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CORNELL ECON 2040 - ps2

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Networks: Fall 2013 Homework 2David Easley and´Eva Tardos Due at 11:15am, Monday, September 23, 2013As noted on the course home page, homework solutions must be submitted by upload tocourse’s Blackboard site. The file you upload must be in PDF format. It is fine to write thehomework in another format such as Word; from Word, you can save the file out as PDF foruploading. You can choose type ”pdf” when you save the file, or print it and choose ”Adobepdf” as your printer. (Changing the file extension from doc, or docx to pdf does not changethe format, only makes the file unreadable)Blackboard will stop accepting homework uploads after the posted due date. We cannotaccept late homework except for University-approved excuses (which include illness, a familyemergency, or travel as part of a University sports team or other University activity).Reading: The questions below are primarily based on the material in Chapters 6 and 8 ofthe book.(1) (8 points) In this question we will consider several two-player games. In each payoffmatrix below the rows correspond to player A’s strategies and the columns correspond toplayer B’s strategies. The first entry in each box is player A’s payoff and the second entryis player B’s payoff. Both players prefer higher payoffs to lower payoffs.(a) Find all pure (non-randomized) strategy Nash equilibria for the game described bythe payoff matrix in Figure 1.Player APlayer BL RU 7, 0 3, 2D 3, 5 1, 8Figure 1: The payoff matrix for Question (1a).(b) Find all pure (non-randomized) strategy Nash equilibria for the game described bythe payoff matrix in Figure 2.Player APlayer BL RU 2, 2 3, 4D 4, 3 5, 1Figure 2: The payoff matrix for Question (1b).(c) Find all Nash equilibria for the game described by the payoff matrix in Figure 3.1Player APlayer BL RU 1, 2 3, 3D 2, 2 1, 1Figure 3: The payoff matrix for Question (1c).[Hint: This game has two pure strategy equilibria and a mixed strategy equilibrium. Tofind the mixed strategy equilibrium let the probability that player A uses strategy U be pand the probability that player B uses strategy L be q. As we learned in our analysis ofpenalty kicks, if a player uses a mixed strategy (one that is not really just some pure strategyplayed with probability one) then the player must be indifferent between two pure strategies.That is the strategies must have equal expected payoffs. ](2) (8 points) In this question we will consider the Prisoner’s Dilemma and a variationon it. There are two players called 1 and 2. Each player has two actions: Confess (C) andNot-Confess (NC). In the payoff matrix below the rows correspond to player 1’s strategiesand the columns correspond to player 2’s strategies. The first entry in each box is player1’s payoff and the second entry is player 2’s payoff. You can interpret these payoffs as thenegative of the number of months each prisoner will spend in jail. They prefer spending lesstime in jail to spending more time in jail.Player 1Player 2C NCC −20, −20 0, −40NC −40, 0 −4, −4(a) Find all Nash equilibria for the version of the Prisoner’s Dilemma game described bythe payoff matrix above.(b) Now let’s suppose that each prisoner feels a bit of the pain of the other prisoner’sjail time. In particular, we suppose that each prisoner’s payoff is the negative of the sumof his own jail time and one-half of the other prisoner’s jail time. So for example, thepayoff to the pair of strategies (C,C) is now (−30, −30); each prisoner spends 20 monthsin jail which yields him a payoff of −20 and his payoff from the other prisoners jail time is(1/2)(−20) = −10, and so his total payoff is −20 − 10 = −30.(i) Write the new payoff matrix for this game.(ii) Find all of the pure strategy Nash equilibria.(iii) This game also has a mixed strategy equilibrium. Find it.2(3) (8 points) In class we talked about a simple game theoretic model of penalty kicks insoccer. In this problem we will think about a similar model of football (though no knowledgeof the game of football is needed to solve the problem). The game is played between twoteams: one playing offense, the other playing defense. The team playing offense has twostrategies to “pass” or to “run”, the team playing defense also has two strategies. They caneither “defend pass” or “defend run”.• To keep the description simple, we will assume that if the offense and defense choosethe same strategy then the defense is always successful. So if the offense plays “run”and the defense chooses “defend run”, then the “run” fails, and similarly, if the offenseplays “pass” and the defense chooses “defend pass”, then the “pass” fails. In eithercase, both teams get a payoff of 0.• Similarly, we will assume that if the offense plays “run” and the defense chooses to“defend pass” then the “run” succeeds, and if the offense plays “pass” and the defensechooses “defend run” then the “pass” succeeds.• A successful “run” is worth 5 points to the offense, while a successful “pass” is worth10 points.• The goal of the defense is to prevent the offense from gaining points, so the value tothe defense is -10 if the offense succeeds in a “pass” and -5 if the offense succeeds in a“run”.(a) Set up the payoff matrix for this game.(b) Is there a pure strategy equilibrium? Find all such equilibria, if there are any.(c) Find the mixed strategy equilibrium.(d) The team on offense has a strategy (to pass) worth more points, and a strategy (torun) worth fewer points. Which one are they playing more at the mixed Nash equilibrium?How would you explain your finding to a friend (who has not taken a class about games andNash equilibria).(4) (8 points) In this problem we will consider an attack-defense game. (This particulargame is a simple version of the “Colonel Blotto“ game which was first proposed by´EmileBorel in 1921.) There is a military battle taking place at two nearby mountain passes, whichwe’ll call A and B, and two Colonels from opposing armies are directing the battle. OneColonel — the attacker — is trying to break through at least one of these mountain passesto the territory beyond, while the other Colonel — the defender — is trying to prevent thisfrom happening.The defender has to decide which of the two mountain passes to reinforce. His possibleactions are to reinforce pass A or to reinforce pass B. He can’t defend both passes.


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