The Math Behind Polymarket: How Prediction Markets Price the Future

On election day 2024, one market on a crypto app was quietly doing something the pollsters could not: it was pricing the future with real money. Polymarket's 'Presidential Election Winner' market traded more than 3.2 billion dollars in volume by that day, and its Trump price — a single number between 0 and 1 — was a live, money-weighted forecast of who would win. Not a pundit's guess. A price, set by thousands of people staking their own cash on being right.

One of them was a French trader known only as 'Théo,' who turned roughly 30 million dollars into about 85 million by trusting that number over the polls. His story is the seductive part. The math underneath it is the useful part — and it is the sibling of the sports-betting article's math: a price is a probability, a fair bet is one where expected value is zero, and even in a market this elegant, the average participant still loses. The difference is that here the house's tax is almost invisible, which makes the whole mechanism easier to see clearly.

On a prediction market you are not betting against a bookmaker. You are buying a share in an outcome. A Polymarket contract pays exactly one dollar if the event happens and zero if it does not. So if 'YES' is trading at sixty cents, the market is telling you it believes the event is about sixty percent likely.

That is the whole trick, and it is worth pausing on. The price is not analogous to a probability. It is one. Rearranged, the implied probability of an outcome is simply its price:

A share at ninety cents encodes a ninety-percent belief; a share at four cents encodes long-shot odds of about one in twenty-five. Every dollar that moves in or out nudges that number, so the price updates continuously as new information arrives — the moment a candidate concedes, the moment a jobs report drops. It is a probability you can watch tick in real time.

To know whether buying a share is a good bet, you need the same tool the sportsbook quietly uses against you: expected value. Suppose you believe the true probability of an outcome is p-true, and the share is trading at some price. Buy one YES share and, on average, you collect p-true dollars against the price you paid up front. Your expected profit per share is the difference:

The consequence is clean. The bet is fair — expected value exactly zero — precisely when the price equals your true probability:

If you think an event is seventy percent likely and the market prices it at sixty cents, each share carries a positive expected value of about ten cents, and buying is rational. If the market prices it at eighty cents, your edge is negative and you should walk away — or sell. Everything a prediction-market trader does reduces to hunting for prices that disagree with a probability they can defend better than the crowd can.

There is a second law holding the whole system together, and it needs no trust in anyone's judgment — only arithmetic. On a binary market exactly one of YES and NO will pay out one dollar. So the two prices must add up to a dollar:

If they did not — say YES traded at sixty cents and NO at thirty — you could buy both for ninety cents, guarantee a one-dollar payout, and pocket ten cents with no risk at all. That is a textbook arbitrage, and in a liquid market traders erase it in seconds, which is exactly what forces the sum back to a dollar. This is why a prediction market's prices behave like a single coherent probability distribution rather than two independent guesses. The no-arbitrage condition is doing the bookkeeping.

Polymarket, founded in 2020 by Shayne Coplan, runs this machinery on the Polygon blockchain, and every position is denominated in USDC, a dollar-pegged stablecoin. Orders are matched by a hybrid order book — the matching happens off-chain for speed, but the settlement is recorded on-chain — so the venue looks and feels like a fast electronic exchange rather than a casino.

The detail that matters for the math is the cost of trading. Polymarket charges no direct trading fee. Your cost is the spread — the small gap between the best price to buy and the best price to sell — plus the blockchain's own transaction costs, which are tiny on Polygon. Compare that to a sportsbook, where the house's cut is baked into the odds on every single wager, and you can see why a prediction-market price sits closer to a true probability than a bookmaker's line does. There is far less tax distorting the number on the screen.

To feel the difference, price the same fifty-fifty event three ways. A fair coin should trade at fifty cents, implying fifty percent, with the two sides summing to exactly a dollar. A sportsbook instead offers the standard minus-110 line on each side, which implies:

Each side implies 52.38 percent, and the two sides sum to 104.76 percent — an impossible probability. That extra 4.76 percent is the overround, the vig, the book's built-in tax:

The book does not need to predict the game; it collects that margin no matter who wins. A prediction market, charging only the spread, keeps its two-sided sum right up against a dollar. The table shows where each venue takes its cut on a fifty-fifty event.

Read down that last column and the series' thesis comes into focus. The lottery is a probability with a colossal tax; a sportsbook line is a probability with a modest one; a fee-light prediction market is about as close to a naked probability as a real market gets. Closer to true does not mean free — but it means the number on the screen is doing more forecasting and less skimming.

