PART A and PART B in detail, Thank you

There are two petrol stations in Auckland, 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 There are 100 residents in Auckland. 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, - If two prices are equal, or - 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 prot = price set by BP X number of residents buying from BP X 10| Caltex's prot = price set by lCaltex X number of residents buying from Caltex X 10 (a) 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. (Show the workings for the payoff matrix.) (b) State whether the following statements are True or False and explain your answers 1) Statement 1: There exists a Nash equilibrium where BP chooses M and Caitex chooses i_. 2) Statement 2: There exists a Nash equiiibr'ium where BP chooses i. and Caitex choose H. 3) Statement 3: There is a Nash equilibrium in pure strategies. (c) Consider a dynamic game where BP moves first, and Caltex moves second. That is, BP sets its price rst following which Caltex sets its price. (i) Draw the extensive form game tree. Remember to cleariy identify the nodes, specify player at each node, and write down the payoffs at terminal nodes (ii) Find the strategy played by each player in the unique subgame perfect Nash equilibrium of the extensive form game. For each subgame, briey explain how you arrived at the best response for the corresponding player