Kelly made simpler
Erik Reuter
6/30/97


The thread called "Kelly made simple" was not simple enough for me.
Whatever David D'Aquin was demonstrating went totally past me. Most of the
problem seemed to be terminology and definitions. So, I've read a Kelly
FAQ to learn some of the jargon and assumptions, and here I will summarize
my understanding. I think I understand the basics, but if I say something
wrong, I hope someone will correct me. This is mostly a check to make sure
I understood what I read, so please feel free to comment or elaborate on
anything I say.

Consider a gambling game where you start with a bankroll B and have a
probability p of winning V units and a probability (1 - p) of losing 1
unit.

The expected value of the win per unit bet is

W = p (V) + (1 - p)(-1) = p (V + 1) - 1

and if you bet a fraction f of the bankroll N times, then the expected
value of the ending bankroll is

E[f, W, N, B] = B (1 + f W) ^ N , 0 <= f <= 1

Assuming W > 0, it seems natural to ask how much to bet on each play of
the game, given that N, W and B are known. Clearly, to maximize E[f] we
should choose f = 1, that is, bet the entire bankroll each time. With f =
1, you will usually go broke for moderate to large values of N, unless you
have a p close to 1. But those times that you don't go broke you will win
a large amount. The latter offsets the former and results in maximum
expected value of the final bankroll for f = 1.

For some people, maximizing E[B, f] is not the most desirable goal. Kelly
considered choosing a (somewhat arbitrary) utility function of which the
expected value is maximized:

u[x] = Log[x]

where x is the bankroll and u[x] is the utility of the bankroll. With this
utility function, as the bankroll grows large the marginal increase in
utility diminishes (you reach a region of diminishing returns). As the
bankroll decreases to near 0, each small decrease in bankroll becomes a
great loss in utility.

The expected value of u[B] for one play of the gambling game is

K[f, V, p, B] = p Log[B (1 + f V)] + (1 - p) Log[B (1 - f)]
              = p Log[1 + f V] + (1 - p) Log[1 - f] + Log[B]

We can find the maximum expected value of utility u[B], that is, maximum
K[f], by taking the derivative of K[f] with respect to f, setting it equal
to zero, and solving for f (the check that it is indeed a maximum and not
a minimum or saddle point is left for the reader!).

K'[f_max] = 0 = p V / (1 + f V) - (1 - p) / (1 - f)

f_max = ( p (V + 1) - 1 ) / V = W / V

So, the fraction of the bankroll to bet which maximizes the expected value
of the logarithm of the bankroll is equal to the expected win of a unit
bet divided by the payoff for a unit bet.

David D'Aquin demonstrated in his post that for a few numerical examples,

B (1 + f) ^ (p N) (1 - f) ^ (N - p N)

was maximized by this choice of f. It can easily be shown that this is
indeed the case by differentiating and solving. I think he claims this is
the median value of the distribution of final bankrolls, but I don't know
how to show this
algebraically. I also don't understand the relevance of this value,
whether it is the median or not, to Kelly betting. Perhaps he could
explain.

--
Erik Reuter, e-reuter@uiuc.edu