So is a money-weighted crowd a good forecaster? For liquid, near-term questions the answer is: usually, and often better than the alternatives. A well-calibrated market is one where events priced at about seventy percent actually happen about seventy percent of the time — call it a hit rate somewhere in the high sixties to low seventies, as an illustration. Skin in the game is the reason: a pundit pays nothing for being wrong, while a trader who misprices an outcome loses real money, and that feedback loop grinds prices toward accuracy.

The failure modes matter just as much, and they are structural. Calibration decays in thin markets, where a single large trader can move a price that few others are correcting, and in long-horizon questions, where there is little information and lots of wishful betting. After the 2024 election, Polymarket's volumes fell roughly eighty-four percent — and a thin market is a noisy forecaster. The crowd is wise mainly when the crowd is large, liquid, and paying attention.

Prediction markets went mainstream in 2024. The single presidential-winner market traded more than 3.2 to 3.3 billion dollars in volume by election day, and roughly 3.6 to 3.7 billion over its life. At one election-day snapshot the Trump side held about 1.3 billion dollars in volume against roughly 827 million on the Harris side, according to Fortune. Election markets came to dominate the platform, accounting for close to ninety percent of its open interest.

That concentration is a double-edged fact. It made Polymarket a genuine real-time barometer of the race, watched by newsrooms that once quoted only polls. It also meant that when the election ended, most of the liquidity left with it — the same eighty-four-percent drop — which is precisely the thin-market risk the calibration math warns about.

Polymarket's legal path in the United States is worth getting exactly right, because it is a case study in how a new market gets absorbed by an old regulator. On January 3, 2022 the CFTC fined the operating entity, Blockratize, 1.4 million dollars and required it to wind down unregistered markets for US users. Then, on November 13, 2024 — days after the platform's biggest week ever — the FBI searched founder Shayne Coplan's apartment; no charges followed. On July 15, 2025, the Department of Justice and the CFTC dropped their probes. Days later, on July 21, 2025, Polymarket closed a 112 million dollar acquisition of QCX and QCEX, a CFTC-licensed exchange and clearinghouse — buying, in effect, a regulated door back into the US market. And on November 25, 2025, the CFTC amended its order of designation, clearing the way for a regulated American return.

Read in order — a fine, a raid, dropped probes, a license — the timeline is the arc of a novel technology moving from the regulatory gray zone into the daylight. The math never changed. The paperwork did.

Here is where the sibling article and this one meet. The mechanism is genuinely beautiful: a live, self-correcting probability, built by strangers staking their own money, taxed so lightly that the price is almost a pure forecast. It is often a better estimate of the future than experts produce, and it costs you almost nothing in vig to read.

And yet the average participant still loses. Not because the market is rigged — it is far fairer than a sportsbook — but because trading it profitably requires you to be systematically better-calibrated than a large, motivated crowd, and almost no one is. Add the spread, add the thin-market noise, add the human pull toward long shots and hometown teams, and the median account drifts negative. The beautiful part and the losing part are the same fact: the price is already about right, so there is little edge left lying around. The people who make money here reliably are the ones who can out-model the crowd — and that is a skill, priced and paid, not a lucky night.

Can you actually beat a prediction market? A few people do, the way a few people beat any market: by being better-calibrated than the crowd on a specific question, or by spotting a price the crowd has not corrected yet. But the market price already encodes the crowd's best money-weighted guess, so your edge has to beat that, after the spread. For almost everyone, it is negative expected value — closer to fair than a sportsbook, but still a loss on average.

Are prediction markets legal in the US now? The picture cleared through 2025. Federal probes against Polymarket were dropped in July 2025, the company acquired a CFTC-licensed exchange that same month, and the CFTC amended its order of designation in November 2025 — the pieces of a regulated US return. Rules still vary and are evolving, so treat any specific claim as time-sensitive.

Is it just gambling? Mechanically it resembles betting, but a fee-light prediction market with a true no-arbitrage price is closer to a forecasting instrument than a casino game, and its prices carry real information that newsrooms and researchers use. That does not repeal the expected-value math: for the average trader it is still a negative-EV activity.

Are markets more accurate than polls? For liquid, near-term questions they are often at least as good, because traders are penalized in cash for being wrong. But the edge shrinks in thin or long-horizon markets, where a few dollars set the price and calibration decays. A market is only as wise as it is deep.