The market-maker problem

Open a brand-new market and you hit a wall immediately. You want to buy a contract — but a trade needs a counterparty, and on a market nobody has touched yet, there is no one on the other side. The order book is empty. No bids, no asks, no price. The cleverest question in the world is worthless if there is nothing to trade against it.

This is the cold-start problem, and it's the quiet reason most prediction markets die before they live. A market needs traders to be liquid, but traders only show up where there's already liquidity. It's a chicken-and-egg standoff, and the order book can't break it on its own — an empty book stays empty.

The fix is to put a robot in the middle that will always quote a price. Not a human market maker who can step away, but an automated market maker (AMM): an algorithm that stands ready to buy or sell any contract at any moment, with a price that's a pure function of how much has been traded so far. From the first second, the market has a counterparty. The whole edifice — the live probability, the hedging, the forecast — rests on this one piece. And it is the hardest piece to get right, because the algorithm has to be willing to lose money to a better-informed trader and still be solvent at the end of the day.

In What is a prediction market? I introduced the canonical AMM — Hanson's scoring rule — in a paragraph. This piece is the deep version, and it runs in two movements. First the machine: the two designs that run today's markets and the math that makes each one work. Then the money: why the always-on quote loses to the informed by construction, who absorbs that loss, and how the answer quietly draws the map of which markets get to exist at all. Because the choice of market maker is not a detail — it is a choice about who pays to make the market exist.

What a market maker has to promise

Before any formula, fix what the robot owes you. A usable AMM has to satisfy three things at once, and they pull against each other.

First, it must always quote — a firm bid and ask for every outcome, no matter how lopsided the book gets. Second, its prices must be coherent: for a binary market the YES and NO prices have to sum to $1, so the quote reads as a clean probability and can't be arbitraged against itself. Third, the operator needs to know its worst case — how much money this thing can bleed if every trade goes against it — before switching it on. An AMM with an unbounded downside isn't a product; it's a way to go broke on schedule.

Two families of mechanism deliver on those promises in very different ways. One was designed for forecasting and bounds the operator's loss by construction. The other was borrowed from crypto exchanges and shifts the cost onto whoever supplies the capital. Start with the one built for this job.

LMSR — the forecaster's market maker

In 2003, Robin Hanson published the Logarithmic Market Scoring Rule, and it remains the standard AMM for prediction markets.¹ LMSR doesn't track a book of bids and asks at all. It tracks one thing: \(q_i\), the total number of shares ever bought on each outcome \(i\). The entire market is summarized by a single cost function:

Read \(C(\mathbf{q})\) as the total amount of money the market maker has taken in once the outstanding shares are \(\mathbf{q} = (q_1, \dots, q_n)\). To buy shares and move the state from \(\mathbf{q}\) to \(\mathbf{q}'\), you pay the difference, \(C(\mathbf{q}') - C(\mathbf{q})\). That's the whole pricing engine — one scalar function, no counterparty required. The robot is happy to take any trade; it just charges the change in \(C\).

The price you see quoted is the slope of that cost function — the marginal cost of the next infinitesimal share. Differentiate \(C\) with respect to \(q_i\) and the instantaneous price of outcome \(i\) falls out as a softmax:

This is a beautiful little result, and two properties of it are worth pausing on. The prices always sum to one — \(\sum_i p_i = 1\) by inspection, since the denominator is the sum of every numerator — so an LMSR market is a coherent probability distribution at every instant, automatically. You can never get the YES and NO quotes to add up to anything but a full dollar, which means there's no internal arbitrage to pick off. And the prices respond the way intuition demands: buy YES, you push \(q_{\text{YES}}\) up, the softmax moves probability toward YES, and the next YES share costs more. The market maker leans against every trade, exactly like a human specialist would.

The liquidity parameter \(b\)

Everything turns on the single constant \(b\), the liquidity parameter. It sets how hard the price resists being moved. A trade pushes the price; \(b\) decides how far. Crank \(b\) up and the book is deep — large orders barely budge the quote, and traders enjoy tight pricing. Shrink \(b\) and the book is thin — even a small buy sends the price flying.

