COMP 3121 Homework 6

Question 1

1000 cars travel from A to B. There are two possible routes: the upper route through C or the

lower route through D. Let x be the number of cars traveling on the edge (A, C) and let y be the

number of cars traveling on the edge (D, B). The directed graph in the Figure indicates that the

travel time of a car on edge (A, C) is x/100 if there are x cars on edge (A, C), and similarly the

travel time of a car on edge (D, B) is y/100 if there are y cars on edge (D, B). The travel time of a

car on edges (C, B) and (A, D) is 12 regardless of the number of cars on these edges.

Each driver wants to minimize his travel time. The drivers make simultaneous choices.

(a) Find the Nash equilibrium of x and y.

(b) Now the government builds a new one-way road from town C to town D, with a travel time

of 0. Find the Nash equilibrium of x and y.

(c) Suppose now that the conditions on edges (C, B) and (A, D) improves so that the travel time

on each edge is reduced to 5. The road from C to D that was constructed in part (b) is still

available. Find the Nash equilibrium of x and y.

Question 2

Suppose we have 3 sellers a, b and c, and 3 buyers x, y and z. Each seller has a car park for sale,

and the valuations of the buyers are as follows.

Buyer Valuation on a’s car

park

Valuation on b’s car

park

Valuation on c’s car

park

X 6 5 2

Y 7 6 3

Z 6 7 6

There are 4 sets of prices for a’s, b’s and c’s car parks.

1. a=0, b=0, c=0

2. a=2, b=1, c=0

3. a=0, b=1, c=0

4. a=3, b=1, c=0

How many sets are market-clearing prices?

A. 0

B. 1

C. 2

D. 3

E. 4

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