Understanding The Kelly Criterion*
Edward O. Thorp
In January 1961 I spoke at the annual meeting of the American Mathematical Society on
“Fortune’s Formula: The Game of Blackjack.” This announced the discovery of
favorable card counting systems for blackjack. My 1962 book Beat the Dealer explained
the detailed theory and practice. The “optimal” way to bet in favorable situations was an
important feature. In Beat the Dealer I called this, naturally enough, “The Kelly
gambling system,” since I learned about it from the 1956 paper by John L. Kelly.
(Claude Shannon, who refereed the Kelly paper, brought it to my attention in November
of 1960.) I have continued to use it successfully in gambling and in investing. Since
1966 I’ve called it “the Kelly criterion”. The rising tide of theory about and practical use
of the Kelly Criterion by several leading money managers received further impetus from
William Poundstone’s readable book about the Kelly Criterion, Fortune’s Formula. (As
this title came from that of my 1961 talk, I was asked to approve the use of the title.) At
a value investor’s conference held in Los Angeles in May, 2007, my son reported that
“everyone” said they were using the Kelly Criterion.
The Kelly Criterion is simple: bet or invest so as to maximize (after each bet) the
expected growth rate of capital, which is equivalent to maximizing the expected value of
the logarithm of wealth. But the details can be mathematically subtle. Since they’re not
covered in Poundstone (2005) you may wish to refer to my article Thorp (2006), and
other papers in this volume. Also some services such as Morningstar and Motley Fool
have recommended it. These sources use the rule: “optimal Kelly bet equals edge/odds”,
which applies only to the very special case of a two-valued payoff.
Hedge fund manager Mohnish Pabrai (2007) gives examples of the use of the Kelly
Criterion for investment situations. (Pabrai won the bidding for the 2008 lunch with
Warren Buffett, paying over $600,000.) Consider his investment in Stewart Enterprises
*
Reprinted revised from two columns from the series A Mathematician on Wall Street in Wilmott
Magazine, May and September 2008. Edited by Bill Ziemba.
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1
(Pabrai, 2007: 108-115). His analysis gave what he believed to be a list of worst case
scenarios and payoffs over the next 24 months which I summarize in Table 1.
Table 1 Stewart Enterprises, Payoff Within 24 Months
Probability
Return
p1  0.80
R1  100%
p2  0.19
R2  0%
p3  0.01
R3  100%
______________________________
Sum=1.00
The expected growth rate of capital g  f  if we bet a fraction f of our net worth is
3
(1)
g  f    pi ln 1  Ri f 
i 1
where ln means the logarithm to the base e. When we use Table 1 to insert the pi values,
replacing the Ri by their lower bounds gives the conservative estimate
(2)
g  f   0.80 ln 1  f   0.01ln 1  f  .
Setting g(f)=0 and solving gives the optimal Kelly fraction f *  0.975 noted by Pabrai.
Not having heard of the Kelly Criterion in 2000, Pabrai only bet 10% of his fund on
Stewart. Would he have bet more, or less, if he had then known about Kelly’s Criterion?
Would I have? Not necessarily. Here are some of the many reasons why.
(1)
Opportunity costs. A simplistic example illustrates the idea. Suppose Pabrai’s
portfolio already had one investment which was statistically independent of Stewart and
with the same payoff probabilities. Then, by symmetry, an optimal strategy is to invest in
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2
both equally. Call the optimal Kelly fraction for each f *. Then 2 f * < 1 since 2 f *  1
has a positive probability of total loss, which Kelly always avoids. So f *  0.50. The
same reasoning for n such investments gives f *  1/ n. Hence we need to know the
other investments currently in the portfolio, any candidates for new investments, and their
(joint) properties, in order to find the Kelly optimal fraction for each new investment,
along with possible revisions for existing investments. Formally, we solve the nonlinear
programming problem: maximize the expected logarithm of final wealth subject to the
various constraints on the asset weights. See the papers in section 6 of this volume for
examples.
Pabrai’s discussion (e.g. pp. 78-81) of Buffett’s concentrated bets gives considerable
evidence that Buffet thinks like a Kelly investor, citing Buffett bets of 25% to 40% of his
net worth on single situations. Since f *  1 is necessary to avoid total loss, Buffett must
be betting more than .25 to .40 of f * in these cases. The opportunity cost principle
suggests it must be higher, perhaps much higher. Here’s what Buffett himself says, as
reported in http://undergroundvalue.blogspot.com/2008/02/notes-from-buffett-meeting2152008_23.html, notes from a Q & A session with business students.
Emory:
With the popularity of “Fortune’s Formula” and the Kelly Criterion, there
seems to be a lot of debate in the value community regarding diversification
vs. concentration. I know where you side in that discussion, but was curious
if you could tell us more about your process for position sizing or averaging
down.
Buffett:
I have 2 views on diversification. If you are a professional and have
confidence, then I would advocate lots of concentration.
For
everyone else, if it’s not your game, participate in total diversification.
So this means that professionals use Kelly and amateurs better off with
index funds following the capital asset pricing model.
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3
