Kelly's betting criterion

For a long sequence of bets, maximise the expectation of s.

Kelly interpreted gambling as a Communication Channel
where the gambler receives faulty information about who is going to win.

Betting

Every day at noon you make a bet.
You give your friend pounds and they flip a coin

1. if it lands on heads, you receive pounds
2. if it lands on tails, you receive pounds

The coin lands on heads with probability
What proportion of your wealth should you bet each day in order to remain in the game long term?

Let be the proportion of bet you are gambling
Let be your wealth after days.
Then

where

Apply Weak Law of Large Numbers

So to maximise we maximise

Then

If don’t bet.
Assume .
If then is decreasing for so don’t bet.
If , take maximum at .