A prediction market can be crowded, liquid, and numerically precise while remaining unable to say what kind of world its prices belong to.

Open one on a busy morning and the future arrives as a clean column of percentages. A rate cut: 60 percent. A recession: 60 percent. A ceasefire: 42 percent. A successful launch: 71 percent. Each number appears to be a small fact, disciplined by money and continuously revised by people with something to lose.

In the simplest binary market, a “Yes” contract pays one dollar if the market resolves Yes and nothing if it resolves No. Under familiar idealizations, a contract trading at sixty cents is read as a 60 percent probability. That interpretation is useful. It is also incomplete.

The price describes one contract. A page full of prices describes many contracts. Neither necessarily supplies a coherent account of how the corresponding events fit together.

A market may estimate that two events are each 60 percent likely while saying almost nothing about whether they will occur together, whether one makes the other more likely, or whether they are alternative expressions of the same underlying force. It can answer each question separately without possessing a model in which all of its answers coexist.

Linear algebra makes this incompleteness precise.

Two prices, three worlds

Suppose there are two contracts. The first asks whether the central bank will cut interest rates before the end of the year. The second asks whether the economy will enter a recession before the end of the year.

Call the first event C and the second R. Suppose both Yes contracts trade at sixty cents. For the moment, set aside spreads, fees, risk preferences, and market structure. Treat each price as an exact probability:

It is tempting to feel that we know quite a lot. In fact, these two numbers are compatible with very different futures.

Figure 1Identical marginal prices can belong to radically different joint distributions. The market has fixed the columns’ totals, not the dependence between the two events.

Every row gives the rate cut a total probability of 0.60. Every row gives the recession a total probability of 0.60. Yet the stories are not remotely equivalent.

In the first model, the events arrive together. Perhaps the recession causes the cut, or both are consequences of the same deterioration. In the second, knowing about one does not change the probability of the other. In the third, the events mostly separate. A cut might avert a recession, or a recession might occur precisely when the bank declines to act.

The probability that both events occur can lie anywhere from 0.20 to 0.60. These limits follow from the classical bounds

The quoted prices fix the margins. They do not fix the dependence.

The missing dependence often carries the explanation. Policy, risk management, and scientific inference depend less on isolated frequencies than on relationships: what travels together, what excludes what, what precedes what, and what changes when another condition is present.

The blind direction

The same example can be written more compactly. There are four possible states of the world:

Let their unknown probabilities be collected in the vector π. The two market prices, together with the requirement that probabilities sum to one, impose three equations:

In short,

The matrix A records what each contract pays in each possible world. The vector p records the observed prices. The unknown vector π is the joint distribution we would need in order to describe the market’s implied world.

But A has fewer independent constraints than π has components. The system does not have a unique solution.

To see exactly what remains hidden, consider the vector

The vector v lies in the null space of the payoff matrix. Starting from the independent model, an entire family of joint distributions can be written as

Increasing t moves probability into the worlds where both events agree, either both occur or neither does. Decreasing t moves probability into the worlds where exactly one occurs. Throughout the entire interval, Aπ(t) = p. The market prices do not move at all.

Figure 2Movement along the null direction redistributes probability among the four possible worlds without changing either quoted marginal. Hover or focus the figure to slow the probe.

Here, “null” refers only to what the available measurements can see. These possibilities may still be important or likely. The null space contains changes in the world-model that leave every quoted contract price unchanged.

This distinction matters because it survives perfect measurement. If traders eliminate the spread, multiply the volume, and estimate both marginal probabilities to six decimal places, the hidden direction remains hidden. More trading in the same contracts sharpens the values inside the market’s existing field of view while leaving the field itself unchanged.

A billion dollars of liquidity could distinguish 0.600000 from 0.600001. It could not, by itself, tell us whether the two events are independent.

Two different failures

Prediction markets are often judged by whether their prices are coherent. Coherence is essential, but it addresses a different failure.

