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*growth*. That is, if we start out with an initial bankroll \(B_0\), we seek to maximize \(\mathrm{E}[g(t)]\), where \(B_t = B_0\cdot e^{g(t)}\).

As a simple example, consider the following choice. We can have a sure $3000, or we can take the gamble of a \(\frac{4}{5}\) chance of $4000 and a \(\frac{1}{5}\) chance of $0. What does Kelly say?

Assume we have a current bankroll of \(B_0\). After the first choice we have \(B_1 = B_0+3000\), which we can write as \[\mathrm{E}[g(1)] = \log\left(\frac{B_0+3000}{B_0}\right);\]for the second choice we have \[\mathrm{E}[g(1)] = \frac{4}{5} \log\left(\frac{B_0+4000}{B_0}\right).\]And so we want to compare \(\log\left(\frac{B_0+3000}{B_0}\right)\) and \(\frac{4}{5} \log\left(\frac{B_0+4000}{B_0}\right)\).

Exponentiating, we're looking for the positive root of \[{\left({B_0+3000}\righ…