Show all your work for full credit. Providing only correct answers is not enough

/in /by

Test 2

ECO 444

Konrad Grabiszewski

• This is an individual take-home test. Prepare the answers on your own.

• Deadline: deliver your exam in class on Monday, April 2, 2018.

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

Exercise 1 (30 points)

For each game, identify all Nash equilibria (pure and mixed strategies). 10 points per game.

Bob

L R

Ann T 4,4 2,2

B 2,2 4,4

Game 1

Bob

L R

Ann T 3,7 2,7

B 1,1 3,4

Game 2

Bob

L R

Ann T 1,1 1,1

B 1,1 1,1

Game 3

Exercise 2 (30 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.

1

• If sA +sB ≤ 100, then the players receive their demands (and the owners eats any leftovers).

In this case, Ann gets sA (and sA is her utility) and Bob gets sB (and sB is his utility).

• 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.

a) (15 points) Depict best-response correspondences of Ann and Bob.

b) (15 points) Find all (pure strategy) Nash equilibria. Hint: You may want to rely on graphical

analysis.

Exercise 3 (40 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)

a) (30 points) Depict best-response correspondences of Ann and Bob.

b) (10 points) Find all (pure strategy) Nash equilibria.

2