# Economics GAME THEORY FINAL EXAM

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Final Exam ECON 414 Winter 2021 1. (35 pts) Player 1 and Player 2 are involved in a dispute. Player 2 is either Weak or Strong. Player 2’s strength is private information. 2 knows his strength; 1 knows that 2 is Strong with probability p and that he is Weak with probability (1 – p). Both players will decide to either Fight or Yield simultaneously. The payoffs in this game are as follows: · If they both choose Yield each earns 0 regardless of 2’s type. · If one chooses Fight and the other chooses Yield, the player who chooses Fight earns 100 and the other player earns 0 regardless of 2’s type. · If they both choose Fight · If Player 2 is Strong then Player 1 earns a payoff of -100 and Player 2 earns a payoff of 100. · If Player 2 is Weak then Player 1 earns a payoff of 100 and Player 2 earns a payoff of -100. (a) (5 pts) Draw the game tree. (b) (15 pts) Suppose p =0.3. Find the Bayes-Nash equilibrium of this game. Defend your answer carefully. (c) (15 pts) Suppose p =0.7. Find the Bayes-Nash equilibrium of this game. Defend your answer carefully. 2. (10 points) Suppose there is a newly developed drug test for an illegal drug. Further suppose the test is 97% accurate in the case of a user of that drug (it produces a positive result with probability .97 in the case that the tested individual uses the drug) and 70% accurate in the case of a non-user (it is negative with probability .7 in the case the person does not use the drug). Suppose it is known that 15% of the entire population uses this drug. (a) (5 pts) Suppose that a randomly selected individual tested negative for drug use. What is the probability that this individual uses the illegal drug? Explain your answer very carefully. (b) (5 pts) Suppose that a randomly selected individual tested positive for the drug use. What is the probability that individual does not use the drug? Explain your answer very carefully. 3. (55 pts) Player 1 and 2 are thinking of starting a new company. Suppose that there are two types for Player 1: Type-O(ordinary) with prob. 3/4 and Type-C(cooperative) with prob. 1/4. Player 1’s type is private information. · At the start of the game Player 1 decides whether to invest or not. If player 1 decides not to invest, game ends. · If he decides to invest, then Player 2 decides whether to invest or not. · If Player 2 decides 2 invest, Player 1 decides to steal or split the profits. · Player 2 observes Player 1 ‘s decision to invest or not before making her investment decision. · Player 1 observes Player 2’s decision before deciding whether to split or steal. Payoffs from different scenarios are summarized in the table below: (a) (5 pts) Draw the game tree. (b) (5 pts) We said that a perfect Bayes-Nash equilibrium has three elements. What are those three elements? (c) (15 pts) Show that this game does not have a separating equilibrium. Use your answer to part c to formulate your answer. (d) (15 pts) Show that this game does not have a pooling equilibrium. Use your answer to part c to formulate your answer. (e) (15 pts) Let p be the probability that Type-O player will invest and let q be the probability that Player 2 will invest. Find the mixed strategy perfect Bayesian Nash equilibrium.

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