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

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Consider an alternative definition of WUDi(X) in which the strategies for player i are not limited to the set Xi :WUDi(X) K {si ∈ Si | si is not weakly dominated in Si × X−i}.Find a normal-form game for which the iterative removal of weakly dominated strategies does not converge but rather
Examine the game pictured here.In this three-player game, player 1 selects between A, B, and C and therefore chooses which matrix will be effective. Players 2 and 3 select between U and D and between M and N, respectively. The actions of the three players are taken simultaneously. Only the payoff
Consider the following game. There are n students. Simultaneously and independently, they each write an integer between 1 and 100 on a slip of paper.The average of all of the students’ numbers is then computed, and the student coming closest to 2>3 of this average wins $20. If two or more
Prove the following assertions made at the end of this appendix: for finite games, (a) Rk is nonempty for each k; (b) Rk ⊂ Rk−1, for each positive integer k; and (c) there exists an integer K such that Rk = RK for all k > K.
Consider the following two-player team production problem. The players simultaneously choose effort levels (a1 for player 1 and a2 for player 2), and revenue is given by 2k[a1 + a2]. Suppose that player i’s cost of effort is a2 iin monetary terms. Revenue is split equally between the players, so
A manager and a worker interact as follows. The manager would like the worker to exert some effort on a project. Let e denote the worker’s effort.Each unit of effort produces a unit of revenue for the firm; that is, revenue ise. The worker bears a cost of effort given by ae2, where a is a
Consider the same setting as that in Exercise 7, except now consider that both players have private information. Player 1’s type is either high or low as well.The players’ types are independently selected by nature with the probability p.(Thus, p2 is the probability that both players are the
Suppose that two people (person 1 and person 2) are considering whether to form a partnership firm. Person 2’s productivity (type) is unknown to person 1 at the time at which these people must decide whether to create a firm, but person 1 knows that, with probability p, person 2’s productivity
Recall the bargaining game from part (b) of this chapter’s Guided Exercise, where player 1 is interested in selling a television to player 2. Player 2’s value of owning the television, v, is privately known to player 2. Player 1 only knows that v is uniformly distributed between 0 and 1.
Compute the PBE for the investment–reputation model discussed in this chapter, under the assumption that nature chooses the cooperative type with probability p (rather than 1>4 as assumed in the text). What happens if p > 1>2?
What is the relation between the PBE of the investment–reputation game in this chapter and the Bayesian Nash equilibria of its normal-form representation? You do not need to draw the normal-form representation to determine the answer, but it will help. Explain why you find this relation.
Consider the extensive-form game of incomplete information in the diagram that follows. There is a firm and a worker. In this game, nature first chooses the “type” of the firm (player 1). With probability p, the firm is of high quality (H) and, with probability 1 − p, the firm is of low
Compute the PBE of the job-market signaling model under the assumption that the worker is a high type with probability 1>2 and a low type with probability 1>2 (rather than probabilities of 1>3 and 2>3 as assumed in the text).
Consider the job-market signaling model analyzed in this chapter. Would an education be a useful signal of the worker’s type if the types had the same education cost? Explain your answer.
Consider the Microsoft–Celera game described in Exercise 5(b) of Chapter 24. Determine whether this game has any pooling or separating equilibria and, if so, describe them.
Find the perfect Bayesian equilibrium in the extensive-form game that follows. To help you analyze the game, player 2’s beliefs are labeled as p, q, and 1 − p − q. These beliefs are not part of the description of the game.(Hint: Start your analysis with player 2.) B L (1/2) bz N H (1/2) D
Suppose that a delivery person named Clifford (player 2) is to deliver a package to a house with a chihuahua (player 1) in the yard. The yard around the house is fenced, but the gate is unlocked. Clifford can either enter the yard and deliver the package (action D) or leave an “attempted
Consider the following game with nature(a) Represent this game in the normal form.(b) Calculate and report the unique Nash equilibrium of this game. (It may be in mixed strategies.)(c) Is there a perfect Bayesian equilibrium with the strategy profile you found in part (b)? If so, what is the
Compute the PBE of the three-card poker game described in Exercise 4 of Chapter 24. (Hint: Start by determining whether there are any information sets at which a player has an optimal action that is independent of his belief about his opponent’s strategy.) Show that the PBE strategy profile is
Consider the “worker status” model of Exercise 3 in Chapter 25. Suppose there are two possible types of worker, H and L. The types differ in the parameters y and q. Specifically, y = 75 and q = 3>5 for the H type, whereas y = 65 and q = 2>5 for the L type.(a) Using your answer from Exercise 3
In the classic Rob Reiner movie The Princess Bride, there is a scene at the end where Wesley (the protagonist) confronts the evil prince Humperdinck.The interaction can be modeled as the following game: Wesley is one of two types: weak or strong. Wesley knows whether he is weak or strong, but the
A defendant in a court case appears before the judge. Suppose the actual harm to the plaintiff caused by the defendant is equal to 1000x dollars, where either x = 0 or x = 1. That is, the defendant is either innocent (x = 0) or guilty of 1000 dollars of damage (x = 1). The defendant knows x and has
Consider an extensive-form game in which player 1 is one of two types: A and B. Suppose that types A and B have exactly the same preferences; the difference between these types has something to do with the payoff of another player. Is it possible for such a game to have a separating PBE, where A
Consider the following game of incomplete information.(a) Does this game have a separating perfect Bayesian equilibrium? If so, fully describe it.(b) Does this game have a pooling perfect Bayesian equilibrium? If so, fully describe it.
