PROBLEM SET 4 L C R T 5,1 0,0 0,0 M 0,0 1,5 0,0 B 0,0 0,0 2,2 A B C D A 1,1 2,2 3,4 9,3 B 2,5 3,3 1,2 7,1 A B C A 1,3 2,-2 0,6 B 3,2 1,4 5,0

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1 PROBLEM SET 4 1. What is the Opponent s Indifference Property? 2. When the Opponent s Indifference Property holds, why should a player choose his appropriate mixture, given that any other probability distribution yields the same expected payoff? 3. Find ALL (Pure and Mixed Strategy) Nash Equilibria of the following games and the expected payoffs in equilibrium if the game has a mixed strategy Nash Equilibrium: L C R T 5,1 0,0 0,0 M 0,0 1,5 0,0 B 0,0 0,0 2,2 A B C D A 1,1 2,2 3,4 9,3 B 2,5 3,3 1,2 7,1 A B C A 1,3 2,-2 0,6 B 3,2 1,4 5,0 L C R T 0,3 2,0 1,7 M 2,4 0,6 2,0 B 1,3 2,4 0,3 L C R T 2,0 1,1 4,2 M 3,4 1,2 2,3 B 1,3 0,2 3,0 A B A 3,5 3,4 B 5,1 1,2 C 2,2 10,3 A B A 0,5 5,0 B 4,1 1,4 C 3,2 2,3

2 A B C D A 2,1 3,3 2,2 2,0 B 0,3 1,3 3,1 2,2 C 0,0 3,3 1,1 2,2 D 2,1 0,3 1,2 3,0 A B C A 1,1 0,0 2,2 B 3,3 0,3 1,1 C 2,0 3,0 1,4 A B C D A 0,1 1,0 1,0 0,1 B 1,0 0,1 1,0 0,1 C 1,0 1,0 0,1 0,1 D 0,1 0,1 0,1 1,0 4. (Generalizing Pure Coordination Games) Consider the following Pure Coordination Game: Where X>0. A B A X,X 0,0 B 0,0 X,X a) What are the Pure Strategy NE of this game? b) What is the Mixed-Strategy Nash Equilibrium (MSNE)? c) What happens to the MSNE if X changes? What is the intuition behind this result? 5. (Generalizing the Stag-Hunt Game) Consider the following Stag-Hunt Game: A B A X,X 0,Y B Y,0 Y,Y Where X>0, Y>0, and X>Y. a) What are the Pure Strategy NE of this game? b) What is the Mixed-Strategy Nash Equilibrium (MSNE)? c) What happens to the MSNE if X increases? What is the intuition behind this result? d) Derive an expression for the frequency of mismatch between these two players in equilibrium. e) How do you minimize coordination in equilibrium? 6. (Generalizing the Battle-of-the-Sexes Game) Consider the following BOS Game:

3 A B A X,Y 0,0 B 0,0 Y,X Where X>0, Y>0, and X>Y. a) What are the Pure Strategy NE of this game? b) What is the Mixed-Strategy Nash Equilibrium (MSNE)? c) How does the MSNE depend on Y and X? What is the intuition behind this result? d) Derive an expression for the frequency of mismatch between these two players in a MSNE. e) What is the condition for maximum coordination? f) Derive an expression for the expected payoff of each player in equilibrium. g) What is the relationship between X and Y and the expected payoffs of the players? 7. (Generalizing the Chicken Game) Consider the following Chicken Game: A B A Z,Z 0,X B X,0 -Y,-Y Where X>0, Y>0, Z>0, X>Z, and Y>Z. a) What are the Pure Strategy NE of this game? b) What is the Mixed-Strategy Nash Equilibrium (MSNE)? c) How does the MSNE depend on Y, Z, and X? What is the intuition behind this result? d) Derive an expression for the frequency of mismatch between these two players in a MSNE. e) What happens to the probability of achieving the worst outcome (B,B) when the three parameters change? f) Derive an expression for the expected payoff of each player in equilibrium. g) What is the relationship between the parameters and the expected payoffs of the players? What is the intuition behind your result? 8. (The role of commitment) In this problem, you are going to analyze the role of commitment in all Coordination Games (Pure Coordination, Battle of the Sexes, Chicken, and Stag-Hunt). Use the payoff matrices described in problems 4, 5, 6, and 7. The rules are as follows: before the two players play the game being analyzed, they have to play the commitment game, in which each player chooses whether to commit to a particular strategy or not. The commitment is credible, in the sense that there is a technology that enforces the choice made by the players in this first stage. You are only free to choose any strategy in the second stage if you choose not to commit in the first stage. For each type of coordination game, find the Nash Equilibrium and the equilibrium payoffs with and without commitment. When is the possibility of mutual commitment beneficial and when is it harmful? 9. (Chickens go to Washington) Consider a lobbying game between two firms. Each firm may lobby the government in hopes of persuading the government to make a decision that is

