博弈论习题集
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PROBLEM SET I OF GAME THEOR Y1. State whether the following gameshave unique pure strategy solutions, and if so whatthey are and how they can be found.2. Draw the normal form gamefor the following gameand identify both thepure-a nd mixed-strategy equilibria. In the mixed-strategy Nash equilibrium determine each firm ' s expected profit level if it enters the market.There are two firms that are con sideri ng en teri ng a new market, and must make their decision without knowing what the other firm has done. Unfortun ately the market is on ly big eno ugh to support one of the two firms. If both firms enter the market, then they will each makea loss of £onlyPlayer 1⑵Player 1⑶Player 1Player 2Player 2one firm enter s the market, th at firm will earn a profit of £ 50m, and the other firm will just break even.3. Con vert the follow ing exte nsive form game into a no rmal form game, and identifythe Nash equilibria and subgame perfect Nash equilibria.Finally, what is the Nash equilibrium if both players maketheir movessimulta neously4. Consider an economy consisting of one government and two people. Let X i be thechoice of the people, where X i € X = {x L, x 叫x H}, and i=1,2, and y the choice of the government, where y € Y={ y L, y M y H}. Thepayoffs to the government-household are given by the values of u i(x 1, X2, y) and u 2(x 1, x 2, y) = u 1 (x 2,x 1, y) . These payoffs are entered in the following table:12government' s policy. Enter the blank with value ranges such that the Nash equilibria are supported.(2) Suppose the government moves first, find Nash Equilibria, the subgame perfect Nashequilibria, and the subgame perfect outcome. Is the outcome efficie nt Why(3) Show whether there exists Nash equilibrium (in pure strategies) forthe one-period economy when households and the government move simultaneously.(4) lf the household choose first, do question (2) again.5. Assume that two players are faced with Rosenthal ' s centipede game.Use Bayes' theorem to calculate the players ' reputation for being co-operative in the follow ing situati ons if they play across.(1) At the beginning of the gameeach player believes that there is a 50/50 chanee thatthe other player is rational or co-operative. It is assumed that a co-operative player always plays across. Furthermore supposethat a rati onal player will play across with a probability of(2) At their second movethe players again moveacross. (Continue to assumethat the probability that a rati onal player plays across rema ins equal .(3) How would the players ' reputation have changed after the first movehad the other player believed that rational players always play across. (Assume all other probabilities rema in the same.)(4) Fin ally, how would the players ' reputati on have cha nged after the first move hadthe other player believed that rati onal players n everplay across. (Aga in assume all other probabilities rema in the same.)6. Assume there are midentical Stackelberg leaders in an industry,indexed j =1,…, m and n identical Stackelberg followers, indexed k=1,…,n. All firms have a constant marginal cost of c and no fixed costs. The market price, Q, is determ ined accord ing to the equatio nP - a - C, where Q is total industry output, and a is a constant. Findthe subgame perfect Nash equilibrium supply for the leaders and the followers.Confirm the duopoly results for both Cour not competiti on and Stackelbergcompetitio n, and the gen eralized Cour not result for n firms derived in Exercise . 7. Assume that there are i =1,…,n identical firms in an industry, each with con sta ntmargi nal costs of c and no fixed costs. If the marketprice, P, is determined by the equation , where Qis totalindustry output and a is a constant, determine the Cournot-Nash equilibrium outputlevel for each firm. Where happe ns as n—、8. Find the separating equilibrium behaviour of the low-cost incumbent in the follow ingtwo-period model. The in cumbe nt has marginal costsequal to either £ 4or £ 2.0nly the incumbent initially knows its exact costs. Theentrant observes the incumbent ' s output decision in thefirst period and only enters the market in the second period if it believes that theincumbent has high marginal costs. If entry does occur, the two firms Cour notcompete, and we assumethat at this stage in the game the incumbent ' s true costs are revealed. Price, P, isdetermined by the following equation 卜「./.：：■-』，where Qis the combinedoutput of the two firms. Finally, it is assumed that the firms ' discount factor is equal to .9. In the text we argued that a weak government can exploit the privatesector ' s uncertainty about the government ' s preferences topartially avoid the inflationary bias associated withtime-i neon siste nt mon etary policy .In this exercise we provided a simple model that illustrates this result.Assume that the government, via its monetary policy, can perfectly con trol in flati on. Furthermore the gover nment can be one of two types. Either it is strong or it is weak. A strong government is only concerned about the rate of in flati on, and so n ever in flates the economy. A weak government, however, is concerned about both inflation and unemployment.Specially, its welfare in time-period t is given by the followingequation :where 严and g are the rates of inflation and unemployment intime-period t respectively, and c, d and e are all positive parameters.It is assumed the gover nment does not disco unt future welfare, and so a weak gover nment attempts to maximize the sum of its per-period welfare over all current and future periods. The constraint facing the government is given by the expectations-augmented Phillips curve. This is written as=山-也#where is the expected rate of in flati on in period t determ ined at thebeg inning of that period, and aga in a and b are positive parameters. The private sector formulates its expectati ons rati on ally in accorda nee with Bayes' Theorem. Finally, it is assumed that this policy game lasts foronly two periods.(1) Determine the subgame perfect path of inflation if it is common knowledge thegovernment is weak.(2) Determine the sequential equilibrium path of inflation if there is incomplete informationand the private sector 's prior probabilitythat the government is strong is . (Hint: initially determine the necessary condition for the weak government to be indifferent between inflating and not inflating theeconomy.)。
