Penn State ECON 402 – Uncertain Termination ABCabc

only with numbers 1 and 2 as soon as possibleName:  ECON402 HW 5Please scan your answers to this homework assignment and upload to Canvas. You must show yourwork in order to receive credit.These homework assignments can be difficult. Feel free to work together by asking questions on thediscussion board. Don’t ask for an answer; let us know where you are stuck and we can help you getun-stuck. Don’t just “give away” the answers, but help your fellow students think through where theyare getting stuck. 1 Uncertain Termination ABCabc (25 points) Consider the following game that is played with uncertain termination. That is, after each round thegame continues with probability ? and ends with probability 1 ? ?, whereupon each player gets a payoffof 0. Players move simultaneously and independently. Then each player is informed about the actionstaken by the other player in the round and, given this, they play it again with probability ?. The payofffor the whole game is the expectation of the sum of the payoffs a player obtains. You may remember aversion of this question from Lesson 8 where the game was played only twice. Note the payoff for (A,c)has changed.ABC a3,22,21,2 b5,04,40,2 c2,21,22,3 a) (5 points) Is it possible for B and b to be played forever, in a subgame perfect equilibrium, even though(B, b) is not a Nash equilibrium in the payoff table above, if ? = 0?b) (5 points) Define a grim trigger strategy for this game.c) (5 points) Reconsider part a) if ? = 0.9.d) (10 points) What is the minimum value of ? needed to sustain cooperation? 2 Certain Termination ABCabc (25 points) Let’s take the game from above and make 2 small changes…the payoff to (B,c) has changed to 1,5 and(A,c) is 1,1. In this case, define the total payoff equal to the sum of payoff from both rounds becauseboth will surely be played. Player 1 ABC Playerab3,2 5,02,2 4,41,2 0,2 2c1,11,52,3 a) (10 points) Is it possible to have B,b as part of a SPNE in the first round of a 2-round game?1 Whyor why not?1hint: begin by specifying the following “strategy:” if (B,b) is observed in the first period, play (A,a); otherwise play(C,c) then see what happens and work from there. ECON 402 HW 5 Page 2 b) (15 points) Is it ever possible to have B,b as part of a SPNE in the first round of a multi-round game?Why or why not? 3 Credible Threats (10 points) Imagine a parent making a threat to a child saying “I brought you into this world, so I can take you outof it.” The implication is that the parent is threatening to kill the child. The probability of a parentactually killing the child after making this threat is very low. Draw a game tree which explains whyparents might make this threat and why they do not execute the threat to kill. Write a short paragraphexplaining your rationale for listing the payoffs and drawing the tree the way you did. 4 Hunt for Red October (15 points) From textbook, Ch. 9, # 4: In Tom Clancy’s novel Hunt for Red October, the Soviet Union has developeda submarine named Red October that can run “silently” and thereby escape detection. On its maidenvoyage, the ship’s captain, Marko Ramius, has decided to defect, because he believes that this technologyrisks war by destroying the balance of power between the United States and the USSR. He has puttogethera set of officers who are loyal to him and have agreed to defect as well. However, the captain is concernedthat an officer may change his mind during the voyage and, furthermore, that an officer may be moreinclined to change his mind if he thinks that other officers will do so. The captain is then consideringwriting a letter — to be delivered to the Soviet government after the submarine has departed from itsbase — stating his plan to defect. The extensive form of the game is shown below. The captain inititallydecides whether or not to send the letter. After revealing his decision to his officers (once they are allout to sea), the officers, which, for tha sake of parsimony are limited to Captain Ramius, Second RankBorodin, and Lieutenant Melekhin, simultaneously decide between continuing with the plan to defect orreneging on the plan and insisting that the submarine return to the Soviet Union. The payoffs are suchthat all three players would likke to defect and would prefer that it be done without the letter being sent(which results in the Soviet government sending out another submarine to sink Red October). ECON 402 HW 5 Page 3 a) (10 points) Derive all SPNE for this game.b) (5 points) Explain why the captain would send the letter. 5 Grim Trigger (25 points) There are two players engaged in a repeated prisoners’ dilemma with a continuation probability of ? = 0.7.Solve for x, where x is the minimum payoff necessary to sustain cooperation in this game if both playersemploy a grim trigger strategy.Player 2cdPlayer 1 C x,x 1,5 D 5,1 2,2