Repeated Games - University of Notre Dametjohns20/gametheory_SP16/slides6.pdf · But what if the...

Repeated Games

Econ 400

University of Notre Dame

Econ 400 (ND) Repeated Games 1 / 48

Relationships and Long-Lived Institutions

Business (and personal) relationships: Being caught cheating leads topunishment or exclusion




Government: We are willing to give up resources now in theexpectation that we will be paid back later (Fiat money, SocialSecurity)




Government: We are willing to give up resources now in theexpectation that we will be paid back later (Fiat money, SocialSecurity)

Social Norms: By fixing what society expects, we can achieve bettercoordination than acting alone



Is cooperation always good?




Is anonymity always bad?





Do we value relationships because of the relationship per se, or for thestream of benefits it provides? Is that bad?





Do we value relationships because of the relationship per se, or for thestream of benefits it provides? Is that bad?

Magellan’s Victoria


Repeated Prisoners’ Dilemma

Suppose two players are going to play prisoners’ dilemma t = 1, 2, ...,Ttimes, where the payoffs are given by

S C

S 1,1 -1,2

C 2,-1 0,0



Suppose two players are going to play prisoners’ dilemma t = 1, 2, ...,Ttimes, where the payoffs are given by

S C

S 1,1 -1,2

C 2,-1 0,0

What are the subgame perfect Nash equilibria of the game?



But what if the other players was known to be a “nice guy/gal”, who playsS as long as you have played S in all previous periods, then chooses C forall future periods once you have chosen C — i.e., they play “nice” untilyou cheat them. Maybe now there will be cooperation for some number ofperiods?



But what if the other players was known to be a “nice guy/gal”, who playsS as long as you have played S in all previous periods, then chooses C forall future periods once you have chosen C — i.e., they play “nice” untilyou cheat them. Maybe now there will be cooperation for some number ofperiods?Consider the payoff from choosing S for the first T − 1 periods, thenchoosing C in the final period to get the 2. Then the payoff from thestrategy (S ,S , ...,S ,C ) is:

1 + δ + δ2 + ...+ δT−1 + 2δT =1− δT

1− δ+ 2δT



But what if the other players was known to be a “nice guy/gal”, who playsS as long as you have played S in all previous periods, then chooses C forall future periods once you have chosen C — i.e., they play “nice” untilyou cheat them. Maybe now there will be cooperation for some number ofperiods?Consider the payoff from choosing S for the first T − 1 periods, thenchoosing C in the final period to get the 2. Then the payoff from thestrategy (S ,S , ...,S ,C ) is:

1 + δ + δ2 + ...+ δT−1 + 2δT =1− δT

1− δ+ 2δT

What if we choose the C one period sooner? Would this improve ourpayoff? This strategy is (S ,S , ...,S ,C ,C ):

1 + δ + δ2 + ...+ δT−2 + 2δT−1 + 0 =1− δT−1

1− δ+ 2δT−1 + 0



But then comparing confessing first at T − 1 to confessing first at time T

gives

(

1− δT−1

1− δ+ 2δT−1

)

−(

1− δT

1− δ+ 2δT

)

= δT−1

(

1 + δ

1− δ+ 2− δ

)

> 0

So it looks like the unravelling is going to occur, even if one player iswilling to cooperate.



But let’s look at the difference in payoffs between confessing for the firsttime in period T − 1 and confessing for the first time in period T again:

δT−1

(

1 + δ

1− δ+ 2− δ

)



But let’s look at the difference in payoffs between confessing for the firsttime in period T − 1 and confessing for the first time in period T again:

δT−1

(

1 + δ

1− δ+ 2− δ

)

As T gets large, δT−1 becomes small, and cooperation is almost anequilibrium of the repeated game early on. But as the end of the gameapproaches, the unravelling motive will kick in.


Repeated Games

Definition

A repeated game is

(i) a terminal date, T , giving the number of times the players interact,where T = 1, 2, 3, .... The calendar date is given by t = 1, 2, ...,T .

(ii) a discount factor, 0 ≤ δ ≤ 1, that represents both how patient theplayers are and how likely the game is to continue.

(iii) a stage game of finite length: A specification of the players,actions, payoffs, and timing, which is usually independent of thecalendar date.


