Exam about game theory in economic application

Test 2

ECO 444

Konrad Grabiszewski

• Deadline: 6:15pm on March 15, 2016. Deliver in class or by email. If you deliver by email,

please send your exam in one file.

• Show all your work for full credit. Providing only correct answers is not enough; I need to

see how you reached your conclusions.

Exercise 1 (25 points)

For each game, identify all Nash equilibria (pure and mixed strategies). 5 points per game.

Bob

L R

Ann T 4,4 2,2

B 2,2 4,4

Game 1

Bob

L R

Ann T 1,2 0,0

B 0,0 2,1

Game 2

Bob

L R

Ann T 4,1 5,0

B 1,0 3,1

Game 3

Bob

L R

Ann T 3,7 2,7

B 1,1 3,4

Game 4

Bob

L R

Ann T 1,1 1,1

B 1,1 1,1

Game 5

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Exercise 2 (15 points)

Ann and Bob are in an Italian restaurant. The owner offers them a free pizza under the following

condition. Each player must simultaneously announce how much of pizza (in terms of percentages)

they would like. That is, each players announces one number from the interval [0, 100] where 0

means no pizza and 100 means the whole pizza.

Let sA and sB denote the strategy of Ann and Bob, respectively. If sA+sB ≤ 100, then the players

receive their demands (and the owners eats any leftovers); that is, Ann gets sA and Bob gets sB.

However, if sA + sB > 100, then the players get nothing; in this case, make an assumption that

the utility of each player is zero.

Assume that a player cares only about how much pizza she/he individually consumes. That is,

Ann does not care how much pizza Bob gets and Bob does not care about how much pizza Ann

gets. Also, assume that each player wants to have as much pizza as possible.

(This is an infinite version of the game you had in Exercise 4, Exercises 1.)

a) (5 points) Depict this game in a normal form.

b) (5 points) Depict best-response correspondences of Ann and Bob.

c) (5 points) Find all (pure strategy) Nash equilibria. Hint: You may want to rely on graphical

analysis.

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Exercise 3 (15 points)

In this exercise, we develop and solve the model of political voting. Given a population of citizens

who vote for political candidates, how should candidates position themselves along the political

spectrum? A cynical view is that politicians care only about getting elected and hence will choose

a platform that maximizes their chances. This is precisely the view that we take in this exercise.

We assume that a political platform is represented as a number on the [0, 1] interval. In other

words, we simplify the real world and suppose that policy space is one-dimensional. We also

assume that there are only two parties, denoted by D and R. These are our players. They

simultaneously choose their political platforms. A platform is a point on the [0, 1] interval. Let

sD and sR be a strategy of player D and player R, respectively.

We assume that parties incur no costs (i.e., no ideological bias) and their objective is to win the

elections. If party i gets more than 50% of votes, then i wins and gains utility 1. The loser gets

0. If each party gets 50% of votes, then the winner is decided by a coin flip (hence, the utility in

this scenario is 0.5).

There is a mass one of voters. Each voter is represented by a point on the [0, 1] interval that is

has his/her ideal policy. We assume that everyone votes. Voters choose a party that is closest to

their ideal policy.

Voters are uniformly distributed on [0, 1]. This means that, for example, 25% of voters are located

between 0 and 0.25, and also between 0.1 and 0.35, and also between 0.7 and 0.95. Knowing that

voters are uniformly distributed allows us to compute the percentage of votes each party gets.

Example 1. Suppose that sD = 0.2 and sR = 0.6. Graphically:

0 1 0.2

sD

0.6

sR

Everyone to the left of 0.2 votes for party D (this is because sD = 0.2 is the closest platform for

these voters). Hence, party D gets at least 20% of votes. Everyone to the right of 0.6 votes for

party R (this is because sR = 0.6 is the closest platform for these voters). Hence, party R gets at

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least 40% of votes. Graphically:

0 1 0.2

sD

0.6

sR

R D

So far, we know that D has 20% of votes and R has 40% of votes. What about the remaining 40%

of votes? These are the people “located” between 0.2 and 0.6. The interval [0.2, 0.6] is split into

halves: H1 = [0.2, 0.4) and H2 = (0.4, 0.6]. People in H1 (20% of votes) vote for D since they are

closer to sD = 0.2 than to sR = 0.6. People in H2 (20% of votes) vote for R since they are closer

to sR = 0.6 than to sD = 0.2. The person at 0.4 is indifferent between D and R (same distance to

both sD and sR). However, what that voter does is not important because the mass of individual

voter is zero. Graphically:

0 1 0.2

sD

0.6

sR

R D

0.4

D R

In our example, party D gets 20% + 20% = 40% of votes, and party R gets 40% + 20% = 60%

of votes. Hence, party R wins. The utility of party D is zero, and the utility of party R is one.

