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Report CopyRight/DMCA Form For : Game Theory Solutions Amp Answers To Exercise Set 1
Add either training or revenue share output 22 000. Both training and revenue share output 25 000, We can then build the payoff matrix with unit of account 0 000. Vera, Revenue sharing Fixed wage, Training 11 5 12 5 9 12. Raquel, No training 11 11 10 10, 2 No there is no equilibrium in dominant strategies because Raquel has no dominant. strategy She prefers to train only if Vera gives revenue sharing while prefers not to. train with a fixed wage , 3 Yes Fixed wage is a dominated strategy for Vera Assuming that players are rational. and that this information is common knowledge Raquel knows that Vera will never. choose a fixed wage Then she will choose to train because No training is a dominated. strategy after the elimination of Vera s dominated strategy . 4 Yes Every equilibrium identified by Iterated Elimination of Dominated Strategies is a. 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, Will, Left Right, Up 9 20 90 0. John Middle 12 14 40 13, Down 14 0 17 2, Will and John 2. Will, Left Centre Right, Up 2 8 0 9 4 3, John, Down 3 7 2 10 2 15. Will and John 3, 2, Will, Left Right, Up 9 86 7 5. John Middle 6 5 10 6, Down 15 75 4 90,Solution ,1 Will and John 1. The reaction functions are the following, John Will. Down Left, John s R F , Up Right, Will John, Left Up. Will s R F Left Middle, Left Down, The Nash equilibrium is defined by mutually consistent best responses therefore down . left is the unique Nash equilibrium of the game ,2 Will and John 2. The reaction functions are the following, John Will. Down Left, John s R F Up Centre, Up Right, Will John. Centre Up, Will s R F , Right Down, The Nash equilibrium is defined by mutually consistent best responses therefore up . centre is the unique Nash equilibrium of the game . 3 Will and John 3, The reaction functions are the following. 3, John Will, Down Left, John s R F , Middle Right. Will John, Left Up, Will s R F Right Middle, Right Down. The Nash equilibrium is defined by mutually consistent best responses therefore middle . right is the unique Nash equilibrium of the game . Exercise 3 by Kim Swales , The table below represents the pay offs in a one shot simultaneous move game with com . plete information Player As pay offs are given first . Player B, Left Middle Right, Top 7 17 21 21 14 11. Player A 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 B. Advertising R D, Advertising 50 25 10 70, Company A. R D 20 40 60 35, 1 Find the mixed strategy equilibrium . 2 What are the expected pay offs for both firms , 4. 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 . Solution , 1 The pay off for firm i is total profits i which equals total revenue T Ri minus. total cost T Ci Therefore for the following sets of strategies . a High price High price Total demand is equal to 10 000 and so each firm sells. 5 000 units , T Ri 5 000 10 50 000, T Ci 5 000 2 10 000. i 50 000 10 000 40 000 i , b Low price Low price Total demand is equal to 18 000 and so each firm sells. 9 000 units , T Ri 9 000 5 45 000, T Ci 9 000 2 18 000. i 45 000 18 000 27 000 i , c High price Low price Firm sells 2 000 and while firm sells 15 000 units . T R 2 000 10 20 000, T C 2 000 2 4 000, 20 000 4 000 16 000. T R 15 000 5 75 000, T C 15 000 2 30 000, 75 000 30 000 45 000. 5, d Low price High price Firm sells 15 000 and while firm sells 2 000 units . T R 15 000 5 75 000, T C 15 000 2 30 000, 75 000 30 000 45 000. T R 2 000 10 20 000, T C 2 000 2 4 000, 20 000 4 000 16 000. The pay off matrix therefore is , Beta, High price Low price. High price 40 40 16 45, Alpha, Low price 45 16 27 27. where Alpha s pay off is first and Beta s pay off is second and both are given in . thousands , 2 Each player has a dominant strategy low price The equilibrium is therefore low price . low price with pay offs 27 27 , 3 It has the two crucial characteristics of the Prisoner s Dilemma game each player has. a dominant strategy low price But where both players play their dominated strategy. high price the outcome 40 40 is a Pareto improvement on the outcome where they. both play their dominant strategies 27 27 It also has the characteristic often found in. Prisoners Dilemma games that the equilibrium outcome is the one that gives the lowest. joint pay off , 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 . Scotland, High fishing activity Low fishing activity. High fishing activity 1 1 3 0, Spain, 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 . 6, Game Theory, Solutions Answers to Exercise Set 2. Giuseppe De Feo, May 10 2011,Exercise 1 Cournot duopoly . Market demand is given by, , 140 Q if Q 140, P Q . 0 otherwise, 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 ,Solution . 1 In a Cournot duopoly the reaction function of Firm A identifies its optimal response to. any quantity produced by Firm B In the presence of private firms the optimal quantity. is the one that maximizes A Firm A s profit where. A P Q qA cqA, 140 qA qB qA 20qA, The first order condition for profit maximization is . A, 140 2qA qB 20 0, qA, 120 qB, q A qB , 2, 1, Since the game is symmetric firms have identical cost functions the reaction function. of firm B is , 120 qA, q B qA , 2, 2 Cournot equilibrium is identified by the quantities that are mutually best responses for. both firms so they are obtained by the solution of the following two equation system . 