6With multiple equilibria
T = {0,1}
3,30,00,0B
0,04,40,5M1
0,05,01,1T
RM2L21
Infinitely repeated Games with
observable actions
• T = {0,1,2,…,t,…}
• G = “stage game” = a finite game
• At each t in T, G is played, and players
remember which actions taken before t;
• Payoffs = Discounted sum of payoffs in the
stage game.
• Call this game G(T).
7Definitions
The Present Value of a given payoff stream π =
(π0,π1,…,πt,…) is
PV(π;δ) = Σ∞t=1 δtπt = π0 + δπ1 + … + δtπt +…
The Average Value of a given payoff stream π is
(1−δ)PV(π;δ) = (1−δ)Σ∞t=1 δtπt
The Present Value of a given payoff stream π at t is
PVt(π;δ) = Σ∞s=t δs-t πs = πt + δπt+1 + … + δsπt+s +…
Infinite-period entry deterrence
Strategy of Entrant:
Enter iff
Accomodated before.
Strategy of Incumbent:
Accommodate iff
accomodated before.
1 2 Enter
X
Acc.
Fight
(0,2) (-1,-1)
(1,1)
8Incumbent:
• V(Acc.) = VA =
• V(Fight) = VF =
• Case 1: Accommodated before.
– Fight =>
– Acc. =>
• Case 2: Not Accommodated
– Fight =>
– Acc. =>
– Fight Ù
Entrant:
• Accommodated
– Enter =>
– X =>
• Not Acc.
– Enter =>
– X =>
Infinitely-repeated PD
• VD = 1/(1−δ);
• VC = 5/(1−δ) = 5VD;
• Defected before (easy)
• Not defected
– D =>
– C =>
– C Ù
1,16,0D
0,65,5C
DC
A Grimm Strategy:
Defect iff someone
defected before.
9Tit for Tat
• Start with C; thereafter, play what the other
player played in the previous round.
• Is (Tit-for-tat,Tit-for-tat) a SPE?
• Modified: Start with C; if any player plays
D when the previous play is (C,C), play D
in the next period, then switch back to C.
Folk Theorem
Definition: A payoff vector v = (v1,v2,…,vn) is feasible
iff v is a convex combination of some pure-strategy
payoff-vectors, i.e.,
v = p1u(a1) + p2u(a2) +…+ pku(ak),
where p1 + p2 +…+ pk = 1, and u(aj) is the payoff
vector at strategy profile aj of the stage game.
Theorem: Let x = (x1,x2,…,xn) be s feasible payoff
vector, and e = (e1,e2,…,en) be a payoff vector at
some equilibrium of the stage game such that xi > ei
for each i. Then, there exist δ < 1 and a strategy
profile s such that s yields x as the expected
average-payoff vector and is a SPE whenever δ > δ.
10
Folk Theorem in PD
• A SPE with PV
(1.1,1.1)?
– With PV (1.1,5)?
– With PV (6,0)?
– With PV (5.9,0.1)?
1,16,0D
0,65,5C
DC
Infinitely-repeated
Cournot oligopoly
• N firms, MC = 0; P = max{1-Q,0};
• Strategy: Each is to produce q = 1/(2n); if any
firm defects produce q = 1/(1+n) forever.
• VC =
• VD =
• V(D|C) =
• Equilibrium Ù
11
0 20 40 60 80 100
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 200 400 600 800 1000
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
12
IRCD (n=2)
• Strategy: Each firm is to produce q*; if any one
deviates, each produce 1/(n+1) thereafter.
• VC = q*(1-2q*)/(1-δ);
• VD = 1/(9(1-δ));
• VD|C = max q(1-q*-q) +δVD
• Equilibrium iff
• Ù
( ) ( )δ
δ
−
+−=
19
4/*1 2q
( ) ( )( ) 9/4/*11*21* 2 δδ +−−≥− qqq
)9(3
59* δ
δ
−
−≥q
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
x
y
x = δ, y = (3-5/3 δ)/(9-δ )
13
Carrot and Stick
Produce ¼ at the beginning; at ant t > 0, produce ¼ if both
produced ¼ or both produced x at t-1; otherwise, produce
x.
Two Phase: Cartel & Punishment
VC = 1/8(1-δ). Vx = x(1-2x) + δVC.
VD|C = max q(1-1/4-q) + δVX = (3/8)2 + δVX
VD|x = max q(1-x-q) + δVX = (1-x)2/4 + δVX
VC ≥ VD|CÙ VC ≥ (3/8)2 + δ2VC + δ x(1-2x)
Ù (1-δ2) VC - (3/8)2 ≥ δ x(1-2x) Ù(1+δ)/8 - (3/8)2 ≥ δ x(1-2x)
VX ≥ VD|CÙ (1-δ)Vx ≥ (1-x)2/4 Ù (1-δ)(x(1-2x) + δ/8(1-δ)) ≥ (1-x)2/4 Ù (1-δ)x(1-2x) + δ/8 ≥ (1-x)2/4
2x2 – x + 1/8 – 9/64δ ≥ 0
(9/4-2δ)x2 – (3-2δ)x +δ/8(1-δ) ≤ 0