Consider the following game with nature:(a) Does this game have any separating perfect Bayesian equilibrium?Show your analysis and, if there is such an equilibrium, report it.(b) Does this game have any pooling perfect Bayesian equilibrium? Show your analysis and, if there is such an equilibrium,
Consider the market game in the Guided Exercise of Chapter 24.(a) Find a separating perfect Bayesian equilibrium for this game. (Report strategies and beliefs.)(b) Find a pooling perfect Bayesian equilibrium for this game. Under what values of q does it exist?
Consider a first-price auction with three bidders, whose valuations are independently drawn from a uniform distribution on the interval [0, 30]. Thus, for each player i and any fixed number y ∈ [0, 30], y>30 is the probability that player i’s valuation vi is below y.(a) Suppose that player 2 is
Consider the common-value auction described in this chapter, where Y = y1 + y2 and y1 and y2 are both uniformly distributed on [0, 10]. Player i privately observes yi . Players simultaneously submit bids and the one who bids higher wins.(a) Suppose player 2 uses the bidding strategy b2(y2) = 3 + y2
Suppose John owns a share of stock in Columbus Research, a computer software firm. Jessica is interested in investing in the company. John and Jessica each receive a signal of the stock’s value v, which is the dollar amount that the owner will receive in the future. John observes x1 , whereas
Suppose that you and two other people are competing in a third-price, sealed-bid auction. In this auction, players simultaneously and independently submit bids. The highest bidder wins the object but only has to pay the bid of the third-highest bidder. Sure, this is a silly type of auction, but it
Consider an “all-pay auction” with two players (the bidders). Player 1’s valuation v1 for the object being auctioned is uniformly distributed between 0 and 1. That is, for any x ∈ [0, 1], player 1’s valuation is below x with probability x. Player 2’s valuation is also uniformly
Consider the two-bidder auction environment discussed in this chapter, where the bidders’ values are independently drawn and distributed uniformly on the interval [0, 900]. Demonstrate that, by setting a reserve price, the auctioneer can obtain an expected revenue that exceeds what he would
Suppose that a person (the “seller”) wishes to sell a single desk. Ten people are interested in buying the desk: Ann, Bill, Colin, Dave, Ellen, Frank, Gale, Hal, Irwin, and Jim. Each of the potential buyers would derive some utility from owning the desk, and this utility is measured in dollar
Complete the analysis of the second-price auction by showing that bidding one’s valuation vi is weakly preferred to bidding any x < vi .
Suppose you and one other bidder are competing in a private-value auction.The auction format is sealed bid, first price. Let v and b denote your valuation and bid, respectively, and let vn and bn denote the valuation and bid of your opponent. Your payoff is v − b if it is the case that b Ú bn.
Regarding the trade game played by Jerry and Freddie that was analyzed in this chapter, are there values of p such that no equilibrium exists? Are there values of p such that the equilibrium entails no trade whatsoever?
Consider a version of the “social unrest” game analyzed in Chapter 8(including Exercise 8) with incomplete information. Two people (players 1 and 2) have to simultaneously choose whether to protest (P) or stay home (H). A player who stays home gets a payoff of 0. Player i’s payoff of
Consider the Cournot duopoly game with incomplete information. First, nature chooses a number x, which is equally likely to be 8 or 4. This number represents whether demand is high (x = 8) or low (x = 4). Firm 1 observes x because this firm has performed market research and knows the demand curve.
