Strategy An Introduction To Game Theory 3rd Edition Joel Watson - Solutions

Suppose the president of the local teachers’ union bargains with the superintendent of schools over teachers’ salaries. Assume the salary is a number between 0 and 1, 1 being the teachers’ preferred amount and 0 being the superintendent’s preferred amount.
A physician and a confectioner are located in adjacent storefront offices. As part of his process for making candy, the confectioner uses a large blender that is very noisy and prevents the physician from effectively consulting with patients. (The blender is so loud that the physician cannot hear
Ashley is negotiating an employment contract with a prospective employer, the La Jolla YMCA. The contract specifies two things: (1) Ashley’s job description, which is surfing instructor or tennis instructor, and (2) Ashley’s salary t. Ashley is better at teaching people to surf than teaching
Discuss a real-world example of negotiation in terms of maximized joint value, bargaining weights, and a disagreement point.
Suppose that you must negotiate with another party and that you have an opportunity either to raise your disagreement payoff by 10 units or raise the maximal joint value v* by 10 units. Which should you choose? Is your choice efficient?
Suppose that you must bargain with another party over how to realize a large joint value v*. Explain why you care about the other party’s disagreement payoff.
Use the standard bargaining solution to find the outcomes of the following bargaining problems. Player 1’s payoff is u1 = v1(x) + t and player 2’s payoff is u2 = v2(x) − t. In each case, graph the maximized joint value and the default outcome; report the chosen x, t as well as the players’
Consider the setting of this chapter’s Guided Exercise. Suppose John can invest some of his free time either enhancing his productivity with the firm(increasing x) or raising his productivity as an individual computer consultant (increasing w). How would you recommend that John spend his free
Calculate the standard bargaining solution for the following variations of the Jerry–Rosemary example in this chapter. In each case, graph the bargaining set, find the maximized joint value, determine the players’ individual values, and compute the transfer t that the players select.(a) dJ = 0,
Consider a board game played on an m × n matrix. Player 1 has an unlimited supply of white chips, and player 2 has an unlimited supply of black chips. Starting with player 1, the players take turns claiming cells of the matrix. A player claims a cell by placing one of her chips in this cell. Once
Consider a three-player version of Chomp. Play rotates between the three players, starting with player 1. That is, player 1 moves first, followed by player 2, then player 3, then back to player 1, and so on. The player who is forced to select cell (1, 1) loses the game and gets a payoff of 0. The
Chomp is a game in which two players take turns choosing cells of an m × n matrix, with the rule that if a cell has been selected, then it and all cells below and/or to the right of it are removed from consideration (graphically, filled in) and cannot be selected in the remainder of the game. That
That is, player 1 moves first, followed by player 2, then player 3, then back to player 1, and so on. Players are allowed to move the rock just as described above. The player who moves the rock into cell (1, 1) wins and gets a payoff of 1;the other two players lose, each obtaining 0. Is there a
Consider a variant of the game in Exercise 1, in which the player who moves the rock into cell (1, 1) wins the game.(a) Does one of the players have a strategy that guarantees him a win? If so, which player has a winning strategy?(b) Now consider a three-player version of the game, whereby play
Consider Marienbad, a variation of the game described in the Guided Exercise of this chapter. As before, players take turns removing balls from the baskets, except that players are allowed to remove as many balls as they wish, provided that each player removes balls from only one of the baskets in
The game Cliff runs as follows. There are two players, each of whom has a pocketful of pennies, and there is an empty jar. The players take turns tossing pennies into the jar, with player 1 moving first. There are two rules:(a) When a player is on the move, he must put between one and four pennies
Consider the following game between two players. The players take turns moving a rock among cells of an m × n matrix. At the beginning of the game (before the first move), the rock is placed in the bottom-right cell of the matrix [cell (m, n)]. Player 1 goes first. At each turn, the player with
Consider the application on dynamic price competition with capacity constraints presented at the end of this chapter. In the game described in the text, is there a subgame perfect Nash equilibrium in which the players choose sales caps of less than 15 in the first period? If so, describe such an
Regarding the dynamic monopoly game, can you find ownership values for Hal and Laurie that make scheme A optimal? Can you find values that make scheme B optimal?
