博弈论习题集

博弈论习题集(总4页)
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PROBLEM SET I OF GAME THEORY
1. State whether the following games have unique pure strategy solutions, and if so what they are and how they can be found. (1) Player 2
Player 1
(2) Player 2
Player 1
(3) Player 2
Player 1
2. Draw the normal form game for the following game and identify both the pure-and 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 considering entering a new market, and must make their decision without knowing what the other firm has done. Unfortunately the market is only big enough to support one of the two firms. If both firms enter the market, then they will each make a
loss of ￡ only one firm enter s the market, that firm will earn a profit of ￡50m, and the other firm will just break even.
3. Convert the following extensive form game into a normal form game, and identify the Nash equilibria and subgame perfect Nash
equilibria. Finally, what is the Nash equilibrium if both players make their moves simultaneously
4. Consider an economy consisting of one government and two people. Let x i be the choice of the people, where x i ∈X = {x L , x M , x H }, and i=1, 2， and y the choice of the government, where y ∈Y= {y L , y M , y H }. The payoffs to the government-household are given by the values of u 1(x 1, x 2, y) and u 2(x 1, x 2, y) = u 1(x 2,x 1, y) . These payoffs are entered in the following table:
12the government ’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 Nash equilibria, and the subgame perfect outcome. Is the outcome efficient Why
(3)Show whether there exists Nash equilibrium (in pure strategies)
for the one-period economy when households and the government move simultaneously.
(4)If 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 following situations if they play across.
(1)At the beginning of the game each player believes that there is
a 50/50 chance that the other player is rational or co-operative.
It is assumed that a co-operative player always plays across.
Furthermore suppose that a rational player will play across with a probability of
(2)At their second move the players again move across. (Continue
to assume that the probability that a rational player plays across remains equal .
(3)How would the players’ reputation have changed after the first
move had the other player believed that rational players always play across. (Assume all other probabilities remain the same.)
(4)Finally, how would the players’ reputation have changed after
the first move had the other player believed that rational players never play across. (Again assume all other probabilities remain the same.)
6.Assume there are m identical 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 determined according to the equation , where Q is total industry output, and ɑ is a
constant. Find the subgame perfect Nash equilibrium supply for the leaders and the followers. Confirm the duopoly results for both
Cournot competition and Stackelberg competition, and the
generalized Cournot result for n firms derived in Exercise .
7.Assume that there are i=1,…, n identical firms in an industry,
each with constant marginal costs of c and no fixed costs. If the market price, P, is determined by the equation , where Q
is total industry output and ɑ is a constant, determine the
Cournot-Nash equilibrium output level for each firm. Where happens as n
8.Find the separating equilibrium behaviour of the low-cost
incumbent in the following two-period model. The incumbent has
marginal costs equal to either ￡4 or ￡2. Only the incumbent
initially knows its exact costs. The entrant observes the
incumbent’s output decision in the first period and only enters the market in the second period if it believes that the incumbent has high marginal costs. If entry does occur, the two firms
Cournot compete, and we assume that at this stage in the game the incumbent’s true costs are revealed. Price, P, is determined by
the following equation , where Q is the combined output
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
private sector’s uncertainty about the government’s preferences to partially avoid the inflationary bias associated with time-
inconsistent monetary policy. In this exercise we provided a
simple model that illustrates this result.
Assume that the government, via its monetary policy, can perfectly control inflation. Furthermore the government can be one of two types. Either it is strong or it is weak. A strong government is only concerned about the rate of inflation, and so never inflates the economy. A weak government, however, is concerned about both
inflation and unemployment. Specially, its welfare in time-period t
is given by the following equation：
,
where and are the rates of inflation and unemployment in time-period t respectively, and c, d and e are all positive parameters. It is assumed the government does not discount future welfare, and so a weak government 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
,
where is the expected rate of inflation in period t determined at the beginning of that period, and again ɑ and b are positive parameters. The private sector formulates its expectations rationally in accordance with Bayes’ Theorem. Finally, it is assumed that this policy game lasts for only two periods.
(1)Determine the subgame perfect path of inflation if it is common
knowledge the government is weak.
(2)Determine the sequential equilibrium path of inflation if there
is incomplete information and the private sector’s prior probability that the government is strong is . (Hint: initially determine the necessary condition for the weak government to be indifferent between inflating and not inflating the economy.)。