Question 2: Game Theory [10 marks; 3,4,3]

There are two petrol stations in Kingsford, Caltex and BP, which sell petrol at one of the three prices:

1. High (H): $1.80 a litre; 2) Medium (M): $1.40 a litre, and 3) Low(L): $1.00
   • if two prices are equal, or

There are 100 residents in Kingsford. Each resident buys 10 litres of petrol each week at one of the two stations. If prices between two stores differ by more than 50 cents per litre, all 100 residents buy petrol from the cheaper petrol station. Otherwise,

• the two prices differ by less than 50 cents per litre, half of them buy at Caltex and the other half at BP.

BP's profit = price set by BP X number of residents buying from BP X 10

Caltex's profit = price set by Caltex X number of residents buying from Caltex X 10

a. [3 marks] Write down the 3 X 3 payoff matrix associated with the game where

• Caltex and BP are two players; each chooses one of the three prices: H, M, or L.
• BP is the row player and Caltex is the column player.

[Calculations for the payoff matrix]

b. [4 marks] State whether the following statements are True or False and explain your answer.

i. Statement 1: There exists a Nash equilibrium where BP chooses M and Caltex chooses L. (1 mark)

ii. Statement 2: There exists a Nash equilibrium where BP chooses L and Caltex choose H. (1 mark)

iii. Statement 3: There is a Nash equilibrium in pure strategies. (2 marks)

(c) [3 marks] Consider a dynamic game where BP moves first, and Caltex moves second. That is, BP sets its price first following which Caltex sets its price.

(i) (1 mark) Draw the extensive form game tree. Remember to clearly identify the nodes, specify player at each node, and write down the payoffs at terminal nodes. [Note: No further explanation needed for this part]

(ii)(2 marks) Find the strategy played by each player in the unique subgame perfect Nash equilibrium of the extensive form game. For each subgame, briefly explain how you arrived at the best response for the corresponding player.