This is the most important picture in the whole subject, so here it is literally. The same fixed-size buy walks the price up a steep ramp in a thin market and a gentle slope in a deep one.

Thin · small b · steep impact Deep · large b · gentle impact

So why not just set \(b\) enormous and give everyone a deep, friendly market? Because depth is not free, and the bill lands on the operator. To see why — and why it's a bill at all, not just a setup cost — you have to look at who the robot is trading against.

Why it loses money — adverse selection

Here is the uncomfortable fact at the center of market-making, the one the cost function alone won't tell you: an automated market maker that always quotes a price will, on average, lose money to the people it trades against. Not because it's badly designed — because it can't tell who's on the other side. The robot quotes the same firm price to everyone. Some of those traders are noise: they're rebalancing, hedging, or trading for fun, and their orders are uncorrelated with where the price is actually heading. Some are informed: they know something the price hasn't caught up to yet — a poll, a filing, a result — and they only trade when the quote is on the wrong side of the truth.

Against the noise traders the maker does fine: they buy and sell roughly at random around the fair price, and the spread it charges is pure profit. Against the informed it is structurally the patsy. The informed trader buys YES precisely when YES is too cheap and sells when it's too dear; every trade they take is one the maker wishes it hadn't. This is adverse selection, and it's the oldest result in market microstructure: a maker quoting a fixed price loses to better-informed flow and profits from uninformed flow, full stop.²

Write it as a ledger. Suppose a fraction \(\alpha\) of the orders that hit the quote come from informed traders and the rest, \(1-\alpha\), are noise. On a noise trade the maker earns the spread; call that gain \(g\). On an informed trade it gets picked off and loses some amount \(\ell\). Its expected profit per trade is

A market maker that wants to break even has to set its spread wide enough that the first term covers the second — that's the Glosten–Milgrom intuition, and it's why every quote has a spread in the first place. But notice what happens when you don't insist on breaking even. If you're willing to let \(\mathbb{E}[\pi]\) run negative — to keep the spread tight, the quotes generous, the market deep and inviting, which is exactly what a bigger \(b\) buys you — then that expected loss is the money you are spending to make the market good. The subsidy and the adverse-selection loss are the same number. Depth isn't free because the informed traders collect for it; liquidity costs precisely what they extract.

So the robot doesn't just spend money to open the market — it bleeds money to run it, by design, to whoever knows more than it does. The obvious worry is that this bleed could be unbounded: a stream of perfectly informed traders, picking the quote off over and over. The thing that makes LMSR safe to run is that it can't.

The bounded loss — \(b \ln n\)

Here is the elegant part, and it's what makes LMSR the right tool for forecasting. The maker bleeds to the informed, as we've just seen — but its maximum loss is capped, and you can compute the cap before you open. For a market with \(n\) outcomes, the most the operator can ever pay out, net of what it collects, is

The intuition is clean. The market starts at \(\mathbf{q} = \mathbf{0}\), where the maker has collected \(C(\mathbf{0}) = b\ln n\) worth of "starting" value. It ends when the truth is known: the winning outcome's shares pay out a dollar each, the losers pay nothing, and the cost function collapses to the winning corner. Work through the accounting and the gap between what the maker takes in and what it pays out is never more than that initial \(b\ln n\) — no matter what traders do, in what order, with what information, including the worst case where every dollar of informed flow lands against the maker. For a simple YES/NO market, \(n = 2\), so the cap is just \(b\ln 2 \approx 0.69\,b\). Pick \(b\) and you've picked exactly how much you're willing to spend to seed the market.

That single number — a known, finite subsidy — is why LMSR fits forecasting so well. The operator is, in effect, buying a public probability for the crowd and paying a fixed, capped price for it. There is a real tension baked into that constant, and it's worth stating plainly:

The right way to read \(b\ln n\), then, is as rent. It is what you pay to rent a liquid market into existence — the price of buying the crowd a public probability — and the rent scales with the depth you want. The picture below is the same adverse-selection bleed drawn over the life of a market: informed flow drives the maker's running P&L down as the price is pushed toward the truth, but the curve flattens against a hard floor at \(-b\ln n\) and goes no further. The gap between zero and the curve is exactly the money the informed collect — and exactly the subsidy the operator agreed to spend. There is no free depth; depth is a number on that floor, and the floor only moves one way as \(b\) grows.