Consider the simple poker game described in Exercise 1 of Chapter 24, where there are just two cards in the deck and one card is dealt to player 1.This game has a single Nash equilibrium (perhaps in mixed strategies). Calculate and report the equilibrium strategy profile. Explain whether bluffing
Consider a simple simultaneous-bid poker game. First, nature selects numbers x1 and x2 . Assume that these numbers are independently and uniformly distributed between 0 and 1. Player 1 observes x1 and player 2 observes x2 , but neither player observes the number given to the other player.
Consider the following static game of incomplete information. Nature selects the type (c) of player 1, where c = 2 with probability 2>3 and c = 0 with probability 1>3. Player 1 observes c (he knows his own type), but player 2 does not observec. Then the players make simultaneous and independent
Find the Bayesian Nash equilibrium of the game pictured here. Note that Exercise 3 of Chapter 24 asked you to convert this into the normal form.
Consider a differentiated duopoly market in which firms compete by selecting prices and produce to fill orders. Let p1 be the price chosen by firm 1 and let p2 be the price of firm 2. Let q1 and q2 denote the quantities demanded (and produced) by the two firms. Suppose that the demand for firm 1 is
Demonstrate that, for the Cournot game discussed in this chapter, the only rationalizable strategy is the Bayesian Nash equilibrium.
Suppose that nature selects A with probability 1>2 and B with probability 1>2. If nature selects A, then players 1 and 2 interact according to matrix“A.” If nature selects B, then the players interact according to matrix “B.”These matrices are pictured here. Suppose that, before the players
Two players simultaneously and independently have to decide how much to contribute to a public good. If player 1 contributes x1 and player 2 contributes x2, then the value of the public good is 2(x1 + x2 + x1x2), which they each receive. Assume that x1 and x2 are positive numbers. Player 1 must pay
Consider the following game. Nature selects A with probability 1>2 and B with probability 1>2. If nature selects A, then players 1 and 2 interact according to matrix “A.” If nature selects B, then the players interact according to matrix “B.” These matrices are pictured here.
Consider a game in which two players take turns moving a coin along a strip of wood (with borders to keep the coin from falling off). The strip of wood is divided into five equally sized regions, called (in order) A, B, C, D, and E. Player l’s objective is to get the coin into region E (player
Consider a game inspired by the long-running television program The Price Is Right. Three players have to guess the price x of an object. It is common knowledge that x is a random variable that is distributed uniformly over the integers 1, 2, . . . , 9. That is, with probability 1>9 the value of x
Most of this exercise does not concern players’ preferences about risk, but it is a good example of a game with moves of nature that can be analyzed by using subgame perfection. Consider a T-period bargaining game like the alternating-offer game discussed in Chapter 19, except suppose that in
A firm and worker interact as follows. First, the firm offers the worker a wage w and a job z; z = 0 denotes the “safe” job and z = 1 denotes the“risky” job. After observing the firm’s contract offer (w, z), the worker accepts or rejects it. These are the only decisions made by the firm
Repeat the analysis of the principal–agent game in this chapter, under the assumption that p is the probability of a successful project with high effort(rather than 1>2 as in the basic model). How does the optimal contract depend on p?
Discuss some examples of jobs about which you know, highlighting the relation between risk, risk aversion, and the use of bonuses.
Persons 1 and 2 are forming a firm. The value of their relationship depends on the effort that they each expend. Suppose that person i’s utility from the relationship is x 2 j + xj − xixj , where xi Ú 0 is person i’s effort and xj is the effort of the other person (i = 1, 2).(a) Compute the
Consider the following strategic setting. Every fall, two neighboring elementary schools raise money for field trips and playground equipment by selling giant candy bars. Suppose that individuals in the surrounding communities love candy bars and care about helping the children from both schools
Consider the prisoners’ dilemma stage game pictured here:Row player’s payoff is listed first.The following questions ask you to consider various discrete-time environments in which people meet and play the stage game. For each environment, you are to determine the extent to which cooperation
Consider an infinitely repeated game with the following stage game:Suppose that this repeated game is the underlying game in a contractual setting with external enforcement. Specifically, an external enforcer will compel transfer a from player 2 to player 1 in every period in which (N, I) is
This exercise addresses the notion of goodwill between generations. Consider an “overlapping generations” environment, whereby an infinite-period game is played by successive generations of persons. Specifically, imagine a family comprising an infinite sequence of players, each of whom lives
Repeat the analysis of the Guided Exercise for the following stage game:In your answer to part (d), discuss the role of bargaining weights and disagreement points in the renegotiation phase.