4 favorable to the firm. The two firms, X and Y, independently and simultaneously decide whether to lobby (L) or not (N). Lobbying entails a cost of 15. Not lobbying costs nothing. If both firms lobby or neither firm lobbies then the government takes a neutral decision (supposedly the one that maximizes social welfare), which yields 10 to both firms. If firm Y lobbies and firm X does not, then the government makes a decision that favors firm Y, yielding 0 to firm X and 30 to firm Y. If firm X lobbies and firm Y does not, then the government makes a decision that favors firm X, yielding a payoff of x-40 to firm X and 0 to firm Y. Assume that x>25. a) What are the Pure Strategy NE of this game? b) Compute the MSNE. c) Given the mixed-strategy NE computed in part (b), what is the probability that the government makes a decision that favors firm X? d) As x rises, does the probability that the government makes a decision favoring firm X rise or fall? Is this good from an economic standpoint? 10. Consider the following game: L M R U X,X X,0 X,0 C 0,X 2,0 0,2 D 0,X 0,2 2,0 Compute ALL Nash Equilibria. 11. Suppose you know the following about a particular two-player game: S1={A,B,C}, S2={X,Y,Z}, u1(a,x)=6, u1(a,y)=0, and u1(a,z)=0, where S1 and S2 are the strategies available to players 1 and 2, and u1 is the utility that player 1 gets when a specific outcome happens. In addition, suppose you know that the game has a mixed-strategy NE in which (a) the players select each of their strategies with positive probability, (b) player 1 s expected payoff in equilibrium is 4, and (c) player 2 s expected payoff in equilibrium is 6. Calculate the probability that player 2 selects X in equilibrium. 12. (Watson, 2008) The famous British spy 001 has to choose one of four routes, a, b, c, or d (listed in order of speed in good conditions) to ski down a mountain. Fast routes are more likely to be struck by an avalanche. At the same time, the notorious rival spy 002 has to choose whether to use (y) or not to use (n) his valuable explosive device to cause an avalanche. The payoffs of this game are represented here. n Y a 12,0 0,6 b 11,1 1,5 c 10,2 4,2 d 9,3 6,0 Are there any routes you would advise 001 definitely not to take? Explain your answer.