If it’s your game, diversification doesn’t make sense. It’s crazy to put
money in your 20th choice rather than your 1st choice. If you have
LeBron James on your team, don’t take him out of the game just to
make room for some else.
Charlie and I operated mostly with 5 positions. If I were running
50, 100, 200 million, I would have 80% in 5 positions, with 25% for
the largest. In 1964 I found a position I was willing to go heavier into,
up to 40%. I told investors they could pull their money out. None did.
The position was American Express after the Salad Oil Scandal. In
1951 I put the bulk of my net worth into GEICO. With the spread
between the on-the-run versus off-the-run 30 year Treasury bonds, I
would have been willing to put 75% of my portfolio into it. There were
various times I would have gone up to 75%, even in the past few years.
If it’s your game and you really know your business, you can load up.
This supports the assertion in Rachel and Bill Ziemba’s (2007) book, that Buffett thinks
like a Kelly investor when choosing the size of an investment. They discuss Kelly and
investment scenarios at length.
Computing f * without considering the available alternative investments is one of the
most common oversights I’ve seen in the use of the Kelly Criterion. It is a dangerous
error because it generally overestimates f * .
(2)
Risk tolerance. As discussed at length in Thorp (2006), “full Kelly” is too risky
for the tastes of many, perhaps most, investors and using instead an f  cf *, with
fraction c where 0  c  1, or “fractional Kelly” is much more to their liking. Full Kelly
is characterized by drawdowns which are too large for the comfort of many investors.1
1
Several papers in this volume in section 3 as do the following two papers in this section, discuss
fractional Kelly strategies.
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4
(3)
The “true” scenario is worse than the supposedly conservative lower bound
estimate. Then we are inadvertently betting more than f * and, as discussed in Thorp
(2006), we get more risk and less return, a strongly suboptimal result. Betting f  cf *,
0  c  1, gives some protection against this. See the graphs in MacLean, Ziemba and
Blazenko (1992) in section 3 for examples illustrating this.
(4)
Black swans. As fellow Wilmott columnist Nassim Nicholas Taleb (2007) has
pointed out so eloquently in his bestseller The Black Swan, humans tend not to appreciate
the effect of relatively infrequent unexpected high impact events. Failing to allow for
these “black swans,” scenarios often don’t adequately consider the probabilities of large
losses. These large loss probabilities may substantially reduce f *. One approach to
successfully model such black swans is to use a scenario optimization stochastic
programming model.2 For Kelly bets that simply means that you include such extreme
scenarios and their consequences in the nonlinear programming optimization to compute
the optimal asset weights. The f * will be reduced by these negative events.
(5)
The “long run.” The Kelly Criterion’s superior properties are asymptotic,
appearing with increasing probability as time increases. For instance:
As time t tends to infinity the Kelly bettor’s fortune will, with probability tending
to 1, permanently surpass that of any bettor following an “essentially different”
strategy.
The notion of “essentially different” has confounded some well known quants so
I’ll take time here to explore some of its subtleties. Consider for simplicity
repeated tosses of a favorable coin. The outcome of the n th trial is  n where
2
There you assume the possibility of an event, specifying its consequences but not what it is. See
Geyer and Ziemba (2008) for the application to the Siemens Austria Pension Fund. Correlations
change as the scenario sets move from normal conditions to volatile to crash which include the
black swans. See also Ziemba (2003) for additional applications of this approach.
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P  n  1  p  12 and P  n  1 is 1  p  q  0. The n  are independent
identically distributed random variables. The Kelly fraction is
f   p  q  E  n   0. The Kelly strategy is to bet a fraction f n  f  at each
trial n  1, 2,
. Now consider a strategy which bets g n , n  1, 2,
at each trial
with gn  f  for some n  N and gn  f  thereafter. The gn  strategy differs
from Kelly on at least one of the first N trials but copies it thereafter, but it does
not differ infinitely often. There is a positive probability that gn  is ahead of
Kelly at time N , hence ahead for all n  N. For example consider the sequence
of the first N outcomes such that n  1 if gn  f  and  n  1 if g n  f  .
Then for this specific sequence, which has probability  q N ,
gn  gains more
than Kelly for each n  N where g n  f , hence exceeds Kelly for all n  N.
What if instead in this coin tossing example we require that gn  f  for infinitely
many n ? This question arose indirectly about 15 years ago in the newsletter
Blackjack Forum when a well known anti Kellyite, John Leib, challenged a well
known blackjack expert with (approximately) this proposition bet: Leib would
produce a strategy which differed from Kelly at every trial but would (with
probability as close to 1 as you wish), after a finite number of trials, get ahead of
Kelly and stay ahead forever. When I read the challenge I immediately saw how
Leib could win the bet.
Leib’s Paradox: Assuming capital is infinitely divisible3, then given   0 there
is an N  0 and a sequence  f n  with f n  f  for all n, such that