Repeated Games

Definition

A repeated game is

(i) a terminal date, T , giving the number of times the players interact,where T = 1, 2, 3, .... The calendar date is given by t = 1, 2, ...,T .

(ii) a discount factor, 0 ≤ δ ≤ 1, that represents both how patient theplayers are and how likely the game is to continue.

(iii) a stage game of finite length: A specification of the players,actions, payoffs, and timing, which is usually independent of thecalendar date.

If st = (st1, st2, ..., s

tN ) is the strategy profile that occurs in period t, the

players’ discounted payoff is

ui(s1) + δui (s

2) + δ2ui (s3) + ...+ δT−1ui (s

T ) =

T∑

t=1

δt−1ui (st)


Histories

Presumably, players will want to keep track of how their opponents havebehaved in previous periods; this allows them to choose strategies thatreward or punish other players for good or bad behavior. But how do wekeep track of what has happened in repeated games?


Histories

Presumably, players will want to keep track of how their opponents havebehaved in previous periods; this allows them to choose strategies thatreward or punish other players for good or bad behavior. But how do wekeep track of what has happened in repeated games?In prisoners’ dilemma, all of the possible outcomes from one repetition ofthe game are

Σ = {(S ,S), (S ,C ), (C ,S), (C ,C )}


Histories

Presumably, players will want to keep track of how their opponents havebehaved in previous periods; this allows them to choose strategies thatreward or punish other players for good or bad behavior. But how do wekeep track of what has happened in repeated games?In prisoners’ dilemma, all of the possible outcomes from one repetition ofthe game are

Σ = {(S ,S), (S ,C ), (C ,S), (C ,C )}When the game goes two periods, however, it becomes

Σ2 = {[(S ,S), (S ,S)], [(S ,C ), (S ,S)], [(C ,S), (S ,S)], [(C ,C ), (S ,S)],

[(S ,S), (S ,C )], [(S ,C ), (S ,C )], [(C ,S), (S ,C )], [(C ,C ), (S ,C )],

[(S ,S), (C ,S)], [(S ,C ), (C ,S)], [(C ,S), (C , S)], [(C ,C ), (C ,S)],

[(S ,S), (C ,C )], [(S ,C ), (C ,C )], [(C ,S), (C ,C )], [(C ,C ), (C ,C )]}

since we need to keep track of what happened in the first period, and thesecond period.


Histories

Let Σ be the set of all the strategy profiles for the stage game. (Forexample, in prisoners’ dilemma, Σ = {(S ,S), (S ,C ), (C ,S), (C ,C )}).


Histories

Let Σ be the set of all the strategy profiles for the stage game. (Forexample, in prisoners’ dilemma, Σ = {(S ,S), (S ,C ), (C ,S), (C ,C )}).

If we want to keep track of the outcomes of a repeated game, we’reinterested in sequences of observations from Σ. For two periods,Σ×Σ = Σ2 is the set of all possible outcomes for two repetitions ofthe game. For three periods, Σ×Σ×Σ = Σ3 is the set of all possibleoutcomes for three repetitions of the game, and so on.


Histories

Definition

Let the set of all strategy profiles for the stage game be Σ. Then the setΣt = Σ×Σ× ...×Σ contains all lists of the possible outcomes in terms ofwhat strategies the players have used in each of the t periods. Then anyelement ht of the set Σt = Ht is a history at time t.


Equilibria in Repeated Games

Definition

A set of strategies is a Subgame Perfect Nash Equilibrium of a repeatedgame if, for any t-period history ht , there is no subgame in which anyplayer has a profitable deviation.



Definition

A set of strategies is a Subgame Perfect Nash Equilibrium of a repeatedgame if, for any t-period history ht , there is no subgame in which anyplayer has a profitable deviation.

Note that no player can have a profitable deviation for any history, eventhough, given the strategies, only one history actually occurs. But it isprecisely because the players know the consequences of their actions thatthe equilibrium history arises.



Suppose the stage game has an equilibrium s∗ = (s∗1 , ..., s∗

N ). Thenthe strategy for the repeated game where each player i plays s∗i afterevery history is a Subgame Perfect Nash Equilibrium of the repeatedgame.



