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Suggested Solutions to Problem Set 6
ECN 416 Game Theory
1
. Suppose you are trying to buy a house and you are bargaining with the current
owner over the sale price. The house is worth
$200
000
to you and
$100
000
to the
current owner. So if the price is between
$100
000
and
$200
000
, then you would both
be better o
f
with the sale. Assume the bargaining takes place with alternate o
f
ers and
that each stage of bargaining (an o
f
er and a response) takes a full day to complete.
If agreement is not reached after
5
days of bargaining, then the opportunity for the
sale disappears (you will have no house and the current owner has to keep the house
forever). Suppose that you and the current owner discount the future according to the
discount factor
per day. The real estate agent has allowed you to decide whether or
not you shall make the
f
rst o
f
er.
(a)
Suppose that
is small. In particular
1
2
. Should you make the
f
rst o
f
er
or let the current owner make the o
f
er?Why?
Asshowninclass,ifyoumakethe
f
rst o
f
er, then your equilibrium payo
f
is
1
−
5
1+
and
if let the other party make the
f
rst o
f
er, then your equilibrium payo
f
is
³
1
−
1
−
5
1+
´
=
+
5
1+
.Soi
f
1
2
,then
1
−
5
1+
+
5
1+
and you are better o
f
making the
f
rst o
f
er.
(b)
Suppose that
is close to
1
;inpart
icu
lar
5
1
2
. Should you make the
f
rst
o
f
er or let the current owner make the
f
rst o
f
er? Why?
If
5
1
2
,then
1
−
5
1+
+
5
1+
and you should let the current owner make the
f
rst
o
f
er.
2.
Consider the following discounted, three period bargaining game. The players
divide
$1
. The common discount factor is
,where
0
1
. In this game, player 1
makes the
f
rst o
f
er. If player
2
rejects this o
f
er, then player
1
makes
another
o
f
er.
If player
2
rejects the second o
f
er, then player
2
makes the
f
nal o
f
er. If no o
f
er is
accepted at the end, both players get
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 Spring '10
 Y.Chen
 Game Theory

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