Question

MY DEAKIN STUDENT ID NUMBER IS 217225221

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.

• Scenario 1: If the 7th digit in your Student ID number is odd (that is, it is either 1, 3, 5, 7 or 9), then assume that locals purchase 800 scoops in total. My Student Id is 217225221 and the seventh number is 2 which is even.

• Scenario 2: If the 7th digit in your Student ID number is even (that is, it is either 0, 2, 4, 6, or 8) then assume that locals purchase 5000 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) What is the 7th digit in your Deakin Student ID number? Based on this, which of the above two scenarios apply to you? 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

Scenario 2 applies.

a. The payoff at various prices are-

		B
	Prices	$2	$4	$5
A	$2	13000,13000	18000,16000	18000,20000
$4	16000,18000	26000,26000	36000,20000
$5	18000,20000	20000,36000	32500,32500

b.

Total scoops purchased by tourists= 8000

Total scoops purchased by locals =5000

Tourists will buy 4000 scoops from each irrespective of price. Locals will buy 2500 scoops from each if price is equal or 5000 from the shop with lower price.

1. When prices are equal, both shops will sell 6500 scoops (4000+2500)-

P_A=P_B= $2

Total Revenue= 6500*2

= $13000

Both A and B earn total revenue of $13000. Their payoff is (13000,13000)

P_A=P_B= $4

Total Revenue= 6500*4

= $26000

Both A and B earn total revenue of $26000. Their payoff is (26000,26000)

P_A=P_B= $5

Total Revenue= 6500*5

= $32500

Both A and B earn total revenue of $32500. Their payoff is (32500,32500)

2. When prices are different, the shop with lower price will sell 9000 scoops (4000 to tourists and 5000 to locals) while the one with higher price will sell 4000 scoops(only to tourists)-

If A set price at $2 and B at $4,

P_A=$2, P_B= $4

TR_A= 9000*2

= $18000

TR_B= 4000*4

= $16000

A earns total revenue of $18000 while B earns total revenue of $16000. Their payoff is (18000,16000)

If A set price at $2 and B at $5,

P_A=$2, P_B= $5

TR_A= 9000*2

= $18000

TR_B= 4000*5

= $20000

A earns total revenue of $18000 while B earns total revenue of $20000. Their payoff is (18000,20000)

If A set price at $4 and B at $2,

P_A=$4, P_B= $2

TR_A= 4000*4

= $16000

TR_B= 9000*2

= $18000

A earns total revenue of $16000 while B earns total revenue of $18000. Their payoff is (16000,18000)

If A set price at $4 and B at $5,

P_A=$4, P_B= $5

TR_A= 9000*4

= $36000

TR_B= 4000*5

= $20000

A earns total revenue of $36000 while B earns total revenue of $20000. Their payoff is (36000,20000)

If A set price at $5 and B at $2,

P_A=$5, P_B= $2

TR_A= 4000*5

= $20000

TR_B= 9000*2

= $18000

A earns total revenue of $20000 while B earns total revenue of $18000. Their payoff is (18000,16000)

If A set price at $5 and B at $4,

P_A=$5, P_B= $4

TR_A= 4000*5

= $20000

TR_B= 9000*4

= $36000

A earns total revenue of $20000 while B earns total revenue of $36000. Their payoff is (20000,36000)

c.

		B
	Prices	$2	$4	$5
A	$2	13000,13000	18000,16000	18000,20000
$4	16000,18000	26000,26000	36000,20000
$5	18000,20000	20000,36000	32500,32500

Examining the strategies for A-

If B sets price at $2, A can maximize revenue by keeping price at $5

If B sets price at $4, A can maximize revenue by keeping price at $4

If B sets price at $5, A can maximize revenue by keeping price at $4

Therefore, A will never choose to keep price at $2 regardless of price chosen by B. $2 is a strictly dominated strategy for A and can be eliminated for A.

		B
	Prices	$2	$4	$5
A
$4	16000,18000	26000,26000	36000,20000
$5	18000,20000	20000,36000	32500,32500

Examining the strategies for B-

If A sets price at $4, B can maximize total revenue by setting price at $4

If A sets price at $5, B can maximize total revenue by setting price at $4

Therefore, B will always keep price at $4 regardless of price set by A. Price strategy $4 strictly dominates the other 2 strategies for B. $2 and $5 price strategies can now be eliminated for B.

		B
	Prices	$4
A
$4	26000,26000
$5	20000,36000

Now that it is known B always choses $4 as his price strategy, A can also set his price accordingly. When price of B is $4, A can maximize total revenue at price of $4.

Therefore, A will only choose price strategy $4 and never $5. As such $5 can also be eliminated for A.

		B
	Prices	$4
A
$4	26000,26000

The only remaing strategy for both player is to keep price at $4.

Therefore, after iterated elimination of strictly dominated strategies, only outcome is ($4,$4) with a payoff of (26000,26000).