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ECON 404 - Games and Economics
Extra Problems for the Final Exam
Juan D. Carrillo
- Burning the Bridge (simple but subtle)
Army 2, of country 2, is occupying an island between countries 1 and 2. Army 1, of country 1, must decide whether to attack army 2. In the event of an attack, army 2 may fight or retreat over a bridge to its mainland. Each army prefers to occupy the island than not to occupy it. A fight is the worst outcome for both armies.
(a) Model this situation as a sequential game and find the Subgame Perfect Equilibrium. (b) Suppose that army 2 can burn the bridge to the mainland before army 1 decides whether to attack (eliminating its option to retreat). Show that it has an interest to do so.
(c) Interpret.
- Repeated Cournot (done in class)
Two firms, i ∈ { 1 , 2 }, compete in quantities. The inverse demand function is P (Q) = a − Q where Q = q 1 + q 2 is the total output produced. Each firm has 0 cost of production. Suppose that at each period, firms 1 and 2 simultaneously decide the quantities produced q 1 and q 2. Firms play this Cournot game an infinite number of times. The stage profit of firm i is πi(q 1 , q 2 ) = (a − q 1 − q 2 )qi. The NPV of profits is the discounted sum of stage profits, where δ ∈ (0, 1) is the discount factor.
Consider the following trigger strategy for each firm: “Produce half of the monopoly quantity in period 1 (call it qM /2). As long as both firms have always produced qM /2 in the past, keep producing qM /2. As soon as one firm produces a quantity different from qM / 2 then produce the Cournot quantity of the static (one-stage) game forever after.”
For which values of δ is this trigger strategy a Subgame Perfect Equilibrium? What is the NPV of payoffs for both firms in this SPE?
- Investment and Competition (mechanical)
Consider the following duopoly model of strategic investment. Firms 1 and 2 currently have a constant marginal cost of production c (> 0), that is, ci(qi) = c qi for i ∈ { 1 , 2 }. Firm 1 (and only firm 1) can install a new technology that will reduce its marginal cost to 0. Installing this technology has a fixed cost f (> 0). Firm 2 will observe whether firm 1 invests or not in the new technology. Once firm 1’s investment decision is observed, the two firms will simultaneously choose output levels q 1 (> 0) and q 2 (> 0) as in any Cournot competition model. Suppose that the inverse demand function is given by P (Q) = a−Q where Q = q 1 +q 2 is the total output produced. Suppose also that a > 2 c (use this information as a hint). Each firm chooses the quantity of output that maximizes its own profit. (a) Suppose that firm 1 decides not to invest on the new technology. What is the output and profits of firms 1 and 2? (b) Suppose that firm 1 decides to invest on the new technology. What is the output and profits of firms 1 and 2? (c) For which values of f firm 1 investing in the new technology is a Subgame Perfect Equilibrium?
- More Quantity Competition (boring)
Consider the Cournot (simultaneous) and Stackelberg (sequential) quantity models dis- cussed several times in class. Suppose there are three firms. Find the Subgame Perfect equilibrium in the following cases. (a) Case 1. Firm 1 chooses q 1. Firm 2 observes q 1 and chooses q 2. Firm 3 observes q 1 and q 2 and chooses q 3. (b) Case 2. Firm 1 chooses q 1. Firms 2 and 3 observe q 1 and choose q 2 and q 3 simultane- ously. (c) Case 3. Firms 1 and 2 choose q 1 and q 2 simultaneously. Firm 3 observes q 1 and q 2 and chooses q 3. (d) Compare the different cases.
- Sharing
One good has to be split between two individuals. Player 1 decides the fraction of the good he keeps (x ∈ [0, 1]), the fraction of the good he gives to player 2 (y ∈ [0, 1 − x]) and the fraction of the good he throws away (1 − x − y). If player 1 keeps x and gives y to player 2, then the utility Ui of player i is given by:
U 1 = x + 2xy + α U 2 and U 2 = y + 2xy + α U 1
where α captures the idea that each player cares about the benefit of the other player.
- Suppose that α = 0. What is player 1’s optimal splitting rule? Does he keep ev- erything? Does he give everything to player 2? Does he throw away part of the good? Interpret.
- Suppose that α ∈ (0, 1). What is player 1’s optimal splitting rule? Does he keep everything? Does he give everything to player 2? Does he throw away part of the good? How do x and y vary with α? Interpret.
- Suppose that α ∈ (− 1 , 0). What is player 1’s optimal splitting rule? Does he keep everything? Does he give everything to player 2? Does he throw away part of the good? How do x and y vary with α? Interpret.
- Nash vs. SPE Consider the following extensive form game. Player 1 either goes hunting with player 2 in which case they engage in the following stag-hunt game
S H S 4,4 0, H 3,0 2, or she chooses to stay home in which case both players receive a payoff of 3. This results in the following extensive form game
(a) Write down the strategic form of this game and find all of its pure strategy Nash equilibria.
(b) Find all the pure strategy subgame perfect equilibria of the game.
- Government and R&D Consider two countries denoted by i = 1, 2, each of which has one firm producing a homoge- nous product only for export, to be sold in the world market. The world’s demand for the product is p(Q) = a − Q, where Q = q 1 + q 2 , and the pre-innovation cost function of each firm is Ci(qi) = cqi, i = 1, 2 (assume 0 < 49 a 6 c < a). Let xi denote the amount of research and development (R&D) sponsored by the government in country i. We assume that when government i undertakes R&D at level xi, the cost function of the firm in country i becomes Ci(qi, xi) = (c − xi)qi, i = 1, 2. Also assume that the total cost to government i of engaging in R&D at level xi is T Ci(xi) = x
(^2) i
- The game takes place in two stages: (1) Governments choose R&D levels xi > 0 simultaneously;
(2) Observing both governments choice of R&D, firms simultaneously choose output levels qi > 0:
(a) Write down firms’ payoff functions.
(b) Assume government cares only about the profit of its company. Write down governments’ payoff functions.
(c) Given (x 1 , x 2 ), what are the Nash equilibrium levels of q 1 and q 2 and the corresponding payoff functions of the firms and the governments (as functions of x 1 and x 2 )?
(d) Using q 1 and q 2 as functions of x 1 and x 2 as found above, find the subgame perfect equilibrium levels of x 1 and x 2.
