Econ 101A — Final Exam
Mo 18 May, 2009.
Do not forget to write Problems 1 and 2 in the
f
rst Blue Book and Problems 3 and 4 in the second Blue
Book
.Goodluckonso
lv
ingtheprob
lems
!
Problem 1. Prisoner Dilemma Game with Altruism (40 points)
Consider the standard Prisoner
Dilemma game that we discussed in class. As you remember, the story is one of two prisoners, each of
which has to choose between defecting (that is, confessing) and not defecting (keeping the mouth shut). The
payo
f
s indicate the number of years in prison:
1
\
2
Defection
No Defection
Defection
−
4
,
−
4
−
1
,
−
5
No Defection
−
5
,
−
1
−
2
,
−
2
1. Write the general de
f
nition of Nash Equilibrium. (3 points)
2. Using this de
f
nition, compute all the
pure-strategy
Nash equilibria of the game (that is, do not allow
for probability distributions). (3 points)
3. Now comes the interesting part of the problem. Unlike in the classroom discussion, the prisoners are
altruistic. The utility function of player 1 is a function both of his own payo
f
in years,
π
1
,
but also
of the payo
f
of player 2,
π
1
.
Thus, player one’s utility is:
U
1
=
π
1
+
απ
2
,
with
α
≥
0
.
Similarly
for player 2:
U
2
=
π
2
+
απ
1
.B
r
i
e
F
y explain why the parameter
α
captures altruism, and discuss the
special cases
α
=0
and
α
=1
.
(4 points)
4. This implies that the matrix can be rewritten in terms of utility as
1
\
2
Defection
No Defection
Defection
−
4(1+
α
)
,
−
α
)
−
1
−
5
α,
−
5
−
α
No Defection
−
5
−
α,
−
1
−
5
α
−
2(1+
α
)
,
−
α
)
Compute all the mixed-strategy equilibria of this new game, as a function of
α,
assuming
α
≥
0
.
Call
u
(for Up) the probability that player 1 defects and
l
(for Left) the probability that player 2 defects.
Discuss intuitively why altruism makes a di
f
erence – or does not make a di
f
erence – in this game.
(15 points)
5. Under what values of
α
is (
s
∗
1
=
No Defection,
s
∗
2
=
No Defection) an equilibrium in dominant strategies
for this game? State clearly the de
f
nition of Dominant Strategy equilibrium and how it di
f
ers from
Nash Equilibrium. (7 points)
6. Now let’s go back to the standard case with no altruism (
α
), but assume that the Prisoner’s
Dilemma game is played twice, instead of once. That is, the game is repeated once. Each time, the
matrix of payo
f
s above applies. Debate this assertion: “If the prisoners meet twice, we will observe the
No Defection equilibrium even in the absence of altruism because the repetition provides an opportunity
for collusion” Use backwards induction and be as precise as you can about your statements. (8 points)
Solution to Problem 1
.