Question

There are two ice cream shops, A and B, in a touristic town. Each shop needs to individually choose the price it charges for a scoop of ice cream, which can be either $2, $4 or $5. Each firm wants to maximize their total revenue from customers via an appropriate choice of price. It is expected that

− 8,000 scoops of ice cream are purchased by tourists, who are split evenly between the two shops regardless of the prices charged by these shops.

− Locals also purchase scoops of ice cream: Assume that locals purchase 800 scoops in total.

− Being more knowledgeable of market conditions, locals purchase their ice cream from the shop with the lowest price. If both shops charge the same price, then the locals are split evenly between the two shops.

(a) Now, write the normal form game between the two ice cream shops where payoffs in each cell of the table represents the total revenue of each shop. Make sure to write your calculated payoff pairs for each cell of the normal form game.

(b) Explain the underlying logic behind your calculated payoffs in the normal form game of part (a) by showing your full workings of payoffs for a sample case where both shops charge the same price. Do the same also for a sample case where firms charge different prices.

(c) Reproduce the normal form game from part (a) and use it to determine the outcome of this game by applying iterated elimination of strictly dominated strategies. Explain your workings.

Answer 1

a. The payoff for the players for each price is-

		B
	Prices	$2	$4	$5
A	$2	8800,8800	9600,16000	9600,20000
$4	16000,9600	17600,17600	19200,20000
$5	20000,9600	20000,19200	22000,22000

b. We know that tourists will buy 4000 scoops from each shop irrespective of the price. Locals will buy 400 scoops from each if price is same or 800 scoops from the shop whose price is lower. Calculating the total revenue for each case-

1. When prices are same both will sell 4400 scoops of ice cream at each price level (4000 to tourists and 400 to locals).

P_A= P_B= $2

TR= 4400*2

= $8800

Both A and B earn a revenue of $8800. Their payoff is (8800,8800)

P_A= P_B= $4

TR= 4400*4

= $17600

Both A and B earn a revenue of $17600. Their payoff is (17600,17600)

P_A= P_B= $5

TR= 4400*5

= $22000

Both A and B earn a revenue of $22000. Their payoff is (22000,22000)

2. If the prices are different, the shop with lower price will sell 4800 scoops (4000 to tourists and 800 to locals) while the one with higher price will sell 4000 scoops (only to tourists).

Lets assume A keeps price at $2 while B keeps price at $4.

P_A= $2, P_B= $4

TR_A= 4800*2

= $9600

TR_B= 4000*4

= $16000

A earns a revenue of $9600 and B earn a revenue of $16000. Their payoff is (9600,16000)

Lets assume A keeps price at $2 while B keeps price at $5.

P_A= $2, P_B= $5

TR_A= 4800*2

= $9600

TR_B= 4000*5

= $20000

A earns a revenue of $9600 and B earn a revenue of $20000. Their payoff is (9600,20000)

Lets assume A keeps price at $4 while B keeps price at $2.

P_A= $4, P_B= $2

TR_A= 4000*4

= $16000

TR_B= 4800*2

= $9600

A earns a revenue of $16000 and B earn a revenue of $9600. Their payoff is (16000,9600)

Lets assume A keeps price at $4 while B keeps price at $5.

P_A= $4, P_B= $5

TR_A= 4800*4

= $19200

TR_B= 4000*5

= $20000

A earns a revenue of $19200 and B earn a revenue of $20000. Their payoff is (19200,20000)

Lets assume A keeps price at $5 while B keeps price at $2.

P_A= $5, P_B= $2

TR_A= 4000*5

= $20000

TR_B= 4800*2

= $9600

A earns a revenue of $20000 and B earn a revenue of $9600. Their payoff is (20000,9600)

Lets assume A keeps price at $5 while B keeps price at $4.

P_A= $5, P_B= $4

TR_A= 4000*5

= $20000

TR_B= 4800*4

= $19200

A earns a revenue of $20000 and B earn a revenue of $19200. Their payoff is (20000,19200)

c.

		B
	Prices	$2	$4	$5
A	$2	8800,8800	9600,16000	9600,20000
$4	16000,9600	17600,17600	19200,20000
$5	20000,9600	20000,19200	22000,22000

Examining possible moves for player A-

If player B sets a price of $2, A can get the maximum revenue by keeping price at $5.

If player B sets a price of $4, A can get the maximum revenue by keeping price at $5.

If player B sets a price of $5, A can get the maximum revenue by keeping price at $5.

Therefore, irrespective of the choice made by B, player A should always keep the price at $5 as it will maximize his total revenue in every situation, i.e $5 strictly dominates the $2 and $4 price strategies.

$2 and $4 can be eliminated from the choices for A-

		B
	Prices	$2	$4	$5
A	$5	20000,9600	20000,19200	22000,22000

Examining choices for B-

Knowing that A will always set his price at $5, B can maximize his total revenue by also keeping his own price at $5. In this case, both will earn $22000. Price strategies $2 and $4 can be eliminated for B as well. The only remaining strategy is for both to keep the price at $5.

		B
	Price	$5
A	$5	22000,22000

Therefore, after applying iterated elimination of strictly dominated strategies, ($5,$5) with payoff (22000,22000) is the outcome.