Eco514|Game Theory
Problem Set 4,Due Tuesday,November 9
1,Machines
Extend Proposition 151.1 (the Perfect Folk Theorem with discounting) to arbitrary mixtures
of payo pro les of the original game G = (N;(Ai;ui)i2N),Allow for both rational and real
weights on the set of pro lesfu(a),a2Ag; note that the statement of the result will involve
an approximation of the payo pro le.
Construct a machine that implements the strategies in your proof.
[Hint,You may wish to refer to the proof of Proposition 146.2]
2,Repeated Prisoner’s Dilemma with a Twist
Consider the following variant of the usual repeated Prisoner’s Dilemma game (Figure 134.1
in OR),There are four players in the population,At even times t,Player 1 plays Prisoner’s
Dilemma with Player 2 and Player 3 plays it with Player 4; at odd times t,1 plays with 3
and 2 plays with 4,There is perfect information,that is,at the end of each stage,every
player can observe the actions chosen by the others,Assume for simplicity that the four
players use the same discount factor,
Determine the set of discount factors for which cooperation at each t is a subgame-
perfect equilibrium outcome,Next,consider the usual version of this repeated game (two
players,1 and 2,who share a common discount factor,play with each other repeatedly)
and determine the set of ’s for which cooperation at each t is a subgame-perfect equilibrium
outcome,Compare the two sets.
3,War of Attrition
Two small grocery stores on the same block are feeling the e ects of a large supermarket
that was recently constructed a half-mile away,As long as both remain in business,each
will lose $1000 per month,On the rst day of every month,when the monthly rent for the
stores is due,each grocer who is still in business must independently decide whether to stay
in business for another month or quit,If one grocer quits,then the grocer who remains
1
will make $500 per month pro t thereafter,Assume that,once a grocer quits,his or her
lease will be taken by some other merchant (not a grocer),so he or she will not be able to
reopen a grocery store in this block,even if the other grocer also quits,Each grocer wants
to maximize the expected discounted average value of his or her monthly pro ts,using a
discount factor per month of =,99.
a,Find an equilibrium of this situation in which both grocers randomize between staying
and quitting every month until at least one grocer quits,Is this the only equilibrium of this
game?
b,Suppose now that grocer 1 has a slightly larger store than grocer 2,As long as both
stores remain in business,grocer 1 loses $1,200 per month,and grocer 2 loses $900 per month.
If grocer 1 had the only grocery store on the block,she would earn $700 pro t per month.
If grocer 2 had the only grocery store on the block,he would earn $400 per month,Find an
equilibrium of this situation in which both grocers randomize between staying and quitting
every month,until somebody actually quits,In this equilibrium,which grocer is more likely
to quit rst?
4,From OR,148.1,152.1,153.2
2
Problem Set 4,Due Tuesday,November 9
1,Machines
Extend Proposition 151.1 (the Perfect Folk Theorem with discounting) to arbitrary mixtures
of payo pro les of the original game G = (N;(Ai;ui)i2N),Allow for both rational and real
weights on the set of pro lesfu(a),a2Ag; note that the statement of the result will involve
an approximation of the payo pro le.
Construct a machine that implements the strategies in your proof.
[Hint,You may wish to refer to the proof of Proposition 146.2]
2,Repeated Prisoner’s Dilemma with a Twist
Consider the following variant of the usual repeated Prisoner’s Dilemma game (Figure 134.1
in OR),There are four players in the population,At even times t,Player 1 plays Prisoner’s
Dilemma with Player 2 and Player 3 plays it with Player 4; at odd times t,1 plays with 3
and 2 plays with 4,There is perfect information,that is,at the end of each stage,every
player can observe the actions chosen by the others,Assume for simplicity that the four
players use the same discount factor,
Determine the set of discount factors for which cooperation at each t is a subgame-
perfect equilibrium outcome,Next,consider the usual version of this repeated game (two
players,1 and 2,who share a common discount factor,play with each other repeatedly)
and determine the set of ’s for which cooperation at each t is a subgame-perfect equilibrium
outcome,Compare the two sets.
3,War of Attrition
Two small grocery stores on the same block are feeling the e ects of a large supermarket
that was recently constructed a half-mile away,As long as both remain in business,each
will lose $1000 per month,On the rst day of every month,when the monthly rent for the
stores is due,each grocer who is still in business must independently decide whether to stay
in business for another month or quit,If one grocer quits,then the grocer who remains
1
will make $500 per month pro t thereafter,Assume that,once a grocer quits,his or her
lease will be taken by some other merchant (not a grocer),so he or she will not be able to
reopen a grocery store in this block,even if the other grocer also quits,Each grocer wants
to maximize the expected discounted average value of his or her monthly pro ts,using a
discount factor per month of =,99.
a,Find an equilibrium of this situation in which both grocers randomize between staying
and quitting every month until at least one grocer quits,Is this the only equilibrium of this
game?
b,Suppose now that grocer 1 has a slightly larger store than grocer 2,As long as both
stores remain in business,grocer 1 loses $1,200 per month,and grocer 2 loses $900 per month.
If grocer 1 had the only grocery store on the block,she would earn $700 pro t per month.
If grocer 2 had the only grocery store on the block,he would earn $400 per month,Find an
equilibrium of this situation in which both grocers randomize between staying and quitting
every month,until somebody actually quits,In this equilibrium,which grocer is more likely
to quit rst?
4,From OR,148.1,152.1,153.2
2
