Link: Rational Decision-Making under Uncertainty: Observed Betting Patterns on a Biased Coin

- Optimal bet size as a fraction of bankroll is 2p-1 where p is the probability of winning
^{1}. You will recognize this as the edge per trial reported as a percent. So a 60% coin has 20% expected return or edge - The formula is a solution to a proportional betting system which implicitly assumes the gambler has log utility of wealth

Imagine tossing a 60% coin 100x and starting with a $25 bankroll

**Arithmetic Mean Land**

The mean of one flip is 20% positive expectancy

Optimal bet size is 20% of bankroll since you have .20 expectancy per toss

Increase in wealth per toss betting a Kelly fraction: 20% of bankroll x .20 expectancy = **4%**

Expected (mean) value of game after 100 flips betting 20% of your wealth each time

$25 * (1+.04) ^ 100 = $1,262

**Median Land**

The median of one flip betting a Kelly fraction is (1.2^.60 * .8^.40 – 1) or **2%**

Median value of game after 100 flips betting 20% of your wealth each time

25 * (1.2^60) * (.8^40) = $187.25!

#### Things to note

- The median outcome by definition is the increase in utility since Kelly betting implicitly assumes the gambler has log utility
- After 100 flips, the median outcome is only about 1/10 of the mean outcome! The median outcome gives an idea of how much to discount the mean payoff. If your utility function is not a log function (ie does quadrupling your wealth make you twice as happy) then a different Kelly fraction should be used