Economics 410-3
Spring 2014
Professor Jeff Ely
Midterm Exam
Instructions: There are three questions with equal weight. You have until 10:50 to
complete the exam. This is a closed-book and closed-notebook exam. Please explain all of
your answers carefully.
1. Consider the following two-person game. The game has 2k+1 stages for some positive
integer k. The two players move in alternating stages, with player 1 moving in stage
1, player 2 in stage 2, etc. At the beginning of the game there is a dollar on the table.
In each stage the player to move can either grab the money on the table or “pass.”
Every time a player passes, the amount of money is doubled. The game ends as soon
as a player grabs the money. The grabbing player leaves with all of the money on the
table and the other player leaves with nothing. If after 2k + 1 stages no player has
grabbed then the two players share equally the money left on the table. Each player’s
payoff equals the amount of money she leaves with. This is a finite extensive-form
game with perfect information.
(a) Represent this game in strategic form. Formally describe the pure strategy sets
of the two players and the strategic-form payoff function.
A strategy for player i specifies each stage at which i moves whether he will grab
or pass. One way to describe them is
S1 = {grab, pass}k+1
and
S2 = {grab, pass}k
The payoff to pure strategy profile σ is determined as follows. Find the earliest
stage z at which some player grabs. If z ≤ 2k + 1 then the grabbing player earns
2z−1 and the other player earns zero. If no one ever grabs then both players earn
22k .
(b) What are the sets of rationalizable (pure) strategies for players 1 and 2? Prove
that every strategy in the set you have identified is rationalizable and every pure
strategy outside of that set is not.
Every strategy is rationalizable for both players. To prove this, note that every
strategy for 2 is a best-response to a strategy for 1 that grabs in the first stage.
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Consider any strategy for 1 that grabs in some stage. Let z be the first stage
in which it grabs. This strategy is a best-response to the strategy for 2 that first
grabs in stage z + 1. Finally consider the strategy for 1 that never grabs. It is
a best-response to the strategy for 2 that never grabs. To see this, if they both
pass until the end then they both get 22k . This is the same payoff 1 would get by
grabbing in the last stage and this is higher than 1 could get by grabbing earlier.
(c) What is the set of all Nash equilibria of the game, both in pure and mixed
strategies?
i. There are two SPE which you can compute by backward induction. The first
is always passing and the second always grabbing (for both players).
ii. There are several other Nash equilibria in pure strategies which involve
player 1 and player 2 grabbing in their first chance and either grabbing or
passing at each node in the future.
iii. Mixed strategies in which player 1 grabs in the first period:
Let Σ̂1 be the set of all such mixed strategies for 1. Consider any mixed
strategy α2 for 2 such that the maximum expected payoff player 1 can earn
against it is less than or equal to 1 i.e.,
max u1 (σ̂1 , α2 ) ≤ 1
σ̂1
(There are many such strategies for 2.) Let Σ̂2 be the set of all such mixed
strategies for 2. This set of Nash equilibria is Σ̂1 × Σ̂2 .
iv. Mixed strategies in which no one grabs:
In order for someone to mix in equilibrium at some node reached with positive probability it better be the case that the expected payoff of passing is the
same of grabbing. Inductively: think of the strategy where both players pass
always except player 1 mixes in the last node choosing a probability greater
than 21 of passing, this is a NE. Now think of the same strategies except
player 1 mixes with probability exactly 21 and player 2 also mixes in her last
turn with any probability at least 21 , this is also a NE.
All of this mixed strategies will be of that form: pass up to a point, then
at one node someone mixes with probability greater than or equal to 12 of
passing, and both players mix with probability exactly 21 at every node after
that.
(d) Select one Nash equilibrium that is not Subgame-Perfect and find a profitable
one-stage deviation for one of the players.
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Any of the strategies in bullet 2 in the previous answer which are not grabbing
always has a profitable one-stage deviation (just pass when the other player is
going to pass after this stage).
2. In the two-player game matching pennies, each player chooses from {H, T } and the
matcher tries to choose the same action as his opponent while the opponent, the mismatcher, tries to choose the opposite action of the matcher. The player who achieves
his objective receives one dollar from the other player. This question is about an
incomplete-information version of matching pennies. It is common knowledge that
one player is the matcher and the other player is the mis-matcher but the incomplete
information is about which player is the matcher.
In particular, it is common knowledge that
• Each player is equally likely to be the matcher
• Player 1 knows who is the matcher (and knows this at the time the players
simultaneously choose actions.)
• Player 2 has no additional information beyond what is given above.
(a) Represent this game of incomplete information as a Bayesian game with a common prior. What is the set of types for each player? What is the common prior?
What is the Bayesian-game payoff function?
Player 1 has two types {t1 , t01 }, player 2 has only 1 type and the common prior
assigns equal probability to the two types of player 1. The payoff function is
represented by the following matrices.
(b) Find a Bayesian Nash equilibrium in which player 1 plays a pure strategy. Player
1 plays H regardless of his type. Player 2 randomizes with equal probability on
{H, T }.