5 A viewer of this epic drama is trying to determine what will happen. Find a NE in which one player plays a pure strategy and the other plays a mixed strategy. Find a different mixed strategy equilibrium in which this same pure strategy is assigned zero probability. Are there any other equilibria? 13. Consider a game with n players. Simultaneously and independently, the players choose between X and Y. The payoff of each player who selects X is 2m 2 x m x 3, where mx is the number of players who choose X. The payoff of each player who selects Y is 4 my, where my is the number of players who choose Y. a) For the case of n=2, find the pure strategy NE; b) Suppose that n=3. What are the pure strategy NE of this game and the MSNE of this game? 14. (The Auditing Game) The goal of the Internal Revenue Service is to either prevent or catch cheating at minimum cost, and it must decide whether to audit a certain class of suspect tax returns to discover whether they are accurate or not. The suspects want to cheat only if they will not be caught. Assume that the benefit of preventing or catching cheating is 4, the cost of auditing is C, where C<4, the cost to the suspects of obeying the law is 1, and the cost of being caught is the fine F>1. If they all move simultaneously, what is the Mixed Strategy Nash equilibrium of this game? Let s say that you found that p=p* and q=q*, where p is the probability that the IRS will audit each person in equilibrium, and q* is the probability that a person will cheat. In this game, what is the difference between auditing each person with probability p* and announcing in advance that a fraction p* of the population will be audited for sure, and the suspects are chosen randomly? 15. Rapoport and Amaldoss (1997) set up a patent race game in which a weak player is given an endowment of 4, any integral amount of which could be invested in a project with a return of 10. However a strong player is given an endowment of 5 and both players are instructed that whichever player invests the most will receive the return of 10 for the patent, and if there is a tie, neither gets the return of 10. What is the mixed strategy equilibrium of this game? What is the payoff of the game for each player? 16. Recall Problem 1 of Problem Set number 3. Now, suppose that general stores can only be set up at locations 0, 1/n,..., (n 1)/n, 1 (multiple stores can occupy the same location). a) What is the mixed strategy Nash equilibrium when n=3? b) What is the mixed strategy Nash equilibrium when n=4? c) What is the mixed strategy Nash equilibrium when n=5? (a software might be necessary) d) What is the mixed strategy Nash equilibrium when n=6? (a software might be necessary) 17. In Santa Fe there is nothing to do at night but look at the stars or go to the local bar, El Farol. Let us define the utility of looking at the stars as 0, and let the cost of walking over to the bar be 1. Suppose the utility from being at the bar is 2 if there is at least one person who does not go to the bar and 1/2 if everybody in Santa Fe goes to the bar. Suppose there are n people in Santa Fe.

6 a) Find the Mixed Strategy Nash Equilibrium of this game. b) What happens to the probability of each person going to the bar as the population increases? Explain. c) What happens to the probability that everybody in Santa Fe goes to the bar as the population increases? Explain. 18. Lucy offers to play the following game with Charlie: Let us show pennies to each other, each choosing heads or tails. If we both show heads, I pay you $X. If we both show tails, I pay you $1. If the two don t match, you pay me $2. What is the largest value of X that Lucy would be willing to pay in the case of both showing heads? 19. You pick a number from 1 to 3. I receive three dollars and I have to guess the number. You respond (truthfully!) by saying high, low, or correct. The game continues until I guess correctly. You receive from me a number of dollars equal to the number of guesses I took. a) What are the Nash strategies for this game? b) How much do you expect to win in this game? c) Find the Nash Equilibrium of a similar game, but with only one difference: you can pick any number from 1 to Each of three firms (1, 2, and 3) uses water from a lake for production purposes. Each has two pure strategies: purify sewage (strategy 1) or divert it back into the lake (strategy 2). We assume that if zero or one firm diverts its sewage into the lake, the water remains pure, but if two or more firms do, the water is impure and each firm suffers a loss of 3. The cost of purification is 1. Find all NE of this game. 21. (Advertising Game) Three firms put three items on the market and can advertise these products either on morning or evening TV. A firm advertises exactly once per day. If more than one firm advertises at the same time, their profits are zero. If exactly one firm advertises in the morning, its profit is 1, and if exactly one firm advertises in the evening, its profit is 2. Firms must make their daily advertising decisions simultaneously. Find all NE of this game. 22. A pedestrian is hit by a car and lies injured on the road. There are n people in the vicinity of the accident. The injured pedestrian requires immediate medical attention, which will be forthcoming if at least one of the n people calls for help. Simultaneously and independently, each of the n bystanders decides whether or not to call for help (by dialing 911 on a cell phone or a pay phone). Each bystander obtains b units of utility if someone (anyone) calls for help. Those who call for help pay a personal cost of c. Find all Nash Equilibria of this game. Then, compute the probability that at least one person calls for help in equilibrium. Imagine now that the 911 policy for emergencies has changed. Because of the increasing number of pranks and false emergency calls, medical attention will be forthcoming only if two people or more call for help. If only one person calls, the pedestrian will not get the necessary medical attention, but this person will pay the personal cost of c. Find all Nash Equilibria of this game.