Suppose the stage game has an equilibrium s∗ = (s∗1 , ..., s∗

N ). Thenthe strategy for the repeated game where each player i plays s∗i afterevery history is a Subgame Perfect Nash Equilibrium of the repeatedgame.

But is this the only equilibrium of a repeated game?


Prisoners’ Dilemma

Consider the following behavior strategy for an infinitely repeatedprisoners’ dilemma:

If the history at time t is {(S ,S), (S ,S), ..., (S ,S)}, play S .

For any other history at time t, player C .


Prisoners’ Dilemma

Consider the following behavior strategy for an infinitely repeatedprisoners’ dilemma:


For any other history at time t, player C .

Is this a subgame perfect Nash equilibrium of the infinitely repeated game?


Prisoners’ Dilemma: Equilibrium Analysis

Well, we have to check all the possible histories to see if there are anyprofitable deviations. There’s really just two cases:

{(S ,S), (S ,S), ..., (S ,S)}

and anything else.



Well, we have to check all the possible histories to see if there are anyprofitable deviations. There’s really just two cases:

{(S ,S), (S ,S), ..., (S ,S)}

and anything else. Let’s start with “anything else”:

Suppose the history at time t is not {(S ,S), (S ,S), ..., (S ,S)}. Thentoday and in all future periods, all my opponents will choose C . ThenI should choose C , since it maximizes my discounted payoff. So thereare no profitable deviations from the proposed strategies for thesehistories.



Suppose the history at time t is {(S ,S), (S ,S), ..., (S ,S)}. Is playingS an optimal strategy, given the behavior strategies of the players? IfI choose S this period, the next period’s history is{(S ,S), (S ,S), ..., (S ,S), (S ,S)}, and given the strategies, the playerswill choose S from then on forever. The payoff from that is

1 + δ + δ2 + δ3 + ... =1

1− δ




1 + δ + δ2 + δ3 + ... =1

1− δ

If I deviate this period, next period’s history is

{(S ,S), (S ,S), ..., (S ,S), (C ,S)} 6= {(S ,S), (S ,S), ..., (S ,S), (S ,S)}so all players will confess, generating a sequence of histories whereplayers confess forever.




1 + δ + δ2 + δ3 + ... =1

1− δ

If I deviate this period, next period’s history is

{(S ,S), (S ,S), ..., (S ,S), (C ,S)} 6= {(S ,S), (S ,S), ..., (S ,S), (S ,S)}so all players will confess, generating a sequence of histories whereplayers confess forever. The payoff from that is

2− δ0 − δ20− .... = 2



So it all comes down to whether it’s better to cooperate than cheat in anyperiod, or

1

1− δ≥ 2 −→ δ ≥ 1

2


Prisoners’ Dilemma: Equilibrium

If δ > 12 , both players using the strategy


For any other history at time t, play C .

is a Subgame Perfect Nash Equilibrium of the infinitely repeated prisoners’dilemma.


Collusion and Bertrand

In the Bertrand pricing game, there are two firms 1 and 2 who havethe same marginal costs c and compete in prices, choosingp1, p2 = {0, 1, ..., c , ..., 8, ..., 10}.



In the Bertrand pricing game, there are two firms 1 and 2 who havethe same marginal costs c and compete in prices, choosingp1, p2 = {0, 1, ..., c , ..., 8, ..., 10}.Fix demand at 1 for all prices less than or equal to 10.




If p1 < p2, all the consumers go to firm 1 and firm 2 gets no business.If p1 = p2, the firms split the market equally. If p2 > p1, allconsumers go to firm 2 and firm 1 gets no business.




If p1 < p2, all the consumers go to firm 1 and firm 2 gets no business.If p1 = p2, the firms split the market equally. If p2 > p1, allconsumers go to firm 2 and firm 1 gets no business.

The profits for firm 1 are:

π1(p1, p2) =

(1)(p1 − c) , p1 < p2(

12

)

(p1 − c) , p1 = p20 , p1 > p2

and similarly for firm 2.



Consider the repeated game:

T = ∞Discount factor: 0 < δ < 1

Stage game: Bertrand Competition



Consider the repeated game:

T = ∞Discount factor: 0 < δ < 1

Stage game: Bertrand Competition

Notice that if there are 11 price increments, there are (22)t possibleoutcomes we might observe by time t. If t = 5, there are 5, 153, 632histories. Even on a very good computer, computing the extensive formand payoffs would take a lot of time.


