Kelly Criterion for Prediction Markets: Why Quarter-Kelly or Less

Where Kelly comes from

In 1956 John Kelly, a physicist at Bell Labs, published "A New Interpretation of Information Rate." The question behind it was practical: given an informational edge over a market, what bankroll fraction maximizes long-run compounded growth? Claude Shannon, two hallways over, helped sharpen the argument. The formula sat in academic use for a decade.

Ed Thorp made it operational. Beat the Dealer (1962) used Kelly to size card-counting bets at blackjack. Thorp then ran Princeton Newport Partners on Kelly-family sizing rules for two decades with minimal drawdowns (Thorp, 2017, A Man for All Markets). The framework moved from casino tables to options desks to, eventually, retail prediction markets.

The formula

For a binary contract with known probability p, payoff odds b per dollar risked, and complement q = 1 − p:

Worked example. A Polymarket contract on a Senate race is trading at 40 cents. The trader's view, anchored to a credible poll aggregate, is that the true probability is 55 percent. A Yes share bought at 0.40 pays 1.50 to 1 if it resolves. So p = 0.55, q = 0.45, b = 1.5.

Full Kelly says to put 25 percent of bankroll on this one contract. Every experienced Polymarket trader reading that number feels the alarm bell. It should be ringing. Keep going.

Why practitioners have always used less than full Kelly

Full Kelly is geometrically growth-optimal under two assumptions: the probability estimate is the true probability, and the return distribution has finite variance. In practice neither assumption holds exactly, and full Kelly produces drawdowns of 30 to 50 percent that are mathematically recoverable but operationally lethal (MacLean, Thorp, Ziemba, 2011, The Kelly Capital Growth Investment Criterion, ch. 1).

The compromise for finite-variance markets is fractional Kelly. Half-Kelly has been the traditional practitioner default for equity long/short books, discussed in Thorp's 1997 Wilmott essay and MacLean-Thorp-Ziemba above. Half-Kelly on the Senate example would size at 12.5 percent of bankroll per contract. Still aggressive for most books, but survivable.

The Polymarket complication: the tail index is 1.28

The Hill tail-index estimator applied to realized Polymarket PnL across the Convexly 8,656-wallet cohort returns α = 1.28, with a 95 percent confidence interval of 1.20 to 1.36 (Edge Score Methodology V1). For any α below 2, the variance of the underlying distribution is formally infinite. The second moment that Kelly sizing implicitly depends on does not exist in sample.

This is not a rounding problem. Taleb makes the argument quantitatively in Statistical Consequences of Fat Tails(2020, ch. 10): under power-law payoffs with α < 2, sample variance estimates diverge as the sample grows, so any sizing rule that depends on variance is estimating a quantity that does not converge. Half-Kelly inherits that instability. On Polymarket, the Taleb critique applies directly.

Ergodicity adds a second problem. Peters (2019, "The ergodicity problem in economics," Nature Physics) shows that for non-ergodic, single- path realizations, the expected-value-maximizing bet size is not the time-average-survival-maximizing bet size. An individual trader experiences the time-average path, not the cross-sectional expectation. Even when Kelly is theoretically optimal in cross-section, it is typically too large along any single trader's realized path.

The operational regime: quarter-Kelly inside a barbell

Two concessions stack. First, the Taleb barbell (Taleb, 2012, Antifragile, ch. 11): cap total Polymarket exposure at 10 to 20 percent of overall bankroll. The speculative leg can be wiped out without impairing the safe leg. This is the primary constraint and it is non-negotiable.

Second, inside that speculative cap, take quarter-Kelly as the ceiling per position. On the Senate example, quarter-Kelly is 6.25 percent of bankroll. If total Polymarket exposure is capped at 15 percent of net worth, that means the contract can absorb at most ~40 percent of the speculative leg. A hard per-event cap of 10 percent of the speculative leg then overrides whenever Kelly wants more.

Why quarter and not half. Under Hill α = 1.28, even half-Kelly is estimating a variance that does not converge. Quarter-Kelly loses roughly 75 percent of the theoretical geometric growth rate in exchange for surviving the fat-tail draws that the half-Kelly formula is implicitly betting do not exist. The Convexly 10K-wallet study shows the top 1 percent of profit wallets capture 36.2 percent of signed profit (10,000 Polymarket Wallets Scored): survival is the precondition for being in the distribution at all.

Garbage in, garbage out: calibration is the prerequisite

Kelly, fractional or not, requires a probability estimate. The sensitivity is severe. If a trader thinks p = 0.55 but the true probability is 0.48, Kelly will size the position as if there is edge when there is none. A sequence of such bets compounds into ruin. Calibration measurement (see the Brier score post) tells the trader whether the probability estimates being fed into Kelly are any good before the position-sizing math can help.

This is also why the Convexly finding that Brier explains only about 2 percent of profit-rank variance on Polymarket does not contradict the importance of calibration. Calibration is a necessary floor. It stops the Kelly formula from amplifying a systematic overconfidence error into a total loss. What it does not do is, by itself, produce profits. That part comes from concentration on the events where the trader has a real gap to consensus.