Question 2 [19 points) Anusha and Bill went to the Royal Easter Show where each of them bought identical toys. Both lost their toys on their way back home. Their parents decided to reimburse them. They do not know the exact price, but they know the price is either $2, or $3. or 34. They proposed the following scheme: a Anusha should write either p... = 2,3 or :L as the price on a piece of paper and hand it to her parents. Bill should do the same, writing :13 = 2,3. or 4, without consulting Anusha. a If the difference between pg and pg is less than 2, then each child will receive the amount they wrote down as their payoff. a If the difference between pi. and 193 is 2. then the child who wrote the higher number will receive 51 less than what their sibling wrote down, and the child who wrote the lower number will receive $1 more than what their sibling wrote down. (a) (5 points} Write down the 3 x 3 payoff matrix associated with the game where Anusha and Hill are the two players. each choosing to write dorm one of three prices: 2.3. or 4. Let Anushe be the row player, and Bill be the column player. (b) For the following statements, assert whether they are true or false. and briey justify your answer. i. (2 points] Writing 4 is a dominant strategy for Bill. True or False? Explain in no l'lltll't! than 25 Winds. ii. (2 points) There exists a Nash equilibrium where both Anusha and Bill write 3. True or False? Explain in no more than 26 words. iii. {3 points) Now consider a reduced version of the game where \"3" is not an option. That is. Anushs and Bill can write either 2 or 4. The rules of this revised game are same as the ones stated at the outset. There is no Nash equilibrium in the revised game. Time or False? Explain in no more than 5|] words. You may use a diagram to aid your explanation if you wish. (c) Now consider a. dynamic game where Anuslta motes rst. and Bill moves second. That i3. Anusha quotes pa rst following which Bill quotes pa. As in part. {a}. Anusha and Bill can choose any of the three numbers: 2', 3. or 4. i. (2 points} Draw the extensive Form game tree. Clearly identify the nodes, specify the planer making the decision at each node. and write down the paynlfs at the terminal nodes. No additional working or explanation is required. ii. [5 points) Fill in the blanks below and nd the strategy played by each player in the unique sub-game perfect Nash equilibrium of the extensive form game. If Anushn chm 2, Bill maximises his payoff by choosing action D and earning a payoff of l:i If Anmha chooses 3. Bill maximises his payo' by choosing action D and cutting a payoff of D. If Anusha chooses 4, Bill maximises his payo' by choosing action D and earning a ParoofD Anticipating Bill's strategy, Anusha will maximise her payo by choosing action D and earning a payo' ofD Hence. the SPNE is |:|