n
n
i 1
i 1


P Vn*  Vn for all n  N  1   where Vn   1  fi i  and Vn*   1  f *i .
Furthermore there is a b  1 such that P Vn / Vn*  b, n  N   1   and
3
The infinite divisibility of capital is a minor assumption and can be dealt with as needed in
examples where there is a minimum monetary unit by choosing a sufficiently large starting
capital.
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6


P Vn  Vn*    1   . That is, for some N there is a non Kelly sequence that
beats Kelly “infinitely badly” with probability 1   for all n  N.
PROOF. The proof has two parts. First we want to establish the assertion for
n  N. Second we show that once we have an  f n , n  N  that is ahead of Kelly
at n  N , we can construct  f n  f * , n  N  to stay ahead.
To see the second part, suppose VN  VN* . Then VN  a  bVN* for some
a  0, b  1. For instance VN  VN*  c  0 since there are only a finite number of
sequences of outcomes in the first N trials, hence only a finite number with
VN  VN* . So
VN  c  VN*  c / 2   c / 2   VN*   c / 2   dMaxVN*  VN*   c / 2   d  1VN*
where dMaxVN*  c / 2 defines d  0 and MaxVN* is over all sequences of the first
N trials such that VN  VN* . Setting c / 2  a  0 and d  1  b  1 suffices. Once
we have VN  a  bVN* we can, for bookkeeping purposes, partition our capital
into two parts: a and bVN* . For n  N we bet f n  f * from bVN* and an
additional amount a / 2n from the a part, for a total which is generally unequal to
f * of our capital. If by chance for some n the total equals f * of our total capital
we simply revise a / 2n to a / 3n for that n. The portion bVN* will become bVN*
for n  N and the portion a will never be exhausted so we have Vn  bVn* for all
n  N. Hence, since P Vn*     1, we have P Vn / Vn*  b   1 from which it
follows that P Vn  Vn*     1.
To prove the first part, we show how to get ahead of Kelly with probability 1  
within a finite number of trials. The idea is to begin by betting less than Kelly by
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7
a very small amount. If the first outcome is a loss, then we have more than Kelly
and use the strategy from the proof of the second part to stay ahead. If the first
outcome is a win, we’re behind Kelly and now underbet on the second trial by
enough so that a loss on the second trial will put us ahead of Kelly. We continue
this strategy until either there is a loss and we are ahead of Kelly or until even
betting 0 is not enough to surpass Kelly after a loss. Given any N , if our initial
underbet is small enough, we can continue this strategy for up to N trials. The
probability of the strategy failing is p N , 12  p  1. Hence, given   0, we can
choose N such that p N   and the strategy therefore succeeds on or before trial
N with probability 1  p N  1   .
More precisely: suppose the first n trials are wins and we have bet a fraction
f *  ai with ai  0, i  1,

1  f *  a1
Vn

Vn*
1 f *

, n, on the i th trial. Then
 1  f  a 
 1  f 
*
n
*

a 
 1  1 * 
 1 f 

an 
 1  a1 
1 
* 
 1 f 
1  an   1   a1 
 an 
where the last inequality is proven easily by induction. Letting a1 
 an  a,
so Vn / Vn*  1  a, what betting fraction f *  b will put us ahead of Kelly if the
next trial is a loss? A sufficient condition is


*

Vn 1 Vn 1  f  b
a
b 
b

or
provided


1

a
1


1

a
1

b

1







*
*
1 a
Vn 1
Vn* 1  f *
 1 f 


b  f * and 0  a  1. If a  12 then b  2a suffices. Proceeding recursively, we
have these conditions on the ai : choose a1  0. Then
an1  2  a1 
 an  , n  1, 2,
f  x   a1 x  a2 x2 
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provided all the an 
1 .
2
Letting
we get the equation
8

f  x   a1 x  2 xf  x  1  x  x 2 

 2 xf  x  / 1  x 



whose solution is f  x   a1  x  2 3n  2 x n  from which an  2a1 3n  2 if n  2.
n2