This exercise will help you think about the relation between inflation and output in the macroeconomy. Suppose that the government of Tritonland can fix the inflation level p>2 − 30. The government and the ASE interact as follows. First, the ASE selects the rate of nominal wage increase. Then the
Consider the following market game: An incumbent firm, called firm 3, is already in an industry. Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 10. First, firm 1 decides whether to enter or not. Then, after observing firm 1’s choice, firm 2
Consider the location game (in Chapter 8) with nine possible regions at which vendors may locate. Suppose that, rather than the players moving simultaneously and independently, they move sequentially. First, vendor 1 selects a location. Then, after observing the decision of vendor 1, vendor 2
Imagine a market setting with three firms. Firms 2 and 3 are already operating as monopolists in two different industries (they are not competitors).Firm 1 must decide whether to enter firm 2’s industry and thus compete with firm 2 or enter firm 3’s industry and thus compete with firm 3.
Consider a slight variation of the dynamic monopoly game analyzed in this chapter. Suppose there is only one high-type customer (Hal) and only one low-type customer (Laurie).(a) Analyze this game and explain why p2 = 200 is not optimal if Hal does not purchase a monitor in period 1. Find the
This is called the limit quantity.(d) Find the incumbent’s optimal choice of output and the outcome of the game in the following cases: (i) F = 18,723, (ii) F = 8,112,(iii) F = 1,728, and (iv) F = 108. It will be easiest to use your answers from parts (b) and (c) here; in each case, compare firm
This exercise extends the analysis of the Stackelberg duopoly game (from Chapter 15) to include fixed costs of production. The analysis produces a theory of limit quantity, which is a quantity the incumbent firm can produce that will induce the potential entrant to stay out of the market.Suppose
Consider a variation of the limit-capacity model analyzed in this chapter.Suppose that instead of the firms’ entry decisions occurring sequentially, the firms act simultaneously. After observing each other’s entry decisions, market interaction proceeds as in the original model. Find the subgame
Continuing with the advertising model, suppose the firms compete on price rather than quantity. That is, quantity demanded is given by Q = a − p, where p is the price consumers face. After firm 1’s selection of the level of advertising, the firms simultaneously and independently select prices
Consider the model of advertising and Cournot competition analyzed in this chapter. Suppose the two firms could write an externally enforced contract that specifies an advertising level a and a monetary transfer m from firm 2 to firm 1. Would the firms write a contract that specifies a = 3? If not,
Suppose players 1 and 2 will play the following prisoners’ dilemma.(a) What values of p1 and p2 are needed to make (C, C) a Nash equilibrium of the induced game?(b) What values of p1 and p2 will induce play of (C, C) and would arise in a subgame perfect equilibrium of the entire game (penalty
Consider the following game.(a) How many proper subgames does this game have?(b) Solve the game by backward induction and report the strategy profile that results.(c) Find the set of strategies that survive iterated conditional dominance.Compare these with the strategy you found for part (b).(d)
This exercise will help you see that subgame perfection does not embody the notion of forward induction presented at the end of this chapter.(a) Consider the game in Figure 15.5. Calculate and report the subgame perfect Nash equilibrium strategy profiles. Are all of these equilibrium strategy
Consider the following two-player game. First, player 1 selects a real number x, which must be greater than or equal to zero. Player 2 observes x. Then, simultaneously and independently, player 1 selects a number y1 and player 2 selects a number y2, at which point the game ends. Player 1’s payoff
Imagine a game in which players 1 and 2 simultaneously and independently select A or B. If they both select A, then the game ends and the payoff vector is (5, 5). If they both select B, then the game ends with the payoff vector(−1, −1). If one of the players chooses A while the other selects B,
Consider a game in which player 1 first selects between I and O. If player 1 selects O, then the game ends with the payoff vector (x, 1) (x for player 1), where x is some positive number. If player 1 selects I, then this selection is revealed to player 2 and then the players play the
Consider a variation of the television station broadcast game of Exercise 4 in Chapter 7. Suppose the stations interact sequentially. First, MBC chooses between 6:00 and 7:00. Then, after observing MBC’s choice, RBC decides between 6:00 and 7:00. Finally, after observing the behavior of both MBC
In the envelope game, there are two players and two envelopes. One of the envelopes is marked “player 1,” and the other is marked “player 2.” At the beginning of the game, each envelope contains one dollar. Player 1 is given the choice between stopping the game and continuing. If he chooses
Calculate and report the subgame perfect Nash equilibrium of the game described in Exercise 3 in Chapter 14.