Consider an infinite-period repeated prisoners’ dilemma game in which a long-run player 1 faces a sequence of short-run opponents. (You dealt with games like this in Exercise 8 of Chapter 22.) Formally, there is an infinite number of players—denoted 21, 22, 23, . . .—who play as player 2 in
Repeat the analysis of goodwill presented in this chapter for the following stage game:
Examine the infinitely repeated tariff-setting game, where the stage game is the two-country tariff game in Chapter 10 (see also Exercise 3 in that chapter).(a) Compute the Nash equilibrium of the stage game.(b) Find conditions on the discount factor such that zero tariffs (x1 = x2 = 0)can be
Consider the Bertrand oligopoly model, where n firms simultaneously and independently select their prices, p1 , p2 , . . . , pn, in a market. (These prices are greater than or equal to 0.) Consumers observe these prices and only purchase from the firm (or firms) with the lowest price p, according
Consider the infinitely repeated prisoners’ dilemma and recall the definition of the grim-trigger strategy. Here is the definition of another simple strategy called Tit-for-tat: Select C in the first period; in each period thereafter, choose the action that the opponent selected in the previous
This is an extension of the previous exercise. Consider the following stage game between a manager (also called the “Principal”) and a worker (the “Agent”). Let the manager be player 1 and the worker be player 2. Simultaneously, the manager chooses a bonus payment p ∈ [0, ) and the
Consider a repeated game between a supplier (player 1) and a buyer (player 2). These two parties interact over an infinite number of periods. In each period, player 1 chooses a quality level q ∈ [0, 5] at cost q. Simultaneously, player 2 decides whether to purchase the good at a fixed price of 6.
Consider the following “war of attrition” game. Interaction between players 1 and 2 takes place over discrete periods of time, starting in period 1. In each period, players choose between “stop” (S) and “continue” (C) and they receive payoffs given by the following stage-game matrix:The
Consider an infinite-period repeated game in which a “long-run player” faces a sequence of “short-run” opponents. Formally, player 1 plays the stage game with a different player 2 in each successive period. Denote by 2t the player who plays the role of player 2 in the stage game in period
Consider the following three-player game.The players make their choices simultaneously and independently. The payoffs are listed in order of the player numbers.(a) Find the (pure-strategy) Nash equilibria of this game.(b) Consider the two-period repeated game in which this stage game is played
Which is more important to achieving cooperation through a reputation, a long history together or a long horizon ahead?
Consider the infinitely repeated game where the stage game is the matrix in Exercise 2(c). Under what conditions is there a subgame perfect equilibrium in which the players alternate between (C, C) and (C, D), starting with(C, C) in the first period? Under what conditions is there a subgame perfect
If its stage game has exactly one Nash equilibrium, how many subgame perfect equilibria does a two-period, repeated game have? Explain. Would your answer change if there were T periods, where T is any finite integer?
Consider the following stage game. 2 X Y A 5.6 0.0 B 8,2 2,2
Find conditions on the discount factor under which cooperation can be supported in the infinitely repeated games with the following stage games.Use the grim-trigger strategy profile.
Consider a repeated game in which the stage game in the following figure is played in each of two periods and there is no discounting.Fully describe a subgame perfect equilibrium in which the players select (U, L) in the first period.
Consider the game shown here, which models a partnership between players 1 and 2. The players work as a team, with player 1 exerting effort x and player 2 exerting effort y on the job. Assume g ∈ [0, 1], w ∈ (−, ), x ∈ [0, ), y ∈ [0,), and t ∈ (−,).At the initial
Suppose that prior to negotiation with a firm, a worker chooses whether to invest (I) or to not invest (N). Investing entails a personal cost of 10 to the worker. Not investing entails a cost of zero. The manager of the firm observes the worker’s investment choice and then negotiates with the
Here is a description of interaction between two players who are considering a possible business partnership. First the players simultaneously choose whether to make an investment. Investment entails a personal cost of 3; not investing costs nothing. The investment choices become common knowledge.
Suppose that you work for a large corporation and that your job entails many hours of working with a computer. If you treat the computer with care, it will not break down. But if you abuse the computer (a convenience for you), then the computer will frequently need costly service. Describe
Describe a real-world setting in which option contracts are used.