Then given   0 and an N such that p N   it suffices to choose a1 so that


aN  2a1 3N  2  min f * , 12 . Q.E.D.
Although Leib did not have the mathematical background to give such a proof he
understood the idea and indicated this sort of procedure.
So far we’ve seen that all sequences which differ from Kelly for only a finite
number of trials, and some sequences which differ infinitely often (even always),
are not essentially different. How can we tell, then, if a betting sequence is
essentially different than Kelly? Going to a more general setting than coin
tossing, assume now for simplicity that the payoff random variables X i are
independent and bounded below but not necessarily identically distributed.
At this point we come to an important distinction. In financial applications one
commonly assumes that the f i are constants that are dependent only on the
current period payoff random variable (or variables). Such “myopic strategies”
might arise for instance, by selecting a utility function and maximizing expected
utility to determine the amount to bet. However, for gambling systems the
amount depend on previous outcomes, i.e. f n  f n  X1 , X 2 ,
, X n1  , just as it
does in the Leib example. As Professor Stewart Ethier pointed out, our discussion
of “essentially different” is for the constant f i case. For a more general case,
including the Leib example and many of the classical gambling systems, I
recommend Ethier’s (2010) book on the mathematics of gambling.
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9
We assume E  X i   0 for all X i from which it follows that fi *  0 for all i. As
n
n
i 1
i 1


before, Vn   1  fi X i  and Vn*   1  fi* X i from which
ln Vn   ln 1  f i X i  and ln Vn*   ln 1  f i * X i  . Note from the definition f *
n
n
i 1
i 1
that E ln 1  f i * X i   E ln 1  f i X i  , where E denotes the expected value, with
equality if and only if fi *  fi . Hence


E ln Vn* / Vn   E  ln 1  fi * X i   ln 1  fi X i    ai
n
i 1
n
i 1
where ai  0 and ai  0 if and only if fi *  fi . This series of non-negative terms
either increases to infinity or to a positive limit M . We say  fi  is essentially
 f  if and only if  a
n
different from
Otherwise,
*
i
i 1
i
tends to infinity as n increases.
 fi  is not essentially different from  fi* .
The basic idea here can
be applied to more general settings.
(6)
Given a large fixed goal, e.g. to multiply your capital by 100, or 1000, the
expected time for the Kelly investor to get there tends to be least.
Is a wealth multiple of 100 or 1000 realistic? Indeed. In the 51½ years from 1956 to mid
2007, Warren Buffett has increased his wealth to about $5x1010. If he had $2.5x104 in
1956, that’s a multiple of 2x106. We know he had about $2.5x107 in 1969 so his multiple
over these 38 years is about 2x103. Even my own efforts, as a late starter on a much
smaller scale, have multiplied capital by more than 2x104 over the 41 years from 1967 to
early 2007. I know many investors and hedge fund managers who have achieved such
multiples. One of the best is Jim Simons, who recently retired form running the
Renaissance Medallion Fund. His record to 2005 is analyzed in section 6 of this book.
The caveat here is that an investor or bettor many not choose to make, or be able to make,
enough Kelly bets for the probability to be “high enough” for these asymptotic properties
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10
to prevail, i.e. he doesn’t have enough opportunities to make it into this “long run.”
Below I explore investors for which Kelly or fractional Kelly may be a more or less
appropriate approach. An important consideration will be the investor’s expected future
wealth multiple.
Using Kelly Optimization at PIMCO
During a recent interview in the Wall Street Journal (March 22-23, 2008) Bill Gross and
I discussed turbulence in the markets, hedge funds and risk management. Bill considered
the question of risk management after he read Beat the Dealer in 1966. That summer he
was off to Las Vegas to beat blackjack. Just as I did some years earlier, he sized his bets
in proportion to his advantage, following the Kelly Criterion as described in Beat the
Dealer, and ran his $200 bankroll up to $10,000 over the summer. Bill has gone from
managing risk for his tiny bankroll to managing risk for Pacific Investment Management
Company’s (PIMCO) investment pool of almost $1 trillion.4 He still applies lessons he
learned from the Kelly Criterion. As Bill said, “Here at PIMCO it doesn’t matter how
much you have, whether it’s $200 or $1 trillion. … Professional blackjack is being
played in this trading room from the standpoint of risk management and that’s a big part
of our success.”
The Kelly Criterion applies to multiperiod investing and we can get some insights by
comparing it with Markowitz’s standard portfolio theory for single period investing.
Compound Growth and Mean–Variance Optimality
Nobel Prize winner Harry Markowitz introduced the idea of mean-variance optimal
portfolios. This class is defined by the property that, among the set of admissible
portfolios, no other portfolio has both higher mean return and lower variance. The set of
such portfolios as you vary return or variance is known as the efficient frontier. The
concept is a cornerstone of modern portfolio theory, and the mean and variance refer to
4
PIMCO is widely regarded as the top bond trading operation in the world.
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11
one period arithmetic returns.5 In contrast the Kelly Criterion is used to maximize the
long term compound rate of growth, a multiperiod problem. It seems natural, then to ask
the question: is there an analog to the Markowitz efficient frontier for multiperiod growth
rates, i.e. are there portfolios such that no other portfolio has both a higher expected
growth rate and a lower variance in the growth rate? We’ll call the set of such portfolios
the compound growth mean-variance efficient frontier.
Let’s explore this in the simple setting of repeated independent identically distributed
returns per unit invested, where the payoff random variables are
 X i : i  1,
, n with
E  X i   0 so the “game” is favorable, and where the non-negative fractions bet at each
trial, specified in advance, are  fi : i  1,
, n. To keep the math simpler, we also
assume that the X i have a finite number of distinct values. After n trials the compound
growth rate per period is G  f i  
1 n
 log 1  fi X i  and the expected growth rate
n i 1
1 n
1 n


