a) Ariana and Beyonce were returning from a game theory class and found a bitcoin on the street. Ariana saw it first and grabbed it telling Beyonce that they should share it. Beyonce suggested apply some of what they have learnt. Beyonce proposed the following to Ariana: B: "How about you make me an offer to split it now? If I accept it, we split it that way, and if I don't accept it, tomorrow I will make you an offer. If you accept my offer, we split it that way, and if you do not, we just send it to lost and found." A: "Although it is fair that each one of us has the opportunity of making an offer, I think that I should let you go first and make me an offer. If I say yes, we split the bitcoin that way and if I say no, tomorrow I will make you an offer, if you accept my offer tomorrow, we split the bitcoin that way and otherwise we take it to lost and found." B: "Ariana, please, you found it so you should go first. It is only fair if you, the person who found it, makes the first proposal." A: "Beyonce, I appreciate your concern but if I go first, I think I should also make the last offer." B: "I do not think I can accept that unless we make the last offer disappear. How about we make each other a proposal on alternate days until one of us accepts? Today is your turn tomorrow is my tum, and so on and so forth. In that way, there is no ending and we never take it to lost and found." A: "That will take us forever so I rather split it in half now. How about that?" B: "OK, it is fair." A: "Fair or not fair, it was going to happen anyway." Assuming that they are very patient (like making 6, how much they both discount tomor- row's utility from today's perspective, really close to 1) and they both want to get as much of the bitcoin as possible, explain using your knowledge and techniques of game theory why they made the comments they made and whether you agree with each comment or not. b) Consider the following situation. Player A has a pie of size 1 and needs to split it with player B and C. They proceed as follows. In the first round, player A makes an offer to player B who must decide whether to accept or reject the offer. The offer is just a portion < 1 of the pie for player B. If player B rejects the offer, each one of the players (A, B, and C) gets 1/3. If he accepts the offer then player B receives a credit for a and needs to wait until player A and C negotiate. A makes an offer to split the remaining pie 1-2 between A and C. If C accepts, then they split it in the way that A suggested. If C rejects the offer, then B's credit disappears and the three of them split the pie in three ways (1/3 each). Question 2 Normal Form Games. (25 marks) a) Find all Nash equilibria of the following game, both in pure and in mixed strategies. L2 C2 R2 L 2,1 1,0 1,1 C 1,0 3,1 2,0 R 3,1 2,0 3,1 b) In this part of the question you will assess the relation between individual options and social dilemmas i) Find all Nash equilibrium in pure and mixed strategies of the following game. Show your work. L2 R2 L1 2,2 1,1 R 1,1 2,2 ii) Find all Nash equilibrium in pure and mixed strategies of the following game. Show your work. L2 C2 R2 L 2,2-1,5 1,1 C 5,-1 0,0 5,-1 R 1,1 -1,5 2,2 iii) Can u find the similarities between the two previous games? Explain how they are similar and how they are different. iv) Using the solution to the previous two games, discuss whether you agree or not with the following statement: "There are some results in economics that clearly are wrong. For example, giving men more opportunities is as if you were making men more free, and since freedom is a superior ideal they can never be worse off in economic terms. This statement is valid independently of the market structure, the moment in time, and the strategic actors as all actors are humans." QuestioIT General Equilibrium & Uncertainty 30 marks) Bella has an orchard that produces 1t of fruit when it's sunny, but nothing when it's rainy. Everybody knows that it is sunny with probability and rainy with with probability 1. Her utility is given by u = 4(x), where xB is the amount of fruit she consumes. a) Calculate Bella's Arrow-Pratt Measure of Absolute Risk Aversion. Show your work. How does this change with the amount of fruit she has? Explain the meaning of this measure in your own words and give an interpretation of your result. b) Bella is considering buying insurance against rain. (5 marks) (10 marks) i) What are her certainty equivalent and risk premium? Explain the meaning of these terms. ii) If the insurance company offers to pay her one unit of fruit (f) in the case of rain at the price of q = , how much insurance would she buy? Show your work and comment on the results. Ao has an orchard that produces 1t of fruit when it's rainy, but nothing when it's sunny. His utility is given by u = 2(4), where x is the amount of fruit he consumes. Bella decides not to buy insurance and instead trades state contingent fruit with Ao before they know the weather. c) State and explain the relevant budget constraints and Bella and Ao's maximisation problems. (5 marks) d) Calculate Bella and Ao's demands for fruit with sun and rain as a function of prices. Show your work. (10 marks) e) State and explain the relevant market clearing conditions. Find the equilibrium prices and allocations. Show your work and explain the result. (5 marks) f) After completing a MSc in Economics at Bristol, the queen wants to maximise the utilitarian social welfare function W = E (u^) + E (u). (15 marks) i) How should she redistribute the fruit to maximise W? ii) Comment on the merits of this particular utilitarian social welfare function and discuss general issues with social welfare functions given the utility functions in this setup. (100-