So LMSR's loss is bounded, but it is also guaranteed to be spent — the subsidy is a cost the operator eats to bring the market into being. That's the right shape for a forecasting tool a company or an exchange wants to stand up. But it raises an obvious question: what if you don't want to subsidize the market at all? What if you could get someone else to fund the liquidity, and pay them for it? That's the second design — and it comes from a different world entirely.

CPMM — liquidity borrowed from DeFi

When prediction markets moved on-chain, they inherited the market maker that crypto exchanges had already converged on: the constant-product market maker, the engine inside Uniswap-style automated exchanges.³ Instead of a cost function, a CPMM holds a literal pool of two reserves — for a prediction market, a pile of YES tokens \(x\) and a pile of NO tokens \(y\) — and enforces one stubborn rule on every trade:

The product of the reserves must stay constant. That invariant is the market maker. Want to buy YES tokens? You add NO tokens to the pool and remove YES tokens, and the only constraint is that \(x\cdot y\) comes out to the same \(k\) you started with. Because pulling YES out shrinks \(x\), you have to add a more-than-proportional amount of \(y\) to hold the product fixed — and that asymmetry is precisely what makes the price move against you as you buy. No order book, no cost function, just a hyperbola the trades have to stay on.

The quoted price is the slope of that hyperbola — the ratio of the reserves. Differentiate \(x\cdot y = k\): along the curve \(y = k/x\), so

The marginal price of YES, in units of NO, is the reserve ratio \(y/x\) — read straight off the pool. When the two reserves are equal the price is 1:1 — a 50/50 market. As traders drain one side, the ratio tilts and the price with it. Here too the quote stays coherent: a YES token and its matching NO token always redeem together for one dollar at settlement, so their prices pin to a dollar by construction, the same coherence LMSR gets from its softmax.

Geometrically the whole mechanism is one curve, and a trade is a slide along it. A small trade barely moves you; a large one drags you up the steepening arm of the hyperbola, and every share you buy is priced worse than the last. That worsening is slippage.

The genius of the CPMM is who supplies the capital. Anyone can be a liquidity provider: deposit matched value into both reserves, and the pool puts your money to work quoting prices. In exchange you collect a fee on every trade that flows through. Liquidity is no longer a subsidy the operator burns — it's a yield-bearing position that the crowd funds, permissionlessly, because they're paid to. That's why CPMMs are capital-efficient and need no central underwriter: the market bootstraps its own depth.

But the liquidity provider is not getting paid for nothing. They're carrying a real risk, and it has a famously gentle name.

Impermanent loss — the price of being the pool

When the price moves, the CPMM rebalances the LP's deposit automatically — and always in the wrong direction. As YES rises, the invariant sells the pool's YES and accumulates NO; the LP ends up holding more of whatever fell and less of whatever rose. Compared with simply holding the two piles of tokens untouched, the LP comes out behind whenever the price has moved in either direction. That shortfall is impermanent loss⁴ — "impermanent" because it closes if the price wanders back, and permanent the moment the LP withdraws while it's open. In a prediction market the price doesn't wander back; it marches to 0 or $1 at resolution. The "impermanent" loss becomes entirely permanent, and the LP is left holding a fat stack of the losing token. The fees have to be rich enough to cover that, or no one provides liquidity for long.

So the two mechanisms answer the same question — who pays to make the market exist? — with opposite answers. LMSR says the operator pays, a fixed and bounded subsidy. The CPMM says the liquidity providers pay, a variable cost they accept in exchange for fees. Neither is free; the cost just shows up in a different place on the ledger.

LMSR

Hanson · cost function · the operator subsidizes

• Prices from a cost function — softmax over outcomes
• Loss bounded by \(b\ln n\), known before launch
• Liquidity is a fixed subsidy the operator funds
• Built for forecasting; favored off-chain

CPMM

Uniswap-style · x·y=k · the LPs supply capital

• Prices from the reserve ratio along a hyperbola
• Permissionless, capital-efficient, self-bootstrapping
• LPs earn fees but bear inventory risk / impermanent loss
• Borrowed from DeFi; favored on-chain

Who picks up the bill

That mechanism cut — operator-subsidized versus LP-funded — is really a cut over who pays. Open it up and there are only three places the cost of liquidity can land. Every prediction market you've ever seen is one of these three, or a blend.