g  f i   E G  f i     E log 1  f i X i    E log 1  f i X   E log 1  fX . The
n i 1
n i 1
last step follows from the (strict) concavity of the log function, where as X has the
common distribution of the X i , we define f 
1 n
 fi and we have equality if and only
n i 1
if fi  f for all i. Therefore if some f i differ from f we have g  fi   g
 f   .
This
tells us that betting the same fixed fraction always produces a higher expected growth
rate than betting a varying fraction with the same average value. Note that whatever f
turns out to be, it can always be written as f  cf * , a fraction c of the Kelly fraction.
Now consider the variance of G  f i  . If X is a random variable with
2
  1  af 
P  X  a   p, P  X  1  q, and a  0 then Var ln 1  fX   pq ln 
 .
  1  f 
5
A comprehensive survey of mean-variance theory is in Markowitz and Van Dijk (2006).
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12
(Compare Thorp, 2006, section 3.1). Note: the change of variable f  bh, b  0, shows
the results apply to any two valued random variable. We chose b  1 for convenience.)
A calculation shows that the second derivative with respect to f is strictly positive for
0  f  1 so Var  ln 1  fX   is strictly convex in f . It follows that
Var G  f i   
1 n
1 n
Var log 1  f i X i    Var log 1  f i X    Var log 1  fX  

