Kelly (1956) — A New Interpretation of Information Rate

Summary

John Larry Kelly Jr.'s 1956 paper "A New Interpretation of Information Rate" (Bell System Technical Journal, Vol. 35, pp. 917–926), available via Princeton's website, is the original publication introducing what became known as the Kelly criterion. Kelly showed that the optimal betting strategy for maximizing the long-term exponential growth rate of wealth is to bet a fraction f* = (bp - q) / b of your bankroll, where b = odds - 1, p = probability of winning, q = 1 - p.

The paper's key insight — connecting Claude Shannon's information theory to gambling — was that a gambler receiving information about a horse race (via a noisy channel) could use that information to achieve exponential capital growth by betting according to the Kelly formula. The growth rate G(f) = p ln(1 + fb) + q ln(1 - f) is maximized at f*.

The paper is mathematically dense but the core formula is accessible. It also includes the original derivation of the geometric growth rate maximization and discusses the relationship between information rate and betting edge.

Key Concepts

  • Information channel analogy: The gambler receives symbols (horse race results) through a noisy channel and uses them to place bets at odds consistent with their probabilities
  • Exponential growth maximization: Kelly maximizes E[ln(wealth)] rather than E[wealth] — this is what gives Kelly its risk-aversion-like properties
  • The Kelly fraction: f* = (bp - q) / b = (p(b+1) - 1) / b for binary outcomes
  • Information rate = growth rate: Kelly showed that the maximum exponential growth rate equals the rate of information transmission over the channel
  • Infinite horizon assumption: Kelly assumes infinite sequence of bets — convergence to the optimal growth rate requires many bets
  • Fair vs. superfair odds: Kelly only recommends betting when b × p > q (positive expected value); with fair odds (b = 1/p - 1), Kelly says don't bet

Formulas

Kelly fraction (binary outcome):
$$f^* = \frac{bp - q}{b} = \frac{p(b+1) - 1}{b}$$

Where b = net odds received (decimal odds - 1), p = win probability, q = 1 - p.

Simplified Kelly (b=1, even money):
$$f^* = 2p - 1$$

A 60% win probability → f* = 0.20 (20% of bankroll).

Geometric growth rate:
$$G(f) = p \ln(1 + fb) + q \ln(1 - f)$$

Kelly maximizes G(f) by setting dG/df = 0.

Capital after N bets:
$$W_N = W_0 \prod_{i=1}^{N} (1 + f_i b_i)^{S_i} (1 - f_i)^{1-S_i}$$

Where S_i = 1 if win, 0 if loss.

Notes

  • This is the original source for the Kelly criterion — the existing kelly-criterion.md note covers the formula and Python implementation; this source provides the academic foundation
  • Kelly's original framing was about information theory (Shannon's channel capacity) applied to gambling — this is why it's called "A New Interpretation of Information Rate"
  • The paper includes the Bernoulli reference (the pre-Kelly gambler's problem) and shows how Kelly's approach generalizes it
  • Key practical insight from the paper: Kelly assumes known probabilities — in practice, our estimated probabilities are never exactly correct, which is why fractional Kelly is used
  • The infinite horizon assumption is critical: Kelly growth rates only converge with many bets; with only 64 World Cup matches, actual results will vary significantly from theoretical Kelly growth
  • The paper also discusses the "doubling scheme" (betting a fixed fraction of current wealth rather than initial wealth) — relevant for understanding why fractional Kelly is often preferred