Suppose the contract for C costs 0.60, while a contract for C ∩ R can be sold for 0.70. This violates the elementary relation

The conjunction cannot occur unless C occurs. If short selling and settlement terms permit it, a trader could sell the conjunction for 0.70 and buy C for 0.60. The trader receives 0.10 initially. If both events occur, the dollar received from C covers the dollar owed on the conjunction. If C occurs without R, the C contract pays while the conjunction does not. If C does not occur, neither contract pays.

The portfolio serves as a certificate: the quoted prices cannot all come from one probability distribution.

Recent empirical work has begun measuring such violations across real platforms. A 2025 preprint identifies both within-market and cross-market inconsistencies in Polymarket data.1 A separate preprint uses language models to find overlapping, correlated, and potentially contradictory contracts.2 These projects address a real problem: markets are authored separately, traded separately, and often connected by relationships that the platform does not formally encode.

But a coherent market can still be incomplete.

Incoherence means that no possible joint distribution fits the prices. Incompleteness means that many joint distributions fit them. Arbitrage tests whether at least one world-model is possible. Null-space analysis asks how many materially different models remain possible.

In a real order book, exact equality is too strict. Prices have bid-ask spreads, trades occur at different times, fees matter, and related contracts may not settle under perfectly identical definitions. A more honest object is therefore an uncertainty set:

Here, εi allows for measurement error, market friction, and model mismatch.

If 𝒬 is empty, the prices are inconsistent with the proposed state space and tolerance. If it contains one point, the joint distribution is identified relative to that state space. If it contains a region, the market admits a family of worlds.

For any untraded proposition with payoff vector g, one can ask for the tightest price bounds implied by the contracts that do exist:

These are ordinary optimization problems. Their answer may be a narrow interval, a wide one, or no stable interval at all. In the two-contract example, the sharp answer is the interval P(C ∩ R) ∈ [0.20, 0.60].

The width of that range is itself the calculation’s most important result.

The contract is a procedure

So far, we have assumed that each row of A corresponds to a well-defined event. Actual contracts introduce another layer of uncertainty.

A market title might ask whether a recession will occur. Settlement follows a rule: a named source, a particular release, a deadline, a numerical threshold, a treatment of revisions, and a procedure for exceptional cases. The human concept of recession, in all its economic and historical ambiguity, lies outside that procedure.

Formally, the traded object is a resolution function

In the simplest case, the function returns one for Yes and zero for No. More complicated rules may specify intermediate values or special handling when a source fails to publish. Void and refund provisions require additional bookkeeping, but the principle is the same: the instrument pays according to a procedure.

Under an idealized risk-neutral interpretation, the contract value is

The market estimates the expected payout produced by the settlement rule. That may differ from the probability of the ordinary-language event suggested by the headline.

The distinction governs actual settlements. Polymarket’s documentation separates the market title from the rules that determine resolution, including the designated source, end date, and treatment of edge cases.3 Kalshi’s guidance for weather markets gives a particularly clean example: settlement may follow a specified National Weather Service report and its time convention even when a consumer weather application or a person’s recollection differs.4

The physical day and the settlement record can diverge. A city may feel hotter than forecast. A phone may display one temperature and the official station another. When those meanings diverge, the designated procedure governs the payout.

The same problem appears in more consequential markets:

  • Does an election outcome count when news organizations call it, when votes are certified, or when a candidate assumes office?
  • Does a spacecraft “launch” when engines ignite, when it clears the tower, or when it reaches a specified orbit?
  • Does a ceasefire exist when negotiators announce it, when the agreement takes effect, or only if violence remains below a defined threshold?
  • Does an economic release count in its first published form, or after later revisions?

The title points toward a concept. The contract must choose a predicate.

We can describe the gap by separating an ordinary event Ei from the settlement function Ri. One candidate measure of semantic basis risk is

For a binary payout, this is the probability, under Q, that the everyday interpretation and settlement outcome disagree. Both the operational definition of the ordinary event and the distribution Q shape its value. The formula nevertheless identifies the right object: the distance between what readers think the question asks and what the security actually pays for.

Semantic precision is therefore part of market design. A badly specified rule creates a customer-service problem at settlement and changes the financial instrument itself.

Contract count and independent information

It might seem that incompleteness will disappear as platforms list more questions. Sometimes it does. The right additional contract can remove an entire blind direction.

