Game Theory Solutions & Answers to Exercise Set 1 AND Set 2 | Equilibrium Concepts | Prisoners’ Dilemma Games.

Game Theory Solutions & Answers to Exercise Set 1 and Set 2 | Equilibrium concepts | Prisoners’ Dilemma games. 1 Equilibrium concepts Exercise 1 (Training and payment system, By Kim Swales) Two players: The employee ( ...

Game Theory Solutions & Answers to Exercise Set 1 and Set 2 | Equilibrium concepts | Prisoners’ Dilemma games. 1 Equilibrium concepts Exercise 1 (Training and payment system, By Kim Swales) Two players: The employee (Raquel) and the employer (Vera). Raquel has to choose whether to pursue training that costs $1; 000 to herself or not. Vera has to decide whether to pay a fixed wage of $10; 000 to Raquel or share the revenues of the enterprise 50:50 with Raquel. The output is positively affected by both training and revenue sharing. Indeed, with no training and a fixed wage total output is $20; 000, while if either training or profit sharing is implemented the output rises to $22; 000. If both training and revenue sharing are implemented the output is $25; 000. 1. Construct the pay-off matrix 2. Is there any equilibrium in dominant strategies? 3. Can you find the solution of the game with Iterated Elimination of Dominated Strategies? 4. Is there any Nash equilibrium? Exercise 2 (Simultaneous-move games) Construct the reaction functions and find the Nash equilibrium in the following normal form games. Will and John 1 John Will Left Right Up 9; 20 90; 0 Middle 12; 14 40; 13 Down 14; 0 17; −2 Will and John 2 John Will Left Centre Right Up 2; 8 0; 9 4; 3 Down 3; 7 −2; 10 2; 15 Will and John 3 2John Will Left Right Up 9; 86 7; 5 Middle 6; 5 10; 6 Down 15; 75 4; 90 Exercise 3 (by Kim Swales) The table below represents the pay-offs in a one-shot, simultaneous move game with complete information. (Player As pay-offs are given first) Player A Player B Left Middle Right Top 7; 17 21; 21 14; 11 Middle 10; 5 14; 4 4; 3 Bottom 4; 4 7; 3 10; 25 • Find the Nash equilibria in pure strategies for the game whose pay-offs are represented in the table above. • What is the likely focal equilibrium and why? Exercise 4 (by Kim Swales) Companies A and B can compete on advertising or R+D. The table below represents the pay-offs measured in profits ($, million) in a one-shot simultaneous move game, with complete information. Company A’s profits are shown first. Company A Company B Advertising R&D Advertising 50; 25 10; 70 R&D 20; 40 60; 35 1. Find the mixed strategy equilibrium. 2. What are the expected pay-offs for both firms? 2 Prisoners’ Dilemma games Exercise 5 (A prisoner’s dilemma game, by Kim Swales) Firms Alpha and Beta serve the same market. They have constant average costs of $2 per unit. The firms can choose either a high price ($10) or a low price ($5) for their output. When both firms set a high price, total demand = 10,000 units which is split evenly between the two firms. When both set a low price, total demand is 18,000, which is again split evenly. If one firm sets a low price and the second a high price, the low priced firm sells 15,000 units, the high priced firm only 2,000 units. Analyse the pricing decisions of the two firms as a non-co-operative game. 1. In the normal from representation, construct the pay-off matrix, where the elements of each cell of the matrix are the two firms’ profits. 2. Derive the equilibrium set of strategies. 3. Explain why this is an example of the prisoners’ dilemma game. Exercise 6 (An example of the Tragedy of Commons, by Kim Swales) Show how the phenomena of overfishing can be represented as a Prisoners’ Dilemma. (hint: set up the game with two players, each of which can undertake low or high fishing activity). Solution. The case of overfishing should be set up in a manner similar to this: Spain Scotland High fishing activity Low fishing activity High fishing activity 1; 1 3; 0 Low fishing activity 0; 3 2; 2 The sustainable fishing catch is higher when both nations undertake low fishing activity. However, there is then an incentive for both to increase fishing. In fact, high fishing is a dominant strategy for both players. We therefore, end up with the worst outcome. Exercise 1 (Cournot duopoly) Market demand is given by P (Q) = ( 140 0 − Q if Q < otherwise 140 There are two firms, each with unit costs = $20. Firms can choose any quantity. 1. Define the reaction functions of the firms; 2. Find the Cournot equilibrium; 3. Compare the Cournot equilibrium to the perfectly competitive outcome and to the monopoly outcome. 4. One possible strategy for each firm is to produce half of the monopolist quantity. Would the resulting outcome be better for both firms (Pareto dominant)? Explain why this is not the equilibrium outcome of the Cournot game. Exercise 2 (Cournot duopoly with asymmetric firms) In a market characterized by the following (inverse) demand function P = 40 − Q two firms compete `a la Cournot. Firm A has production cost described by the cost function c A (qA) = 20qA, while firm B’s cost function is cB (qB) = qB2 . 1. Which firms has increasing marginal cost? Which one has constant marginal cost? 2. Define the reaction functions of the firms. 