Infinitely repeated games
Prisoners’
dilemma
Discounted infinitely repeated game
u2
D
C
D 1, 1 3, 0
3
2
C 0, 3 2, 2
1
Can repetition discipline
1
the players to cooperate?
Deviating yields a short run gain.
Deviating yields a loss of reputation
that undermines future cooperation.
2
u1
3
Yes, if gain
now  PV of
future loss
• (Average) payoff in a repeated game
(1   )t 1  t 1vt

• Discounting
p: Probability that the game ends.
1 p

1 r
r: Rate of time preference.
Doubling of detection lag decreases   2
• Let G  N, (Ai), (ui) be a normal form game,
where Ai is the set of actions for player i and ui is
the payoff function for player i.
• A -discounted infinitely repeated game of G
is an extensive game where G is played infinitely
many times, where players can observes the
actions of previous rounds and where the payoff
function for player i is given by
ui (( a t ))  (1   )t1 t 1ui (a t )
Simple strategy profiles (Abreu, 1988)
Consider the following paths:
(a (0) t )  a (0)1 , a (0) 2 , a (0) 3 , a (0) 4 , 
(a (1)t )  a(1)1 , a(1) 2 , a (1)3 , a(1) 4 , 

(a (n)t )  a(n)1 , a (n) 2 , a(n)3 , a (n) 4 , 
• Why multiply with (1  )?
Suppose that vt  v for all t. Then

(1   )t 1  t 1vt  (1   )(1     2  )v  v
Special case 1: v1  v , but vt  0 for all t  1.

(1   )t 1  t 1vt  (1   )v
Special case 2: v1  0 , but vt  v for all t  1.

(1   )t 1  t 1vt  v
The one-deviation property
• A strategy profile is a subgame
perfect equilibrium for the discounted infinitely repeated game
of G if and only if there is no history
after which a player can gain by
deviating for a single period.
Rules: Start with (a(0)t).
If i  N (and only i) deviates from (a(j)t), then (re)start (a(i)t).
Write:
 (a(0)t ), (a (1)t ),  , (a(n)t ) 
u2
• The set of payoff profiles that
can be realized in subg. perf. equilibria for the -disc. infinitely repeated game of G is compact. m(2)
• Consider the -disc. infinitely
Set of subgame
perfect equilibrium
payoffs
u1
m(1)
repeated game of G. For each
player i  N, there is (a) a minimal payoff m(i) that
can be realized as a subg. perf. equilibr. payoff, and
(b) a subg. perf. equilibr. path (aˆ (i)t ) s.t. ui ((aˆ (i )t ))  m(i )
• Consider the -disc. inf. repeated game of G.
Then (at) is a subg. perf. equilibr. path if and only if
t
t
t
the simple strategy profile   (a (0) ), (aˆ (1) ), , (aˆ (n) ) 
is a subg. perf. equilibrium.
1
Cooperation in infinitely rep.
Prisoners’ Dil.
u2
" Trigger strategy" :
2
Play D if D has been used earlier;
1
otherwise play C . (Start with C .)
u2
“Getting Even”
3
1
2
u1
3
If cooperation breaks down, it will never be restarted.
Subgame perfect equilibrium (no profitable 1-per. dev.)?
If cooperation has broken down: No profitable 1-per. dev.
If cooperation has not broken down:
Short-run gain  PV of long-run loss
1
 (1   )(3  2)   (2  1)  1   2   
2
3
Play C , unless " permitted" to play D.
2
" Permitted" to play D if the opponent
1
played D w/o " perm" in previous period.
1
If 1 deviates, then 1 is punished in the next period.
Short-run gain
 PV of loss in next period
If cooperation has not broken down:
1
 3  2   ( 2  0)  1   2   
2
If cooperation has broken down:
1
 1  0   ( 2  0)  1   2   
2
Repeated Prisoners’ Dil.: Conclusions
Repeated Bertrand comp.
Consider the following paths:
P (Q )  60  Q
q1m  q2m  15
u1
2
3
2
Coop.: Both set monopoly price:
(a t )  (C , C ), (C , C ), (C , C ),
(nt )  ( D, D), ( D, D ), ( D, D),
(a (1)t )  (C , D), (C , C ), (C , C ),
(a (2)t )  ( D, C ), (C , C ), (C , C ),
  (a t ), (nt ), (nt )  is a subg. perf. equilibrium if   12
  (a t ), (a (1)t ), (a (2)t )  is a subg. perf. equilibrium if   12
Repeated Bertrand comp.: Conclusions
Consider the following paths:
( p t )  (30,30), (30,30), (30,30),
(nt )  (0, 0), (0, 0), (0, 0),
  ( p t ), (nt ), (nt )  is a subg. perf. equilibrium if   12
p m  30  p1m  p2m
 1m   2m  450
1
If deviation, coop. breaks down:
b
b
b
b
b
b
p1  p2  0
1   2  0
q1  q2  30
Subgame perfect equilibrium (no profitable 1-per. dev.)?
If cooperation has broken down: No profitable 1-per. dev.
If cooperation has not broken down:
Short-run gain
PV of long-run loss

1
 (1   )(900  450)   (450  0)  450   900   
2
Repeated Cournot comp.
2
Coop.: In total, monopoly quantity.
P (Q )  60  Q
q1m  q2m  15
m
p  30
 1m   2m  450
If deviation, coop. breaks down:
1
 1c   2c  400
q1c  q2c  20
p c  20
Subgame perfect equilibrium (no profitable 1-per. dev.)?
If cooperation has broken down: No profitable 1-per. dev.
If cooperation has not broken down:
Short-run gain  PV of long-run loss
 (1   )(506.25  450)   (450  400)
9
 56.25   106.25   
17
2
Harsher punishments?
2
Coop.: In total, monopoly quantity.
P (Q)  60  Q
q1m  q2m  15
m
p  30
 1m   2m  450
If deviation, price = 0 in 1 period:
p0
1   2  0
q1  q2  30
“Getting Even”
1
Short-run gain  PV of loss in next period
If cooperation has not broken down:
1
 506.25  450   (450  0)  56.25   450   
8
If cooperation has broken down:
1
 225  0   ( 450  0)  225   450   
2
2
Coop.: In total, monopoly quantity.
q1m  q2m  15
 1m   2m  450
1
If 1 deviates, 2 takes over for 1 period:
p  30
q1  0 & q2  30
 1  0 &  2  900
P (Q )  60  Q
p m  30
 PV of loss in next period
Short-run gain
If cooperation has not broken down:
1
 506.25  450   (450  0)  56.25   450   
8
If cooperation has broken down:
1
 225  0   ( 450  0)  225   450   
2
Repeated Cournot comp.: Conclusions
Consider the following paths:
(q t )  (15,15), (15,15), (15,15),
(nt )  (20, 20), (20, 20), (20, 20),
(qˆ t )  (30,30), (15,15), (15,15),
(q (1)t )  (0,30), (15,15), (15,15),
(q (2)t )  (30, 0), (15,15), (15,15),
  (q t ), (nt ), (nt )  is a subg. perf. equilibrium if   179
  (q t ), (qˆ t ), (qˆ t )  is a subg. perf. equilibrium if   12
  (q t ), (q (1)t ), (q (2)t )  is a subg. perf. equilibrium if   12
3