A Bold Strategy Is Not Always Optimal in the Presence of Inflation
Author(s): Robert W. Chen, Larry A. Shepp, Alan Zame
Source: Journal of Applied Probability, Vol. 41, No. 2 (Jun., 2004), pp. 587-592
Published by: Applied Probability Trust
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J. Appl. Prob. 41, 587-592 (2004)
Printed in Israel
© AppliedProbabilityTrust2004
A BOLD STRATEGY IS NOT ALWAYS OPTIMAL
IN THE PRESENCE OF INFLATION
ROBERTW. CHEN,*** Universityof Miami
LARRYA. SHEPP,***Rutgers University
ALAN ZAME,* Universityof Miami
Abstract
A gambler,with an initial fortune less than 1, wants to buy a house which sells today
for 1. Due to inflation, the price of the house tomorrowwill be 1 + ca, where ca is a
nonnegativeconstant,and will continueto go up at this rate, becoming (1 + a)n' on the
nth day. Once each day, he can stake any amountof fortunein his possession, but no
morethanhe possesses, on a primitivecasino. It is well knownthat,in a subfairprimitive
casino withoutthe presenceof inflation,the gamblershouldplay boldly. The presenceof
inflationwould motivatethe gamblerto recognize the time value of his fortuneand to try
to reachhis goal as quickly as possible; intuitively,we would conjecturethatthe gambler
should again play boldly. However,in this note we will show that, unexpectedly,bold
play is not necessarilyoptimal.
Keywords:Gamblingproblem;primitivecasino; optimalstrategy;bold strategy
2000 MathematicsSubjectClassification:Primary60G40
Secondary60K99
1. Introduction and formulation of the problem
Suppose that you have x dollars and you want to buy a house which sells today for y dollars,
>
x > 0. Due to inflation, the price of the house tomorrow will be (1 + a)y (where a > 0),
y
and will continue to go up at this rate each day, so as to become (1 +a)ny on the nth day. Once
each day, you can stake any amount of money in your possession, but no more than you possess,
on a 'primitive casino', i.e. one in which only one type of bet is available. If you make a bet,
you gain r times your stake with probability w and lose your stake with the complementary
probability w = 1 - w, where r is a positive constant. How much should you stake each day
so as to maximize your chance of eventually catching up with inflation and being able to buy
the house?
When w < 1/(1 + r), the primitive casino is subfair and, when w = 1/(1 + r), the primitive
casino is fair. As is well known, in a subfair or fair primitive casino without the presence of
inflation, the gambler should play boldly since there is no other strategy that provides him with
a higher probability of reaching his goal (being able to buy the house). In [3], Chen proved
that the gambler should play boldly when r = 1 (when r = 1, a primitive casino is called
'red-and-black'). The presence of inflation would intuitively motivate the gambler to recognize
the time value of his fortune and to try to reach his goal as quickly as possible. We would
Received 20 June 2003; revision received 21 October2003.
* Postal address:Departmentof Mathematics,Universityof Miami, CoralGables, FL 33124-4250, USA.
** Email address:[email protected]
***Postal address: Departmentof Statistics, Rutgers University, 110 FrelinghuysenRoad,
Piscataway,NJ 08854,
USA.
587
R. W.CHENETAL.
588
suspect that the gamblershould again play boldly. However,in this note we will show that,
surprisingly,bold play is not necessarily optimal. This result is a counterexampleto a 1909
assertionby Coolidge. Coolidge [5] stated:
Theplayer'sbestchanceof winninga certainsumata disadvantageous
gameis to stakethe
sumthatwill bringhimthatreturnin one play,or,if thatbe notallowed,to makealways
thelargeststakewhichthebankerwill accept.
Different counterexamplesto Coolidge's statementhave also been provided in [2], [3], [8],
and [9].
To makethis new gamblingproblemfit moreclearlyandeasily into the gamblingframework
of DubinsandSavage [6], we can assumethatthe priceof the house is fixed at 1 butthe value of
money will be discountedstep by step by the fixed discountrate ( 1 + a)-1, i.e. the currentone
dollarwill be worthonly (1 + a)-1 dollarsat the next step. Therefore,our gamblingproblem
can be formallyformulatedas a game whose set of fortunes,utilityfunction,andset of available
gambles are respectivelyas follows:
F =[0, oo),
u(f)
u(f) =
O
I
if 0
<l,
iff >, 1,<
iff
'y(f,s)
(
-s
0<s<f,
y(f,s)=ws +ww(f
( +rs)
1+ta
< f
if
r(f) =
< f < 1,
+
(Af)
, y(f,s) =Wy(f+) w(
f
rs) +
( f-s
if l < f <o.
