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## Final Lecture

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The Stage Game

- Prisoners’ dilemma structure applies in many situations
- Lovers or roommates
- Colluding oligopolists
- Arms control agreements
- Common-pool resources
- Cooperating vampire bats
- …many more

Stage Game

- In the stage game, Defect is a dominant strategy for both players.
- So the only Nash equilibrium has them both playing defect and each getting a payoff of 0
- Both would be better off if they both cooperated.
- Can cooperation be sustained in repeated play?

Repeated Play

- Suppose that after each round of play, players are told their payoff on the previous round and with probability d>0, they go on to play another round.
- Can we get cooperative play by having each player threaten to punish a defection.

Punishment and forgiveness

- Grim trigger: (No forgiveness) I will cooperate until you defect, but If you ever defect, I will defect in all future rounds.
- Conditional N-period punishment. If you defect, I will start to defect and will keep defecting until I have seen you cooperate N times in a row. Then I will cooperate so long as you do not defect.

Symmetric SPNE with Grim Trigger

- Suppose that the other player is playing Grim Trigger.
- If you play Grim Trigger as well, then you will both cooperate as long as the game continues and and you will each receive an expected payoff of R×(1+d +d2 + d3 + d4 + ….+ )=R/(1-d)

When does grim trigger sustain cooperation?

- If you defect against Grim Trigger, you get a payoff of T>R the first time that you defect. After this, the other guy will always play defect. The best you can do then is to always defect as well.
- You both get zero when you both defect, so expected payoff from defecting is just T+0=T
- So both paying grim trigger and always cooperating is a SPNE if T<R/(1-d)
- For example if d=.9, grim trigger sustains cooperation if T<10R.

What did we learn?

- Cooperation can be sustained if T<R/(1-d)
- Equivalently if (1-d)T<R
- This is the case if temptation is not too big

and if the probability d of playing again is not too small.

Getting by with it

- Suppose that the other player gets only partly accurate signals of what you do.
- Simple example: If you defect, other player finds out about it with probability c<1.
- To simpllify this example, assume that cooperation otheris always observed accurately.
- When can cooperation be sustained by the grim trigger strategy?

Grim trigger with faulty detection

- Each player will cooperate so long as he or she does not catch the other defecting.
- If you are caught defecting, other player will defect forever.
- If the other player plays grim trigger and so do you, your payoff is R/(1-d).

Payoff to sneaking

- If you always defect, your payoff is T so long as you are not caught and 0 forever after you are caught.
- The probability that you are not caught on the first round is 1-c. The probability that the game goes on for at least N rounds and you haven’t been caught is d N-1(1-c)N-1

So your discounted payoff from playing always defect is

T(1+(1-c)d+(1-c)2d2+…(1-c)NdN…)=T/(1-(1-c)d)

Comparing payoffs

- If you play always cooperate with grim trigger your expected payoff is R/(1-d).
- If you play always defect against grim trigger your expected payoff is T/(1-c(1-d))
- Cooperation can be sustained by grim trigger only if R/(1-d)>T/(1-(1-c)d).
- Equivalently, only if T<R(1-d+cd)/(1-d).
- For example if d=.9, this would imply

T<1+9c.

In general, cooperation can be sustained only if the probability of getting caught is large enough and the temptation T is small enough.

Is the grim trigger too unrelenting?

- In the games we have just considered, would it make sense to forgive defection?
- Simple answer for these games. No.
- If there is an equilibrium with eternal cooperation supported by grim trigger threat, the threat never has to be carried out.
- If a player cheats in response to a forgiving strategy, that player will cheat again after being forgiven.

Real world considerations

- When why might it be a bad idea to have an unforgiving punishment?
- Would you use grim trigger to motivate a child or a pet?
- Why not?

Reasons to forgive?

- What if you get noisy signals about other’s behavior and mistakenly believe other defected.
- What if other player made a one-time mistake or was subjected to unusual temptation
- These questions are much wrestled with in religion and in politics.

Forgiveness and religion

- There is a tension in religious prohibitions.
- To make people act as the priests would like them to, it might seem useful to tell them that they will be eternally punished for actions the priests don’t like.
- But if they do that, people who have violated the rules might as well continue to do so, since they are damned anyway.
- Hence religions often claim a “forgiving” deity in hopes of bringing lost sheep back into the herd.

Collusion, and price wars

- Companies that have been colluding but for some reason miscoordinate sometimes get into “price wars” of limited duration.
- These can be thought of as punishment strategies for perceived violations of past agreements.
- Price wars typically do not last forever, but often companies “forgive each other” and return to collusion rather than revert to perpetual competition.

What if temptation varies?

- Suppose that in the stage game of repeated prisoners’ dilemma, the temptation T varies randomly from one period to another.
- Suppose also that each player knows his or her own T in any round but doesn’t know the other’s.
- One can model a game with punishment and forgiveness where people cooperate so long as the other player is not in disgrace and so long as their own temptation is not too high.
- Model is too elaborate to treat here.

False blame

- Suppose that sometimes you cooperate but the other player occasionally gets the signal that you defected.
- Without forgiveness, cooperation would break down.
- Cooperation can be restored by strategies that impose finite term of punishment for perceived defections.

The story

- Vampire bats make a living by sucking blood from large mammals—cows, horses, etc
- They need blood every day to stay alive.
- Vampire bats may have a good day or a bad.
- Those who have a good day sometimes share blood with those who had a bad day.
- Text discusses two-player game in which bats are motivated by reciprocity.
- This model is not only about bats.

Payoffs from the game

- If you had a good day, you can either share your blood with the other bat or keep it for yourself. If you keep the blood for yourself your payoff today is 10. If you share, your payoff today is 8.
- If you had a bad day and the other bat shares with you, your payoff for the day is 4 and if the other bat doesn’t share, your payoff is -1.

Payoffs

- The probability that you have a good day is s and the probabiltiy that youhave a bad day is 1-s.
- In the stage game the only Nash equilibrium is to not help when you have a good day, but if the game is repeated, the grim trigger strategy might sustain sharing.

Suppose each bat shares when he has a good day and the other is needy, so as long as the other has always shared when the reverse is true—If ever the other had a good day and didn’t share when you had a bad day, you will never share with him.

Payoff from sharing or not

- Let V be expected payoff from this strategy.

V=s(10s+8(1-s))+(1-s)(4s- (1-s))+dV

V=(-3s2+14s-1)/1-d

- If you don’t share, other won’t either and your payoff is 10s-1(1-s)=11s-1

If you don’t share

- Suppose other bat is using grim trigger and you don’t share.
- Then if you have had a good day, your expected payoff if you don’t share is

10+(11s-1)(d+d2=…+)=10+(11s-1)/(1-d)

- Sharing is sustained if
- (-3s2+14s-1)/1-d>10+(11s-1)/(1-d)

Interpretation

- There is a cooperative solution for intermediate values of s (the probability of finding blood).
- If s is very large, you don’t share because the chances that you will need help is small.
- If s is very small, you don’t share because the chance that if you need help the other will be able to provide it is small.

Final Exam

- Questions may relate to any of the topics we have covered during the term.
- More emphasis on post midterm chapters.
- You don’t need bluebooks. Do not bring calculators, cell phones, crib sheets, or vampire bats.

May all your subgames be happy..

If not always regular and proper.

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