博弈论试题3

Introduction to Game Theory Problem Set 3

Professor Derek Liu

1.Congress has delegated a number of policy decisions to the EPA, making it,

in many people’s eyes, one of the most powerful government agencies. Like all agencies, however, the EPA is supposed to be monitored by a

subcommittee in Congress, which can punish it (with budget cuts and audits, for example) if the agency’s policy choices are contrary to congressional intent. The EPA can choose either a weak or a strict pollution standard. The EPA has a utility value for the strict standard of 7, and a value for the weak standard of 5. The subcommittee values the weak standard at 7, and the

strict one at 5; these values reflect the preferences of the subcommittee’s constituents. After the EPA chooses a standard, the subcommittee decides whether to punish the EPA or not. Punishment costs the EPA 4 units of

utility. If the subcommittee punishes the EPA for setting a strict standard, it gains 1 unit of utility (because its constituents approve of the punishment).

If the subcommittee punishes the EPA for setting a weak standard, it loses 1 unit (because its constituents disapprove.) Punishment does not affect the standard –the EPA’s choice remains in effect whether it is punished or not.

a.Draw the game tree that represents the strategic interaction between

the EPA and the subcommittee. For each player, list all strategies.

b.Find the equilibrium in this game.

c.Transform this game into normal form and solve it.

d.If we change this sequential game into a simultaneous game, in

which both players make decision at the same time, how to draw it

in normal form and solve it?

e.Does your answer to part (a) change if the amount of utility that the

EPA loses when punished is lower? What is the minimum amount

that the EPA must care about punishment in order to get the answer

in part (a)?

2. A prisoner is trying to escape from prison. He can attempt to climb over the prison

wall or dig a tunnel under the prison wall. The warden can prevent the prisoner

from climbing over the wall by posting guards at the wall and he can prevent the prisoner from tunneling under the wall by having regular inspections of cells, but he has only enough guards to do one or the other and not both.

a.Choose (and justify) some simple numerical payoffs for this game, and

then write the game in normal form. Be sure to label the strategies and

players in your game matrix. Is this game constant sum or non-constant

sum? Explain/justify your answer.

b.Now express this game in extensive form, assuming the prisoner and

warden make their decision at the same time. (Hint: remember the

information set for the second mover in the game tree!)

c.What is the mutual best response (Nash equilibrium) for the simultaneous

move version of this game, as you have depicted in parts a or b? Give your

reasoning.

d.Now express the same game in extensive form assuming that the warden

makes his decision first and the prisoner, after seeing the warden’s strategy,

moves second. Use backward induction to find the solution in the