How much to wager when you have edge? (Hint: median not mean outcomes!)

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 winning1. 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

  1. General Kelly Formula: edge / odds. Formula: (bp-q)/b where p is the probability of winning, q is the probablity of losing (q = 1 – p) and b is the odds you are being offeredIn this case, it’s a 1 to 1 payout when you win so b= 1

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