博弈论习题集

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 £10m.If only one firm enter s the mark et, 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 u1(x1, x2, y) and u2(x1, x2, y) = u1(x2,x1, y) . These payoffs are entered in the following table:

One-period payoffs to the government-household (values of u (x1, x2, y))

(1)Suppose (*) denotes that (x1, x2) is a Nash equilibrium given y, the

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 0.2

(2)At their second move the players again move across. (Continue to assume

that the probability that a rational player plays across remains equal 0.2).

(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 4.1.