n i 1
n i 1
with equality if and only if fi  f for all i. Since every admissible strategy is therefore
“dominated” by a fractional Kelly strategy it follows that the mean-variance efficient
frontier for compound growth is a subset of the fractional Kelly strategies. If we now
examine the set of fractional Kelly strategies
 f   cf *
we see that for 0  c  1, both
the mean and the variance increase as c increases but for c  1, the mean decreases and
the variance increases as c increases. Consequently  f *  dominates the strategies for
which c  1 and they are not part of the efficient frontier. No fractional Kelly strategy is
dominated for 0  c  1. We have established in this limited setting:
Theorem. For repeated independent trials of a two valued random variable, the meanvariance efficient frontier for compound growth over a finite number of trials consists
precisely of the fractional Kelly strategies cf * : 0  c  1 .
So, given any admissible strategy, there is a fractional Kelly strategy with 0  c  1 which
has a growth rate that is no lower and a variance of the growth rate that is no higher. The
fractional Kelly strategies in this instance are preferable in this sense to all the other
admissible strategies, regardless of any utility function upon which they may be based.
This deals with yet another objection to the fractional Kelly strategies, namely that there
is a wide spread in the distribution of wealth levels as the number of periods increases. In
fact this eventually enormous dispersion is simply the magnifying effect of compound
growth on small differences in growth rate and we have shown in the theorem that in the
two outcome setting this dispersion is minimized by the fractional Kelly strategies. Note
that in this simple setting, a one-period utility function will choose a constant fi  cf
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which will either be a fractional Kelly with c  1 in the efficient frontier or will be too
risky, with c  1, and not be in the efficient frontier.
As a second example suppose we have a lognormal diffusion process with instantaneous
drift rate m and variance s 2 where as before the admissible strategies are to specify a set
of fixed fractions
 fi  for each of n
unit time periods, i  1,
, n. Then for a given f
and unit time period VarG  f   s 2 f 2 as noted in (Thorp, 2006, eqn. (7.3)). Over n
n
n
i 1
i 1
periods VarG  f i   s 2  f i 2  s 2  f 2 with equality if and only if fi  f for all i.
This follows from the strict convexity of the function h  x   x2 . So the theorem also is
true in this setting. I don’t currently know how generally the convexity of Var[ln(1+f X)]
is true but whenever it is, and we also have Var[ln(1+f X)] increasing in f , then the
compound growth mean variance efficient frontier is once again the set of fractional
Kelly strategies with 0  c  1. In email correspondence, Stewart Ethier subsequently
showed that Var[ln(1+f X)] need not be convex. Example (Ethier):
Let X assume values -1, 0 and 100 with probabilities 0.5, 0.49 and 0.01,
respectively. Then, on approximately the interval [0.019, 0.180] the second
derivative of the variance is negative, hence the variance is strictly concave on
that interval. The first derivative of Var[ln(1+f X)] equals 2Cov(ln(1+fX), X/(1+f
X)), which is always nonnegative because the two functions of X in the
covariance are increasing in X. Thus Var[ln(1+f X)] is always increasing in f. The
second derivative of Var[ln(1+fX)] equals 2Var(X/(1+f X))-2Cov(ln(1+f X)-1,
X2/(1+f X)2). However, the covariance term sometimes exceeds the variance
term.
Samuelson’s Criticisms
The best known “opponent” of the Kelly Criterion is Nobel Prize winning economist Paul
Samuelson, who has written numerous polemics, both published and private, over the last
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40 years. William Poundstone’s book Fortune’s Formula gives an extensive account
with references. The gist of it seems to be:
(1) Some authors once made the error of claiming, or seeming to claim, that acting to
maximize the expected growth rate (i.e. logarithmic utility) would approximately
maximize the expected value of any other continuous concave utility (the “false
corollary”).
Response: Samuelson’s point was correct but, to others as well as me, obvious the first
time I saw the false claim. However, the fact that some writers made mistakes has no
bearing on an objective evaluation of the merits of the criterion. So this is of no further
relevance.
(2) In private correspondence to numerous people Samuelson has offered examples and
calculations in which he demonstrates, with a two valued X (“stock”) and three utilities,
H W   1/ W , K W   log W , and T W   W 1 2 , that if any one who values his
wealth with one of these utilities uses one of the other utilities to choose how much to
invest then he will suffer a loss as measured with his own utility in each period and the
sum of these losses will tend to infinity as the number of periods increases.
Response: Samuelson’s computations are simply instances of the following general fact
proven 30 years earlier by Thorp and Whitley (1972, 1974).6
Theorem 1. Let U and V be utilities defined and differentiable on  0,   with
U   x  and V   x  positive and strictly decreasing as x increases. Then if U and
V are inequivalent, there is a one period investment setting such that U and V
have distinct sets of optimal strategies. Furthermore, the investment setting may
6
The first Thorp and Whitley paper is reprinted in this book in section 4 where three of
Samuelson’s papers are reprinted and discussed in the introduction to that part of this book.
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be chosen to consist only of cash and a two-valued random investment, in which
case the optimal strategies are unique.
Corollary 2. If the utilities U and V have the same (sets of) optimal strategies
for each finite sequence of investment settings, then U and V are equivalent.
Two utilities U1 and U 2 are equivalent if and only if there are constants a and b
such that U 2  x   aU1  x   b  a  0 , otherwise U1 and U 2 are inequivalent.
Thus no utility in the class described in the theorem either dominates or is dominated by
any other member of the class.
Samuelson offers us utilities without any indication as to how we ought to choose among
them, except perhaps for this hint. He says that he and an apparent majority of the
investment community believe that maximizing U  x   1/ x explains the data better
than maximizing U  x   log x. How is it related to fractional Kelly? Does this matter?
Here are two examples which show that this utility can choose cf * for any 0  c  1,
c
1
2
, depending on the setting.
For a favorable coin toss and U  x   1/ x we have f *   /

p q

2
which
increases from  / 2 or half Kelly to  or full Kelly as p increases from ½ to 1, giving
us the set
1
2
 c  1. On the other hand if P  X  A  P  X  1  1 2 describe the
returns and A  1 so   0,    A  1 / 2 and the Kelly f *   / A. For U  x   1/ x


we find U maximized for f  2 A  4 A2  A  A  1

2 1/ 2

/ A  A  1 which is
asymptotic to A1/ 2 as A increases, compared to the Kelly f * which is asymptotic to
1/ 2 as A increases, giving us the set 0  c  1 2 .
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In the continuous case the relation between c, g  f  and   G  f   is simple and the
tradeoff between growth and spread in growth rate as we adjust c between 0 and 1 is
easy to compute and it’s easy to visualize the correspondence between fractional Kelly
and the compound growth mean-variance efficient frontier. This is not the case for these
two examples so the fact that U  x   1/ x can choose any c, 0  c  1, c 
1
2
doesn’t
necessarily make it undesirable.7 I suggest that a useful way to look at the problem for
any specific example involving n period compound growth is to map the admissible


