Sizing the Bet & the Real Risks

You now know the whole machine: a share that pays $1 if you’re right and $0 if you’re wrong, an order book where it trades, an optimistic oracle that decides who was right, a calibration plot that scores the crowd, the favorite–longshot bias that warps it, and the binary arbitrage that mops up free dollars. What’s missing is the part that actually moves money out of your account and (you hope) more money back in: how much to bet.

This is where your other two expert courses come collect their debt. Bayesian-finance taught you to build a probability from a prior and evidence; kelly-and-cagr taught you to size a repeated edge so it compounds instead of busting you. This lesson welds them onto a live binary contract: form your probability $p$ the Bayesian way, compare it to the price to read off your edge, size the position with Kelly on a binary share, then deliberately bet a fraction of that — and finally face the risks that no formula prices for you. By the end you can take an opinion about the world and turn it into a position that won’t quietly walk you off a cliff.

From belief to edge — build your probability, then beat the price

Analogy. Before a poker pro shoves chips in, they don’t ask “do I like my hand?” — they ask “is my hand better than the price the pot is offering me?” A prediction market is identical. The price isn’t your opponent’s bluff; it’s the crowd’s carefully-weighted estimate, posted in cents. You don’t profit by being right. You profit by being more right than a price that is already pretty good.

Definition. Your edge on a YES share is the gap between your probability $p$ that the event happens and the market’s price $c$ (in dollars, so 60¢ = $0.60$, which is also the crowd’s implied probability). If $p > c$, YES is underpriced for you and buying it is positive expected value; if $p < c$, NO is the +EV side. Concretely, a YES share costs $c$ and returns $1$ if the event happens, so your expected profit per share is

$\text{EV}_{\text{YES}} = p \cdot (1 - c) - (1 - p)\cdot c = p - c.$

The whole EV collapses to $p - c$: your probability minus the price. Clean. But it’s only real if your $p$ is genuinely better-calibrated than the crowd’s — otherwise the “edge” is just your overconfidence wearing a lab coat.

Forming $p$ the Bayesian way. From the bayesian-finance course: start with a prior from base rates, then update on the specific evidence to get a posterior. Suppose you’re pricing “Will the incumbent win re-election?” Base-rate prior: incumbents in this office have won about 60% of the time, so $P(\text{win}) = 0.60$ before you look at anything specific. Now the evidence: a batch of high-quality polls shows the incumbent ahead. Polls like these have historically pointed the right way about 75% of the time when a candidate is winning, and about 35% of the time when they’re losing ($P(\text{poll lead}\mid \text{win}) = 0.75$, $P(\text{poll lead}\mid \text{lose}) = 0.35$). Bayes:

$P(\text{win}\mid \text{poll lead}) = \frac{0.75 \times 0.60}{0.75 \times 0.60 + 0.35 \times 0.40} = \frac{0.45}{0.45 + 0.14} = \frac{0.45}{0.59} \approx 0.76.$

Your posterior is about 76%. But here’s the move most people skip: the market price is itself a strong prior you should update toward. A liquid crowd at 60¢ has aggregated information you may not have. So don’t bet your raw 76% blindly — shrink it toward the price (the same shrinkage logic as the bayesian-finance course). Land somewhere around 70%, treat that as your $p$, and compare to the 60¢ price: edge $p - c = 0.70 - 0.60 = +0.10$ per share. Positive — YES is the +EV side.

Kelly on a binary outcome share

Analogy. Same casino, new chip. In the Kelly course you bet on a coin that paid $b$-to-1. A prediction-market share is just that coin with the odds written in the price tag: pay $c$, win $1 - c$ if it hits, lose $c$ if it doesn’t. So the payoff odds are baked into the cents — and Kelly slides right in.

Definition. Buy a YES share at price $c$ with your probability $p$. With probability $p$ you stake $c$ to win $1 - c$ (net); with probability $1 - p$ you lose your $c$. That’s a binary bet whose net odds are $b = (1 - c)/c$. Drop it into the Kelly formula $f^* = (pb - q)/b = p - q/b$ from the kelly-and-cagr course and the cents cancel beautifully:

$f^* = \frac{p - c}{1 - c} \quad (\text{buying YES}).$

This is just edge ÷ odds again — the numerator $p - c$ is your edge per share, the denominator $1 - c$ is the profit a winning share pays, i.e. the odds. Nothing new; it’s the binary Kelly you already derived, specialized to a $1/$0 contract.