Bertrand: Stage Game Equilibrium and Trigger Strategies

We know from the first part of the class that p∗1 = p∗2 = c is a Nashequilibrium of the stage game. Let’s use this as the “punishment” for abreakdown in cooperation, but otherwise have the players usep1 = p2 = 10.


Bertrand: Stage Game Equilibrium and Trigger Strategies

We know from the first part of the class that p∗1 = p∗2 = c is a Nashequilibrium of the stage game. Let’s use this as the “punishment” for abreakdown in cooperation, but otherwise have the players usep1 = p2 = 10. Consider the strategies:

If the history is {(10, 10), (10, 10), (10, 10), ..., (10, 10)}, play 10 thisperiod.

For any other history, play c this period.


Bertrand: The Optimal Deviation

If your opponent adopts these strategies and “plays nice” up to some datet = 0, 1, ..., what is the best way to stab him in the back?



If your opponent adopts these strategies and “plays nice” up to some datet = 0, 1, ..., what is the best way to stab him in the back? (If you’re goingto ruin a relationship, at least do it optimally)



If your opponent adopts these strategies and “plays nice” up to some datet = 0, 1, ..., what is the best way to stab him in the back? (If you’re goingto ruin a relationship, at least do it optimally) By charging the price just

below the cutoff point, 9− 1, the deviator captures the whole market andonly losses a dollar on each unit:

πd = (1)(9 − 1− c)


Bertrand: Equilibrium Analysis

Once a deviation has occurred, the players use p∗1 = p∗2 = c in allfuture periods. In any of these scenarios, there are no profitabledeviations, because if your opponent is playing c in these periods,your best response is c .

The payoff to cooperating is

10− c

2+ δ

10 − c

2+ δ2

10− c

2+ ... =

10− c

2

1

1− δ

The payoff to optimally deviating is

(9− c) + δ0 + δ20 + ... = 9− c



Then cooperating is better than deviating if

10− c

2

1

1− δ≥ 9− c

or

δ ≥ 8− c

19− 2c




10− c

2

1

1− δ≥ 9− c

or

δ ≥ 8− c

19− 2c

For example, if c = 2, we have δ ≥ 6/15. But if c = 6, we have δ = 2/7.


Bertrand: Equilibrium

Then as long as

δ ≥ 8− c

19− 2c

the strategies



are a Subgame Perfect Nash Equilibrium of the infinitely repeatedBertrand game.


Bertrand: Equilibrium

Then as long as

δ ≥ 8− c

19− 2c

the strategies



are a Subgame Perfect Nash Equilibrium of the infinitely repeatedBertrand game. So collusive is possible in the infinite-horizon version ofthe repeated game. (What about the finite version of the repeated game?)


The Pattern

Solve for all of the equilibria of the stage game. (Competitive Play)


The Pattern


Find a strategy profile that gives all the players a higher payoff.(Cooperative Play)


The Pattern



Enforce cooperation through trigger strategies: If all players havepreviously cooperated, continue cooperating. If any player haspreviously defected, play competitively.


The Pattern



Enforce cooperation through trigger strategies: If all players havepreviously cooperated, continue cooperating. If any player haspreviously defected, play competitively.

For sufficiently high values of the discount factor δ, this will be anequilibrium of the repeated game.


Examples of Equilibria in Repeated Games

For any game, playing the equilibrium of the stage game forever is aSubgame Perfect Nash Equilibrium.

In prisoners’ dilemma, as long as the players didn’t discount theirpayoffs too much (δ > .5), one Subgame Perfect Nash Equilibrium ofthe game was to be silent in all periods unless your opponent hadpreviously confessed at some point, and then to confess forever.

In Cournot, as long as the players didn’t discount their payoffs toomuch (δ > .377), one Subgame Perfect Nash Equilibrium of the gamewas to collude in all periods unless your opponent had previouslyplayed competitively, and then to play the Cournot quantity forever


Examples of Equilibria in Repeated Games

For any game, playing the equilibrium of the stage game forever is aSubgame Perfect Nash Equilibrium.

