Department of Economics
The Ohio State University
Econ 817—Game Theory
Fall 2007
Prof. James Peck
Homework #1 Answers
1. O-R, exercise 19.1.
Answer:
There are
n
players, and each player
i
has the action set,
A
i
=
{
out
}
∪
[0
,
1]
. Each player prefers an action pro
f
le with more votes than any
other player than one in which he/she ties for the most votes; prefers to tie than
to be out; and prefers to be out rather than lose.
With two players, the unique equilibrium is for both players to choose the
median of the distribution,
m
=
F
−
1
(
1
2
)
. Th
i
si
saNE
,becau
sebo
th
f
rms
tie, and any deviation will cause that
f
rm to lose or be out. To see that this
NE is unique, there cannot be a NE in which one of the players is out, because
that player could guarantee at least a tie by choosing the right position. There
cannot be a NE in which the players choose di
f
erent positions, because a player
standing to lose could guarantee at least a tie, and a player standing to tie
could move closer to the other player and thereby win. There cannot be a NE
in which the players choose the same position other than the median, because
they would stand to tie, but a player could deviate closer to the median and
win.
With three players, there cannot be a NE. If all three are out, then one
player could choose a position and win.
If two are out, then one of those
players could guarantee at least a tie by choosing the right position.
If one
player is out, then the other two must be choosing the median voter position; in
that case, the player that is out could choose a position close to the median voter
position and receive almost half the votes, thereby winning. Finally, suppose
all three voters choose positions.
They must tie for
f
rst, because otherwise
being out is preferred by a loser.
If the players choose three distinct points,
then one of the outside players could move closer to the middle and win. If two
of the players choose the same position, then the other player could move closer
to them and win.