3. Compute the Cournot equilibrium quantities and price. Bertrand oligopoly Exercise 3 (Competition `a la Bertrand) Market demand is given by P (Q) = ( 100 0 − Q if Q < otherwise 100 Suppose that two firms both have average variable cost c = $50. Assuming that firms compete in prices, then: 1. Define the reaction functions of the firms; 2. Find the Bertrand equilibrium; 3. Would your answer change if there were three firms? Why? Exercise 4 (Bertrand game with differentiated products) If two firms have the same constant marginal cost, they earn zero profits in the Bertrand equilibrium. This depends crucially on the feature that the goods involved are perfect substitutes. If products are differentiated instead, then the Bertrand equilibrium can lead to positive profits. The products are differentiated when consumers consider them only imperfect substitutes Whilst a consumer may be unwilling to buy the product of one producer, she will have the incentive to do this if the price of their favourite product becomes too high. To model this we allow the demand for each good to depend not only on its own price but also on the price of the other good. Assume for example that the demand for the good produced by Firm1, q1, and the demand for the good produced by Firm2, q2, are described by the following functions: q1 = 180 − p1 − (p1 − p¯) q2 = 180 − p2 − (p2 − p¯) 6where p¯ is the average price that is taken over the prices of the two firms. Each firm has average (and marginal) cost c = 20. Suppose the firms can only choose between the three prices f94; 84; 74g. 1. Compute the profits of the firms under the 9 different price combinations that are possible in the model. 2. Using you answer to the previous point, construct the 3x3 payoff matrix for the normal form game where the payoffs are given by the profits of the firms 3. Find the (Bertrand-)Nash equilibrium of this game. What are the profits at this equilibrium? Exercise 1 (Sustainable cooperation in the long run) Two farmers, Joe and Giles, graze their animals on a common land. They can choose to use the common resource lightly or heavily and the resulting strategic interaction may be described as a simultaneous-move game. The payoff matrix is the following: Joe Giles light heavy light 40, 40 20, 55 heavy 55, 20 30, 30 1. Find the Nash equilibrium of the game and show that it is an example of “Prisoners’ Dilemma” games. 2. Suppose that the same game is repeated infinitely. Is the {light, light} outcome a Nash equilibrium if both players play a Grim strategy and have a discount factor of 0.7? Exercise 2 Distinguish simultaneous-move games and dynamic games in terms of information. Explain why in dynamic games Nash equilibria may not be subgame perfect. Using examples, show how non-credible threats are ruled out using backward induction. 1 Finitely repeated games Exercise 3 (Entry Deterrence) A market is characterized by a demand function Q = 1 − P and by a single firm with constant marginal cost c. The monopolist is facing potential entry from a new firm having the same marginal cost but an additional fixed cost of entry F = 0.1. If the incumbent accepts the entry passively, then Cournot competition is played. However, the monopolist can threaten to produce the competitive output (i.e., the quantity such that P = c) so that the new entrant will make losses if it enters the market. If the new firm does not enter, the incumbent behaves as a monopolist. 1. Assume that c = 0 and compute the payoffs of both firms (i.e., profits) in the cases of Monopoly, Cournot duopoly, and aggressive behaviour. 2. Using the extensive form (game tree) representation, describe this entry game as a twostage game where in the first stage the new entrant decides whether to enter or not, and in the second stage the incumbent firm decides whether to be passive or aggressive in case of entry, while it does nothing in case the new firm does not enter the market. 3. Is the threat of aggressive behaviour by the monopolist credible? Answer the question by finding the subgame perfect equilibrium of the game. 4. Describe the dynamic game using the normal form (pay-off matrix) representation and find the Nash equilibria of the game. Does the game have any Nash equilibria that are not subgame perfect? 3Assume that incumbent is monopolist in 10 different national markets and it faces the threat of entry in all the markets sequentially (i.e., in stage 1 of the game the monopolist plays the game with a potential entrant in the Belgian market; in stage two the same game is played with another potential entrant for the Dutch market, and so on up to 10 rounds). 5. Find the SPNE of the entire dynamic game by backward induction. 6. Is there room for building a reputation? Explain your answer. Exercise 4 (Centipede Game) Trinny and Susannah are playing the ”Centipede” game. At each node in the game the player can either move down, which means the game stops, or move right, which means that the game is passed to the other player. The two cycle game is depicted in Figure 1. Trinny’s pay-off is given first. R R R D D D (2,2) (1,0) (0,2) (3,1) T1 S1 T2 Figure 1: 1. Solve the game by backward induction 2. Show that the outcome is inefficient. 