Suppose that an entrepreneur is deciding whether or not to build a new highspeed railroad on the West Coast. Building the railroad will require an initial sunk cost F. If operated, the new railroad will generate revenue R. Operating the railroad will cost M in fuel and nw in wages, where n is the
This exercise asks you to combine the investment and hold-up issue from this chapter with the “demand” bargaining game explained in Exercise 4 of Chapter 19. Consider an investment and trade game whereby player 1 first must choose an investment level x Ú 0 at a cost of x2. After player l’s
Recall that “human capital” refers to skills and expertise that workers develop. General human capital is that which makes a worker highly productive in potential jobs with many different employers. Specific human capital is that which makes a worker highly productive with only a single
Estelle has an antique desk that she does not need, whereas Joel and his wife have a new house with no furniture. Estelle and Joel would like to arrange a trade, whereby Joel would get the desk at a price. In addition, the desk could use restoration work, which would enhance its value to Joel.
A bicycle manufacturer (the “buyer,” abbreviated B) wishes to procure a new robotic system for the production of mountain-bike frames. The firm contracts with a supplier (S), who will design and construct the robot. The contractual relationship is modeled by the following game:
This question asks you to compare two games, one of which has a joint decision.(a) Find the pure-strategy negotiation equilibria of the extensive-form game shown in the illustration that follows. Note that y is restricted to be in the interval [0, 4] and t ∈ (−, ). Is there an equilibrium
A manager (M) and employee (E) interact as follows. The manager can choose whether to offer a written contract or simply shake hands with the employee. If the manager offers a written contract, the parties bargain over a wage and a bonus. Then the employee decides whether to work or shirk.Under the
This question leads you through the computation of the negotiation equilibrium of the game described in Exercise 5.(a) Start by analyzing the joint decisions made at the end of the game. For given production levels q1 and q2 , calculate the players’ payoffs by using the standard bargaining
Construct a game with joint decisions by augmenting the Cournot duopoly game in the following way: Suppose the two firms have a profit-sharing pact that is enforced by the government (legal collusion). First, the firms simultaneously select quantities, q1 and q2 , at a marginal cost of 10. The
Consider a contractual setting with two players, where the underlying strategic interaction is given by the following “partnership game”:This matrix describes the technology of the relationship. Suppose that before noncooperative interaction, the players sign an externally enforced contract
A manager (M) and a worker (W) interact as follows: First, the players make a joint decision, in which they select a bonus parameter p and a salary t. The salary can be any number (positive or negative). The bonus parameter p must be between 0 and 1; it is the proportion of the firm’s revenue
Consider another version of the game between Carina and Wendy, where Carina selects any effort level e on the job. Assume e Ú 0. The revenue of the firm is equal to $800e. Carina’s disutility of effort is e2(measured in dollars). Carina and Wendy interact as before; first, they jointly determine
A manager (M) and a worker (W) interact as follows: First, the players make a joint decision, in which they select a production technology x, a bonus paymentb, and a salary t. The variable x can be any number that the players jointly choose. The bonus is an amount paid only if the worker exerts
Consider a bargaining game among three political parties—the Conservative Party, the Liberal Democratic Party, and the Labor Party (each party is a player). There are T periods of bargaining, numbered t = 1,c, T.In each period, one party is the “proposer,” one is the “responder,” and the
In experimental tests of the ultimatum bargaining game, subjects who propose the split rarely offer a tiny share of the surplus to the other party. Furthermore, sometimes subjects reject positive offers. These findings seem to contradict our standard analysis of the ultimatum game. Many scholars
Consider a three-player bargaining game, where the players are negotiating over a surplus of one unit of utility. The game begins with player 1 proposing a three-way split of the surplus. Then player 2 must decide whether to accept the proposal or to substitute for player 1’s proposal his own
Consider the following discounted, three-period bargaining game. The discount factor isd, where 0 < d < 1. In this game, player 1 makes the first offer. If player 2 rejects this offer, then player 1 makes another offer. If player 2 rejects the second offer, then player 2 makes the final offer. In
[Nash’s demand game6] Compute and describe the Nash equilibria of the following static bargaining game. Simultaneously and independently, players 1 and 2 make demands m1 and m2. These numbers are required to be between 0 and 1. If m1 + m2 … 1 (compatible demands, given that the surplus to be
Consider the alternating-offer bargaining game with T stages, in which the players discount payoffs received in successive stages according to the discount factord. That is, the two players take turns making offers. In each stage, one of the players makes an offer, followed by acceptance or
Suppose that you are attempting to buy a house, and you are bargaining with the current owner over the sale price. The house is of value $200,000 to you and $100,000 to the current owner; so, if the price is between $100,000 and$200,000, then you would both be better off with the sale. Assume that

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

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