In prisoners’ dilemma, as long as the players didn’t discount theirpayoffs too much (δ > .5), one Subgame Perfect Nash Equilibrium ofthe game was to be silent in all periods unless your opponent hadpreviously confessed at some point, and then to confess forever.

In Cournot, as long as the players didn’t discount their payoffs toomuch (δ > .377), one Subgame Perfect Nash Equilibrium of the gamewas to collude in all periods unless your opponent had previouslyplayed competitively, and then to play the Cournot quantity forever

Today, we want to make the argument that “As long as players arepatient, they can cooperate in infinitely repeated games in ways thataren’t possible in finitely repeated games”.


The Nash Threats Folk Theorem

Theorem

Consider any N-player infinitely repeated game with a stage game

equilibrium s∗ = (s∗1 , s∗

2 , ..., s∗

N ) yielding payoffs u∗ = (u∗1 , u∗

2 , ..., u∗

N ).Suppose there is another strategy profile s = (s1, s2, ..., sN ) yielding payoffs

u = (u1, u2, ..., uN ), where, for every player i ,

ui ≥ u∗i

If the players are sufficiently patient, then there is a Subgame Perfect Nash

Equilibrium in which the players use s in every period of the infinitely

repeated game.


The Folk Theorem

Let’s “fill in the blanks” for the prisoners’ dilemma game:

Theorem

Consider prisoners’ dilemma with a stage game equilibrium s∗ = (C ,C )yielding payoffs u∗ = (0, 0). Suppose there is another strategy profile

s = (S ,S) yielding payoffs u = (1, 1), where, for every player i ,

1 ≥ 0


Equilibrium in which the players use (S ,S) in every period of the infinitely

repeated game.


The Folk Theorem

Let’s “fill in the blanks” for the prisoners’ dilemma game:

Theorem

Consider prisoners’ dilemma with a stage game equilibrium s∗ = (C ,C )yielding payoffs u∗ = (0, 0). Suppose there is another strategy profile

s = (S ,S) yielding payoffs u = (1, 1), where, for every player i ,

1 ≥ 0


Equilibrium in which the players use (S ,S) in every period of the infinitely

repeated game.

What do we need to think about in proving the Folk Theorem?


The Folk Theorem: Trigger Strategies

Consider the following trigger strategy for player i :

If the history at t is ht = (s, s , ..., s), play si in period t.

For any other history at time t, play s∗i in period t.


The Folk Theorem: Trigger Strategies

Consider the following trigger strategy for player i :

If the history at t is ht = (s, s , ..., s), play si in period t.

For any other history at time t, play s∗i in period t.

This is called a “trigger strategy” because it starts in “cooperative” mode,but after any defection by any player, it switches to “punishment” or“competitive” mode, and they play the stage game strategies forever.


The Folk Theorem: Optimal Deviations

Since u is presumably not a Nash equilibrium of the stage game, there areat least some players for whom

udj > uj ≥ u∗j

i.e., while they prefer cooperating to the equilibrium of the stage game,they prefer defection to cooperation.


The Folk Theorem: Optimal Deviations

Since u is presumably not a Nash equilibrium of the stage game, there areat least some players for whom

udj > uj ≥ u∗j

i.e., while they prefer cooperating to the equilibrium of the stage game,they prefer defection to cooperation. Notice that if a player is tempted todeviate, the above inequality implies that

udj − u∗j ≥ udj − uj


The Folk Theorem: Cooperating and Deviating

The payoff to cooperating to player j is

uj + δuj + δ2uj + ... =1

1− δuj




uj + δuj + δ2uj + ... =1

1− δuj

The payoff to deviating to player j is

udj + δu∗j + δu∗j + ... = udj + u∗j δ(1 + δ + δ2 + ...) = udj + u∗jδ

1− δ




uj + δuj + δ2uj + ... =1

1− δuj

The payoff to deviating to player j is

udj + δu∗j + δu∗j + ... = udj + u∗j δ(1 + δ + δ2 + ...) = udj + u∗jδ

1− δ

Then cooperating is better than deviating for player j if

1

1− δjuj ≥ udj + u∗j

δj1− δj

or

δj ≥udj − uj

udj − u∗j

But 1 ≥ δj , by the work on the previous slide.Econ 400 (ND) Repeated Games 32 / 48

The Folk Theorem: Equilibrium:

Letδ∗ = {δ1, δ2, ..., δN}

so we have selected the highest discount factor for which cooperating isbetter than deviating, for all the players.