120 qB, , qA 2, 120 qA, qB 2, The equilibrium quantities are. qA qB 40, Q 80 and the equilibrium price is, P Q 140 80 60. and firms profits are , B A 60qA 20qA 1600, 3 The competitive equilibrium outcome is characterized by P Q c 20 So total. quantity should be , P Q 140 Q 20 Q 120, In such a case firms profits are zero . The quantity produced by a monopoly is obtained by the usual first order condition . M 140 qM qM 20qM, M, 140 2qM 20 0, qM, qM 60, The price under monopoly is P qM 140 60 80 and profits are. M 80 60 20 60 3600, So profits under monopoly are higher than the sum of firms profits under cournot. competition i e M A B , q, 4 If the each firm agreed to produce half of the monopolist quantity M 2 40 their. M, profits would be 2 1800 larger than the Cournot profits So a Pareto improvement. with respect to Cournot equilibrium would be possible However this cannot be an. equilibrium since firms strategies are not mutually consistent best response that is. q q, q i M 6 M i A B, 2 2, 2, 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. cA 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 . Solution , cA qA , 1 The marginal cost of firm A is qA 20 that is constant and independent of qA . cB qB , The marginal cost for firm B is qB 2qB that is incresing in qB . 2 In a Cournot duopoly the reaction function of Firm A identifies its optimal response to. any quantity produced by Firm B The optimal quantity is the one that maximizes A . Firm A s profit where, A P Q qA c qA , 40 qA qB qA 20qA. The first order condition for profit maximization is . A, 40 2qA qB 20 0, qA, 20 qB, q A qB , 2, The reaction function of firm B identifies the quantity qB that maximizes firm s B profits. helding constant qA Firm s B profit is given by , B P Q qB c qB . 40 qA qB qB qA2, The first order condition is , B. 40 2qB qA 2qB 0, qB, 40 qA, q A qB , 4, 3, 3 Cournot equilibrium is identified by the quantities that are mutually best responses for. both firms so they are obtained by the solution of the following two equation system . 20 q, , qA 2 B, 40 q, qB 4 A, The equilibrium quantities are. 40 60, qA qB , 7 7, 100, Q 7 and the equilibrium price is. 100 180, P Q 40 , 7 7, and firms profits are , 180 1600. A q 20qA , 7 A 49, 180 7200, B q qB 2 , 7 B 49,1 Bertrand oligopoly. Exercise 3 Competition a la Bertrand , Market demand is given by. , 100 Q if Q 100, P Q , 0 otherwise, 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 . Solution , 1 The construction of the reaction function in a competition a la Bertrand proceeds in the. following manner , Since firms compete in prices we need to use the direct demand function where Q is a. function of P From the inverse demand , , 100 P if P 100. Q P , 0 otherwise, 4, There are two firms firm 1 and firm 2 Consider now the effect of the price choice of. any firm i on its own profits for any given price chosen by firm j with i j 1 2 and. i 6 j , a Pi Pj i 0, In this case firm 1 sells no output and therefore gets zero profits . 1, b Pi Pj i 2 Pj 50 100 Pj , This is where firm i shares the total profits in the industry The total profits of the. industry are here determined in the following way Price minus average cost gives. the profit per unit of output Pj 50 and this is multiplied by the total output. 100 Pj , c Pi Pj i Pi 50 100 Pi , here firm i gets the whole profits of the industry . In order to firm the reaction function of firm i we can distinguish 3 different cases . a , P i Pj 50 if Pj 50, In such a case any price Pi Pj will make negative profits so firm i will prefer to. lose the race rather than beating j on price More precisely we can say that firm i. will never set a price below c 50 This is usually portrayed as the strategy that. if Pj 50 Pi 50 1, b , P i Pj Pj if 50 Pj 75 P M, where P M is the monopolistic price If Pj 50 there is potential for positive. profits as long as Pj 100 at which point quantity demanded falls to zero so. that profits would be zero too The first issue is should firm i match the price of. firm j or attempt to undercut its rival Intuition suggests that the firm should. undercut its rival rather than match Pj This can be shown formally by comparing. the profits for firm i if it matches Pj A B, i with the profits i it gets if undercuts. Pj by a small amount so that Pi Pj , 1, A, i Pj 50 100 Pj . 