Worked example — the 70-vs-60 bet. Your $p = 0.70$, the price $c = 0.60$:

$f^* = \frac{0.70 - 0.60}{1 - 0.60} = \frac{0.10}{0.40} = 0.25.$

Full Kelly is 25% of your bankroll on YES. A 70/30 belief priced at 60/40 is a fat edge, and Kelly is happy to size it large — which, foreshadowing wildly, is exactly why you’ll cut it down in a moment.

The NO-side version. If you think the event is less likely than the price says, you buy NO. A NO share costs $1 - c$ and pays $1$ if the event doesn’t happen. By symmetry — replace “win prob” with $1 - p$ and “price” with $1 - c$:

$f^*_{\text{NO}} = \frac{(1 - p) - (1 - c)}{1 - (1 - c)} = \frac{c - p}{c}.$

Example: you think an event priced at 80¢ ($c = 0.80$) is really only 60% likely ($p = 0.60$). YES is overpriced, so buy NO: $f^*_{\text{NO}} = (0.80 - 0.60)/0.80 = 0.20/0.80 = 0.25$. Same 25% — the mirror image of the YES math.

The growth curve below is the very same Kelly hump from the kelly-and-cagr course — drag the win probability and odds to watch where the peak (your $f^*$) sits and where the over-betting cliff begins. For our bet, think of it as $p = 0.70$ with payoff odds $b = (1-c)/c = 0.4/0.6 \approx 0.67$.

Why fractional Kelly here, especially

Analogy. Full Kelly assumes you typed the odds in correctly. In a casino you basically can — the wheel’s geometry is fixed. On Polymarket your “odds” come from a probability you estimated, against a sharp competitor (the crowd) who set the price. It’s like flooring a car toward a guardrail using a speedometer you built yourself out of guesswork. The cliff is real; your instrument is fuzzy. Slow down.

Definition. The kelly-and-cagr course nailed the asymmetry: the growth curve is a gentle hump with a sheer drop on the over-betting side. Past roughly twice Kelly your long-run growth goes negative — you go broke with certainty despite a real edge. Crucially, overestimating your edge has the same effect as over-betting it: if your true $p$ is lower than you think, your “full Kelly” is secretly already past the peak, drifting toward the cliff. Because prediction-market edges are small and your $p$ is noisy, the fix is fractional Kelly — deliberately bet $\tfrac12$ or $\tfrac14$ of $f^*$.

Worked example — half-Kelly on our bet. Full Kelly on the 70-vs-60 bet was 25%. Half-Kelly is

$\tfrac12 \times 0.25 = 0.125 \Rightarrow \textbf{12.5\% of bankroll}.$

From the Kelly course you know the trade this buys: half-Kelly keeps roughly three-quarters of the growth for half the volatility — and, more importantly here, it leaves a margin of safety so that if your true edge is smaller than your estimate, you’re still comfortably on the safe side of the hump. Quarter-Kelly (6.25% here) is even more conservative and is a perfectly reasonable default when your $p$ is little more than an educated guess.

Resolution & oracle risk — being right and still paying $0

Analogy. You bet on a horse, it crosses the line first, and the stewards disqualify it on a technicality you never saw coming. You were “right” and your ticket is worthless. On a prediction market the stewards are the oracle, and the rulebook is the market’s resolution wording — and both can go sideways.

Definition. Resolution risk is the chance that a market settles against the plain-common-sense outcome because of how it’s adjudicated, not what happened in the world. Three flavors: ambiguous wording (the title says one thing, the fine-print resolution criteria say another), an oracle dispute that resolves the wrong way (recall UMA’s optimistic oracle — a proposer posts an outcome, and a disputed market escalates to a token-holder vote that can land on the “wrong” side), and a malicious, well-funded proposal that posts a false outcome and out-bonds honest disputers. The key property: this risk is largely not priced into the share the way ordinary uncertainty is.