Consider the following game.(a) Solve the game by backward induction and report the strategy profile that results.(b) How many proper subgames does this game have?
Compute the Nash equilibria and subgame perfect equilibria for the following games. Do so by writing the normal-form matrices for each game and its subgames. Which Nash equilibria are not subgame perfect?
Consider the following extensive-form games.Solve the games by using backward induction.
Consider a variant of the game described in Exercise 4. Suppose that the firms move sequentially rather than simultaneously. First, firm 1 selects its quantity q1, and this is observed by firm 2. Then, firm 2 selects its quantity q2, and the payoffs are determined as in Exercise 4, so that firm
Consider the ultimatum-offer bargaining game described in this chapter and recall the cutoff-rule strategy for player 2.(a) Suppose that player 1 selects the strategy p = 50 and player 2 selects the cutoff-rule strategy with p– = 50. Verify that these strategies form a Nash equilibrium of the
Draw the extensive form of the following game, a version of the Cournot duopoly game (analyzed in Chapter 10). Two firms compete in a homogeneous good market, where the firms produce exactly the same good. The firms simultaneously and independently select quantities to produce. The quantity
Consider the following two-player game: First, player 1 selects a number x, which must be greater than or equal to zero. Player 2 observes x. Then, simultaneously and independently, player 1 selects a number y1 and player 2 selects a number y2, at which point the game ends. Player 1’s payoff is
Given an extensive-form game, prove that each pure-strategy profile induces a unique path through the tree.
Do normal-form games exhibit perfect or imperfect information? Explain.
Suppose that Shtinki Corporation operates a chemical plant, which is located on the Hudson River. Downstream from the chemical plant is a group of fisheries. The Shtinki plant emits some byproducts that pollute the river, causing harm to the fisheries. The profit Shtinki obtains from operating the
The worker’s payoff is zero minus his disutility of effort(zero if he exerted low effort), and the manager’s payoff is her revenue minus her disutility of effort.(a) Draw the normal-form matrix that represents the underlying game.(b) If there is no external enforcement, what would be the
Suppose a manager (player 1) and a worker (player 2) have a contractual relationship with the following technology of interaction. Simultaneously and independently, the two parties each select either low (L) or high (H) effort.A party that selects high effort suffers a disutility. The worker’s
The technologies of interaction for two different contractual relationships are given by matrices A and B.
Which remedy is more likely to achieve efficiency: expectation damages or restitution damages? Explain.
Which is more critical to effective external enforcement: breach or verifiability? Explain.
Reconsider the pedestrian-injury example of Exercise 8 in Chapter 11. Suppose you are a prominent legal expert. The government has asked for your ?
Consider a setting of complete contracting in a discretionary environment, where the court will impose transfers as specified by the players. For each of the following two underlying games, how much should the players be jointly willing to pay to transform the setting from one of limited
Consider a contractual setting in which the technology of the relationship is given by the following partnership game:Suppose the players contract in a setting of court-imposed breach remedies.The players can write a formal contract specifying the strategy profile they intend to play; the court
Consider a contractual setting in which the technology of the relationship is given by the following underlying game:Suppose an external enforcer will compel transfer a from player 2 to player 1 if (N, I) is played, transfer b from player 2 to player 1 if (I, N) is played, and transfer g from
Prove that any zero-sum game is also strictly competitive with respect to mixed strategies.
Give an example of a game that is strictly competitive with respect to pure strategies but is not strictly competitive with respect to mixed strategies.
The guided exercise in this chapter demonstrates that not all security strategies are rationalizable. Find an example in which player 1’s security strategy is dominated (in the first round of the rationalizability construction).For your example, what is the relation between player 1’s security
For a two-player game, two pure-strategy Nash equilibria (s1 , s2) and (t1 , t2)are called equivalent if ui(s) = ui(t) for i = 1, 2; they are called interchangeable if (s1 , t2) and (t1 , s2) also are Nash equilibria. Mathematically prove that any two pure-strategy Nash equilibria of a two-player,
Give some examples of games people play for entertainment that are strictly competitive.
Find the players’ security strategies for the games pictured in Exercise 1.
