[Ganchrow]

Originally Posted by sports_quant99

The thing i'm not following you on is "expected growth". It doesnt make sense ... expected *value* is how your bankroll will truly grow in the long run.

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The example you gave in your Rx thread, whereby raising your bet to 25% of bankroll, your expected growth rate then becomes negative, defies common sense. The only thing that changes is your volatility, but your long term bankroll will grow.

First off, welcome to SBR.

I take it you've already read the first article in this series where I tried to illustrate the difference between expected value and expected growth? Guess I didn't do such a good job. huh?

The difference between expected value and growth is the same as that between an arithmetic and a geometric mean.

You can think of the expected value of a bet as the arithmetic mean of all outcomes were you to repeat the same dollar-value bet an infinite number of times.

Expected growth, on the other hand, corresponds to the geometric mean outcome you'd obtain were you to repeat the same percentage-of-bankroll bet an infinite number of times.

This is a subtle but extremely importance difference.

The best way to see the difference is by considering a bet of 100% of one's bankroll. The win probability and payout odds are irrelevant to the discussion, just long as they're less than 100% and infinity respectively. For the sake of this discussion we'll assume the win probability is 99% and the bet is made at +100 (decimal: 2.0000). This bet has an expected value of 2*99% - 1 * 100% = 98% of bankroll, and corresponds to expected growth of (1+(2-1)*100%)^99% * (1-100%)^1% = -100% (this latter figure implies that as your number of sequential bets increases, your probability of going bankrupt approaches certainty). We'll assume the starting bankroll is $100.

After 1 bet, there's a 99% probability of winning and ending up with a bankroll of $200, and a 1% probability of ending up with a bankroll of zero (in which case betting would stop as the player would have no more money with which to play).

After 10 bets, there's a 1-99% ≈ 90.4% probability of ending up with a bankroll of 2¹⁰* $100 = $102,400 and a roughly 9.6% probability of ending up bankrupt.

After 1,000 bets, there's a 1-99%^1,000 ≈ 0.00431712% probability of ending up with a bankroll of 2^1,000* $100 ≈ $1.07151 × 10³⁰³ and a 99%^1,000 ≈ 99.995683% probability of ending up bankrupt.

What we see is that the expected return per bet is always constant at 98%. However, the expected average growth rate per bet is -100%.

Another way to think of expected growth after a large number of bets is that it represents the single most likely outcome. Expected value, on the other hand, represents the average outcome regardless of its relative likelihood.