7 Then, compute the probability that the pedestrian will get medical attention, and compare this result with the previous case. 23. (Silverman s game) Each of two players chooses a positive integer. If player i s integer is greater than player j s integer and less than three times this integer, then player j pays $1 to player i. If player j s integer is at least three times player i s integer, then player i pays $1 to player j. If the integers are equal no payment is made. Each player s preferences are represented by her expected payoff. What is the NE of this game? 24. (The Economics of Crime) A population is divided into potential victims and potential criminals. Victims carry an amount of money equal to m, and when they encounter a criminal, they have to choose either to comply to the mugger s request or to fight. Muggers have to choose either to use a fake gun or a real gun. The victims cannot distinguish between a real gun and a fake gun. It is costless for muggers to acquire fake guns, but real guns have a cost equal to Cg. If a victim complies the mugger always gets the victim s money. If a victim fights the mugger, and the mugger is using a fake gun, assume that the victim does not lose the money and that the mugger is caught by the police and goes to prison, which has a cost of Cp. If a victim fights the mugger, and the mugger is using a real gun, then you will get shot, which has a cost of Cs, and you will lose the money. The mugger escapes with the money. a) Find ALL Nash equilibria of this game; b) If the victim is carrying more money, what will change in equilibrium? Explain. c) If real guns become more expensive, what will change in equilibrium? d) If the penalty for mugging becomes harsher, what will change in equilibrium? e) If real guns become more fatal, what will change in equilibrium? f) Assume that the government wants to minimize the criminal s payoff. What could the government do? g) Now assume that the government just wants to minimize the probability that a victim will get shot. What could the government do to achieve this goal? 25. (Smoking Bans) Two bars have to select simultaneously and independently whether to allow smoking or not. Two customers (one smoker and one non-smoker) observe the choices made by the bars, and then decide simultaneously and independently whether to go to Bar 1 or Bar 2. Obviously, the non-smoker never smokes, regardless of the smoking regime. The smoker always smokes if smoking is allowed, and never smokes if smoking is not allowed. Assume that α is the smoker s benefit of smoking, and β is the non-smoker disutility of having to face the smoke if the smoker is in the same bar and is smoking. The customers also enjoy talking to each other. Specifically, they get a benefit of γ from the social interaction if they happen to select the same bar. Assume that α < β < γ. The customers choose which bar to go in order to maximize their payoffs, and the bars choose the smoking regime to maximize the expected number of customers each day. Social welfare is equal to the sum of the customers expected payoffs. a) Find the Subgame Perfect Nash equilibrium of this two-stage game; b) Is it possible that one bar will allow smoking and the other will not? Explain. c) What happens to the probability of both bars allow smoking when the disutility of smoking increases? Explain.

