The Kelly Criterion in Gambling, Investment and Innovation

The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money. i.e. how much to bet. In stock markets the problem is similar but more complex. [...] In both these settings we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm wealth.

THE KELLY CRITERION IN BLACKJACK, SPORTS BETTING, AND THE STOCK MARKET by E.O. Thorp

Every risk taking could eventually fail. Even if you know of a positive expected outcome the random process can possibly work against you.
Throw a coin, bet on head - there is a 1:1 chance for head. The expected outcome is even - 50%. This is the margin of a positive expectation, less then 50% and the "odds" are against you.

Well, the betting odds are a different thing: the experts bookmakers or social aggregated estimation of probability on the outcome of an event. Lets say a bookie cannot measure the properties of a coin directly, he therefore needs to estimate the bias of the coin. It is unbaised 50% or does it tend in a certain percentage to head or tail?

The estimated propensity of an outcome, given by the bookie - or in case of an investment by the market are reflected in the reward rate, the odds.

If the bookie thinks the coin is baised so that in 100 throws, head comes up 60 times he could set the marginal odds to reward a tail with a 60/40-1 = 50% grand. A won 1€ bet will grant you 50 cents in addition to your stake of 1€. On average tail will come up 40 times in hundred, granting you a reward of 40% * 1.50 € - 60% * 1€ = 0 €. Neither the bookie nor you win a cent. The book is balanced. A interesting thing about averages is that they are a fictive measure in the shortrun. In our coin game, a fortunous series of hundred tails is still possible. A gambler who feels a different propensity of the game, just by experience or because of optimistic mood could assume a higher chance of win. Something could make me think that the coin is baised differently. Maybe I have better information than the bookie which makes me believe that head comes up 50 times in 100 throws. If I'm right this will give me on average an edge of 10%.

Wenn betting on head I gain 25 cent per throw.

50% 1.50€ - 50% 1€ = 25 cents

Still no one says that I cannot loose everything by pure bad luck.

Let's say I have a bankroll of 5€. Since I know I have an advantage of 10% I'm confident and put all in. 5€ on head. My wealth grows to 7.50 if in one case, I'm bankrupt in the other. I'm out of the game then. Most people then would say: don't bet now you know better. But if I'm not betting at all, we have the other extreme, 5€ for me no gain.

Somehow between loosing all and keeping my five euro there is the chance of multiplying my bankroll. If I bet long enough I should be able to take advantage of the 10% edge over the bookie.

The Kelly criterion finds the fraction of the bankroll that you need to invest to gain the maximum growth rate while mitigating the risk of going bankrupt.

The math shows that the optimal growth rate is the geometric mean of the probability distribution of returns.

A simplified Kelly formula is edge/odds, in ourcase this would result in 0.1/0.50= 0.2 as the rate of investment

Others have mapped the Kelly Criterion to Innovation, because its reasonable to see the funding of innovation as a game of chance.