3. If the game was extended to 100 cycles, with the pay-offs increasing in a similar manner, as represented in Figure 2, how would this affect the outcome of the game? Exercise 1 Define the concepts of expected value, expected utility and explain the relationship between this two concepts. Exercise 2 Mark wants to maximise his expected utility. His preferences are represented by the utility function U(y) = y12 where y is a monetary payoff. Mark is offered the following bet on the toss of a coin by Amanda; • If the coin comes up tails Amanda pays Mark $1; 000 • If the coin comes up heads Mark pays Amanda $1; 000 Mark’s initial capital is $10; 000 which he retains in its entirety if he does not take the bet. 1. What is Mark’s expected utility if he accepts the bet? 2. Will he accept the bet? Explain your answer. 3. Is Mark risk averse, risk neutral or risk loving? Explain your answer. 1Amanda offers Mark an alternative bet whereby if the coin comes up tails Amanda gives him $10; 000 but if the coin comes up heads Mark gives Amanda his entire $10; 000. 4. Show that Mark does not accept this bet. Amanda offers Mark yet another alternative bet whereby Mark still loses his entire $10; 000 if the coin comes up heads, but if the coin comes up tails Amanda pays him$50; 000. 5. Does Mark accept this new alternative? Explain your answer. 6. Given that Mark loses his entire$10; 000 if the coin comes up heads, what is the smallest amount that Amanda has to pay Mark in the event of tails in order to persuade him to take the bet? Exercise 3 The local government wants to hire a manager to undertake a public project. If the project fails, it will lose $20; 000. If it succeeds, the project will earn $100; 000. The manager can choose to \work" or to \shirk". If she shirks, the project will fail for sure. If she works, the project will succeed half of the times but will still fail half of the times. Measured in monetary terms, the manager’s utility is $10; 000 lower if she works than if she shirks. In addition, the manager could earn $10; 000 in another job (where she would shirk). Assume further that both parties are risk neutral. 1. Describe the dynamic game using the extensive form representation. 2. Which is the compensation scheme that maximizes government’s utility? Which is the expected wage of the manager in any equilibrium of the game? 3. Do you think the solution will change with a risk-averse manager? 4. Assume that the government prefers to sell the project to the manager. What should be the fee paid to the government? Exercise 4 Discuss the trade-off between insurance and incentives in the presence of moral hazard when the agent is risk-averse. Do you think it is true that a risk-neutral principal should pay more to hire a risk-averse agent than a risk neutral agent? Explain your answer. Exercise 1 (MIT Sloan School of Management) Consider the market for health insurance. Suppose that the market is comprised of 4 groups of people of differing risk categories. There are a large and equal number of people in each group, but insurers cannot tell which group a person belongs to (i.e. this is a situation of asymmetric information). Each group faces a risk of requiring medical treatment of value $10,000. Suppose that the willingness to pay of people in each group is as follows: Risk 0.2 0.4 0.6 0.8 willingness to pay 2,500 5,200 6,800 8,500 Actuarially fair premium Risk Premium a. Complete the table of actuarially fair insurance premiums that could be charged to each group separately by an insurance company large enough to diversify the risks. How do these compare to the willingness to pay? b. Suppose now that the risk category is private information. What is the average riskiness of a person seeking insurance? What premium would an insurance company have to charge to break even? c. Will all the agents participate at this price? If not what would be the composition of risks facing the insurer? Would the premium found above be sufficient to cover the risks taken by the insurer? d. Continue with this logic. What will be the price of insurance in the equilibrium and which groups will participate? e. Is this an efficient outcome? Exercise 2 (Hindriks and Myles, 2006, 9.3) Are the following statements true or false? a. An insurance company must be concerned about the possibility that someone will buy fire insurance on a building and then set fire to it. This is an example of moral hazard. b. A life insurance company must be concerned about the possibility that the people who buy life insurance may tend to be less healthy than those who do not. This is an example of adverse selection. c. In a market where there is separating equilibrium, different types of agents make different choices of actions. d. Moral hazard refers to the effect of an insurance policy on the incentives of individuals to exercise care. e. Adverse selection refers to how the magnitude of the insurance premium affects the types of individuals that buy insurance. Exercise 3 (Watson, 2008, p.319) 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 observe c. Then players make symultaneous and independent choices and receive payoffs as described by the following matrix. Player 1 Player 2 X Y A 0; 1 1; 0 B 1; 0 c; 1 a. Draw the Bayesian normal form matrix of the game b. Compute the Bayesian Nash equilibrium Exercise 4 (Watson, 2008, p.346) 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 and B behave differently?