Here w = 1 - w, 3(x) denotes the probabilitymeasurewhich assigns probability1 to {x) for
x E [0, oo), and a is the inflationrate (a device often encounteredin dynamicprogramming
models) for the gambler.The reasonthat3(f) is in F(f) for f > 1 is that, when the gambler
has a fortune f > 1, he has reached his goal already and can buy the house immediately.
Plainly,if a > r, a gamblerwith initial fortunef < 1 can neverreachhis goal, so we assume
thata < r.
The gamblingproblemformulatedaboveis a modificationof the primitivecasino considered
by Dubinsand Savage [6]. The modificationis designedto handleinflationandto motivatethe
gamblerto recognizethe time valueof money andcompletethe game as quicklyas is consistent
with reachingthe goal. To distinguishit from the 'primitivecasino', this modified game will
be called the 'primitivecasino in the presenceof inflation'. When a = 0, the modifiedgame is
identicalto the 'primitivecasino' consideredby Dubins and Savage in [6].
2. The utility of modified bold strategies
We say that the gamblerstakes s when he uses the game y(f, s) at f. The modifiedbold
stake at f is definedby
s(f) =
min
0
+,
rI
f,
if0<
i
if/ f
f < 1
1,
589
A boldStrategyis natalwaysoptimal
where r is a positive constant. A gambler uses the modified bold strategyif he stakes the
modifiedbold stake s(f) wheneverhe has fortunef.
Let Q(f) be the probabilitythatthe gambler,startingfromf E (0, 1) andusing the modified
bold strategy,reaches his goal, [1, oo). It is obvious that Q(0) = 0, Q(f) = 1 if f > 1, and,
if 0 < f < 1, then, consideringone game,
'Q(f) =wQ( f + rs(f)
1+
+ wfQ( f-s(f)
1±l+a
Therefore,
Q(f) =
0
iff <0,
wQ(f)
if
<
w+WQ(
if
<f<1,
< B,
(1)
)
if f>1,
1
where/ = (1 + a)/(1 + r) < 1 as noted in Section 1.
Lemma 1. The boundedsolution of () is unique.
Proof. The proof is essentially the same as thatin [6, p. 99] and is omitted.
Let {Xn}n>i be a sequence of i.i.d. randomvariablessuch that P(X1 = 0) = w = 1 P(X1 = 1). For each integer n > 1 and each integer k E [l,n), let Sn,n = Xn and
Sn,k = /BXk+ fi[1 + (r - l)Xk] min{1, Sn,k+l-} Since Sn,1 is seen to be nondecreasing in n,
the function G(x) = limn,, P{Sn,1< x} is well definedfor all x.
-
Lemma 2. For all x, G(x) < 1 and thefunction G satisfies (1).
Proof. The proof is similarto thatin [3, pp. 295-296] and is omitted.
In view of Lemmas 1 and 2, we have the following theorem.
Theorem 1. Thereis one and only one boundedfunctionQ on the interval[0, oo) that satisfies
(1). Moreover,thefunctionQ is rightcontinuouson the interval[0, oo) and is strictlyincreasing
on the interval [0, 1].
3. The nonoptimality of modified bold strategies
For each integerk > 0, let Ak be the set of all fortunesf which satisfy the following three
conditions:
(i) There exist k + 1 positive integers 1 < no < n < ..
< nk such that
k
f = E rj,ni.
/=0
j=o
(ii) We have 0 < f < /no-1.
(iii) For each i E {O, 1, 2,...,
k - 2},
k-i-I
E
j=O
rjpni+l+j-ni
< pni+l-ni-1
R.W.CHENETAL.
590
Theorem 2. If f
=
Ij=o
rJfinj is in Ak, then
k
Q(f)=
E Wiwnrij=0
Proof. This can be provedby recursiveapplicationof (1) and mathematicalinduction.
Let
ln(w)
_ln(w)
where [-I is the interger-partfunction,and let
a=
if 1K > ,
if K = 0.
maxO, (1 + r)r-(l+l/K)
0
Lemma 3. Supposethat r > 1 and 0 < w < 1. If the inflationrate a is in the interval [a, r),
then the modifiedbold strategyis not optimal.
Proof We shall show that, if the initial fortuneis f as defined in (2) below and the initial
stake is s as definedin (3) below, then
WQ
::(:)
I+aa
> Q() .
l+aa
Since s < s(f), where s(f) is the modifiedbold stake at f, Theorem2.14.1 of [6, pp. 32-33]
shows thatthe modifiedbold strategyis not optimal.
Since f < 1, we can choose a positive integer y such that 0 < r6Y < 1 and (1 + a)fY <
B<
1 - rfY. Let f be defined by
f
= -2{1 + rpY + r2 2Y + ... + rKPKy + rK+l,P(K+l)y
+ rK+21(K+2)y}
(2)
and s be definedby
s = f-
(1 + a)fiBY
. . +rK-2f(K-l)Y
+.