8 d) If there is one equilibrium in which both bars allow smoking, would a smoking ban increase or decrease welfare? Explain. 26. The host of a TV show asks general knowledge questions on air, and viewers can call the show from their homes to attempt to answer the questions. All callers pay a fee of c, but only one random caller will be selected to participate in the show and be asked to answer the question. There are n people watching the show (assume this is common knowledge), and a correct answer means that the person wins a prize of b. Assume for now that b and c are given, and the producers of the show can only choose the level of difficulty of the question, x. This level can be interpreted as follows: if the level of difficulty of a question is x, then 1-x is the fraction of viewers who know the correct answer to the question. The producers of the TV show want to maximize profit (total revenue generated by the calls minus b in case of a correct answer). a) Assume that all players are rational, in the sense that they know whether they know the correct answer to the question or not. Only those who know the correct answer will think about calling the show (but that does not mean that all of them will call). Find the subgame perfect Nash Equilibrium of this game. b) Is it possible to find the equilibrium if the producers of the show are free to choose b, c, and x simultaneously? c) What if the producers are again only free to choose x, but some viewers are not perfectly rational? Those who know the answer know that they know, and will behave rationally, whereas a fraction y of the viewers who do not know the correct answer will think that they know, even though they do not. And those who do not know but think they know always call. It is plausible to assume that y also depends on x. Feel free to assume a functional form that you think captures the true relationship between y and x, and find the SPNE. 27.(Penalty kicks in soccer) In soccer, if your opponent commits a foul on someone from your team punishable within their own penalty area (also known as the box or the 18 yard box), your team will be awarded with a penalty kick, which is a free kick taken from twelve yards (11 meters) out from your opponent s goal with only the goalkeeper and you (the penalty taker). Due to the short distance between the penalty spot and the goal, there is very little time for the goalkeeper to react to the shot. Because of this, the goalkeeper will usually start his or her dive before the ball is actually struck. In effect, the goalkeeper must act on his or her best prediction about where the shot will be aimed. Model this as a simultaneous game, in which a successful outcome is worth 1 and a bad outcome is worth 0. The payoffs are reversed if the goalkeeper defends the penalty kick. The players have three strategies available to them: Left (L), Middle (M), and Right (R). If they choose different strategies, the kicker will score a goal. If the choose the same strategy, then this means that the goalkeeper guessed correctly. But a correct guess guarantees a successful defense only if they both choose the Middle strategy. Some kickers are so good, that even if the goalkeeper chooses the correct side, it is possible that a goal will be scored. Call the quality of the kicker x, which represents the probability that a goal will be scored when the kicker chooses one of the two sides (Left or Right), and the goalkeeper guesses correctly. Additionally, it is well-known that fans and teammates do not like when the goalkeeper guesses incorrectly by staying in the Middle. This is a well-known psychological

9 propensity called Action Bias. Assume that the magnitude of this bias is c. The payoff matrix then becomes (the kicker is the Row player): Left Middle Right Left x, 1-x 1, -c 1, 0 Middle 1, 0 0, 1 1, 0 Right 1, 0 1, -c x, 1-x a) Find the MSNE. Interpret the results. b) Now assume that the kicker developed a new maneuver called the paradinha, in which the kicker pretends to kick the ball, with the intention of making the goalkeeper jump off to one side, leaving the kicker free to kick the ball into the opposite lower corner for an easy goal. If the goalkeeper jumps off to one side, the paradinha is always successful. If the goalkeeper stays in the middle and the kicker uses the paradinha, then a second stage is initiated. The kicker will have to choose one of the three strategies, and the goalkeeper will wait for the kick and will choose the correct side (the goalkeeper has an incentive to do this now because an unsuccessful paradinha causes the shooter to kick the ball less hard). New parameters have to be added. Assume that αx (where 0 α 1) represents the probability that a goal will be scored when the paradinha was unsuccessful and the shooter chose one of the two sides (a kick to the middle followed by an unsuccessful paradinha will never be a goal). Also, assume that m > 0 is the level of shame that a goalkeeper feels when the paradinha is successful (soccer fans are aware that a goal scored using the paradinha is humiliating for the goalkeeper). The new payoff matrix is: Left Middle Right Left x, 1-x 1, -c 1, 0 Middle 1, 0 0, 1 1, 0 Right 1, 0 1, -c x, 1-x Paradinha 1, -m αx, 1- αx 1, -m Find the MSNE. 28. (Online Dating Market) Let s analyze a simplified version of an online dating market. Two persons have created their profiles on a website. Person 1 is less attractive than Person 2. Person 3 and Person 4 also created profiles, and Person 3 is less attractive than Person 4. Persons 3 and 4 have to ask either Person 1 or Person 2 for a date. The choices are made simultaneously. If only one person asked you, you will date this person. If both Person 3 and Person 4 chose to date you, you will choose to date Person 4, since Person 4 is more attractive. Person 3 gets a payoff of x if he/she dates Person 1, and a payoff of αx if he/she dates Person 2, where α > 1. Person 4 gets a payoff of x if he/she dates Person 2, and a payoff of βx if he/she dates Person 1, where 0 < β < 1. These payoffs are capturing the following preferences: an unattractive person dating another unattractive person generates the same benefit as when an attractive person dates another attractive person; and an unattractive person dating an attractive person is happier than when an attractive person dates an unattractive person (notice that we are only interested in the payoffs of Persons 3 and 4).

