Game Theory take-home midterm. Basically a problem set.
Econ 4415 Midterm Exam 10:00AM 03/15/202110:00AM 03/16/2021 Instructions: 1. Do not contact your TAs or the instructor regarding the problems during the exam; if you have di¢ culty understanding any of the problems, use your own interpretation, and rest assured that we will exercise a good faith judgement. 2. You have 24 hours on the exam. 3. The exam is open book. You should work independently. 4. Submit the problem set on Gradescope. 1. Here is the description of a game, call it Tac-Toe. There are four boxes, labeled a; b; c; d as follows. a b c d Player 1 must
rst place an Xin box a, then Player 2 places a Yin an open box, then Player 1 places an Xin an open box, and
nally Player 2 places a Yin the remaining box. If Player 1 has covered a diagonal with his Xs, he gets paid $1 by Player 2; similarly if Player 2 covers a diagonal with Ys, he is paid $1 by Player 1. You can take payo¤s as monetary values. (a) Is this a game of perfect information? Why or why not? [5pts] (b) Draw an extensive form for the game (with a complete tree, players at each information set, and payo¤s etc.). [5pts] (c) How many pure strategies does Player 2 have? [5pts] 2. Suppose that Anne and Bobmust simultaneously name a number in the set f1; 2; : : : ; 10g. If they name the same number, the each get a payo¤ of 1; if they name di¤erent num- bers, they each get a payo¤ of 0. (a) Find all (pure and mixed strategy) Nash equilibria of this game. [10 pts] (b) How many are there? Explain why. [5pts] (Hint: there are a lot of them!) 3. Nash Equilibrium and Subgame Perfection. Consider the following extensive- form game (on the next page) (a) Solve for all pure strategy Nash equilibria of this game. [10pts] (b) Solve for all pure strategy subgame perfect equilibria of this game. [5pts] 1 Problem 3: Nash Equilibrium and Subgame Perfection. 4. Bargaining. Two players bargain over a pie of size 1: There are two rounds. In the
rst round, both players simultaneously propose an o¤er/demand. If the proposals are consistent with each other (i.e., the sum of demands is no larger than 1), they split the pie accordingly. Otherwise, they continue to the next round.1 In the second round, the size of the pie shrinks to 0:7 (i.e. 0:3 of the original pie is taken away by a third party). In the second round, they bargaining di¤erently: player 1 proposes a division, and player 2 decides either to accept or reject it; If player 1s proposal is rejected by player 2, the game ends and both get 0; if the proposal is accepted, the pie is allocated accordingly. Consider the concept of subgame perfect equilibria. (a) In the second round, what will be the equilibrium outcome? [5pts] (b) Can there be a subgame perfect equilibrium such that an agreement is reached in the
rst round with a division (0:6; 0:4) ; i.e., 0:6 to player 1 and 0:4 to player 2? Briey explain. [10pts] (c) Can there be a subgame perfect equilibrium in which there is a disagreement in the
rst round? Briey explain. [10pts] 5. Consider two spiders in the desert trying to lay eggs within a web. There is only one web. The conict is settled when one spider concdes, leaving the other in sole possession of the web. Biologists studied the following game between two spiders. concede
ght concede 5,5 0,10
ght 10,0 x; y 1For example, if player 1 demands 0:6 and player 2 demands 0:5; there is no agreement in this round because the sum of the demand is more than 1: If player 1 demands 0:8 and player 2 demands 0:2; they split the pie accordingly because the sum of the demand is exactly 1. If player 1 dmeands 0:6 and player 1 demands 0:3; then they get what they ask for, and the remaining 0:1 is left on the table. 2 The row player is spider 1 and the column player is spider 2. If one spider chooses to
ght and the other player concedes, the
ghting spider receives a larger payo¤. If both spiders choose to
ght, their payo¤s are x and y, respectively. (a) Assume x = y > 0: Solve for all Nash equilibria of this game. [5pts] (b) Assume x = y < 0:="" solve="" for="" all="" nash="" equilibria="" of="" this="" game.="" [5pts]="" (c)="" assume="" x="">< 0="">< y="" (spider="" 2="" is="" larger="" in="" size).="" solve="" for="" all="" nash="" equilibria="" of="" this="" game.="" [5pts]="" 6.="" two="" swimmers="" ="" players="" 1="" and="" 2="" ="" are="" to="" participate="" in="" a="" runo¤.="" each="" player="" has="" the="" option="" of="" using="" a="" performance-enhancing="" steroids="" (s)="" or="" not="" using="" it="" (n).="" the="" two="" players="" are="" equally="" good.="" a="" player="" will="" win="" if="" he="" is="" the="" only="" one="" to="" use="" the="" drug.="" the="" payo¤="" matrix="" is="" as="" follows:="" s="" n="" s="" 0;="" 0="" 1;�1="" n="" �1;="" 1="" 0;="" 0="" (a)="" analyze="" this="" game="" and="" explain="" why="" an="" anti-doping="" policy="" is="" needed.="" [5pts]="" (b)="" suppose="" the="" international="" olympics="" organization="" (ioc,="" player="" 3)="" intervene.="" let="" us="" further="" assume="" that="" ioc="" can="" only="" test="" one="" swimmer.="" ioc="" has="" two="" actions:="" testing="" player="" 1="" (t1)="" and="" testing="" player="" 2="" (t2).="" the="" doping="" player="" who="" is="" caught="" by="" ioc="" will="" lose="" the="" game="" and="" pay="" a="" penalty="" of="" b=""> 0; and the IOC in this case will improve its image. The payo¤s of the three-player game are as follows. s n s �1� b; 1; 1 �1� b; 1; 1 n �1; 1; 0 0; 0; 0 t1 ; s n s 1;�1� b; 1 1;�1; 0 n 1;�1� b; 1 0; 0; 0 t2 1. Explain why it is not a Nash equilibrium for IOC to always test player 1. [5pts] 2. Solve for one Nash equilibrium of this game (you dont have to construct all Nash equilibria of this game). [5pts] 3