1. Game Theory
Fall Mike Shor Topic 5

2. “Доверяй, Но Проверяй.” (Trust But Verify) – Old Russian Proverb
Game Theory © Mike Shor 2018

3. Predicting likely outcome of a game
Review
Predicting likely outcome of a game
Sequential (Look forward and reason back)
Simultaneous (Look for simultaneous best replies)
What if interaction is repeated?
When is cooperation possible?
Game Theory © Mike Shor 2018

4. Firm 2 Firm 1 Equilibrium $54K Low High 54 , 54 72 , 47 47 , 72
Prisoner’s Dilemma
Equilibrium
$54K
Firm 2
Low
High
Firm 1
54 , 54
72 , 47
47 , 72
60 , 60
Cooperation
$60K
Game Theory © Mike Shor 2018

5. “Sometimes rational decisions aren't sensible!” – Ian Stewart
mathematician
& science fiction author
Game Theory © Mike Shor 2018

6. Private rationality  Collective irrationality
Prisoner’s Dilemma
Private rationality  Collective irrationality
Dominant strategies makes everyone worse off
Goal:
Mutually beneficial cooperative outcome
Overcome incentives to cheat
(A note about tacit collusion)
Game Theory © Mike Shor 2018

7. Repeated Games
Interaction between the same players playing the same (or similar) game multiple times

8. People think forward if:
Finite Interaction
Cooperation is impossible if the relationship between players is for a fixed and known length of time.
People think forward if:
Game length is uncertain
Game length unknown
Game length too long
Game Theory © Mike Shor 2018

9. Long-Term Interaction
No last period, so no rollback
Use history-dependent strategies
Game Theory © Mike Shor 2018

10. Cooperate as long as the rivals do Upon observing a defection:
Trigger Strategies
Begin by cooperating
Cooperate as long as the rivals do
Upon observing a defection:
Immediately initiate a period of “punishment” – play non-cooperatively for a specified period of time
Game Theory © Mike Shor 2018

11. Grim trigger strategy (GTS)
Trigger Strategies
Grim trigger strategy (GTS)
Cooperate until a rival deviates
Once a deviation occurs, play non-cooperatively forever
Tit-for-tat (TFT)
Cooperate if rival cooperated in the most recent period
Don’t cooperate if rival didn’t in the most recent period
Game Theory © Mike Shor 2018

12. “Is cooperation possible?”
Trigger Strategies
Tit-for-Tat is
Most forgiving
Shortest memory
Proportional
Credible,
but lacks deterrence
“Is cooperation easy?”
Grim Trigger is
Least forgiving
Longest memory
MAD
Adequate deterrence,
but lacks credibility
“Is cooperation possible?”
Game Theory © Mike Shor 2018

13. Why Cooperate against GTS?
Firm 2
Low
High
Firm 1
54 , 54
72 , 47
47 , 72
60 , 60
Cooperate: 60 today, 60 next year, 60 … 60
Defect: 72 today, 54 next year, 54 … 54
Game Theory © Mike Shor 2018

14. Payoff streams (GTS) 72 cooperate 60 54 defect t t+1 t+2 t+3 time
Game Theory © Mike Shor 2018

15. $1 tomorrow is worth less than $1 today If the interest rate is r:
Discounting
$1 tomorrow is worth less than $1 today
If the interest rate is r:
Invest $1 today to get $(1+r) next year
$1 next year is worth $1/(1+r) today
Annuity paying $1 today and $1 every year
is worth $1+1/r today
Game Theory © Mike Shor 2018

16. Aside: Infinite Discounting
Why?
Game Theory © Mike Shor 2018

17. Why Cooperate against GTS?
Cooperate if the present value of cooperation is higher than the present value of defection.
PV(cooperation)
60…60…60…60… /r
6/r r
>
<
PV(defection)
72…54…54…54… /r 12
6/12 = 50%
Game Theory © Mike Shor 2018

18. Why Cooperate against GTS?
Cooperate if the present value of cooperation is higher than the present value of defection.
In this case, cooperation is sustainable using grim trigger strategies as long as r < 50%
Or, as long as $1 invested today does not return more than $1.50 next period
Game Theory © Mike Shor 2018

19. Payoff streams (TFT) 72 defect once cooperate 60 54 defect t t+1 t+2
time
Game Theory © Mike Shor 2018

20. Why Cooperate against Tit-for-Tat?
PV(cooperation)
60…60…60…60…
/(1+r)
13/(1+r) 13 r
>
and
<
PV(defect forever)
PV(defect once)
72…47…60…60…
/(1+r) 12 r
1/12 = 8.3%
Game Theory © Mike Shor 2018

21. Why Cooperate against Tit-for-Tat?
Cooperate if the present value of cooperation is higher than the present value of defection.
In this case, cooperation is sustainable using grim trigger strategies as long as r < 8.3%
Much harder to sustain than grim trigger
In this case, cooperation may not be likely
Game Theory © Mike Shor 2018

22. Grim Trigger and Tit-for-Tat are extremes Balance two goals:
Trigger Strategies
Grim Trigger and Tit-for-Tat are extremes
Balance two goals:
Deterrence
GTS is adequate punishment
Tit-for-tat might be too little
Credibility
GTS hurts the punisher too much
Tit-for-tat is credible
Game Theory © Mike Shor 2018

23. Main Ideas from Axelrod
Not necessarily tit-for-tat
Doesn’t always work
Don’t be envious
Don’t be the first to cheat
Reciprocate opponent’s behavior
Both cooperation and defection
Don’t be too clever
Game Theory © Mike Shor 2018

24. Finite interaction versus infinite interaction Folk theorem
Theoretical Asides
Finite interaction versus infinite interaction
Folk theorem
Game Theory © Mike Shor 2018

25. Challenge of Cooperation Requires deterrence Requires credibility
Summary
Challenge of Cooperation
Higher payoffs today versus future relationship
Requires deterrence
A clear, provocable policy of sufficient punishment
Requires credibility
Must incorporate forgiveness
Looking ahead:
How to be credible?
Game Theory © Mike Shor 2018