portfolios into the   G  fi  , g  fi  plane, analogous to the Markowitz one period
mapping into the (standard deviation, return) plane. Then examine the efficient frontier
and decide what tradeoff of growth versus variability of growth you like. Professor Tom
Cover points out that there is no need to invoke utilities. Adopting this point of view,
we’re simply interested in portfolios on the compound growth efficient frontier whether
or not any of them happen to be generated by utilities. The Samuelson preoccupation
with utilities becomes irrelevant. The Kelly or maximum growth portfolio, which as it
happens can be computed using the utility U  x   log x, has the distinction of being at
the extreme high end of the efficient frontier.
For another perspective on Samuelson’s objections, consider the three concepts
normative, descriptive and prescriptive. A normative utility or other recipe tells us what
portfolio we “ought” to choose, such as “bet according to log utility to maximize your
own good.” Samuelson has indicated that he wants to stop people from being deceived
by such a pitch. I completely agree with him on this point. My view is instead
prescriptive: how to achieve an objective. If you know future payoff for certain and want
to maximize your long term growth rate then Kelly does it. If, as is usually the case, you
only have estimates of future payoffs and want to come close to maximizing your long
term growth rate, then to avoid damage from inadvertently betting more than Kelly you
7
MacLean, Ziemba and Li (2005) reprinted in section 4 of this book, show that for lognormally
distributed assets, a fractional Kelly strategy is uniquely related to the coefficient 0in the
negative power utility function w via the formula c=1/(1-) so ½ Kelly is -1/w. However,
when assets are lognormal this is only an approximation and, as shown here, it can be a poor
approximation.
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need to back off from your estimate of full Kelly and consider a fractional Kelly strategy.
In any case, you may not like the large drawdowns that occur with Kelly fractions over ½
and may be well advised to choose lower values. The long term growth investor can
construct the compound growth efficient frontier and choose his most desirable geometric
growth Markowitz type combinations.
Samuelson also says that U  x   1/ x seems roughly consistent with the data. That is
descriptive, i.e. an assertion about what people actually do. We don’t argue with that
claim – it’s something to be determined by experimental economists and its correctness
or lack thereof has no bearing on the prescriptive recipe for growth maximizing.
I met with economist Oscar Morgenstern (1902-1977), coauthor with John von Neumann
of the great book, The Theory of Games and Economic Behavior, at his company
“Mathematica” in Princeton, New Jersey, in November of 1967 and, when I outlined
these views on normative, prescriptive and descriptive he liked them so much he asked if
he could incorporate them into an article he was writing at the time. He also gave me an
autographed copy of his book, On the Accuracy of Economic Observations, which has an
honored place in my library today and which remains timely. (For instance, think about
how the government has made successive revisions in the method of calculating inflation
so as to produce lower numbers, thereby gaining political and budgetary benefits.)
Proebsting’s Paradox
Next, we look at a curious paradox. Recall that one property of the Kelly Criterion is that
if capital is infinitely divisible, arbitrarily small bets are allowed, and the bettor can
choose to bet only on favorable situations, then the Kelly bettor can never be ruined
absolutely (capital equals zero) or asymptotically (capital tends to zero with positive
probability). Here’s an example that seems to flatly contradict this property. The Kelly
bettor can make a series of favorable bets yet be (asymptotically) ruined! Here’s the
email discussion through which I learned of this.
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From: Todd Proebsting
Subject: FW: incremental Kelly Criterion
Dear Dr. Thorp,
I have tried to digest much of your writings on applying the Kelly Criterion to
gambling but I have found a simple question that is unaddressed. I hope you find
it interesting:
Suppose that you believe an event will occur with 50% probability and somebody
offers you 2:1 odds. Kelly would tell you to bet 25% of your capital.
Similarly, if you were offered 5:1 odds, Kelly would tell you to bet 40%.
Now, suppose that these events occur in sequence. You are offered 2:1 odds, and
you place a 25% bet. Then another party offers you 5:1 odds. I assume you
should place an additional bet, but for what amount?
If you have any guidance or references on this question, I would appreciate it.
Thank you.
From: Ed Thorp
To: Todd Proebsting
Subject: Fw: incremental Kelly Criterion
Interesting.
After the first bet the situation is:
A win gives a wealth relative of 1 + 0.25*2
A loss gives a wealth relative of 1 - 0.25
Now bet an additional fraction f at 5:1 odds and we have:
A win gives a wealth relative of 1 + 0.25*2 + 5f
