Consider a strategic interaction between firms 1 and 2. In Stage 1, firm 1 chooses the level of advertising a > 0. In stage 2, firm 2 learns about firm 1's amount of advertising a. Subsequently, firms 1 and 2 compete in a Cournot fashion. That is, firms 1 and 2 simultaneously and independently choose their outputs: q1 and q2,
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- There are two firms in the market (duopoly). These two firms are competingsimultaneously. The first firm chooses its output level (x) by predicting the second firm’soutput (y). Let c denote the total cost function c(x) = x and c(y) = y. Also, let’s assumethat the inverse demand function is p(Y) = 7 - Y where Y = x + y. (1) Obtain the reactionfunction of the first firm. (2) Find the equilibrium (output and profit of each firm) whentwo firms simultaneously competeProblem 3. Consider the following game with three firms. First, firms 1 and 2 si- multancously choose quantities q1 and q2 respectively. After observing firm 1 and 2's quantities, firm 3 chooses its quantity q3. There is no production cost and the inverse demand function is p= 12 – (91 +2 + 93). (a) Compute the SPNE of this game. (b) Give an example of Nash equilibrium s* with s = 4 and s, = 6 , that is not subgame perfect. game theory question8. Suppose there are two firms (Fr and F2) producing identical product competing for 20 market share and each of which would like to dominate the other, if possible. They faced a choice between defending and cooperating. When either defends or bou cooperate, neither is able to dominate the other. Assuming these preferences are reflected in their profit pay-offs. If both the players choice to defend, their profit will be 1.500 each. When one Firm defends and the other cooperates their profit level will be 5.000 an 1.000 respectively. Similarly, when both cooperate they end up with profit level of 3.000 each. With this in mind: a. Represent the above game in normal form/strategic form. b. Identify the dominant strategy for both firms and the dominant strategy equilibrium. c. Is the above equilibrium Nash equilibrium? Is it Pareto efficient allocation? Why? d. Assuming the game is one-shoot game and Firm 1 moves first represent it in extended form
- Pay-offs (in terms of profit) for the two firms are give Firm2 startegies Firm1 Strategies C B W C [4, 4] [0, 5] [-1, 5] [5, 0] [2, 2] [-1, 1] W [5, -1] [1, -1] [0, 0] a)What is a dominant strategy for each player? b)What are the possible pure strategy Nash equilibrium/equilibria to the one-shot play of this game? c) Explain the likely out the game(2) Cournot oligopoly with N firms. Suppose that, instead of the two-firm model discussed in class, we solve for the Nash equilibrium when N firms compete on quantity simul- tancously. Let Q = a – Q and marginal cost is a constant c for all firms. = q1 + ... + qn, where q; is the production of firm i. As before, (a) Find the Nash equilibrium quantities. (b) What is the market price? (c) What happens to prices and quantities as N → ?The payoff matrix below is for two firms, A and B, deciding the quantity of their output levels. What is the dominant strategy of each firm? icrosc Firm B Strategy High output Low output High output 100, 80 0, 125 Firm A Low output 65, 0 40, 65 Both firms produce low levels of output. DO cGill Both firms produce high levels of output. Temp Firm A's dominant strategy is to produce low levels of output, but Firm B does not have a dominant strategy. Order Article O Firm B's dominant strategy is to produce low levels of output, but Firm A does not have a dominant strategy. Neither firm has a dominant strategy. oy 00 halysis
- Amazon faces the other group (Other), which consists of e-book manufacturers other than Amazon, in a game in which the players choose a format (either Amazon's format, AZW, or the other group's format, EPUB), as the profit matrix shows. Other What are the pure-strategy Nash equilibria if the firms choose their formats simultaneously and are free to choose either format? Is there a mixed-strategy equilibrium? AZW EPUB Determine the pure-strategy Nash equilibria for this game. 3 -3 A. The Nash equilibria are for Amazon and the other group to select the same format. AZW O B. The Nash equilibrium is for Amazon and the other group to select the AZW format. O C. The Nash equilibrium is for Amazon and the other group to select the EPUB format. Amazon O D. This game has no Nash equilibria. -3 O E. The Nash equilibria are for Amazon and the other group to select different formats. EPUB Determine the mixed-strategy Nash equilibrium for this game. The mixed-strategy Nash equilibrium is for Amazon…1. Best responses in a Cournot Oligopoly Firm A and Firm B sell identical goods Total market demand for the good is: The inverse demand function is therefore 1 P(QM) = 780 -Q=780 -0.02222QM 45 QM is total market production (i.e., combined production of firm's A and B. That is: Q(P) = 35, 100- 45P 2M = A +QB As a result, the inverse demand curve for each firm is: P(QA, QB) = 780- -1/32₁-752 45 Unlike the example in class, the two firms have different costs. = 4000A TCA (QA) TCB (QB) = 260QB = 780 -0.022220A -0.02222QB a. Using the demand function and the cost functions above, what is firm A's profit function. b. Using the profit function above and assuming that firm B produces Qg, calculate what firm A's best response is to firm B’s decision to produce QB- Note: Firm A's best response should be a function of BSave Answer Consider two cigarette companies, PM Inc. and Brown Inc. If neither company advertises, the two companies spit the market and earn $60 million each. If they both advertise, they again split the market, but profits are lower by $20 million since each company must bear the cost of advertisirlg. Yet if one company advertises while the other does not, the one that advertises attracts customers from the other. In this case, the company that advertises earns $70 million while the company that does not advertise earns only $30 million. What will these two companies do if they behave as individual profit maximizers? Neither company will advertise, and PM Inc. earns $60. One company will advertise, the other will not. Brown Inc. earns $70. Both companies will advertise, and PM Inc. earns $40. Both companies will advertise, and PM Inc. earns $60.
- (2) Two competing firms are each planning to introduce a new product. Each will decide whether to produce Product A, Product B, or Product C. They will make their choices at the same time. The resulting payoffs are shown below. Firm 2 A В C -10, -10 10, 0 20, 10 0, 10 -20, -20 15, -5 10, 20 -5, 15 -30, -30 А Firm 1 B C a. Are there any Nash equilibria in pure strategies? If so, what are they? b. If both firms use maxmin strategies, what outcome will result? c. If Firm 1 uses a maxmin strategy and Firm 2 knows this, what will Firm 2 do?1. Two firms (A and B) play a competition game (i.e. Cournot) in which they can choose any Qi from 0 to ¥. The firms have the same cost functions C(Qi) = 10Qi + 0.5Qi2, and thus MCi = 10 + Qi. They face a market demand curve of P = 220 – (QA + QB). a. Assume firm A chooses quantity first. Frim B observes this choice and then chooses its own quantity. What is Frim B's profit as a function of QA and QB? b. Firm B has MRB = 220 – 2QB – QA. What is firm B’s best response to an arbitrary QA selected by firm A? c. Given that firm A expects firm B’s best response, what is firm A’s profit as a function of QA? (Hint: the only unknown variable in the profit function should be QA) d. Firm A has MRA = 150 – 4QA/3. What are the equilibrium QA and QB selected in this game? e. What is the equilibrium price, and how much profit does each firm collect?Consider the following game: ● ● U P1 M D P2 C R L 1, 24, 36, 5 1,5 2,7 4, 3 2, 43, 25, 2 Identify the undominated strategies of P1 and P2. Which strategies are rationalizable? Find all the Nash equilibrium of this game. Illustrate your answer by a clearly labelled best response graph, with P1's strategy on the x-axis and P2's strategy on the y-axis.