Determine which of the following games are strictly competitive.
Consider a variant of the Bertrand game with capacity constraints that was analyzed in this chapter. Suppose that firm 1’s capacity constraint is c1 and firm 2’s capacity constraint is c2 , where c1 , c2 Ú 5. That is, firm 1 can produce at most c1 units, and firm 2 can produce at most c2
Consider a game between a police officer (player 3) and two drivers (players 1 and 2). Player 1 lives and drives in the Clairemont neighborhood of San Diego, whereas player 2 lives and drives in the Downtown area. On a given day, players 1 and 2 each have to decide whether or not to use their cell
Player 1 (the “hider”) and player 2 (the “seeker”) play the following game.There are four boxes with lids, arranged in a straight line. For convenience, the boxes are labeled A, B, C, and D. The administrator of the game gives ?
Consider the following three-player team production problem. Simultaneously and independently, each player chooses between exerting effort (E) or not exerting effort (N). Exerting effort imposes a cost of 2 on the player who exerts effort. If two or more of the players exert effort, each player
Consider a game with n players. Simultaneously and independently, the players choose between X and Y. That is, the strategy space for each player i is Si = {X, Y}. The payoff of each player who selects X is 2mx − m2 x + 3, where mx is the number of players who choose X. The payoff of each player
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 (x)
Does the rock–paper–scissors game have any pure-strategy Nash equilibria?Find and report all of the mixed-strategy Nash equilibria of this game. (Take an educated guess to find one, verify it mathematically, and then search for others.) If you forget the representation of this game, refresh
Prove that every 2 × 2 game has a Nash equilibrium (in either pure or mixed strategies). Do this by considering the following general game and breaking the analysis into two categories: (a) one of the pure-strategy profiles is a Nash equilibrium, and (b) none of the pure-strategy profiles is a
Consider the following social problem.5 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
Compute the mixed-strategy Nash equilibria of the following games. (First convert the games into the normal form.) x 2 8,8 A Y 0,0 x 2.2 B C Y 6,6 5,5 I 2 I U 4.-1 0 D 1. 1 -1,0 3,2
Determine all of the Nash equilibria (pure-strategy and mixed-strategy equilibria) of the following games. 1 2. H T 2 D D H 1,-1 -1, 1 C 2,2 0.3 (a) (b) T -1,1 1,-1 D 3,0 21 1, 1 2. 2 1 H D 1 A B H 2.2 3.1 A 1,4 2,0 (c) (d) D 3.1 2,2 B 0,8 3.9 2 1 A B 1 L M R A 2,2 0,0 U 8,1 0,2 4,3 (e) (f) B 0,0
This exercise explores how, in a mixed-strategy equilibrium, players must put positive probability only on best responses. Consider the game in the following figure.
Compute the mixed-strategy equilibria of the following games. 2 2 1 A B L M R A 2,4 0,0 U 8,3 3,5 6,3 B 1,6 3,7 C 3,3 5,5 4,8 D 5,2 3.7 4.9
Consider another version of the lobbying game introduced in this chapter.Suppose the payoffs are the same as presented earlier, except in the case in which firm X lobbies and firm Y does not lobby. In this case, suppose the government’s decision yields x to firm X and zero to firm Y. Assume that
Do you have enough information to calculate the probability that player 2 selects X in equilibrium? If so, what is this probability?