+r,82y +r2p3y
+
rK-1IKY
+ rKi(K+l)Y}.
It can be verifiedthat
+ rs
<f
0<S<f<
=
l+a
+ K+2(K+2)y+l <
and that
f -s
1+a
=
{f
Since f < f, s(f)
+ r 2Y + r2p3y
= f > s. Since f
Q(f) = w2{1 + ww+ ...
+...
+
+
rK-2p(K-I)Y
e AK+2
and (f-
rK-IpKY
+ rK(K+l)Y}.
s)/(1 + a) E AK,
+W2W-2(-1)
+ WKWK(y-1)
+ WK+lw(K+l)(y-1)
+ 1K+2w(K+2)(y-1)}
(3)
591
A bold Strategyis not always optimal
and
Q( f-)
=
2{ WY-1
+
+ -..
22(-
K+
K()
+
(K+l)(y_l)}.
Since (f + rs)/(1 + a) = ,B + rK+2fl(K+2)Y+l < 1,
wQ(
+r)
= w2+ ww Q(rK+lfl(K+2)Y).
Therefore,
Q( f +±rs)
Q
-
Q(f)
-
= Wt{Q(rK+lfi(K+2)Y)
-K+lw(K+2)(y-1)+I}.
We now proceedto show that
wQ (+
) + wQ
+
)-
Q(f) > O,
which will complete the proof of Lemma 3. To do this, it is sufficientto show that
Q(rK+IP(K+2)y) _ WK+lw(K+2)(y-l)+l
>
.
If K > 1, then rK+lpK > 1 since a > (1 + r)r-(l+l/K) - 1. Therefore,
Q(rK+I(K+2)y)
_ WK+lw(K+2)(y-l)+l
> Q(p(K+2)y-K) _ WK+lw(K+2)(y-l)+l
= w(K+2)y-K
_ WK+lw(K+2)(y-1)+
= W(K+2)(y-1)+l(W - WK+1) > o
since 0 < w < 1 and w > wK+1. If K = O, then rf2Y > f2y and Q(rf2y) > Q(f2y)
1 > since w < w.
-w
since r > 1. Therefore, Q(rf2y) - ww2y-1 > w
= w2y
Theorem 3. For each r > 1, the modifiedbold strategy is not optimal in a primitive casino
(subfair,fair, or superfair)if the inflationrate a is in the interval [a, r).
Theorem 4. For each r > 1, if the inflationrate a is in the interval[1/r, r), then the modified
boldstrategyis notoptimalforanysubfairorfairprimitivecasino, thatis,for w (0, 1/(1 +r)].
Proof Since r > 1 and w < 1/(1 + r), we have K > 1 and a < 1/r. Hence, ifa > 1/r,
a is certainlygreaterthana. By Lemma 3, the modifiedbold strategyis not optimal.
Remark 1. For subfairand fairprimitivecasinos, Theorem3 is surprisingand seems counterintuitive: when the inflationratea is 0, the bold strategyis optimal, so a positive inflationrate
would seem to imply that it is to the gambler'sdisadvantageto prolong the game. However,
Theorem3 reveals thatit may not be optimalto play boldly.
Remark 2. This papershows thatthe modifiedbold strategyis (usually) not optimalfor these
cases of 'gamble if you must', and this raises the questionof what is the optimal stategy when
the modifiedbold strategyis not the optimalsolution. The authorshave calculatedapproximate
numericalvalues for the trueoptimal-valuefunctionandit appearsto be so complicatedthatan
exact solution seems unlikely even in the interestingcase when a = w = -and r = 2, when,
592
R. W. CHEN ETAL.
if the gambleralways stakes his entire fortune,the sequence of his renormalizedfortunesis
a martingale(note this is not the modified bold strategybecause the gamblerwill sometimes
overshootlevel 1 if he follows this greedy strategy).The authorsalso looked at the same case
when w = 0.3 and again it appearsthat the optimal strategyis too complex to guess. More
continuousproblemsof optimal stochastic control often have cleaner and simpler solutions;
the difficultyof the problemstudiedin this paperis that the primitivecasino being studiedis
'too discrete'.
Conjecture 1. In [3], Chenproved that, in a subfairorfair 'red-and-black',i.e. when r = 1,
the modifiedbold strategyis optimal. Theauthors'experimentalcomputationsseem to indicate
that, in a subfairorfair primitivecasino, the modifiedbold strategyis optimal if O < r < 1.
Conjecture 2. The authors'experimentalcomputationsalso seem to indicate that Theorem3
is sharp, i.e. the modifiedbold strategyis optimalif the inflationrate a is in the interval[0, a).
Acknowledgement
We would like to sincerely thankthe referee for his invaluablesuggestions and comments
which improvedthe presentationof this papersignificantly.
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