PSet 3 Solutions.pdf - Problem Set 3 Solutions Econ 2020b...

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Problem Set 3 Solutions Econ 2020b / HBS 4011 / API-112 1. Representing the Extensive Form in the Normal Form (a) PI: UT, UM, DT, DM PII: X, Y PIII: A, B (b) By backwards induction, player I chooses X; player I chooses U and M; player III chooses A. Hence, the backward induction Nash equilibrium is {UM; X; A}. (c) X Y X Y UT 2,0,2 2,0,2 UT 1,2,4 1,2,4 UM 2,0,2 2,0,2 UM 2,4,0 -6,2,8 DT 1,5,5 1,5,5 DT 1,2,4 1,2,4 DM 1,5,5 1,5,5 DM 2,4,0 -6,2,8 (d) There are two pure strategy Nash equilibria in the normal form: {UM; X; A} and {UT; Y: B}. (e) {UT; Y: B} is a Nash equilibrium in the normal form but is not selected by the backward induction procedure in the extensive form. 2. A Patent Race (a) Suppose that the position is (4, 3) and it is A ’s turn to move. What is the backward induction solution to the game? If A takes a step towards the finish line with this move, then B will finish the work next period. So instead, A should avoid costs by not moving. In fact, by this reasoning, A should never take a further step. Recognizing this, B can achieve the patent at minimum cost by taking one step in each of the next three moves. The payoff in this backward induction solution is (0, 7). (b) Now suppose that the position is (4, 4) and it is A ’s turn to move. What is the backward induction solution to the game? A can induce the desirable outcome in part (a) by taking one step at this move to get to III B A
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(3, 4), which yields payoff (7, 0). Including the cost of the current step for A, the payoff for the remainder of the game is then (6, 0). (c) Now suppose that the position is (5, 4) and it is A ’s turn to move. What is the backward induction solution to the game? If A takes one step, then B will take one step to get to (4, 3), which is a winning position for B and a losing position for A. But if A takes two or more steps, A can achieve a winning position. So A should take two steps to get to (3, 4), which has payoff (7-4, 0) = (3, 0). For parts (d) to (f), d efine the “trigger zone” as all the points where the firm moving first can ensure victory. (That is, it will get the patent and achieve a positive payoff.) and the “safety zone for A ” as all the points where, regardless of which firm moves first, A can ensure victory. Define the safety zone for B similarly. (d) Consider all the points where both firms are at a distance 5 or less from the finish line. Which points lie in the trigger zone, which lie in the safety zone for A , and which lie in the safety zone for B . If one firm is 3 steps or less from the finish and the other is more than 3 steps from the finish, then the first firm will win the game. So (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) are in the safety zone for A and similarly (4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3) are in the safety zone for B. If both firms are 4 or 5 steps from the finish, then the next firm to move can make a move that leaves a position in its safety zone (at cost at most 4). So (4, 4), (4, 5), (5, 4), (5, 5) are in the trigger zone.
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