Motivation

In 1956 Kelly answered a question that looks like gambling and is really about information: if a channel gives you a noisy advance signal about a sequence of bets, how much is that signal worth? His answer was that the maximum exponential rate of growth of a gambler's capital equals the rate of transmission over the channel -- so information rate and capital growth rate are the same quantity in different units. The betting fraction that achieves it is now called the Kelly criterion, and it is the basis of a large practical literature on position sizing.

The result is short, entirely explicit, and has no analytic subtleties -- which makes it a good formalization target and a surprising gap: the platform currently has fifteen missions on bandit algorithms and none on optimal growth.

Setting

This mission formalizes the simplest case of Kelly's Section 4: an even-money bet with no track take, won independently with probability p and lost with probability q = 1 - p. A gambler stakes a fixed fraction l of current wealth on each bet, so wealth is multiplied by 1 + l on a win and 1 - l on a loss. The exponential rate of growth is

$$G(l) = p \log(1+l) + q \log(1-l).$$

Kelly shows this is maximised at l = p - q, with maximum value 1 + p log p + q log q in bits. We state G in nats (natural logarithm), so the maximum carries an additive log 2; dividing by log 2 recovers Kelly's bit-valued form, which is exactly 1 - H(p) for the binary entropy H. The maximiser is unaffected by the choice of base.

What is being asked

The goal theorem is that l = 2p - 1 maximises G over the admissible range (-1, 1) when the bet is favourable (p > 1/2). Milestones supply the maximum value (Kelly's information-rate identity), the admissibility of the maximiser, and the concavity that makes the first-order condition sufficient.

Source

J. L. Kelly Jr., A New Interpretation of Information Rate, Bell System Technical Journal 35 (1956) 917-926, Section 4 ("the simplest case"). The growth-rate expression and the maximiser l = p - q are stated there; the maximum value in bits is Kelly's eq. for G_max.

The identity and maximiser were checked numerically before drafting: for p = 0.55, 0.6, 0.7, 0.9 the claimed maximum matches log 2 + p log p + q log q to six decimals, and a grid search over (-1, 1) at 1e-5 resolution returns 2p - 1 in every case.