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.In addition, suppose you know that the game has a mixed-strategy Nash equilibrium in which (a) the players select each of their strategies with positive
Consider the following normal-form game.(a) Determine the set of rationalizable strategies for this game.(b) The game has only one Nash equilibrium, and it is a mixed-strategy Nash equilibrium. Compute this equilibrium. Q W z Z X 1,7 1,5 3,4 Y 2,3 0.4 0,6
An island has two reefs that are suitable for fishing, and there are twenty fishers who simultaneously and independently choose at which of the two reefs (1 or 2) to fish. Each fisher can fish at only one reef. The total number of fish harvested at a single reef depends on the number of fishers who
Suppose n students are on a committee to decide the fraction of student fees to spend on student activities. The minimum fraction they can select is 0, and the maximum fraction is 1. Any amount they do not spend will go back into the university’s general financial aid budget.Each committee member
Consider a game in which, simultaneously, player 1 selects a number x ∈ [0, 6]and player 2 selects a number y ∈ [0, 6]. The payoffs are given by:u1(x, y) = 16x y + 2 − x 2 u2(x, y) = 16y x + 2 − y 2.(a) Calculate each player’s best-response function as a function of the opposing
Consider a strategic setting in which two geographically distinct firms(players 1 and 2) compete by setting prices. Suppose that consumers are uniformly distributed across the interval [0,1], and each will buy either one unit or nothing. Firm 1 is located at 0 and firm 2 is located at 1. Assume
Suppose that the speed limit is 70 miles per hour on the freeway and that n drivers simultaneously and independently choose speeds from 70 to 100.Everyone prefers to go as fast as possible, other things equal, but the police ticket any driver whose speed is strictly faster than the speeds of a
Consider a game that has a continuum of players. In particular, the players are uniformly distributed on the interval [0, 1]. (See Appendix A for the definition of uniform distribution.) Each x ∈ [0, 1] represents an individual player; that is, we can identify a player by her location on the
Consider the strategic voting example discussed at the end of this chapter, where we saw that the strategy profile (Bustamante, Schwarzenegger, Schwarzenegger) is a Nash equilibrium of the game. Show that (Bustamante, Schwarzenegger, Schwarzenegger) is, in fact, the only rationalizable strategy
Recall the candidate location game discussed in this chapter, whose analysis led to the median voter theorem. Consider a variant of the game in which some of the voters have to be motivated to vote. In particular, suppose that the policy space [0, 1] is divided into three regions: [0, 1 For
Consider an asymmetric Cournot duopoly game, where the two firms have different costs of production. Firm 1 selects quantity q1 at a production cost of 2q1. Firm 2 selects quantity q2 and pays the production cost 4q2. The market price is given by p = 12 − q1 − q2 . Thus, the payoff functions
Imagine that a zealous prosecutor (P) has accused a defendant (D) of committing a crime. Suppose that the trial involves evidence production by both parties and that by producing evidence, a litigant increases the probability of winning the trial. Specifically, suppose that the probability that the
In the years 2000 and 2001, the bubble burst for many Internet and computer firms. As they closed shop, some of the firms had to liquidate sizable assets, such as inventories of products. Suppose eToys is going out of business and the company seeks a buyer for a truckload of Elmo dolls in its
Consider the game between a criminal and the government described in this chapter.(a) Write the first-order conditions that define the players’ best-response functions and solve them to find the best-response functions. Graph the best-response functions.(b) Compute the Nash equilibrium of this
Consider the tariff game described in this chapter.(a) Find the best-response functions for the countries.(b) Compute the Nash equilibrium.(c) Show that the countries would be better off if they made a binding agreement to set lower tariffs (than in equilibrium). You do not need to speculate how
Consider a more general Bertrand model than the one presented in this chapter. Suppose there are n firms that simultaneously and independently select their prices, p1 , p2 ,c, pn in a market. These prices are greater than or equal to zero. The lowest price offered in the market is defined as p =
Consider a more general Cournot model than the one presented in this chapter. Suppose there are n firms. The firms simultaneously and independently select quantities to bring to the market. Firm i’s quantity is denoted qi , which is constrained to be greater than or equal to zero. All of the
Suppose you know the following for a particular three-player game: The space of strategy profiles S is finite. Also, for every s ∈ S, it is the case that u2(s) = 3u1(s), u3(s) = [u1(s)]2, and u1(s) ∈ [0, 1].(a) Must this game have a Nash equilibrium? Explain your answer.(b) Must this game have
Heather and David (players 1 and 2) are partners in a handmade postcard business. They each put costly effort into the business, which then determines their profits. However, unless they each exert at least 1 unit of effort, there are no revenues at all. In particular, each player i chooses an
Consider the following n-player game. Simultaneously and independently, the players each select either X, Y, or Z. The payoffs are defined as follows.Each player who selects X obtains a payoff equal to g, where g is the number of players who select Z. Each player who selects Y obtains a payoff of
Consider a two-player game and suppose that s* and t* are Nash equilibrium strategy profiles in the game. Must it be the case that {s*1 , t*1 } × {s*2 , t*2 }is a weakly congruous strategy set? Explain why or why not.

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Strategy An Introduction To Game Theory 3rd Edition Joel Watson - Solutions

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