pset3_2023

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Texas A&M University *

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308

Subject

Political Science

Date

Dec 6, 2023

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pdf

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3

Uploaded by crawfell

POLS 308: Problem Set 3 September 27, 2023 Write-up due at the beginning of lecture on October 18 . Read the following instructions carefully: All answers must be typed up or written legibly and stapled. Late submission of the write-up will not be accepted. You are encouraged to work in groups, but you should write up your answers alone and tell us who you worked with. Include your detailed derivation for all intermediate steps. SHOW YOUR WORK 1
In this problem set, we will be learning how to solve for both pure and mixed strategy nash equilibria. Problem 1 (50 pts) You and a friend are spending two days in the woods; you both enjoy hiking. HOWEVER, each of you believes with probability π a bear in the woods may have ingested a brick of cocaine. If a bear is high on cocaine, anyone hiking will surely be attacked. Each of you receives − c for being attacked by a bear on blow, 0 to staying at your cabin, and 1 for a day’s worth of hiking ( c > 0). If either of you is attacked by the cocaine bear on the first day, then you both know the cocaine bear exists and whoever hikes will surely be attacked on the second day, and hence no one will hike, receiving 0. If at least one person hikes and is not attacked on the first day then you know the forest is safe for sure and will hike on the 2nd day, earning 1. If neither of you hike, then on the second day you have the same belief as the first ( π ), and hike if and only if − πc + 1 − π ≥ 0 (i.e., your expected utility from hiking is higher than not hiking). (HINT: you need to consider both the case where − πc + 1 − π ≥ 0 and where − πc + 1 − π ≤ 0. It will be easiest to model as a grid twice). You will be considering ONLY what your decision on the first day will be, knowing that (as described above) it will affect your decision on the second day. Make sure when modeling to consider the utility of both. In otherwords, use the actions in period 1 to find the expected utility in BOTH periods, then put the sum of those expected utiltiies together in the 2x2 grid. 1. (10 pts) Model this situation as a normal form game in which you and your friend decide to hike or not on the first day. 2. (25 pts) Find the mixed strategy Nash equilibrium to this game. 3. (15 pts) Consider the same game, but where you are hiking alone (so it is a 2x1 game, with no other player). Compare the outcomes of the two games. Does the existence of a friend make it more or less likely that you hike on the first day? 2
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