1. Game Theory
Lecture 8

2. problem set 8 from Osborne’s Introd. To G.T.
Ex. (426.1), 428.1, 429.1, 430.1, 431.1, (431.2)

3. Not in a finitely repeated Prisoners’ Dilemma.
A reminder
Repeated games
The grim (trigger) strategy
Begin by playing C and do not initiate a deviation from C
If the other played D, play D for ever after.
i.e. is the pair (grim , grim) a N.E. ??
Is the grim strategy a Nash equilibrium?
Not in a finitely repeated Prisoners’ Dilemma. C D
2 , 2
0 , 3
3 , 0
1 , 1
Punishment does not seem to work in the finitely repeated game.

4. > > > Proof: player 1 … C D …. player 2 ? player 1 … D ….
Every Nash equilibrium of the finitely repeated P.D. generates a path along which the players play only D
Proof:
Consider the last time that any of the players plays C along the Nash Equilibrium path. (assume it is player 1)
After that period they both play D
player 1 … C D ….
player 2 ?
>
>
>
He is better off here
If he switches to play D:
player 1 … D ….
player 2 ?

5. A reminder infinitely An infinitely repeated prisoners’ Dilemma
sub-games 1 2 C D D 1 2 1 2 1 2 1 2 1 C D 1 C D

6. An infinitely repeated game
A history at time t is: { a1, a2, ….. at }
where ai is a vector of actions taken at time i
ai is [C,C] or [DC] etc.
A strategy is a function that assigns an action for each history.

7. An infinitely repeated game
The payoff of player 1 following a history
{ a0, a1, ….. at,...… }
is a stream { G1(a0), G1(a1), ….. G1(at)...… }

8. An infinitely repeated game
If the payoff stream of a player is a cycle (of length n):
w0,w1,w2,……wn-1,w0,w1,w2,……wn-1, w0,w1,w2,……wn-1, ………
his utility is:

9. An infinitely repeated game

10. Not in a finitely repeated Prisoners’ Dilemma.
An infinitely repeated game
Is the pair (grim , grim) a N.E. ??
Not in a finitely repeated Prisoners’ Dilemma.
(grim,grim) is a N.E. in the infinitely repeated P.D. if the discount rate is sufficiently large
i.e. if the future is sufficiently important

11. Assume that player 2 plays ‘grim’:
If at some time t player 1 considers deviating from C (for the first time)
time …
t-2
t-1 t
t+1
t+2
player 1 C D
player 2
time …
t-2
t-1 t
t+1
t+2
player 1 C
player 2
time …
t-2
t-1 t
t+1
t+2
player 1 C
player 2 D …. ? D ….
while if he did not deviate:
time …
t-2
t-1 t
t+1
t+2
player 1 C ….
player 2

12. time … t-2 t-1 t t+1 t+2 player 1 C D …. player 2 time … t-2 t-1 t t+1
2 , 2
0 , 3
3 , 0
1 , 1
The payoffs:
time …
t-2
t-1 t
t+1
t+2
player 1 C D ….
player 2
time …
t-2
t-1 t
t+1
t+2
player 1 C D ….
player 2
time …
t-2
t-1 t
t+1
t+2
player 1 C ….
player 2
time …
t-2
t-1 t
t+1
t+2
player 1 C ….
player 2

13. (grim,grim) is a N.E. if the discount rate is sufficiently large
2 , 2
0 , 3
3 , 0
1 , 1
Player 1 will not deviate if:
(grim,grim) is a N.E. if the discount rate
is sufficiently large
i.e. if the future is sufficiently important

14. (grim,grim) is not a Sub-game Perfect equilibrium of the game
However,
(grim,grim) is not a Sub-game Perfect equilibrium of the game
Assume player 1 follows the grim strategy,
and that in the last period C,D was played
player 1 … C
player 2 D D …. C D …. D
....
Player 1’s (grim) reaction will be:
If Player 1 follows grim:
but he could do better with :

15. Strategies as Finite Automata
A finite automaton has a finite xxxxnumber of states (+ initial state)
Each state is characterized xxxxby an action
Input changes the state of the xxxxautomaton

16. C
C,D C D D
The grim strategy
A state and its action
Inputs : The actions of the other player { C,D }
The transition: How inputs change the state
Initial State

17. Modified Grim 1 2 3 4 Some more strategies C D C,D C,D C C D D D D D

18. Some more strategies
Tit for Tat C D D C D C

19. Axelrod’s Tournament (Nice !!!) Tit for Tat
Robert Axelrod: The Evolution of Cooperation, 1984 C D
Robert Axelrod
Tit for Tat
(Nice !!!) C C D D C D

20. Modified Tit for Tat Tit for Tat Can you still bite ??? D C C C D D C
‘simpler’ than D D D C

21. a strategy that exploits the weakness of
Can you still bite ??? D C C C D C D C C D D
a modification
a strategy that exploits the weakness of C C
C,D D D D C

22. What payoffs are N.E. payoffs of the infinitely repeated P.D. ??
The FEASIBLE payoffs as π1 π2 C D
2 , 2
0 , 3
3 , 0
1 , 1

23. The folk theorem ???? (R. Aumann, J. Friedman)
What payoffs are N.E. payoffs of the infinitely repeated P.D. ??
Clearly, Nash Equilibria payoffs are ≥ (1,1)
????
All feasible payoffs above (1,1) can be obtained as Nash Equilibria payoffs π1 π2
The folk theorem
(R. Aumann, J. Friedman) C D
2 , 2
0 , 3
3 , 0
1 , 1
(1,1)

24. All feasible payoffs above (1,1) can be obtained as Nash Equilibria payoffs
Proof:
choose a point in this region π1 π2
it can be represented as: C D
2 , 2
0 , 3
3 , 0
1 , 1
(1,1)

25. The coefficients αi can be approximated by rational numbers
Proof:
The coefficients αi can be approximated by rational numbers π1 π2 C D
2 , 2
0 , 3
3 , 0
1 , 1
(1,1)

26. Proof: { { { { π1 π2
If the players follow this cycle, their payoff will be approximately the chosen point when the discount rate is close to 1. C D
2 , 2
0 , 3
3 , 0
1 , 1
(1,1)

27. Follow the sequence of the cycle as long as the other player does.
Proof: { { { {
A strategy:
Follow the sequence of the cycle as long as the other player does.
If not, play D forever. π1 π2 C D
2 , 2
0 , 3
3 , 0
1 , 1
(1,1)

28. Follow the sequence of the cycle as long as the other player does.{ { { {
A strategy:
Follow the sequence of the cycle as long as the other player does.
If not, play D forever. π1 π2
This pair of strategies is a N.E. C D
2 , 2
0 , 3
3 , 0
1 , 1
(1,1)

29. One Deviation Property and Agent Equilibria
A player cannot increase his payoff in a sub-game in which he is the first to move, by changing his action in that node only.
A Theorem
A strategy profile is a sub-game perfect equilibrium in an extensive game with perfect information iff both strategies have the one deviation property
(no proof)