2, B, i Pj 50 100 Pj , B, i 2 A, i 150 2Pj 2 , 1, From a more formal game theoretic point of view Pi 50 is only one of the possible infinite best responses. to Pj 50 However for the sake of finding the Bertrand equilibrium there is no loss in considering only the. strategy Pj 50 as best response to Pj 50 , 5, This means . As 0 that is as the P ii gets closer and closer to P ij B A. i 2 i if firm i sets, its price just below P ij its profit will be roughly double the profit from matching. Pj However the fact that firm i will do better by just undercutting firm j than it. does by matching firm j does not mean that just undercutting is always the best. strategy When the firm has the whole market its optimal price is the monopoly. price Therefore when P ij is greater than the monopoly price firm j should set. the monopoly price In this example the monopoly price P M 75 Following this. discussion , c , P i Pj 75 if Pj P M, The reaction function of any firm i is then the following . , 50, if Pj 50, P i Pj Pj if 50 Pj 75, , 75 if Pj 75. , 2 The Nash equilibrium is where the two reaction functions intersect which is where. P1 P2 50 In such a case the best response of the two firms are mutually consistent . This is actually the only point in which this is so Please check this result by your own . Can you find another Nash equilibrium This is the same as the competitive solution . Price is set equal to average marginal cost and zero profits are made . 3 No because an increase in competition does not change the price set by the firms that. cannot be lower than marginal cost This is the essence of the so called Bertrand paradox . two firms are enough to achieve the competitive outcome . 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 substi . tutes If products are differentiated instead then the Bertrand equilibrium can lead to positive. profits The products are differentiated when consumers consider them only imperfect substi . tutes 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 F irm1 q1 and the demand. for the good produced by F irm2 q2 are described by the following functions . q1 180 p1 p1 p , q2 180 p2 p2 p , 6, where 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 94 84 74 , 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 equilib . rium , Solution , The profits of both firms will depend on both prices and can be written in the following. way , , p1 p2, 1 p1 p2 180 p1 p1 p1 20 , 2, , p1 p2. 2 p1 p2 180 p2 p2 p2 20 , 2, Substituting for the different firms price choices profits are obtained and reported in the. payoff matrix of the Bertrand game with discrete choices It is easy to show that both firms. Firm 2, p2 74 p2 84 p2 94, p1 74 5724 5724 5994 5824 6264 5624. Firm 1 p1 84 5824 5994 6144 6144 6464 5994, p1 94 5624 6264 5994 6464 6364 6364. have a pi 84 as a dominant strategy and so the Bertrand Nash equilibrium is given by. the pair of strategies 84 84 and by the payoffs 6144 6144 . 7, Game Theory, Solutions Answers to Exercise Set 3. Giuseppe De Feo, May 10 2011, 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 . Giles, light heavy, light 40 40 20 55, Joe, 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 ,Solution , 1 The Nash equilibrium is heavy heavy with a payoff of 30 for both players Indeed both. Joe and Giles have a dominant strategy to use the common land heavily In addition the. Nash equilibrium is Pareto dominated by the outcome 40 40 arising when both players. choose the dominated strategy light The two features are characteristic of Prisoners . Dilemma games The game describes the so called tragedy of commons in which the. users of a common resource have an incentive to over use it . 2 When the game is repeated infinitely the trigger strategy is the following . Start playing cooperatively light , Play cooperatively as long as the other player chooses light. 1, Whenever the other player chooses heavy switch to heavy and play it forever . The couple of strategies light light can be sustained as Nash equilibrium when both. players play a Grim trigger strategy only if they care enough about future payoffs i e . if the discount factor is high enough , In fact the cooperative strategy will be preferred against a grim trigger strategy if the. net present value of cooperation is larger than the net present value of deviation . If Joe always plays light against Giles who follows a grim trigger strategy Joe s payoff. will be 40 forever The present value of cooperation is then the present value of this. infinite sequence i e , 40, coop 40 40 2 40 3 40 , 1 . By deviating from cooperation and playing heavy Joe will get 55 in the first period but. will face the reaction of his opponent From the following period Gilles will always play. heavy So after his own deviation the best option for Joe is to stick with heavy forever . The net present value of this deviating strategy is then . dev 55 30 2 30 55 30 30 2 30 , , 30, 55 , 1 , Playing cooperatively against a player adopting a grim trigger strategy is sustainable as. a Nash equilibrium from Joe s viewpoint only if coop dev . 40 30, 55 , 1 1 , Multiplying both sides by 1 the inequality becomes. 40 55 1 30, 40 55 25 , By adding 25 40 to both sides the inequality becomes. 25 15, 15, 0 6 1 , 25, Since the game is symmetric condition 1 holds for both Joe and Giles Given that. their discount factor is 0 7 the cooperative outcome light light is sustainable. as a Nash equilibrium of the infinitely repeated game when both players play the grim. trigger strategy , This threat of punishment therefore may represent a solution to the tragedy of com .
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