To Kelly or not to Kelly

Kelly criterion is a popular bet sizing formula that maximizes the expected logarithmic growth of your bankroll. The formula is simple:

\[ f^* = p - \frac{q}{b} = p + \frac{p-1}{b} \]

where:

• \( f^* \): is the fraction of the current bankroll to wager

• \( p \): is the probability of a win

• \( q \): is the probability of a loss \( (q = 1 - p) \)

• \( b \): is the proportion of the bet gained with a win (\( 1 \) means you will gain the same amount you bet)

Applying Kelly to prediction market gambling

Given a prediction market with a mispriced outcome, you can use the Kelly formula to determine the optimal bet size. For example, if you believe a market is mispriced at 60% when you think the true probability is 70%, the net odds of buying in at 60c are b = (1 - 0.6) / 0.6 = 2/3, so you should bet

\[ f^* = 0.7 + \frac{0.7-1}{\frac{2}{3}} = 0.25 \]

Intuitively, betting 25% of your bankroll on a single market still seems too risky, as a single loss wipes out a quarter of your bankroll. Due to the risk of ruin by slight probability misestimations, many will bet a fraction of the Kelly amount, with popular fractions being 1/2 Kelly (aka half-Kelly) or 1/4 Kelly (aka quarter-Kelly).

However, I’d argue that even fractional Kelly still has significant drawbacks and that a simple extension to the concept can boost risk adjusted returns while still offering a framework to take on extra risk whenever needed.

Drawbacks of Kelly

The biggest drawback of Kelly is its sensitivity to probability misestimation. If you misestimate the probability of winning, the results can be catastrophic. However, most people are aware of this, hence the popularity of fractional Kelly.

Take the 60c market from the example above. Here is how bet size changes your expected bankroll growth per bet, depending on the true probability:

Even when your estimate is right, the top of the curve is flat: betting 15% of your bankroll instead of 25% barely dents your growth. This is exactly why fractional Kelly works as well as it does. But when your estimate is wrong, the curve falls off a cliff. If the true probability turns out to be the market’s own 60%, every positive bet size has negative expected growth, and the full Kelly bet grinds your bankroll down at about 2% per bet.

Sometimes estimations can be very uncertain. For example, there are markets like “Trump calls Nicki Minaj hot again in 2026?”, yes I know, this is a real market, and I can shamelessly say that I have bet on it. There’s no reasonable way to estimate the probability of this event, however, it’s possible to have a feel for the probability with respect to the market price. In the absence of certainty, the Kelly formula can be hard to apply.

Ideally, you want to be able to bet even when you are uncertain of your own edge. Ways to do this include fixed bet sizing: a fixed amount, a fixed fraction of your bankroll, or a fixed number of shares.

Fixed fraction bet sizing

The simplest alternative is to risk the same amount on every bet no matter the price or your perceived edge: either a fixed amount (e.g., $100 per bet) or a fixed fraction of your bankroll (e.g., 2% per bet).

The appeal is obvious:

• No probability estimates are required, so you can bet even on markets where you cannot quantify your own edge.
• Dollar risk is identical across bets and trivial to reason about. You always know the worst case for a single bet.
• The fixed fraction version scales automatically with your bankroll.

The problem is that it treats a 5c contract and a 95c contract as the same bet. $100 buys 2,000 shares of a 5c contract but only about 105 shares of a 95c contract. Your share exposure is massively skewed towards the cheapest contracts, which, as we will see, is exactly the wrong place to be skewed.

Here is how many shares the same bet buys at every price, a fixed $100 stake versus a fixed 100 shares:

Fixed shares bet sizing

The alternative I use is to buy the same number of shares on every trade. For example, I might buy 100 shares no matter the market.

Since every share pays out $1 on a win, the money at risk is simply the price times the number of shares: 100 shares cost $5 at 5c and $95 at 95c, and the payout is always $100.

A few practical notes:

• Share counts are easy to reason about and compare across markets. “I own 100 shares” is cleaner than “I own $23.70 worth of some contract”.
• Dollar risk does not scale with your bankroll. As your bankroll grows, step your share count up in tiers instead of adjusting it continuously.
• Dollar risk does scale with price. A 95c contract ties up 19 times the capital of a 5c contract for the same share count, so the rule implicitly demands much more confidence in expensive contracts.

Why I prefer using fixed shares

Fixed fraction sizing puts the same number of dollars into every bet. Fixed shares sizing puts more dollars into high-priced contracts and fewer dollars into low-priced ones. This matters because of a well documented risk premium: high-priced contracts are generally undervalued.

A recent study of Polymarket examines 588 million trades by 2.48 million wallets and documents a classic favorite-longshot bias: the cheap contracts that everyone loves to buy are overpriced, and the expensive contracts that nobody wants are underpriced.

• Purchases below 10c lose 19.3 cents per dollar wagered, while purchases at or above 90c earn 0.83 cents per dollar.
• Weighting every market equally instead of every dollar, longshots still lose 6.3 cents per dollar while favorites earn 0.28 cents per dollar.

