please answer the following questions with game theory.
1 Invest with the Best - 5 points You hold a stock, the price of which is rising. You believe that the stock is overvalued. You would like to sell it, but want to wait until the price is near its peak. In other words, you want to get out of the market just before other investors do. Other investors are thinking in the same manner. What is your dominant strategy? Justify your reasoning. 2 Calculus, Optimization, and Probability Review - 9 points Solve the following problems (3 points each). (a) Maximize U = efx, subject to I = piri + p2$2. (b) What is the maximum value of f (x) = 2-5x - 12? (c) Your utility function for income is U(1) = /. You are considering entering a lottery which pays $10,000,000 with probability 1/3, $0 with probability 1/3, and -$5,000,000 with probability 1/3. A lottery ticket costs you $10. Should you enter this lottery? 3 Doping FTW - 5 points Suppose that 5% of athletes take performance-enhancing drugs. The International Olympic Committee introduces a drug test that is 95% accurate. Show that if an athlete tests positive, then the probability of his being a drug user is 50% (Hint: use Bayes' rule).4 Smoking Guns - 5 points Three legislators vote over tum different policies: full gun legalization {A}. and partial gun legalization {E}. Full legalization {A} gives Legislator 1 a payoff of 1. while legislators 2 and 3 get I). Partial legalization {B} gives Leslators 2 and 3 a payoff of 1. Legislator 1 gets a payoff of I]. The policy that receives 2 or more votes wins. Find this game's pure-strategy Nash equilibria (Hint: there are three). Justify your reasoning. Does this game have a weakly dominant strategy? 5 You Take the High Road. I'll Take the Low - 5 points Two players are deciding how to split $1M}. Players simultaneously announce the shares. 31 and 22. that they would like to receive. with [i 5 21.32 5 l. if 21+ .92 5 10']. then the players receive their named shares. Otherwise. the players receive nothing. What are this game's pure-strategy Nash equilihria? Justify your reasoning. 6 The Matrix is a System, Neo- 9 points Figure 1: A Yuge Game Player 2 x y z T. a 1.2 2.2 5.1 E1 E b 4.1 3.5 3.3 c 5.2 4.2 2.2 d 2.3 2.4 3.2 Number IEEI: {right} of comma refers to row [oolunm's] Paraff- lEonsider the game in Figure 1. Justify your reasoning for each answer. {a} Find the strictly dominant strategy. {3 points} {12} When Player A plays of. what is Player 2's best response? {3 points} {c} What is this game's pure-strategy Nash equilibrium? {3 points) 7 You Say You Want a Revolution - 8 points Consider the following collective action game. There are 10,000,000 players, each of whom has two actions: Revolt, or Not. Payoffs are as follows: (1) If at least 2,000,000 people revolt, the revolution succeeds, and everyone gets a payoff of 1. (2) If fewer than 2,000,000 people revolt, and individual i revolts, then i gets -1. (3) If fewer than 2,000,000 people revolt, and individual i does not, then i gets 0. Remember to justify your reasoning in your responses. (a) What is this game's pure-strategy Nash equilibrium? (4 points) (b) Karl Marx, in German Ideology (1845), writes: This communist consciousness and... the alteration of men on a mass scale is [sic] necessary... [for] revolution. How would you change this game to introduce Marx's 'class consciousness'? What is the new pure-strategy Nash equilibrium? (4 points)
