–1–
Prof. William H. Sandholm
Department of Economics
University of Wisconsin
March 10, 2005
Midterm Solutions – Economics 713
1. Let
v
i
be player
i
’s minmax value:
v
i
=
min
!
"
i
#
$
A
"
i
max
a
i
#
A
i
u
i
(
a
i
,
a
"
i
)
.
Let
F
= convex hull({
u
(
a
):
a
∈
A
}) and
IR
= {
v
:
v
i
>
v
i
for all
i
}. The Classic Folk
Theorem says that if
v
∈
F
∩
IR
, then for all
δ
close enough to one, there is a Nash
equilibrium of
G
!
(
"
)
with payoffs
v
.
(
ii
)
Let
u
(
ˆ
a
) =
v
. Suppose each player follows this grim trigger strategy: “Play
ˆ
a
i
so long as there has never been a period in which exactly one player has deviated; if
player
j
was the first player to unilaterally deviate, then do your part in minmaxing
j
forever.” Then on the equilibrium path, player
i
cannot benefit from deviating if
v
i
>
(1
!
)max
a
#
A
u
i
(
a
)
+
v
i
.
Since
v
i
>
v
i
, this inequality holds so long as
is close enough to 1.
2. (
i
)
v
∈
K
if and only if
v
= (
u
1
(
T
,
σ
2
),
u
1
(
B
,
2
)) for some
2
∈
Δ
S
2
.
(
ii
)
c
* = 2. This is player 1’s minmax value: by playing
1
2
b
+
1
2
c
, the strategy that
generates (2, 2), player 2 ensures that player 1 cannot obtain a payoff higher than 2. If
you draw a picture of
K
, you will see that player 2 cannot restrict 1 to a lower payoff.
(
iii
)
p
* = (
2
3
,
1
3
) and
d
* = 2.
(
iv
)
Let
1
* =
p
*. Then (
p
* ·
v
≥
2 for all
v
∈
K
) is equivalent to (
u
1
*(
1
*,
2
)
≥
2 for all
2
∈
S
2
). Thus, player 1’s maxmin payoff is at least 2; by part (
ii
), it must be exactly 2.
(
v
).
Define
n
-strategy analogues of the sets
J
,
K
, and
L
(
c
). Let
c
* be the largest
value of
c
such that int(
L
(
c
)) and int(
K
) do not intersect. (Notice that if, for example,
player 1 has a dominated strategy,
L
(
c
*)
∩
K
may not include (
c
*, … ,
c
*); this is why we
need the set
L
(
c
*).) If player 2 chooses a
2
that generates a point in