The first answer: the exchange or a sponsor eats it. An operator funds the subsidy directly — runs the LMSR market maker, accepts the \(b\ln n\) loss — because it wants the market to exist. Maybe it's a loss-leader to pull in volume and brand; maybe the sponsor genuinely wants the number the market will discover, the way a company stands up an internal market to forecast a launch date, or a research funder underwrites a replication market to price whether a result will hold. The subsidy buys information the sponsor values more than the cost.

The second answer: liquidity providers eat it. This is the on-chain, constant-product route. Nobody central underwrites the book; instead the crowd deposits capital into the pool and is paid fees for it. The LPs earn the spread the way the maker did — and they bear the same adverse selection, which in a CPMM shows up as the impermanent loss we just walked through. In a market that resolves to 0 or $1 that loss isn't impermanent at all; the price marches to a corner and the LP is left holding the losing token. The cost is still being paid — just by a distributed set of capital providers who took the trade because the fees, they hope, out-earn the bleed.

The third answer: the traders eat it, through the spread and the fees. Tighten nothing, subsidize nothing — just quote a wide enough spread and charge enough fees that the noise traders collectively cover what the informed extract. Here the market funds its own liquidity out of the pockets of the people using it. It works, but only where there's enough trading volume for those small per-trade charges to add up to the cost of carrying the informed. That last clause is the whole game.

The sponsor

Exchange · research funder · loss-leader

Funds the subsidy outright to bring the market into being — a loss-leader for volume, or because it wants the price the market will discover.

Cost → a known, capped \(b\ln n\)

The providers

CPMM LPs · permissionless capital

Crowd-funded liquidity, paid in fees. The pool rebalances against them; in a binary that resolves, impermanent loss becomes permanent.

Cost → fees vs. impermanent loss

The traders

Spread · per-trade fees

The market funds itself: noise flow pays the spread that covers the informed. Works only where volume is thick enough to foot the bill.

Cost → wider spread, higher fees

Notice that none of these makes the cost disappear. They relocate it. The sponsor pays it as a budget line, the LPs as a drag on yield, the traders as a worse price — but the adverse-selection loss is conserved. It is the gas bill of the whole apparatus, and someone's name is always on it.

Cold start, and why the cost draws the map

Step back, put the two halves together, and the economics of the category falls out. A market needs a subsidy to be liquid. A market can fund that subsidy from its own fee volume only if it has enough trading. But trading shows up where there's already liquidity — the cold-start standoff we opened with, now with a price tag on it. So a niche market on an obscure question is trapped: too little volume to pay for its own depth, and too little depth to attract the volume that would pay for it. Unless a patron — a sponsor, a hedger, an advertiser, a protocol budget — is willing to cover the subsidy out of pocket, the market stays thin, its spread stays wide, and its price stays a weak signal.

This is the real reason the liquid prediction markets are the ones you can name. Big elections and major sports clear the bar because their fee volume is enormous: millions of noise traders, trading for the thrill, throw off more than enough spread to pay for the informed sharpening the line.⁵ A market on whether a specific bill clears a specific committee next spring does not — there simply isn't the flow to fund the quote, so it never gets deep, and the thin ones stay thin for the structural reason this whole piece has been building toward: no one will pay the liquidity bill on them. It's not that the question is uninteresting. It's that no one will pay the liquidity bill on it.

Which is why the always-on price is not just the product — it's the binding constraint on the whole map of what's tradable. A prediction market is sold as a clever idea: a price that's a probability, a forecast with money behind it. But the idea is inert until something is willing to quote a price when the book is empty, and to keep quoting through every lopsided, adversarial, last-minute trade. The always-on price is the product, and everything else in this series — the calibration, the accuracy, the hedge — only exists downstream of a market maker that refused to go silent. The catch is that the refusal has a price, and the price has to land on a willing payer.

