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working paper
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TIME CONSISTENT OPTIMAL STOPPING
Lones Smith
No.
97-5
March, 1997
massachusetts
institute of
technology
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Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
TIME CONSISTENT OPTIMAL STOPPING
Lones Smith
No.
97-5
March, 1997
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
MEMORIAL DRIVE
CAMBRIDGE, MASS. 021329
50
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X.
INSTITUTE
j997
Time-Consistent Optimal Stopping
Lones Smith^
Department of Economics
Massachusetts Institute of Technology
Cambridge,
March
JEL
MA
17,
Classification
02139
1997
Number: C44
Keywords: time-consistent, von Neumann Morgenstern
utility, affine,
time preference
Abstract
This paper investigates time consistent preferences where the strategy space
consists of a stopping time,
and the decision-maker
Despite the requirement of identical
vNM
also acts
preferences,
it is
under uncertainty.
shown that
in addi-
tion to the exponential discounting class identified by Strotz (1956), negative
affine discounting also yields
time consistent behavior.
*e-mail address: lones@lones.init.edu
tl thank David Laibson for a clarifying conversation,
sci econ. research.
.
and acknowledge useful feedback from
INTRODUCTION
1.
There
The
is
a rich literature starting with Strotz (1956) examining time-consistent planning.'
insight here
that even in a world of perfect certainty, time preference in a dynamic
is
environment must assume a very
form
specific functional
if
individuals are to have an
incentive to carry out their previously-conceived optimal plans. In a nutshell, current
how
future 'selves' might well disagree on
The
cited literature
is
the wisest course of action for the future
time
t
Strotz'
>
of future consumption at time t
t
main
finding
some
optimal stopping under uncertainty.
/3
consistent preferences
>
is
must meet a sharper
show that discounting may be
is
weighted by 7
—
/3(r
—
know
the
full
—
t,
The
This paper asks this same question
0.
test:
gamut
is
Individuals must enjoy the identical von
as their future selves. In other words, a harsher
in Strotz context.
either exponential or negatively affine
t)
There are two reasons to care about
to
the discounting
exponential discounting:
standard must be met by a smaller class of payoff functions than
consumption
if
But because of the stochastic environment, time-
Neumann-Morgenstern (vNM) preferences
I
that
Here, identical preferences alone, or sooner
preferred to later, has no cutting power.
balance,
—
space and payoff structure that arises in a microeconomics context
for a simpler action
—
e~^^'^~^\ for
is
is
only depends on the horizon length t
then preferences are time-consistent exactly when there
consumption weight
self.
interested in behavior described by a very rich action space
often infinite-horizon consumption paths.
at
and
as of time
(i.e.
On
time-r
t).
this result. First,
of possible time-consistent
it is
useful for
economic theorists
dynamic objective functions under
uncertainty. Second, the precise functional equations required by time-consistency can be
easily
understood
in the simple
optimal stopping context, and qualitatively
differs
from
the deterministic infinite horizon consumption framework.
2.
A
A. The Model.
decision
THE ANALYSIS
maker has a choice
set
each non-stopping action a G ^, a stochastic signal y G
be a
list
of places to look each period,
the sought-for prize.
signals, strategy Ut
is
As a function
a choice of a
new
/i*
in every period. After
F is observed.
and y the simple
of the history
A U {stop}
For instance,
fact that the process hasn't
to time
t
of actions,
A may
found
and observed
(possibly stopping) action, which will stop the
^For a recent and quite encompassing bibliography, see Donoghue and Rabin (1996).
process with chance
stopping at time
>
t
two assumptions:
The
7r(/i*).
goal
is
to stop the process as soon as possible.
weighted by a factor w{t
s is
first strict
— s)
that
is
decreasing in
time preference (stopping sooner
is
t.
Also,
This embeds
preferred to later), and
second time symmetry (since more generally one could envision weighting by w{s,t)).
An
optimal strategy at time
Us{a)
where the expectation
having yet stopped.
B.