Example 2. Suppose that sD = 0.4 and sR = 0.4. That is, parties have identical platforms. In

this case, each voter is indifferent between D and R. Hence, party D get 50% of votes and party

R gets 50% of votes. This means that each party gets expected utility 0.5× 1 + 0.5× 0 = 0.5.

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Here is this game in a normal form.

• N = {D,R} is the set of players;

• Si = [0, 1] is the set of strategies of player i; a strategy si is a political platform;

• ui is the utility function of player i defined as

ui(sD, sR) =

 1 if i wins

0.5 if there is a tie

0 if i loses

(1)

Your task is to find all (pure strategy) Nash equilibria.

Hint: In this game, there is only one Nash equilibrium. First, try to guess the profile that is a

Nash equilibrium. Second, prove that, this profile is, indeed, a Nash equilibrium. Finally, show

that this profile is the only Nash equilibrium. You may want to revisit the handout on Bertrand

oligopoly.

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Exercise 4 (20 points)

There are two firms—A and B—which compete in prices. That is, firms’ strategies are their prices.

Let pA and pB denote price of Firm A and Firm B, respectively. Assume that prices must not be

negative; that is, pA ≥ 0 and pB ≥ 0. Firms choose their prices simultaneously. In addition, there

is no communication between the firms. (That is, we are dealing with a static game.)

We assume that firms face different demand functions. In particular, we have

Firm A’s demand: qA = a− pA + pB (2)

Firm B’s demand: qB = a− pB + pA (3)

where qA and qB are the resulting quantities that Firm A and Firm B sell, respectively, and a > 0

is a constant. (These equations indicate that goods produced by our players are substitutes.)

We assume that the firms incur no costs. This means that profit equals revenue. Keep in mind

that revenue is defined as qA · pA for Firm A and qB · pB for Firm B. We also assume that each

firm maximizes its profit.

Hint: You solve and analyze this game using the same approach we applied to the Cournot model

and Tullock model.

a) (5 points) Depict this game in a normal form.

b) (5 points) Depict best-response correspondences of players A and B.

c) (5 points) Find all (pure strategy) Nash equilibria.

d) (5 points) Comparative statics analysis: Determine the impact of parameter a on (i) equi-

librium strategies of A and B, and (ii) equilibrium utilities of A and B.

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Exercise 5 (15 points)

Consider the Tullock contest with two different players and common value (see Handout 10).

There are 2 players – Ann and Bob – competing for one prize. Ann and Bob assign the same value

to the prize, v > 0 (common value). Not winning the contest results with utility zero.

Players compete in efforts and these efforts determine the probability of winning the contest.

Effort of Ann is denoted by sA and effort of Bob is denoted by sB. We assume that sA, sB ≥ 0.

The probability of Ann winning the contest is sA sA+sB

. The probability of Bob winning the contest

is sB sA+sB

.

Efforts are costly. We assume linear cost functions. That is, CA(sA) = cAsA is Ann’s cost and

CB(sB) = cBsB is Bob’s cost. We assume the following.

cA > cB > 0 (4)

Each player maximizes her/his expected utility. Hence, we obtain utility of Ann uA and utility of

Bob uB.

uA(sA, sB) = v sA

sA + sB − cAsA (5)

uB(sA, sB) = v sB

sA + sB − cBsB (6)

Remark: There is no need to depict this game in a normal form.

a) (5 points) Depict best-response correspondences of Ann and Bob.

b) (5 points) Find all (pure strategy) Nash equilibria.

c) (5 points) Comparative statics analysis: Determine the impact of parameters cA and cB on

(i) equilibrium strategies of Ann and Bob, (ii) equilibrium probability of Ann wining, and

(iii) equilibrium probability of Bob wining.

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Exercise 6 (10 points)

Two students – Ann and Bob – simultaneously download music over the campus computer network.

Let si ≥ 0 represent the total size of student i’s music download, which student i chooses on his/her

own. The more data being downloaded, the slower the network functions. The total time it takes

for student i’s songs to download depends on both the size of his/her download, and on the total

amount of data that the network has to deliver.

Each student benefits from the size of his/her music download but is hurt by the time spent

waiting for the music download to finish. In particular, we assume that the total amount of time

it takes for player i to download his/her music is given by ci(sA, sB) = si(sA + sB); this is a cost

function of player i. At the same time, player i enjoys benefit si. Hence, the utility function of

player i is described by the following equation.

ui(sA, sB) = si − si(sA + sB) (7)

Remark: There is no need to depict this game in a normal form.

a) (5 points) Depict best-response correspondences of Ann and Bob.

b) (5 points) Find all (pure strategy) Nash equilibria.

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