There are two papers of the same title:

@inproceedings{
    title = "Optimal gambling systems for favorable games",
    author = "L. Breiman",
    year = 1961,
    booktitle = "Fourth Berkeley Symposium on Probability and Statistics",
    volume = 1,
    pages = "65--78",
}

@article{
    title = "Optimal gambling systems for favorable games",
    author = "E. O. Thorp",
    booktitle = "Review of the International Statistical Institute",
    volume = 37,
    number = 3,
    pages = 273--293,
    year = 1969,
}

Breiman (1961) paper

The 1961 paper is more mathematically rigourous. It discussed the following model of gambling:

Let outcome be taken on a set such that . There is a class of subsets (part of σ-algebra) of : such that and carry the payoffs respectively. We bet for each of them and in case the outcome is , we receive . In order to allow we hold part of our fortune in reserve, we define . Let be the fortune after plays. If there is a gambling strategy such that a.s., we say the game is favorable.

The paper assume the game is favorable and asked (1) the strategy to minimize the expected number of trials needed to win or exceed an fixed amount ; (2) the size of fortune after some fixed number of plays .

It turns out, the two problems solved by the same optimal strategy , which maximizes . The betting system allows to grow exponentially and the optimal strategy is to maximize the rate of growth.

Thorp (1969) paper

This paper provides more examples than the 1961 paper but avoids some detailed derivations. It gives the following example as “favorable games”:

  • Blackjack
  • Baccarat, if side bets permitted
  • Roulette, using “Newtonian method” and if the roulette wheel is tilted for 0.2 degree or more
  • Wheel of fortune
  • Warrant hedging in stock market (short a call warrant and buy stock)

It mentioned that Blackjack can be modelled as a coin toss (Bernoulli trial) with probability of success is selected independently on each trial and announced before each trial. Probability may even be dependent in short consecutive group. The model is as follows: let be the initial capital and the capital after trial be . Let the bet on trial be (the sequence is the bettig strategy) and the gain (or loss) will be . The trial result is modeled as , then

The betting strategy

is equivalent to maximizing for each .

On stock market, the paper model (call) warrants as follows. Price of warrant , of stock , exercise price . Normally . The option pricing is empirically modelled as (note, the paper dated before Black-Scholes):

where with the number of months until expiration. The paper found that historically warrants are overpriced until close to expiration (34.5% annualized overpriced).

Warrant hedging means to simultaneously short sell overpriced warrant and long common stock in a fixed ratio (commonly 1-3 unit of warrants per stock). This can reduce variance and retain high expectation. Let be warrant price at of now and at expiration respectively, then

If we make the stock price per share of now and at expiration, the gain of shorting a warrant is

and the gain of buying a share is . Therefore the gain of hedging is

where are the number of shares of common stock to buy and unit of warrants to short, are margin used to fund the stock and warrants, respectively. If we do not borrow, i.e. 100% margin, . We can further use the lognormal distribution to model stock price:

Proportional betting: for a constant . If we let number of success and failure in Bernoulli trials to be respectively, then

Thus we have

which the logarithm in the last equation above models the rate of increase per trial in average.

J. L. Kelly (1956) proposes to maximize

which then defined the growth function:

and it is described by the following theorems:

(Theorem) If , attains maximum at with , and

  • is increasing in , and decreasing in
  • There is a that
  • for , and for
  • for

(Theorem) is the bet to maximize because

(Theorem) Maximizing is equivalent to maximizing

Modeling gambler’s ruin situation

There is a formula mentioned in the paper and I prove it here.

Define the gambler’s ruin to be after some . Let the bet be of fixed size for , , and initial fortune . The gamble will stop at gambler’s ruin or reached a predefined goal, . Let the Bernoulli probabily be . The ruin probability is

Prove: Let be the if . Then there is a probability to win and it will be equivalent to , similarly, with a probability it will be equivalent to . Then

The above is valid for and by definition of the model, and . Now,

So for ,

Theorem: For , is a strictly decreasing function of . And for , is a strictly increasing function of .