A loss gives a wealth relative of 1 - 0.25 - f
The exponential rate of growth g(f) = 0.5*ln(1.5+5f) + 0.5*ln(0.75-f)
Solving g'(f) = 0 yields f = 0.225 which was a bit of a surprise until I thought
about it for a while and looked at other related situations.
From: Todd Proebsting
To: Ed Thorp
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19
Subject: RE: incremental Kelly Criterion
Thank you very much for the reply.
I, too, came to this result, but I thought it must be wrong since this tells me to bet
a total of 0.475 (0.25+0.225) at odds that are on average worse than 5:1, and yet at
5:1, Kelly would say to bet only 0.400.
Do you have an intuitive explanation for this paradox?
From: Ed Thorp
To: Todd Proebsting
Subject: Re: incremental Kelly Criterion
I don't know if this helps, but consider the example:
A fair coin will be tossed (Pr Heads = Pr Tails = 0.5). You place a bet which
gives a wealth relative of 1+u if you win and 1-d if you lose (u and d are both
nonnegative). (No assumption about whether you should have made the bet.)
Then you are offered odds of 5:1 on any additional bet you care to make. Now
the wealth relatives are, each with Pr 0.5, 1+u+5f and 1-d-f. The Kelly fraction is
f = (4-u-5d)/10. It seems strange that increasing either u or d reduces f. To see
why it happens, look at the ln(1+x) function. This odd behavior follows from its
concave shape.
From: Todd Proebsting
To: Ed Thorp
Subject: RE: incremental Kelly Criterion
Yes, this helps. Thank you.
It is interesting to note that Kelly is often thought to avoid ruin. For instance, no
matter how high the offered odds, Kelly would never have you bet more than 0.5
of bankroll on a fair coin with one single bet. Things change, however, when
given these string bets. If I keep offering you better and better odds and you keep
applying Kelly, then I can get you to bet an amount arbitrarily close to your
bankroll.
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Thus, string bets can seduce people to risking ruin using Kelly. (Granted at the
risk of potentially giant losses by the seductress.)
From: Ed Thorp
To: Todd Proebsting
Subject: Re: incremental Kelly Criterion
Thanks. I hadn't noticed this feature of Kelly (not having looked at string bets).
To check your point with an example I chose consecutive odds to one of A_n:1
where A_n = 2^n, n = 1,2,... and showed by induction that the amount bet at each
n was f_n = 3^(n-1)/4^n (where ^ is exponentiation and is done before division or
multiplication) and that sum{f_n: n=1,2,...} = 1.
A feature (virtue?) of fractional Kelly strategies, with the multiplier less than 1,
e.g. f = c*f(kelly), 0<c<1, is that it (presumably) avoids this.
In contrast to Proebsting’s example, the property that betting Kelly or any fixed fraction
thereof less than one leads to exponential growth is typically derived by assuming a series
of independent bets or, more generally, with limitations on the degree of dependence
between successive bets. For example, in blackjack there is weak dependence between
the outcomes of successive deals from the same unreshuffled pack of cards but zero
dependence between different packs of cards, or equivalently between different
shufflings of the same pack. Thus the paradox is a surprise but doesn’t contradict the
Kelly optimal growth property.
References
Ethier, S. (2010) The Doctrine of Chances, Springer-Verlag, Berlin.
Geyer, A and W. T. Ziemba (2008) The Innovest Austrian pension fund financial
planning model InnoALM. Operations Research 56 (4): 797-810.
MacLean, L. C., W. T. Ziemba and G. Blazenko (1992) Growth versus security in
dynamic investment analysis. Management Science 38: 1562-1585.
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Markowitz, H. M. and E. van Dijk (2006) Risk return analysis, in S. A. Zenios and W.
T. Ziemba (eds), Handbook of Asset and Liability Management, Volume I:
Theory and Methodology, North Holland, 139-197.
Pabrai, M. (2007) The Dhandho Investor, Wiley.
Poundstone, W. (2005) Fortune’s Formula, Hill and Wang.
Taleb, N. N. (2007) The Black Swan: The Impact of the Highly Improbable, Barnes and
Noble.
Thorp, E. O. (2006) The Kelly Criterion in Blackjack, Sports Betting and the Stock
Market, in S. A. Zenios and W. T. Ziemba (eds), Handbook of Asset and
Liability Management, Volume I: Theory and Methodology, North Holland, 385428 (also available on my website www.edwardothorp.com and reprinted in this
volume).
Thorp, E. O. and R. Whitley (1972) Concave utilities are distinguished by their optimal
strategies, in Colloquia Mathematica Societatis Janos Bolyai 9 and reprinted in
this book.
Thorp, E. O. and R. Whitley (1974) Progress in Statistics, in Proceedings of the
European Meeting of Statisticians, Budapest. North Holland, pp. 813-830.
Ziemba, R.E.S and W. T. Ziemba (2007) Scenarios for Risk Management and Global
Investment Strategies, Wiley.
Ziemba, W. T. (2003) The Stochastic Programming Approach to Asset Liability
Management, AIMR
Ziemba, W. T. and R. G. Vickson, eds. (2006) Stochastic Optimization Models in
Finance, 2nd Edition,World Scientific.
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