In a Cournot duopoly there are two firms, 1 and 2. The market price is given by p=2-9₁-92. The cost to firm i of producing any quantity is zero. Each firm's payoff is defined as its profit. Describe the normal form of this game by expressing the strategy spaces and writing the payoffs as functions of the strategies.
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- Imagine two Pizzerias are playing a static game where the firms have to choose their strategies simultaneously, selecting a strategy that maximizes its profit given what it believes the other firm will do. Their decision is how many pizzas to produce. They can choose between producing 100, 150 or 200 pizzas each. The profit for respective firm is showed in the payoff matrix below. a) What is the solution (if there is one) and also describe the characteristic features of this game. b) If this game was played repeatedly, could that potentially change the outcome of the game? In your answer, you are expected to motivate and describe how.Two identical firms each have a cost function TC = 2y2 and the market demand for their output is P = -4Q+192a) Write the âbest responseâ function for each firm.b) Find the Nash equilibrium in this model c) Show that if each firm produces 1 fewer units than the result in (b), both firms make more profit. Use this information to construct a normal form game. Explain why this game is a prisonerâs dilemma.Consider a Stackelberg duopoly:There are two firms in an industry with demand Q = 1 − Pd.The “leader” chooses a quantity qL to produce. The “follower” observes qL and chooses a quantity qF.Suppose now that the cost function is Ci(qi) = qi2 for i = L, F. (a) Find the subgame perfect equilibrium. (b) Compare the equilibrium you found with the Nash equilibrium if the game was simultaneous (i.e., Cournot competition). Is the Nash equilibrium of the Cournot game also a Nash equilibrium of the sequential game? Why or why not?
- Two firms are competing on price. If they have the same price, they share the market otherwise the one with the lowest price captures all demand Market demand follows Q(P)=100-3P Cost is C(Q)=10Q Firms can only choose between the following prices: 9, 10, 11, 12. In the Nash equilibrium of this game, what prices are charged? Suggestion: calculate the profits they obtain for each of the price combinations, write down the game in its normal form (payoff matrix), and then use the underlining method to match best responses. U ப U 9 12 10 110 11The following market is a duopoly populated only by the companies Alpha and Beta. The pay-off matrix immediately below shows the combinations of pricing strategies available to the two companies. The numbers represent millions of dollars in profit. (The negative sign indicates a loss.) Assuming Beta has a first mover advantage, in a one-shot game, what is liekly to be the Nash equilibrium? Explain your answer. Alpha High price Low price High price 250, 200 200, 100 Beta Low price 50, 150 100, 250a. b. Each firm has four alternative strategies, and a certain profit/payoff is associated with each strategy. The numbers in the payoff matrix denote firm A's profit (in thousands of dollars). The total amount of profit that can be earned by the two firms together is $20000. (This is called a "constant sum game.") Firm B's profit is therefore $20000 minus firm A's profit. What strategies will the two firms select? Is the game strictly determined? If so, how much does each firm gain? B's strategies A's strategies ↓ Increase Advertising Decrease Price Increase Price Alter Product Increase Advertising 0 11 8 11 Decrease Price 8 10 6 2 Increase Price 7 12 15 Alter Product 4 15 3 12 Suppose now that due to a change in consumer preferences, firm A's "Increase Price" strategy pays off better than before when firm B elects to "Decrease Price," that is, the payoff rises from 6 to 14. What strategies will the two firms now select? Is the game strictly determined? If so, how much does each firm…
- Two firms are competing on price. If they have the same price, they share the market - otherwise the one with the lowest price captures all demand Market demand follows Q(P)=100-3P Cost is C(Q)=10Q Firms can only choose between the following prices: 9, 10, 11, 12. In the Nash equilibrium of this game, what prices are charged? Suggestion: calculate the profits they obtain for each of the price combinations, write down the game in its normal form (payoff matrix), and then use the underlining method to match best responses. 12 11 9 10Two firms are competing in an infinitely repeated Bertrand duopoly. The marginal cost of both is equal to 6. There are no fixed costs. The demand function for the good is q = 10 – p, where q is the total quantity demanded and p the price. (a) Find the equilibrium in the infinitely repeated game in which the two firms share the market equally. (b) Suppose that firm l's marginal cost falls to 0. The other firm's marginal cost is still 6. Is there an equilibrium in the infinitely repeated game in which the two firms share the market?Two firms produce Bliffs. They compete by simultaneously choosing prices in a single period. The demand for Bliffs is given by P(Q) = 100-2Q where Q is market quantity and P is market price. Firm 1 has costs C1(q1) = 20q1 and Firm 2 has costs C2(q2) = 10q2. Which statement is true? In the Nash equilibrium to the game, both firms play dominated strategies None of the other answers are correct O In the Nash equilibrium to the game, both firms play dominant strategies In the Nash equilibrium to the game, both firms slowly lower prices towards marginal costs O In the Nash equilibrium to the game, both firms set price equal to marginal cost
- Let G be the following static game. Rose Colin A A (3,4) B (4,2) B (1,5) (0, 1) (a) Determine the pure Nash equilibria of G. (b) Rose A, B, Colin A, B is a mixed Nash equilibrium of H. (You do not have to verify this.) Determine the expected payoffs of both players given that they choose their strategies according to this Nash equilibrium. (c) Draw the payoff polygon of H. Indicate on your diagram the points corresponding to the Nash equilibria of H, and the points corresponding to Pareto optimal solu- tions of H.Two firms face decisions with two alternatives that can be modeled using Game Theory. For each of the situations described below, CIRCLE the ("Nash") equilibrium (or equilibria) for the "game" given the profit payouts (firm 1, firm 2) for the choices for the firms (or mark "None"). a. Sell One Medium-Size Item or Sell One Large Item and One Small Item Firm 1 Choices Firm 2 Choices Medium-Size Item Large and Small Items Medium-Size Item 31,27 25, 33 Large/Small Items 42,24 30,28 b. Use "9" Price Endings or Use "5" Price Endings Firm 2 Choices Use "9" Price Endings | Use "5" Price Endings Firm 1 Choices Use "9" Endings Use "5" Endings 45,28 42,30 c. Hire a Celebrity Endorser or Donate to Charity 27,25 25,28 Firm 2 Choices Firm 1 Choices Hire Celebrity Endorser Donate to Charity Celebrity Endorser 40, 29 30, 33 Donate to Charity 37,36 48, 32 d. Locate Downtown or Locate in Suburban Strip Mall Firm 2 Choices Firm 1 Choices Locate Downtown Locate in Strip Mall Locate Downtown 32,31 28, 34…Game Theory. Consider a Stackelberg competition game with three firms. Firm 1 chooses q1 first. Firm 2 observes q1 and chooses q2. Firm 3 observes both and chooses q3. These three firms are the only firms in the market, so the sum of their outputs is equal to total market supply, i.e. q1+q2+q3=Q. Suppose demand is given by P=12-Q. For simplicity of calculation, suppose each firm has marginal costs of 0, i.e. c1(q1)=0, c2(q2)=0 and c3(q3)=0. (1) What quantity does Firm 1 produce in the SPNE of the game? (2) What quantity does Firm 2 produce in the SPNE of the game? (3) What quantity does Firm 3 produce in the SPNE of the game?