The Folk Theorem: Equilibrium:

Letδ∗ = {δ1, δ2, ..., δN}

so we have selected the highest discount factor for which cooperating isbetter than deviating, for all the players. Then if all players are sufficientlypatient, in the sense that each of their discount factors are greater thanδ∗, then the trigger strategies are a subgame perfect Nash equilibrium ofthe infinitely repeated game, and they will play s in every period.


The Folk Theorem: Geometry


Solving for Equilibria in Repeated Games

1. Solve for all equilibria of the stage game. (Competition)




2. Find a strategy profile (equilibrium or not) where all the players doat least as well as in the stage game. (Cooperation)





3. Design trigger strategies that support cooperation and punish withcompetition.






4. Find the optimal deviation for any player. Compute the minimumdiscount factor for which cooperating is an equilibrium.







5. Conclude that the trigger strategies are an equilibrium of theinfinitely repeated game as long as all the players are sufficientlypatient.







5. Conclude that the trigger strategies are an equilibrium of theinfinitely repeated game as long as all the players are sufficientlypatient.

Note that things are a little more complicated than this: We don’t provedirectly that this is a Subgame Perfect Nash Equilibrium (which is just alittle bit more work), we appeal to the Nash Threats Folk Theorem andthat takes care of the “boilerplate” details.


Trade and Bonuses/Tips

Why do people tip for services or firms pay bonuses to workers?


Equilibria with Forgiveness

The grim trigger strategies of the Nash Threats Folk Theorem are prettyharsh: Mess up once, and cooperation is cut off forever. What if punish byplaying the stage game equilibrium K rounds and then return tocooperative mode, instead?



The grim trigger strategies of the Nash Threats Folk Theorem are prettyharsh: Mess up once, and cooperation is cut off forever. What if punish byplaying the stage game equilibrium K rounds and then return tocooperative mode, instead?Then for Prisoners’ Dilemma, cooperating is better than deviating if

1 + δ + δ2 + ... ≥ 2 + δ0 + δ20 + ...+ δK0 + δK+1 + δK+2 + ...



The grim trigger strategies of the Nash Threats Folk Theorem are prettyharsh: Mess up once, and cooperation is cut off forever. What if punish byplaying the stage game equilibrium K rounds and then return tocooperative mode, instead?Then for Prisoners’ Dilemma, cooperating is better than deviating if

1 + δ + δ2 + ... ≥ 2 + δ0 + δ20 + ...+ δK0 + δK+1 + δK+2 + ...

1

1− δ≥ 2 + δK+1 1

1− δ

By making K sufficiently large and taking δ sufficiently close to 1, theequality will hold.




2δ ≥ 1 + δK+1

or

K ≥ log(2δ − 1)

log(δ)− 1




2δ ≥ 1 + δK+1

or

K ≥ log(2δ − 1)

log(δ)− 1

If you compute the limit as δ → 1 (use L’Hopital’s rule twice), you getthat the minimal punishment period is 0. So players that are sufficientlypatient will never cheat on each other.


Collusion and Cournot

Recall the Cournot game: Two firms simultaneously choose quantitiesq1, q2 > 0, where the market price is p = A− q1 − q2 and the firms haveno costs. (Step 1) The Nash equilibrium of the stage game is

q∗1 = q∗2 =A

3

giving profits

π∗

1 = π∗

2 =

(

A

3

)2



Recall the Cournot game: Two firms simultaneously choose quantitiesq1, q2 > 0, where the market price is p = A− q1 − q2 and the firms haveno costs. (Step 1) The Nash equilibrium of the stage game is

q∗1 = q∗2 =A

3

giving profits

π∗

1 = π∗

2 =

(

A

3

)2

What if the firms made an agreement to work together to improve theirprofits? What could they achieve?



Summing the firms’ profits, we get

π1 + π2 = (A− q1 − q2)q1 + (A− q1 − q2)q2 = (A− q1 − q2)(q1 + q2)

orΠ = (A− Q)Q



Summing the firms’ profits, we get

π1 + π2 = (A− q1 − q2)q1 + (A− q1 − q2)q2 = (A− q1 − q2)(q1 + q2)

orΠ = (A− Q)Q

maximizing gives

Q∗ =A

2, Π∗ =

(

A

2

)2

Since Π∗ =A2

4>

2A2

9= π∗

1 + π∗

2, collusion is potentially profitable.



