A New Interpretation of Information Rate (1956)

This ten-page Bell System Technical Journal paper is the primary source behind the Kelly criterion. It studies a gambler who receives information through a noisy channel and can repeatedly reinvest capital. The archived PDF was read page by page and has SHA-256 d422b93805694672f64a1daacbc7396504d4718f55d115c45cd29afbc5a477bb.

Durable contribution #

Kelly's central warning is that maximizing expected capital after each bet can recommend staking everything. When the signal is fallible, this policy eventually loses all capital with probability one even though its expected terminal value can grow. Maximizing expected logarithmic growth instead balances growth against multiplicative ruin.

In the paper's even-money binary example, if a signal is correct with probability $q$ and wrong with probability $1-q$, the optimal signed fraction is the difference between those probabilities. That formula belongs to its exact payoff convention. It must not be pasted unchanged into a prediction-market contract bought at arbitrary price $p$.

For a YES contract bought at $p$, if $f$ is the fraction of bankroll spent on contracts and the trader's probability is $q$, the wealth multipliers are

$$W_{YES}=1+f\frac{1-p}{p}, \qquad W_{NO}=1-f.$$

Maximizing

$$g(f)=q\log W_{YES}+(1-q)\log W_{NO}$$

gives

$$f^*=\frac{q-p}{1-p}$$

when $q>p$, before costs and under the stated independent, repeated, correctly specified model. If $q\le p$ and shorting is excluded, the optimum for buying YES is zero.

Complete coverage #

• Pages 1–2 introduce the communication-channel betting interpretation.
• Pages 3–4 contrast expected capital with almost-sure logarithmic growth and solve the binary fair-odds case.
• Pages 5–7 generalize to multiple symbols and arbitrary fair odds, linking maximum growth to information rate.
• Pages 7–9 introduce a track take, capital held back, and the resulting constrained allocation.
• Page 10 states limitations and the dominance claim only over fixed allocation strategies, then gives references.

Source claims audit #

Visual audit #

Every page was rendered and inspected. The paper contains no plots, diagrams, photographs, or information-bearing tables. Its displayed equations are the load-bearing visuals; they are represented as native mathematics in GTK rather than raster excerpts. Notation and inequality direction were checked against the page images because OCR alone is not authoritative for formulas.

Editorial role #

GTK uses Kelly as a position-sizing framework, not a promise of profit. The trading chapter derives the formula for a $0/1$ contract, recommends explicit limits and fractional use when $q$ is uncertain, and treats correlated positions as one aggregate bet rather than independent Kelly opportunities.