Here are the same numbers charted across the three ways the study aggregates returns: weighting each market equally, weighting each dollar equally, and weighting each parent event equally (a parent event is Polymarket’s umbrella listing that groups related contracts, such as “Presidential Election Winner 2024” holding one contract per candidate):

To back this up with my own data, I pulled every resolved Polymarket market since 2023 with more than $25k of volume: 32,918 contracts from 17,487 markets. For each contract I compared the last traded price at least 24 hours before resolution with how often the outcome actually happened:

Every bucket below 50c sits below the line, paying out less often than its price implied, and every bucket above 50c sits above it. The error bars show 95% intervals, so any single bucket is noisy, but ten buckets in a row on each side is no coincidence. This is a market level view rather than the study’s purchase level view, and it only looks a day ahead, but the shape is the same: too optimistic on cheap contracts, too pessimistic on expensive ones.

In other words, boring expensive contracts carry a risk premium for whoever is willing to hold them, while lottery ticket longshots are systematically overpriced.

Fixed shares exploits this automatically. For a fixed share count, dollar exposure is proportional to the price, so the bulk of your capital ends up in the undervalued high-priced contracts. With fixed fraction sizing, a 5c longshot and a 95c favorite receive the same capital, which means buying 19 times more shares of the longshot, massively overweighting the worst performing part of the price distribution.

To be clear, this is a premium earned on average, not a free lunch on every trade. The same study shows that longshot returns flip sign when related contracts are grouped by parent event, and that the two-sided pattern is robust in Crypto and Politics but surprisingly absent in Sports. Fixed shares simply makes sure that the average dollar you wager sits on the right side of the bias.

When to break the rules

Fixed share sizing is a default, not a religion. There are two situations where I happily deviate from it.

Markets where you have real conviction. If you have done the work and genuinely believe a market is mispriced, it is fine to bet larger than your standard size. This is especially true when the edge is structural rather than a matter of opinion, like the near-arbitrage on Musk net worth markets I wrote about earlier. Standard fixed sizing treats a random market you have never looked at and the trade of the year exactly the same way, which leaves money on the table when you truly have an edge. The whole reason we avoid sizing by estimated edge is that those estimates are usually unreliable; when the estimate is trustworthy, there is nothing wrong with betting accordingly.

Super long shot bets. For super longshots, it is also fine to use larger share counts. The return profile is greatly asymmetric: a 2c contract risks 2c per share for the chance at $1, a 50x payout. Even a large share count keeps the absolute dollars at risk small, while a single win pays for a great many losses. If the longshot is genuinely mispriced in your favor, the upside dwarfs the stake.

A word of caution: bet correlations

One thing bet sizing rarely account for is correlation between bets.

Ten standard size bets that are all really the same bet is one bet of ten times the size. Markets on the same theme, such as a cluster of “Trump does X by date Y” markets or several Musk net worth markets, resolve together and lose together. Size each of them as if they were independent and your true exposure is much larger than your position list suggests.

Kelly theory at least has a portfolio version that handles dependence. Fixed sizing has no theory at all, so you have to track aggregate exposure yourself:

• Cap your total exposure per theme or underlying risk factor.
• Treat contracts under the same parent event as one position. The Polymarket study I mentioned earlier finds that measured longshot returns change drastically depending on whether contracts are grouped by parent event, which is a good reminder that these contracts are linked.
• Correlated longshots are the most dangerous case: a dozen small cheap positions on the same underlying story can all expire worthless on the same day.

For more on how wrong independence assumptions can go, see my earlier post on examining parlays in prediction markets.

Why the human touch is important

It is tempting to turn all of the above into a rigid process: always buy N shares, never deviate, never think. I would argue against that. Developing a rigid process ignores nuances such as trading dynamics, market structure, and resolution mechanics, all of which can matter much more than your bet size.

A few examples:

• Market structure. A contract quoted at 5c might only be fillable at 7c with a bid at 3c. On a longshot, a spread like that can eat a large fraction of your entire expected edge. Thin order books also mean that your own order moves the price.
• Trading dynamics. How a market trades tells you things no formula can: who is on the other side, whether a price move is noise or information, and whether liquidity will still be there when you want to exit.
• Resolution risk. An edge is worthless if the market resolves against you for rules reasons. Reading the resolution criteria carefully is part of sizing a bet, even though no formula has a term for it.

Conclusion

Kelly is a beautiful theoretical result and a dangerous practical tool. It maximizes growth only when your probability estimates are right, and it punishes you severely when they are not. Fractional Kelly dampens the punishment but does not fix the underlying problem.

My answer is to stop estimating probabilities for sizing purposes at all and buy a fixed number of shares instead. It is simple, it needs no edge estimates, and it quietly exploits the risk premium in high-priced contracts. When you have real conviction, bet bigger. On super long shots with wildly asymmetric payoffs, buy more shares. Watch the correlations between your positions, and use your own judgement.

Bet sizing is not about maximizing the growth rate on paper. It is about survival and making enough repetitions for your edge to show up.