So we usually ask the wrong question. We ask which questions deserve a market — which outcomes are interesting, forecastable, important. The right question is: for which questions is someone willing to pay to discover the price? A market exists wherever a hedger needs to offload a risk, an advertiser wants the attention, a casino-style house wants the spread, a research sponsor wants the forecast, or a protocol will subsidize the depth. Where no one will fund the liquidity, there is no market — however good the question. The set of markets that exist is exactly the set of prices someone is paying to discover, and change who's willing to pay — a new regulation that lets a hedger on, an exchange that decides a price is worth subsidizing, a sponsor with a budget — and the map redraws. This is why I think liquidity, not the interface or the brand, is where these markets are won: whoever supplies cheap, deep liquidity wins, because liquidity is the flywheel the entire mechanism spins on, and the bill for it is the thing that decides which questions ever get asked.

What's still hard

None of this is solved. The market-maker problem is the part of prediction markets that stays genuinely difficult, and it's worth being honest about the open edges — on the mechanism and on the money both.

Subsidy versus LP risk. Neither funding model is comfortable. LMSR's bounded loss is bounded but real, and it doesn't shrink with ambition: every market you stand up is a fresh \(b\ln n\) you expect to spend, and across a long tail of thousands of niche questions that line item compounds into something serious. You cannot subsidize the entire space of interesting questions; you can only fund the few whose value clears their cost. The CPMM dodges the operator's subsidy but only by handing the risk to liquidity providers, whose impermanent loss in a market that resolves to 0 or $1 is not impermanent at all. If fees don't out-earn that loss, the LPs leave and the book thins. Both models are a bet that the value of the market exceeds the cost of making it — and that bet isn't always good.

Thin books, and the dial you can't escape. The cold-start problem never fully goes away. An AMM guarantees a quote, but a guaranteed quote in a market no one trades is a wide, shallow one, and most markets on most platforms are thin — a weak signal, easy to move, slow to inform, barely worth reading. The trap is that tightening the spread to attract traders widens the subsidy: they're the same dial. Generous, deep, inviting markets are exactly the ones that bleed the most to the informed; cheap-to-run markets are exactly the thin, jumpy ones nobody wants to trade. There's no setting that gives you a deep market for free, which is what makes bootstrapping genuine depth on a long tail of niche questions an unsolved, expensive problem rather than a merely annoying one — whichever mechanism you pick.

Adverse selection — the deep one. An automated market maker quotes a fixed formula; it cannot tell a noise trader from someone who knows. So the informed trader picks it off — buys just before the news confirms, sells just before it breaks — and the maker is structurally on the wrong side of every smart trade. LMSR's \(b\ln n\) bound caps how bad this can get; a CPMM's LPs eat it as a drag on returns. Either way the market maker is the patsy at the table by design, and the fee or the spread has to be wide enough to survive the people who know more than it does. Set the price of liquidity too low and the informed bleed you dry; set it too high and no one trades. Threading that needle is the real craft, and no formula does it for you.

That last problem is also the seam where the work I spend my days on lives. The cleanest way to make a subsidy affordable is to make adverse selection cheaper — to shrink the \(\alpha\,\ell\) the informed extract without widening the spread on everyone else. That means leaning on everything around the formula: seeing who is trading and why, surveillance, position limits, the machinery that lets a market maker survive the people who know more than it does, inside the rules of a real jurisdiction.⁶ The mechanism is universal and copyable. Lowering its cost — so more markets clear the bar, more prices become worth discovering, and adverse selection stays survivable under real money — is the part that takes a company.

So the market maker is not a piece of plumbing you bolt on at the end. It's the first decision and the hardest one — the algorithm that turns a question into a market, and the place where the cost of doing so is quietly paid. Get it right and the price is live, deep, and worth trusting. Get it wrong and you have the cleverest empty room on the internet. And next time a market quotes you a price for free, remember that it isn't: the bill was paid upstream, by whoever decided that price was worth knowing. Find out who that is, and you understand why the market exists. Find all the prices no one will pay for, and you've found the map of the markets that never will.