本科博弈论2014年第二次作业

Homework twoNote:1. You can write in Chinese or English, printed or hand written.2. No late homework will be accepted. Please staple your homework and write your name and ID number on the cover.3.The deadline is Nov.12.1) 5.4 Splitting Pizza: You and a friend are in an Italian restaurant, and the owner offers both of you a free eight-slice pizza under the following condition. Eachof you must simultaneously announce how many slices you would like; thatis, each player i ∈{1, 2} names his desired amount of pizza, 0 ≤si≤8. Ifs1+ s2 ≤8 then the players get their demands (and the owner eats any leftover slices). If s1+ s2 > 8, then the players get nothing. Assume that you each careonly about how much pizza you individually consume, and the more the better.a. Write out or graph each player’s best-response correspondence.b. What outcomes can be supported as pure-strategy Nash equilibria?2) 5.9 Tragedy of the Roommates: You and your n − 1 roommates each have five hours of free time you could spend cleaning your apartment. You all dislike cleaning, but you all like having a clean apartment: each person’s payoff is thetotal hours spent (by everyone) cleaning, minus a number c times the hoursspent (individually) cleaning. That is, vi(s1, s2, . . . , sn)=−c . si+ n j=1sj . Assume everyone chooses simultaneously how much time to spend cleaning.a. Find the Nash equilibrium if c < 1.b. Find the Nash equilibrium if c > 1.c. Set n = 5 and c = 2. Is the Nash equilibrium Pareto efficient? If not,can you find an outcome in which everyone is better off than in the Nash equilibrium outcome?3) 5.16 Hotelling’s Price Competition: Imagine a continuu m of potential buyers, located on the line segment [0, 1], with uniform distribution. (Hence the“mass” or quantity of buyers in the interval [a, b] is equal to b − a.) Imaginetwo firms, players 1 and 2, who are located at each end of the interval (player1 at the 0 point and player2 at the 1 point). Each player i can choose its pricepi , and each customer goes to the vendor who offers him the highest value. However, price alone does not determine the value; distance is important aswell. In particular each buyer who buys the product from player i has a netvalue of v − pi− di , where di is the distance between the buyer and vendor iand represents the transportation costs of buying from vendor i. Thus buyera ∈[0, 1] buys from 1 and not 2 if v − p1− d1>v − p2 − d2 and if buyingis better than getting zero. (Here d1 = a and d2 = 1− a. The buying choicewould be reversed if the inequality were reversed.) Finally, assume that thecost of production is zero.a. What is the best-response function of each player?b. Assume that v = 1. What is the Nash equilibrium? Is it unique?c. Now assume that the transportation costs are 1/2di , so that a buyerbuys from 1 if and only if v − p1− 1/2d1> v − p2 − 1/2d2. Write downthe best-response function of each player and solve for the Nashequilibrium.d. Following your analysis in (c), imagine that transportation costs areαdi , with α∈[0, 1]. What happens to the Nash equilibrium as α→0?What is the intuition for this result?4) Comparative Statics of Mixed Strategy Equilibria. C onsider the following two-player game.(a) Find all the Nash equilibria, pure and mixed. Explain how you know you have found all the equilibria.(b) Suppose that the payo_ of the column player u2(D; l) is reduced from 8 to 6, but all other payoffs remain the same. Again, find all the pure- and mixed-strategy Nash equilibria.(c) Compare the mixed-strategy equilibria in parts (a) and (b). Did this worsening in one of player 2's payoffs change player 2's equilibrium mixed strategy? Did it change player 1's? Give some intuition.5) 6.6 Declining Industry: C onsider two competing firms in a declining industrythat cannot support both firms profitably. Each firm has three possible choices,as it must decide whether or not to exit the industry immediately, at the end ofthis quarter, or at the end of the next quarter. If a firm chooses to exit then itspayoff is 0 from that point onward. Each quarter that both firms operate yieldseach a loss equal to −1, and each quarter that a firm operates alone yields ita payoff of 2. For example, if firm 1 plans to exit at the end of this quarterwhile firm 2 plans to exit at the end of the next quarter then the payoffs are(−1, 1) because both firms lose −1 in the first quarter and firm 2 gains 2 inthe second. The payoff for each firm is the sum of its quarterly payoffs.a. Write down this game in matrix form.b. Are there any strictly dominated strategies? Are there any weaklydominated strategies?c. Find the pure-strategy Nash equilibria.d. Find the unique mixed-strategy Nash equilibrium. (Hint: you can useyour answer to (b) to make things easier.)6) 6.12 The Tax Man: A citizen (player 1) must choose whether to file taxes honestly or to cheat. The tax man (player 2) decides how much effort to invest in auditing and can choose a ∈[0, 1]; the cost to the tax man of investing at alevel a is c(a) = 100a2. If the citizen is honest then he receives the benchmark payoff of 0, and the tax man pays the auditing costs without any benefit fromthe audit, yielding him a payoff of−100a2. If the citizen cheats then his payoff depends on whether he is caught. If he is caught then his payoff is −100 andthe tax man’s payoff is 100 −100a2. If he is not caught then his payoff is 50 while the tax man’s payoff is −100a2. If the citizen cheats and the tax man chooses to audit at level a then the citizen is caught with probability a and isnot caught with probability (1−a).a. If the tax man believes that the citizen is cheating for sure, what is hisbest-response level of a?b. If the tax man believes that the citizen is honest for sure, what is hisbest-response level of a?c. If the tax man believes that the citizen is honest with probability p,what is his best-response level of a as a function of p?d. Is there a pure-strategy Nash equilibrium for this game? Why orwhy not?e. Is there a mixed-strategy Nash equilibrium for this game? Why or whynot?。