Question

There are two grocers. Each grocer charges its own price for a loaf of bread, either $1, $4, or $5. The cost of obtaining and selling the bread can be neglected.

• 8000 loafs of bread per month are bought from a grocer by seniors who will choose randomly from one of the two grocers.
• 4000 loafs of bread per month are bought by local residents who go to the grocer with the lower price, and split evenly in both grocers offer the same price.

Create a matrix using the quantity of loafs of bread, then create another matrix showing the revenue of the bread sold.

1. Draw the payoff matrix on the quantity of loafs of bread consumed.
2. Draw the payoff matrix on the revenue of loafs of bread consumed.
3. Find all the Nash Equilibria, if any. Otherwise, write “No Nash Equilibria”.
4. At what price will the grocers offer a loaf of bread?

Answer 1

a) Payoff matrix on quantity of loaves of bread consumed

		Grocer 2
		1	4	5
Grocer 1	1	6000,6000	8000,4000	8000,4000
4	4000,8000	6000,6000	8000,4000
5	4000,8000	4000,8000	6000,6000

The inherent assumption is that the senior citizen has equal probability of selecting any of the grocers. So the total quantity consumed by the senior citizen will be divided equally between the 2 grocers.

b) Payoff matrix on the revenue of loaves of bread consumed

		Grocer 2
		1	4	5
Grocer 1	1	6000,6000	8000,16000	8000,20000
4	16000,8000	24000,24000	32000,20000
5	20000,8000	20000,32000	30000,30000

c) We see that for the Revenue Matrix, we select each of the possible combinations to find the Nash Equilibria

Both select (1,1): There is an incentive for both players to shift their strategy for a better payoff

Grocer 1 Selects $1 Grocer 2 selects $4: Both grocers have an incentive to change strategy and improvise payoff

Grocer 1 Selects $1 Grocer 2 selects $5: Grocer 1 has an incentive to change strategy for better payoff

Grocer 1 Selects $4 Grocer 2 selects $1:Both grocers have an incentive to change strategy and improvise payoff

Grocer 1 Selects $4 Grocer 2 selects $4: Neither grocer has any incentive to change prices. So this is a NASH Equilibrium

Grocer 1 Selects $4 Grocer 2 selects $5: Grocer 2 has an incentive to change strategy for a better payoff

Grocer 1 Selects $5 Grocer 2 selects $1: Grocer 2 has an incentive to change strategy for a better payoff

Grocer 1 Selects $5 Grocer 2 selects $4:Grocer 1 has an incentive to change strategy for better payoff

Grocer 1 Selects $5 Grocer 2 selects $5: Both grocers have an incentive to change strategy and improvise payoff

So there is 1 Nash Equilibrium. When both Grocers select $4 as a price.

d) Since the Nash equilibria is when both grocers select $4, so both the grocers will price their loaf of bread at $4. They can get a better payout by priceing at $5. But then both have an incentive to shift strategies and it will not be stable in the long run.

So both grocers will charge $4 per loaf of bread.

If you found this helpful, please rate it so that I can have higher earnings at no extra cost to you. This will motivate me to write more.