Econ 251
Fall 2006
Dynamic Choice:
American Options and Backward Induction
1
Plans and Actions
How can somebody know what to buy and sell (or how to act) at time
n
unless he has a clear view
about the consequences of his actions for time
n
+1
? But how can he know what the consequences
are for time
n
+ 1
unless he has thought out what he needs for time
n
+ 2
?
To put the problem in other words, how can you know what choice to make today until you
know what choices you will make tomorrow? And tomorrow, how can you know what choice to
make without knowing what choice to make today?
Choices in
fi
nance that occur over di
ff
erent time periods are often called American options.
So how does a trader know how to use his American option?
1.1
Marriage Problem
Consider a politically incorrect man who
fi
gures he will sequentially meet 1000 women in his life.
He is sure that any woman he asks to marry him will say yes, but the trouble is, once he has
passed over a girl, he can never go back and ask her later. He assumes that the suitability of the
women are chosen randomly, independently and uniformly distributed on [0,1], but that he only
sees the suitability of a woman by meeting her. Thus the suitability of girl 3,
v
3
, is a number
drawn at random somewhere between 0 and 1, with equal probability of being at any point. After
seeing her, the man knows
v
3
,
but he does not yet know
v
4
or any other value
ν
t
with
n >
3
before he must decide whether to propose to lady 3. What should he do? Let us assume that the
man wants to pursue the strategy that maximizes the expected suitability of the woman he ends
up with. He cannot guarantee anything for sure, but let us suppose he is indi
ff
erent between a
girl with suitability .5 and a girl whose quality he does not know, because on average the latter
girl will also have quality .5.
After seeing the quality of a candidate, the man has an option: take the woman or not. On
what basis should he decide? All he knows is her suitability, and how many women are left. So
his strategy must be a threshold level changing over time. Let
θ
(
t
)
represent the threshold level
for woman
t
. If her suitability is above
θ
(
t
)
,
he should take her. If her suitability is below
θ
(
t
)
,
he should go on.
optimal strategy
if
v
t
≥
θ
(
t
)
stay with miss
t
if
v
t
<
θ
(
t
)
move on
How high should
θ
(
t
)
be? And how should
θ
(
t
)
depend on
t
? It is clear that if the man has
fewer women left to see, he is in a worse situation. So he must lower
θ
(
t
)
as
t
grows. But can we
say anything more de
fi
nite?
Clearly
θ
(
t
)
should be the quality that the man can expect to end up with if he plays the
game with
1000
−
t
women left. If he can expect to get
θ
(
t
)
(on average) from his choice among
the remaining
1000
−
t
women, there is no point in stopping with a girl before that with quality
1
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less than
θ
(
t
)
.
Conversely, if lady
t
has suitability better than or equal to the
θ
(
t
)
he can expect
by continuing, he should stop with her.
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 '09
 GEANAKOPLOS,JOHN
 Graph Theory

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