ECC2610 ASSIGNMENT 2 Due Date: Friday October 8., 8PM [AEST) Instructions: Read all questions carefully; answer all questions (each ques- tion is worth 20 marks); show all the (nontrivial) steps of your derivations; for your answers, handwriting is acceptable {as long as it is easy to decipher} but typing is preferred; submissions must be made through l't-[oodle, using the drop box in the assignment folder; submit a unique PDF le (multiple les will not be accepted). Good luck! QUESTIONS Q1 Consider the strategicform game depicted below: a b c a 7, 2 2, T 3, 6 b 2, T T, 2 4,5 Let p102) denote the probability with which player 1 (the row player} plays strategy a, and let 331(6) be the probability with which she plays strategy 5. Let p;{a] be the probability with which player 2 [the column player} plays strategy a, pg\") the probability with which he plays strategy b, and 113(6) the probability with which he plays strategy c. a) Show that there is no mixedstrategy Nash equilibrium where p2(o} >- 0, 33205} '2 '0, and 103(3) 3- 0. b) Show that there is no mixedstrategy Nash equilibrium where p2(o} = U, 332(k) )5 '0, and 103(6) '3} 0. c) Show that there is no mixedstrategy Nash equilibrium where pzo} >- 0, 332(5) '9 0, and pc) = 0. d) There is a {unique} mixedstrategy Nash equilibrium where pa} > D, jig-{b} = 0, and 132(6) > 0. Compute this equilibrium. Q2 Let in >- 0 denote your wealth. Your wealth at the beginning ofthe {unique} period under consideration is E > 0. There is a safe asset whose rate of return is zero. There is also a risky asset whose rate of return is 1'9 0. Compute the expected utility from investing A. c) For which of the above two utilities you are more riskaverse? d) What is your optimal investment A if your utility is abs} = ln(w)'? What is your optimal investment A if your utility is u{w} = a + bw can2 (with parameters as in part b}? Q3 Consider the following inverse demand function, p(Q) = 5 7Q, Q = q] + 9'2, where .1}, denotes rm e's output, 1? = 1, 2. Assume that the total cost of rm i is egg-f2, with c I) 0. Firms choose quantities simultaneously and non cooperatively. The game described above is innitely repeated. Firms use grim trigger strategies {innite Nash reversion). Firms discount future prots at a rate r b D. {-1) Compute the cartel prots. b) Derive the critical discount factor above which full cartelization [joint prot maximization) is sustainable as a Subgame Perfect Nash Equilibrium {SPNE) of the innitely repeated game. Consider now the following prisoner's dilemma game: 0 N 0 9,9 0,12 N 12,0 3,3 The above game is repeated a random number of times. After each stage is played, the game ends with probability in. Players discount future payofs at a rate r 2;\" U. c) Compute the threshold of p below which (0,0) is a SPNE of the repeated game that ends after a random number ofrepetitions (hint: think how the discount factor is amected by 33). Comment. d) In repeated games, if players discount future payofs at a different rate, is it possible for cooperation to be sustainable? Explain. Q4 Consider the following normal form game: L R U D, 0 4, -'.1 D 21:, 2:c 2, 2 Assume that :r > 0. Moreover, assume that Player Row chooses U with probability p and Player Column chooses L with probability 4;. a) Derive and plot players' best response functions (p on the horizontal axis and g on the vertical axis}. In) Find all the Nash equilibria [pure and mixed strategies) ofthe above game. Illustrate your answer in a graph {3: on the horizontal axis and g on the vertical axis). Comment. Consider now the following twoplayer simultaneousmove game, called the rockpaperscissorslizard game. R stands for rock, P for paper, S for scissors, [Q and L for lizard. R beats S but loses against P and L; P beats R but loses against S and L; S beats P and L but loses against E; L beats R and P but loses against S. The payoff for winning is 1 and that for losing is 1; when both players choose the same strategy they each get 0. Assume that Player How chooses R with probability 7', P with probability p, and S with probability .9 [similarly for Player Column). c) Write down the normal form representation of the game. d) Find all the Nash equilibria {pure and. mixed strategies) of the game. Comment. Q5 Suppose that the minimum price a seller will sell her boat for is $5, and that the maximum price the buyer will pay for the boat is $3, with B > S > 0. Suppose that both players have the exact same discount factor, (3 = 0.7. Suppose that there are two rounds of bargaining, and that the buyer makes a proposal in the rst round, and the seller makes a proposal in the second round. a) How much the buyer must oer the seller to get the boat? Suppose now that the bargaining can go on forever. I!) How much the buyer must o'er the seller to get the boat? Assume from now on that the seller is the rst proposer {the bargaining can go on forever). 1:) What is the nal price at which the boat will be sold for? How is the surplus split? The seller has now a discount factor equalto 0.8 and the buyer has a discount factor equal to 0.7 [the bargaining can go on forever). d) What is the nal price at which the boat will be sold for? How is the surplus split? Comment