Department of Economics
The Ohio State University
Econ 805—Homework #3 Answers
Prof James Peck
Winter 2008
1.
Mas-Colell, exercise 7.D.1.
Answer:
A strategy must specify what actions are taken at each of a
player’s information sets. Thus, the number of strategies is the product,
N
Y
n
=1
M
n
.
2.
Consider the following game. The players are three voters, who must
vote either YES or NO (no abstentions).
The referendum will succeed if at
least two of the voters vote YES, and it will fail otherwise. Suppose that player
1 receives a payo
f
of
2
if the referendum succeeds, and a payo
f
of
−
2
if it fails.
Players 2 and 3 receive a payo
f
of
−
1
if the referendum succeeds, and a payo
f
of
1
if it fails. Assume that we have a private ballot, where players cannot see
how the other players vote until the ballots are counted.
(a) What is the normal form of this game? That is, for
i
=1
,
2
,
3
,write out
the strategy sets,
S
i
,andthepayo
f
functions,
π
i
.
(b) Express the normal form game using matrices.
(For a three player
game, you can write it as a collection of matrices, one matrix for each strategy
of player 3. Player 3 chooses the matrix, player 1 chooses the row, and player 2
chooses the column. Each box of each matrix has three numbers, corresponding
to the payo
f
s of the three players.)
(c) Represent this game in extensive form.
Answer:
(a) For
i
,
2
,
3
,wehave
S
i
=
{
yes,no
}
. The payo
f
functions
are given by
π
1
(
yes, yes, yes
)=2
π
1
(
yes, yes, no
π
1
(
yes,no,yes
π
1
(
yes,no,no
)=
−
2
π
1
(
no, yes, yes
π
1
(
no,yes,no
−
2
π
1
(
no, no, yes
−
2
π
1
(
no, no, no
−
2
1