10 a) Find the Nash Equilibrium. b) Now assume that Person 5 also has an online profile and is added to the market. He/She also has to choose between Persons 1 and 2. Person 5 has the same level of attractiveness as Person 4 (attractive). The rules are the same for Persons 1 and 2: if only one person chooses them, they will date this person; if more than one person chooses them, they will choose the more attractive one; and if both Person 4 and Person 5 choose you, you will choose one randomly. Find the Nash Equilibrium, and the payoffs in equilibrium. Can Person 3 benefit from the addition of Person 5 to the market? c) Now assume that Person 6 was added to the market, and Person 6 has the same level of attractiveness as Person 3 (so we have two unattractive people and two attractive people). What is the Nash equilibrium now? 29. Consider an all-pay auction for a good worth 1 to each of the two bidders. Each bidder can choose to offer a bid from the unit interval. Players care only about the expected value they will end up with at the end of the game. Find the Nash Equilibrum. What if the number of players is n? Can you find the Nash Equilibrium? 30. A citizen must choose whether to file taxes honestly or to cheat. The tax man decides how much effort to invest in auditing and can choose a [0,1]; the cost to the tax man of investing at a level a is c(a) = 100a 2. If the citizen is honest then he receives the benchmark payoff of 0, and the tax man pays the auditing costs without any benefit from the audit, yielding him a payoff of 100a 2. If the citizen cheats then his payoff depends on whether he is caught. If he is caught then his payoff is -100 and the tax man s payoff is a 2. If he is not caught than his payoff is 50 while the tax man s payoff is 100a 2. If the citizen cheats and the tax man chooses to audit at level a then the citizen is caught with probability a. Find the Nash Equilibrium. 31. Completely categorize the set of correlated equilibria for these games and give one example of a correlated equilibrium for each one of them: GAME 1 A B A 8,0 0,1 B 0,1 1,0 GAME 2 A B A 5,5 0,10 B 10,0 2,2 GAME 3 A B A 3,1 0,0 B 0,0 1,3 GAME 4 A B A 8,8 0,3 B 3,0 3,3

11 GAME 5 A B A 7,7 4,9 B 9,4 0,0 32. Answer the following questions about the five games that were analyzed in question 33: a) Is a Nash Equilibrium also a Correlated Equilibrium? b) Is a Correlated Equilibrium also a Nash Equilibrium? c) Why is the set of correlated strategies larger than the set of mixed strategies? d) For each game, what is the correlated equilibrium that maximizes the sum of the players expected payoffs? e) Now assume that the Coreographer is a good friend of the Row player. Find the correlated equilibrium that maximizes Row s expected payoffs. f) For each game, what is the correlated equilibrium that minimizes the sum of the players expected payoffs? 33. Consider the following game: A B C A 4,3 0,0 1,4 B 0,0 2,1 0,0 A correlated strategy is being implemented in this game. The information given is that (A,B) is chosen by the Coreographer with the same probability as (B,C). What is the maximum probability that the choreographer can choose these two outcomes and that still generates a Correlated Equilibrium? 34. Revisit the game from question 3. For each game, find the efficient correlated equilibrium, i.e., the one that maximizes the sum of expected utilities. 35. (k-rationalizability) Analyze the games from Question 3. For levels of rationality ranging from 0-3 (feel free to assume a plausible non-strategic rule of thumb for the level-0 players), find the outcomes of all those games. 36. (Cognitive Hierarchy) Do the same thing using the Cognitive Hierarchy Model (from levels of rationality ranging from 0-3). Assume that λ = 2. Then, imagine that you are the Row player in each of those games. As a rational player who understands that others may behave irrationally, specify what is the optimal strategy for you.