The Main
<
Lemma
One's
functions a
>
vNM
and
Thus,
it is
vNM
given the strategy a, conditional on not
exists that stops in finite expected time.
necessary and sufficient that an individual at any
preferences over actions at
+ t) =
a{s)w{t)
vNM
w
behaves
+
exist functions d
>
and
b of (si, S2)
w{t
Since this must hold
nonconstant in
t, it
Thus, a(si,
=
S2)
if
-
si)
we replace
>
Si.
there exist
(2)
=
it is
a(s2
solving (2)
ae"^*
The
+ 7,
necessary and sufficient that there
such that
a{si, S2)w{t
(si, S2, s)
by
-
(si
S2)
+
+
6(si, S2)
r, S2
+
r, s
—
Si)
(3)
+
r),
and because
w
is
= d(si + r, §2 + t) and 6(si S2) = b{si + r, S2 + r)
and 6(si, 82) = b{s2 — Si). Equivalently, putting s = S2 — 5i,
D
(2) for w follows.
,
,
Proposition (Time-Consistent Characterization)
w
vNM utility function. By the
vNM preferences are constant iff
a
like
follows that d(si S2)
the characterizing equation
Proof:
iff
S2
b{s)
Expected Utility Theorem,
these utilities are positively affinely related. Hence,
=
>
Since separate stopping times are mutually exclusive states of the world, with
uniqueness part of the
w{t)
t
such that
b
a pyoff realized just once, the weight
weights
times
all
preferences over future stopping times are constant
w{s
Proof:
(1)
In every period, one's actions affect the evolution of the future
have identical
S2
(yt)
assume that a strategy
I
lotteries over public signals.
Si
= £{Zr=s<t-sMh')W)
over future signals
is
Result.
two times
one that maximizes
s is
and yielding a well-defined
or negatively affine w{t)
functional equation (2)
is
The only monotonic decreasing
objective function (1) are exponential
= — /?t + 7
in time,
where a,
generally solved in Corollary
/?
1 in
>
0.
section 3.1.3 of
Aczel (1966). That result also admits the possibility of a negatively exploding exponential
function {a,
a
finite
P <
0) for
which the objective
(1) will
not be defined. Conversely, assuming
expected stopping time, any vanishing exponential or negative afl&ne function will
D
yield a well-defined objective.
C. Discussion of an Example.
Suppose that one wishes to
get 'as soon as possible' given partial information of its location.^
preferences describe this objective function?
WLOG
set
=
7
in the Proposition
an optimal search strategy
e~^* or
is
—t.
The
is
-t
=
is
lim;3_o(e-^*
It is
for
it
time-consistent
is
is
What
tar-
time-consistent
just single payofi",
iff
we may
the payoff" from discovery in period
t
is
where the player seeks to minimize
the most natural interpretation of the stated goal.
the limit of the exponential weighting as the interest rate
-
moving
clearly will not affect behavior. In this case,
latter negative affine weighting,
the expected discovery time,
also be seen as
—
Since there
find a
/3
It
can
vanishes:
l)//3.
tempting to draw inferences about more general stopping stopping games such as
price search or alternating
oflfer
bargaining. In those cases, where there
general terminal payoff, or intermediate payoffs, the
Lemma
fails,
is
either a
more
as the negatively affine
solution vanishes. This brings us back to the Strotz solution (but with uncertainty).
References
Aczel,
J.,
Lectures on Functional Equations and their Applications,
Academic
New
York:
Press, 1966.
Donoghue, Ted and Matthew Rabin, "Doing
it
Now
or Later," 1996. Berkeley
Mimeo.
Smith, Lones, "Equilibrium and Efficiency in the Rendezvous Problem," 1994. MIT
Mimeo.
Strotz, R., "Myopia and Inconsistency in Dynamic Utility Maximization," Review of
Economic Studies, 1956, 23, 165-180.
^This problem
3
08
is
9
quite relevant: See Smith (1994) for a compelling coordination
i;
3
3
game with
this flavor.
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