(Step 2) Suppose the two firms are playing the Cournot game an infinitenumber of times, and they share a discount factor δ. Let

q =A

4, π =

(

A

4

)2

This is half the monopoly quantity, A/2, and strictly less than the Cournotequilibrium quantity A/3.



(Step 2) Suppose the two firms are playing the Cournot game an infinitenumber of times, and they share a discount factor δ. Let

q =A

4, π =

(

A

4

)2

This is half the monopoly quantity, A/2, and strictly less than the Cournotequilibrium quantity A/3. (Step 3) Consider the strategy:

If the two firms have both used q in all previous periods, use q = A/4this period.

If either firm ever did anything besides q, play the stage Cournotquantity q∗ = A/3.

Is this a subgame perfect Nash equilibrium of the infinitely repeated game?


Equilibrium Analysis: The Optimal Deviation

To decide if this is an equilibrium, we need to know the payoff fromoptimally stabbing your partner in the back (if we’re going to break up arelationship, we might as well do it optimally, right?). (Step 4) Then weneed to solve

maxq′

(A− q − q′)q′ = maxq′

(

A− A

4− q′

)

q′

Maximizing gives

q′ =3A

8, π′ =

(

3A

8

)2


Equilibrium Analysis: Cooperation

Cooperating in some period t after a history in which all the playerspreviously cooperated keeps the industry in collusive mode (remember,given the proposed strategies), giving a payoff of

π + δπ + δ2π + ... = π1

1− δ

which equals(

A

4

)2 1

1− δ


Equilibrium Analysis: Deviation

Deviating optimally in some period t after a history in which all thepreviously players cooperated switches the game from collusion tocompetition (remember, given the proposed strategies), giving a payoff of

π′ + δπ∗ + δ2π∗ + ... =

(

3A

8

)2

+ δ

(

A

3

)2(

1 + δ + δ2 + ...)

Which is(

3A

8

)2

+ δ

(

A

3

)2 1

1− δ


Equilibrium Analysis


(

A

4

)2 1

1− δ≥

(

3A

8

)2

+ δ

(

A

3

)2 1

1− δ

or1

16

1

1− δ≥ 9

16+

1

9

δ

1− δor

δ ≥ 17

45

So if δ ≥ 1745 , there are no profitable deviations as long as the players have

previous cooperated.


Cournot and Collusion

(Step 5) So if the players are sufficiently patient, or

δ ≥ 17

45

then the strategies

If the two firms have both used q in all previous periods, use q = A/4this period.

If either firm ever did anything besides q, play the stage Cournotquantity q∗ = A/3.

are a subgame perfect Nash equilibrium of the infinitely repeated Cournotgame.


Money as a long-lived institution

Suppose each calendar date t, a generation of new people are born, and aprevious generation die. Each generation lives for two periods. In the firstperiod, they receive a wage of 4 during their “working years”, but receiveno wage in period two of their lives. Their utility function overconsumption pairs is u(c1, c2) =

√c1 +

√c2, where c1 is the amount

consumed in period 1 and c2 is the amount consumed in period 2.



Suppose that there is a “dollar” that each generation hands to the nextone. The dollar obligates the young to give the old y units of their wealthso that the old have consumption in old age.



Suppose that there is a “dollar” that each generation hands to the nextone. The dollar obligates the young to give the old y units of their wealthso that the old have consumption in old age. Suppose the players adoptthe following strategies: “If all previous generations have accepted thedollar and paid y to the old, do so this period. If any previous generationrefused to pay, do not pay.”




Suppose the economy only lasts for T periods. For what values of yis it an equilibrium for the young to honor the dollar?




Suppose the economy only lasts for T periods. For what values of yis it an equilibrium for the young to honor the dollar?

Suppose the economy goes on infinitely. For what values of y is it anequilibrium for the young to honor the dollar?


Repeated Games - University of Notre Dametjohns20/gametheory_SP16/slides6.pdf · But what if the...

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