11/20/2008
1
The Prisoner’s Dilemma
It’s so famous, the payoffs have names!
T>R>P>S
Clyde Barrow
Cooperate
Defect
T – temptation
R – reward
P – punishment
S – sucker
Bonnie
Parker
Cooperate
R , R
S , T
Defect
T , S
P, P
Finitely Repeated Prisoner’s Dilemma
Clyde Barrow
Cooperate
Defect
Bonnie
Parker
Cooperate
3 , 3
1 , 4
Defect
4 , 1
2, 2
“Solving” the Prisoner’s Dilemma means
finding ways of achieving cooperation
Tit-for-Tat invented by Anatol Rapoport
Begin by playing Cooperate (Nice strategy)
Each round, do what opponent did last round
Finitely Repeated Prisoner’s Dilemma
Tit-for-Tat performed well in Axelrod’s
tournament
Tit-for-Tat vs. Tit-for-Tat is not an SPE in a
finitely repeated PD, no matter how many
times it is repeated.
Tit-for-Tat vs. Tit-for-Tat is not a Nash
Equilibrium in the finitely repeated PD
Axelrod’s results are not robust
What if we repeated the PD infinitely many
times?
Repeated Play of Prisoner’s Dilemma
How do we represent preferences over
outcomes of repetitions of a game?
The Present Value of $x n years from now:
PV = $
x/(1+r)
n
PV
$
x/(1 r)
We assume that a player “discounts” future
payoffs the same way we discount future money
Let
denote the discount factor for future
payoffs.
Here,
is like 1/(1+r)
We value a string of payoffs 2, 2, 3, 2 as
1
0
<
<
δ
2
3
2
2
3
2
δ
δ
δ
+
+
+
δ
Repeated Play of Prisoner’s Dilemma
Discounted sums:
1
0
<
<
δ
n
n
S
δ
δ
δ
+
+
+
+
=
"
2
1
1
2
1
1
+
+
+
+
+
+
=
n
n
S
δ
δ
δ
"
Solving for
yields
Exercise: What is
n
n
n
S
S
⋅
+
=
+
=
+
δ
δ
1
1
n
S
δ
δ
−
−
=
+
1
1
1
n
n
S
n
n
T
δ
δ
δ
+
+
+
=
"
2
Repeated Play of Prisoner’s Dilemma
Answer:
, so that
1
)
(
1
−
=
=
−
n
n
n
S
S
T
δ
δ
δ
δ
−
−
=
+
1
1
n
n
T

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