ECON 117: Introduction to Game Theory
Problem Set 3
Sequential games with strategic uncertainty: Backward induction and
subgame perfect Nash equilibrium
Prof. Ignacio Esponda
UC Santa Barbara
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Incumbent vs. entrant
Consider the following game between an incumbent and a potential entrant. The entrant first
decides whether or not to enter the market and then the incumbent decides whether to be tough or
accommodating. The payoffs and the strategic form of the game are the following:
Entrant
Enter
Out
Incumbent
Tough Accommodate
-2,-1
1,2
0,5
0,5
(a) Find all Nash equilibria (in pure strategies) of the strategic form.
(b) Draw the extensive form of the game and solve by backward induction. Show that there is a
Nash equilibrium that is not part of the solution by backward induction and explain intuitively why
that is the case.
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Product proliferation
Bank-matic is planning to enter the ATM business in a particular town. Suppose that the town
has one main street and Bank-matic must decide how many ATMs to locate along this street. Let’s
represent the street by the [0,1] interval. Customers are uniformly distributed along the interval, and
they are willing to travel at most 0.25 in order to use an ATM. Suppose that any firm that serves a
proportion p of the customers gets a profit of p. In addition, placing an ATM at any location along
the [0,1] line costs 0.20. For example, if Bank-matic decides to place two ATMs, one at x = 0 and
another at x = 1, then it serves a proportion 0.25 + 0.25 = 0.50 of customers, making a profit of
0.50 but paying an entry cost of 2 × 0.20 = 0.40. Net profits are then 0.10.
(a) Where will Bank-matic decide to place its ATMs? How many ATMs will be placed, and what
are the net profits?
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Things get a little bit more difficult now that a close competitor, ReadyCash, comes to town.
ReadyCash has the same cost of 0.20 per each ATM it decides to locate, and customers don’t really
care which ATM they use: they always choose to go to the ATM that is nearest to them (and, as
before, decide not to use an ATM if there is no ATM within 0.25 of where they live).
Bank-matic has been studying the market for a longer time, and is therefore in a position to
choose the location(s) of its ATMs before ReadyCash gets to make this choice.
(b) Suppose that Bank-matic made the decision you suggested in (a). Where would ReadyCash
decide to locate, after seeing Bank-matic’s decision? What are the net profits for each firm?
(c) If Bank-matic can anticipate that ReadyCash will come to town after its location decision,
it is likely that Bank-matic will want to do something different than (a). Where do you recommend
Bank-matic to locate in this case? What will happen when ReadyCash comes to town? What will
be the profits of both firms?
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Stackelberg model of duopoly
In 1934, Stackelberg proposed a dynamic model of duopoly in which a dominant (or leader) firm
moves first and a subordinate (or follower) moves second. At some points in the history of the U.S.
automobile industry, for example, General Motors has seemed to play such a leadership role. Following Stackelberg, we will develop the model under the assumption that the firms choose quantities,
not prices, as in the Cournot model.
The timing of the game is as follows:
1. Firm 1 chooses a quantity q1 ≥ 0.
2. Firm 2 observes q1 and then chooses a quantity q2 ≥ 0.
3. The payoff to firm i is given by the profit function
πi (qi , qj ) = qi [P (Q) − c] ,
where P (Q) = a − Q is the market-clearing price, Q = q1 + q2 is the aggregate quantity, and
c is a constant marginal cost of production (fixed costs are zero).
Find the solution of the game by backward induction.
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Agency vs. wholesale pricing
We are going to compare two different pricing models that are common in the manufacturer-retailer
relationship and are known as agency and wholesale pricing. There is one publisher and one retailer.
The inverse demand function for an eBook is given by P = 16–Q, where P is the price consumers
pay for an eBook and Q is the quantity they purchase. eBooks are assumed to cost nothing to
distribute.
Agency pricing: The publisher sets a price P and the retailer must charge that price P for
eBooks. The publisher gets some share 1 − s of the profits and the retailer gets share s.
Wholesale pricing: The publisher sells eBooks at a price p to the retailer. Then, based on this
price p, the retailer chooses a price P to charge consumers.
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(a) What is the price consumers pay under agency pricing? (Hint: Maximizing share (1-s) of the
profits is equivalent to maximizing the entire share of profits. Therefore, this problem is your basic
monopoly pricing problem with a linear demand function.)
(b) What is the price consumers pay under wholesale pricing? (Hint: The game between the
publisher and the retailer can be solved by backward induction. First, determine the optimal decision
of the retailer as a function of the publisher’s price. Then solve for the optimal publisher price. In
each case, you have to solve a simple monopoly problem with a linear demand function.)
(c) Under which pricing model is consumer price the highest? Why?
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Capacity as a Strategic Barrier to Entry
Premise: You are asked to analyze a sequential game illustrating the strategic role of capacity. The
goal is to have a game theoretic understanding of how excess capacity can limit entry in an industry,
that is, how the incumbent can build an endogenous barrier to entry.
Consider an industry whose (inverse) demand is given by P = 900 − Q1 − Q2 , where P is the
market price, Q1 is the output of firm 1 and Q2 is the output of firm 2. To enter the industry, a
company must build a production facility. Capacity is often lumpy. This is captured by allowing the
firms to build one out of two possible facilities: a small facility, or a large facility. A small facility
costs $50,000 and it allows the firm to produce as many as 100 units at (for sake of simplicity) zero
marginal cost. Alternatively, the firm can pay $175,000 to construct a large facility that will allow
the firm to produce as many as 700 units at zero marginal cost. You are asked to analyze two games.
(a) SIMULTANEOUS CAPACITY CONSTRUCTION. Each firm decides secretly and simultaneously which facility to construct. Once the facilities are built (and they can observe each other’s
capacity), the firms decide simultaneously the output to produce. Then, given the resulting total
output, the market price is determined by the industry (inverse) demand. The goal of both firms is
to maximize their profits. Find the subgame perfect Nash equilibrium of the game.
(b) SEQUENTIAL CAPACITY CONSTRUCTION. Consider now the case in which Firm 1
moves first, and it must decide whether to construct a small or a large facility. Then, after having
observed the facility built by Firm 1, Firm 2 decides whether to stay out of the industry, enter
the industry with a small facility, or enter the industry with a large facility. Finally, after having
observed the facilities built, both firm secretly and simultaneously decide the production levels. The
goal of both firms is to maximize their profits. Find the subgame perfect Nash equilibrium of the
game.
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Agenda control in the senate
The U.S Senate is considering three alternative versions of a bill, which we’ll denote A, B, and C.
The preferences of the 100 senators are shown below:
Type of senator
I
II
III
1st choice
A
B
C
2nd choice
B
C
A
3rd choice
C
A
B
Number of senators of each type
10
44
46
For example, ten senators (type I) prefer bill A to bill B to bill C.
The procedure for selecting a bill for passage is a three-stage process:
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• Stage 1: The leader of the Senate sets the agenda by selecting two bills for a vote. His choice
is then either A and B, A and C, or B and C.
• Stage 2: Given the pair of bills selected by the Senate leader, the 100 members vote simultaneously for either one or the other.
• Stage 3: Whichever bill receives more votes in stage 2 is then matched with the third bill, and
the Senate votes simultaneously on those two bills, with the majority winner being the bill
that is passed.
Suppose that you are the Senate leader and can set the agenda. Which two bills would you select
for a vote in Stage 1 if you have the preferences of a type I senator?
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