3. This question is about price competition between two firms under complete information.
(a) Each firm can produce a single unit of output at cost c > 0. There is a single
consumer who views the two firms’ products as perfect substitutes. The consumer has a maximum willingness to pay of v > c for a single unit of output sold
by either firm. The consumer has no additional value for any additional units.
The two firms will simultaneously make price offers to the consumer. The prices
can be any non-negative real number. The consumer observes the price offers p1
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and p2 and then decides whether to purchase from firm 1 at price p1 , purchase
from firm 2 at price p2 or make no purchase at all. The chosen firm produces
its unit and incurs cost c, the other firm does not produce and incurs no costs.
Then the game ends. The consumer’s payoff is v − p if he purchases a unit at
price p and the consumer’s payoff is zero if he does not purchase any unit. Firm
i’s payoff is pi − c if the consumer purchases from firm i and zero otherwise.
This is an extensive-form game with perfect information and simultaneous moves.
Prove that all subgame-perfect equilibria yield the same payoffs for all players:
0 for each of the two firms and v − c for the consumer.
(b) Now there are n distinct buyers who arrive in sequence (buyer 1 arrives in period
1, buyer 2 arrives in period 2, etc.) and the same two firms will compete in each
period to sell to the buyer arriving in that period. All buyers are identical to the
buyer described above: maximum willingness to pay v > c and a demand for at
most a single unit. Each of the two firms has a total production capacity (over
the whole duration of the game) of n meaning that each unit the firm produces
up to the nth can be produced at unit cost c. Competition in each period works
exactly as in the single-period version above: the firms simultaneously quote
prices and the consumer active in that period chooses which firm to purchase
from (or to make no purchase). That consumer then exits the game and the next
stage begins with the arrival of the next consumer. At the beginning of each
stage, all players observe the prices quoted and purchase decisions in all previous
stages. Each firm’s payoff in the overall game is the sum of payoffs it earns in the
n stages. This is a n + 2-player finite-horizon extensive-form game with perfect
information and simultaneous moves. Prove that all subgame-perfect equilibria
yield the same payoffs for all players: 0 for each of the two firms and v − c for
each consumer.
(c) Now there are again n stages but firm 1 has capacity n − 1 while firm 2 has
capacity n. Each firm can produce any number of units up to its capacity
at unit cost c. Once a firm has produced to capacity, that firm cannot produce
additional units and exits the game leaving the other firm in a monopoly position.
(A monopoly firm quotes a price and the consumer chooses whether to purchase
from the monopoly firm at that price or not purchase at all.) Again the firm’s
payoffs are the sum of payoffs earned in the n stages. All of the subgame perfect
equilibria of this game yield the same payoffs for the firms. What are the firms’
subgame perfect equilibrium payoffs?
The idea is that if Firm 1 spends all its capacity in the first n − 1 rounds then
in the last round Firm 2 will be able to extract monopoly profit v − c. But if
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Firm 1 does not make a sale in any of the first n − 1 rounds, capacity constraint
stops being binding so firms compete as in 3(b) again and both make zero profits
thereafter. Thus, Firm 2 wants to ensure that Firm 1 makes sales in all n − 1
first rounds. If the buyers are not strategic (randomize between firms when indifferent) the only SPNE are:
Firm 1: In the round k = 1 . . . n − 1 if Firm 1 has been making sales in all
previous rounds make an offer v. If Firm 1 did not make a sale in any of the
previous rounds, make an offer c. In the last round, if there is capacity left,
make an offer c.
Firm 2: In the round k = 1 . . . n − 1 if Firm 1 has been making sales in all
previous rounds make an offer > v. If Firm 1 did not make a sale in any of the
previous rounds, make an offer c. In the last round, if Firm 1 has capacity left,
make an offer c, and v otherwise.
All buyers: buy from the firm offering the lowest price (if the price is ≤ v).
If offers are the same, randomize between firms (choosing Firm 2 with positive
probability).
To see why this is a subgame-perfect equilibrium note that in any of the first
n − 1 stages, Firm 2 can earn at most strictly less than v − c by a one-stage
deviation. (He must price below v to make a sale and if he does so we will
reach a subgame in which the continuation strategies earn both firms zero in all
remaining stages.)
In addition, if the buyers are strategic (don’t randomize when indifferent), the
following are also SPNE:
Firm 1: In the round k = 1 . . . n − 1 if Firm 1 has been making sales in all
previous rounds make an offer pk ∈ [c, v]. If Firm 1 did not make a sale in any
of the previous rounds, make an offer c. In the last round, if there is capacity
left, make an offer c.
Firm 2: In the round k = 1 . . . n − 1 if Firm 1 has been making sales in all
previous rounds also make an offer pk . If Firm 1 did not make a sale in any of
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the previous rounds, make an offer c. In the last round, if Firm 1 has capacity
left, make an offer c, and v otherwise.
All buyers: buy from the firm offering the lowest price (if the price is ≤ v). If
offers are the same, buy from Firm 1.
Thus, if buyers are not strategic Firm 1 P
gets (n − 1)(v − c), Firm 2 gets v − c.
If buyers are strategic Firm 1’s payoff is n−1
k=1 (pk − c), pk ∈ [c, v], and Firm 2’s
payoff is v − c. In particular, if all pk = c Firm 1 gets 0.
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