Question

Two shops A and B attract two types of customers: price-sensitive customers, who shop at the lower-priced shop, and time-sensitive customers, who shop at whichever shop is on their side of the street. A customer of either type purchases on average 10 goods. Assume there are 800 price-sensitive customers and 500 time-sensitive customers buying goods on any given day. The wholesale price the shop pay per good is $3. For simplicity, assume that they can charge $3.50, $4.00, or $4.50 per good.

a. Complete the matrix depicting each station’s profit given the possible strategy profiles.

b. Are there any dominant strategies or dominated strategies? Explain briefly.

c. Are there any pure-strategy Nash equilibrium? If so, give the strategy profiles and corresponding outcomes.

Answer 1

Both shops A and B have 3 strategies available of pricing it at 3.50, 4.00 and 4.50

Since they will each buy on average 10 goods, the profit for each shop will be total number of customers who visit the shop*10*price of the good they charge - the total cost they pay for the goods = 10*n*p - 10*n*3 = 10*n*(p-3)(where p is the price, n the number of customers that visit and 3 the whole sale price)

Now both A and B will have 250 of the time-sensitive customers, since they don't care about price, and the one with lower price will get all the 800 price-sensitive customers, and 400 each if they both price equally.

The matrix showing how many customers visit will thus be

A\B	3.50	4.00	4.50
3.50	650, 650	1050,250	1050,250
4.00	250,1050	650,650	1050,250
4.50	250,1050	250,1050	650,650

a) Therefore the payoff matrix showing the profit is

A\B	3.50	4.00	4.50
3.50	3250, 3250	5250,2500	5250,3750
4.00	2500,5250	6500,6500	10500,3750
4.50	3750,5250	3750,10500	9750,9750

b) A strategy is dominated strategy when it doesn't lead to the highest payoff in any case.
In this case the strategy of pricing the good at 3.50 is a dominated strategy never leads to the highest profit for any shop in any case, regardless of the other shops' actions.

For example, if the other shops price it at 3.50, your shop is better off pricing it at 4.50, if the other shop prices it at 4, your shop is better off at pricing it at 4, if the other shop prices it at 4.50, you are again better off pricing it at 4.

Thus in no case of the other shops' strategy, will you have the highest profit by using the strategy of pricing it at 3.50. Thus it is a dominated strategy.

There is no dominant strategy which leads to the highest profit in any case, strategy of pricing it at 4.50 gives the optimal profit in 1 case(When the other shop prices it at 3.50) and the strategy of pricing it at 4 gives the optimal profit in 2 other cases(When the other shop prices it at 4 or 4.50)

c) A strategy profile is a Nash equilibrium if, given the other shops' strategy, their strategy leads to the highest payoff possible i.e. neither has any incentive to deviate from the nash equilibrium given the strategy of other shop remains the same.

In this case, we have the pure -strategy Nash equilibrium of the strategy profiles of shop A pricing it at 4.00 and shop B pricing it at 4.00, the corresponding outcomes are that both have a profit of 6500.

Now, neither has any incentive to deviate, as if any shop reduced their price to 3.50 while the other keeps their strategy of 4.00, their profit will reduce to 5250, and if the shop increased their price to 4.50 while the other keeps their strategy of 4.00, their profit will reduce to 3750, so no shop has any incentive to deviate, hence it is a Nash equilibrium.

Hope it helps. Do ask for any clarifications if required.