Worked example — haircut your probability. Suppose your honest, Bayesian probability that the event happens is $p = 0.70$. But you read the resolution criteria and judge there’s a 5% chance the market resolves contrary to the real-world outcome — ambiguous wording, a live dispute, whatever. Your effective probability of the share paying you is

$p_{\text{eff}} = p \times (1 - 0.05) = 0.70 \times 0.95 = 0.665.$

Re-run Kelly with $p_{\text{eff}} = 0.665$ against the 60¢ price: $f^* = (0.665 - 0.60)/0.40 = 0.065/0.40 \approx 0.16$ — a noticeably smaller full Kelly than the 0.25 you’d get ignoring resolution risk (and you’d then take a fraction of that). The haircut both shrinks your bet and, near a coin-flip price, can erase the edge entirely.

Liquidity & correlated-bet risk

Analogy. A thin order book is a narrow mountain road: easy to get a car or two through, but try to move a convoy and you scrape the walls on the way in and on the way out. And many “separate” bets are secretly the same bet wearing different hats — like insuring ten houses on the same fault line and calling it diversification.

Definition — liquidity risk. A thin book has little depth, so a meaningful order slips the price against you on entry, and — worse — you may not be able to exit at all before resolution. Your capital can be locked in the position until the market settles, turning a “trade” into a hold-to-maturity bond whose maturity you don’t control. Slippage hits you twice (in and out), and the spread is a real cost on top of fees, all of which shrink the edge Kelly thinks you have.

Definition — correlated-bet risk. Many prediction-market positions that look independent are driven by a single underlying event. “Candidate A wins state X,” “…state Y,” “…the election,” and “…the Senate flips” all move together when one national wave arrives. Sizing each at Kelly independently secretly overbets the cluster: from the multi-asset Kelly / risk-of-ruin intuition in the kelly-and-cagr course, correlated positions stack their downside, so five “12.5% of bankroll” bets that all resolve on the same election aren’t a diversified book — they’re closer to one 62.5% bet on that election. That’s deep into over-betting territory, and a single bad night can take the whole stack.

Worked example. You find five YES shares you’d each size at half-Kelly (12.5% of bankroll). If they were truly independent, fine — Kelly on independent bets roughly adds. But all five resolve on the same election. Treat the cluster as one correlated position: your real exposure to that one event is up to $5 \times 12.5\% = 62.5\%$ of bankroll — well past twice the Kelly you’d ever set for a single event. The fix: size the theme, not each ticket. Decide how much of your bankroll the election as a whole deserves (say, 12.5%), then split that across the five correlated shares.

Regulatory & practical risk

Analogy. Even a perfectly-sized, well-reasoned bet runs on plumbing you don’t own: a chain, a stablecoin, and a legal environment — any of which can be turned off or spring a leak regardless of whether you called the event correctly.

Definition. Beyond the bet itself, prediction markets carry structural risks. Regulatory and geofencing risk: access to prediction markets varies by jurisdiction and changes over time — some venues restrict or block certain users or regions, and the legal treatment of these markets differs from place to place and evolves. Smart-contract risk: the market lives in code on-chain (recall Polygon), and a bug or exploit in that code can affect funds independent of any market outcome. Stablecoin risk: settlement is in USDC, so you inherit whatever risk that stablecoin carries — a depeg or issuer problem would hit the dollar value of your position and winnings even if you were right about the event.

Worked example — sizing for the plumbing. None of these show up in the share price or your edge calculation, so the only defense is, again, bet smaller and don’t concentrate. If you judge there’s a small but non-zero chance the venue itself fails you (geofencing change, contract bug, stablecoin wobble) in a way that costs you the position, that’s another reason your effective probability of getting paid is below your event probability — and another reason fractional Kelly, capping total exposure to any one platform, is the sane default. Think of platform risk as a permanent extra haircut on $p$ that you can shrink but never fully remove.

Putting it together — and the safety dial

Three of these levers all push the same way: when you’re unsure of your edge (fractional Kelly), when resolution might betray you (probability haircut), or when bets are correlated (size the theme), the correct response is to bet less. Prediction-market sizing is one big argument for humility.

That’s the last piece of the machine: you can now read a price as a probability, build a better probability of your own, measure the edge between them, size it with binary Kelly, cut it down with a fraction and a haircut, and respect the risks no formula prices. What’s left is to prove it. The course closes with the Final Exam — a single graded, locked run across everything you’ve learned, from the order book and minting through UMA resolution, calibration and the favorite–longshot bias, binary arbitrage, and the Bayesian-Kelly sizing of this lesson. One question at a time, no going back, no retries; each answer locks the moment you submit, and your pass/fail score lands only at the end. Bring the whole course — it’s all fair game.