paid
the cost of exerting effort, but nothing is accomplished. Payoffs for this game are given in the following
table. Assume that c reflects the cost of exerting effort, such that 0 < c < 1.
2
no effort effort
1
no effort 0 , 0 0 , -c
effort -c , 0 1 – c, 1 – c
(a) Identify any pure strategy Nash equilibria of this game.
(b) Identify any mixed strategy Nash equilibria of this game.
(c) Plot the best response functions for both players and clearly label all equilibria you identified in
parts (a) and (b).
3. Consider the following two player extensive form game of complete information. In the first stage, player
1 can choose whether to take a hostage or not. If they do not take a hostage, the game ends. If they do
take a hostage, player 2 has a choice of whether to pay the ransom or not. Then, player 1 has a choice
of whether to continue to hold the hostage or release the hostage, regardless of whether the ransom was
paid or not. The game and associated payoffs are provided in the following game tree.
1
1 1
2
3
5
4
1
5
3
x
2
y
4
(a) Assume x = 1 and y = 2. What is the unique subgame perfect Nash equilibrium of the game?
(b) Assume x = 2 and y = 1. What is the unique subgame perfect Nash equilibrium of the game?
(c) How do the values of x and y affect player 1’s decision to take a hostage? According to this simple
model, what is something that would lead to fewer hostages being taken?
4. Consider the following two-person normal form game in which players can choose to cooperate (C) or
defect (D). Assuming both α > 0 and β > 0, demonstrate the conditions under which Grim Trigger
constitutes a subgame perfect equilibrium of the infinitely repeated game. For the purposes of this
question, you can assume that each player will always be willing to carry out the punishment strategy
of the game — that is, you need only check whether given a history in which both players have always
cooperated, whether (or under what conditions) they will each continue to cooperate.
2
C