In the two-event example, a contract on C ∩ R adds the payoff row [1, 0, 0, 0]. Once the probability of the conjunction is known, the remaining state probabilities follow from the two marginals. If the conjunction trades at 0.36, the independent model is selected:

But a hundred restatements of the rate-cut question may add almost no independent information. Contracts can be duplicated, nested, or nearly equivalent. In matrix language, rank matters more than the number of rows in A. It counts how many independent directions in the state distribution the contracts can constrain.

With noisy prices, exact rank is not enough. Singular values measure how strongly each direction is observed. A very small singular value indicates a nearly blind direction: the market technically contains information about it, but small errors in prices can produce large changes in the reconstructed world-model. This is the familiar problem of an ill-conditioned inverse.

Volume can reduce uncertainty along a direction the market already measures. A direction missing from the span of its contracts remains invisible.

The combinatorics become severe very quickly. With n binary events, there are 2n possible joint states. One hundred binary events generate roughly 1.27 × 1030 worlds. No platform can list a separate security for each one, and no analyst can estimate an arbitrary probability for every state.

A tractable world-model therefore needs structure. It might use a factor graph or Bayesian network in which local relationships represent conditional dependencies. Research on graphical-model market makers has shown how a combinatorial market can aggregate a joint distribution over many related events without explicitly enumerating every world.5

Reconstructing a model from today’s fragmented markets is harder. The contracts were not written as coordinated measurements of a shared graph. Their language overlaps imperfectly. Their time windows differ. Their resolution sources may disagree. Their prices carry market noise. Before the probabilities can be assembled, the propositions themselves must be aligned.

This changes how one should think about market expansion. Topicality and trading volume are poor guides to a new contract’s information value. The most useful payoff cuts directly across the market’s current uncertainty. In experimental-design terms, the aim is to add a measurement that reduces the feasible set 𝒬, improves the weakest singular directions, or maximizes expected information gain.

The right new question may be a conjunction, a conditional, or an explicit test of dependence. It may look less exciting on a homepage. Mathematically, it may be the question that finally lets the existing answers belong to one world.

What a price leaves out

Even a complete and semantically exact contract system would leave another inverse problem.

We observe prices while their causes remain latent.

A trader may buy because of new public evidence, private information, a need to hedge another position, confidence copied from a prominent account, or a simple shortage of offers near the current price. A well-capitalized trader and a poorly capitalized expert do not exert equal force. Market makers manage inventory. Fees discourage small corrections. Strategic traders may conceal what they know.

The observed price is therefore better written as

We see pt and attempt to reason backward through ℱ. The inverse is generally not unique. A move from 0.40 to 0.60 could reflect a major piece of evidence. It could also reflect one large order in a thin book, a change in risk tolerance, or the temporary disappearance of sellers.

A recent preprint formalizing prediction markets as Bayesian inverse problems studies this issue through latent trader types, price-volume histories, identifiability, and posterior uncertainty.6 Its central methodological lesson is broader than any single model: information aggregation should not be confused with transparent information transmission. The market mechanism transforms the information before we observe it.

There are therefore at least three distinct maps to keep straight:

Figure 3A market price arrives only after semantics, expectations, and market structure have each compressed information. Reconstructing a world-model requires reasoning backward through all three transformations.

Different worlds can produce the same settlement outcome. Different joint beliefs can produce the same set of expected contract values. Different populations of traders can produce the same price history. By the time a percentage reaches the screen, several forms of uncertainty have already been compressed into one number.

Compression gives the number its usefulness. The error comes when the compressed statistic is treated as if it preserved every feature of the system that produced it.

What an honest market map would show

A more complete prediction-market interface would retain the clarity of a price while exposing the structure around it.

For each contract, it could show:

  • the ordinary-language proposition;
  • the exact settlement predicate, source, and time window;
  • logical relations to other contracts, such as implication, exclusion, conjunction, and equivalence;
  • the current bid, ask, depth, and time of observation;
  • coherence constraints implied by those relations;
  • uncertainty bounds for important propositions that are not directly traded;
  • semantic warnings where plausible interpretations lead to different payouts;
  • the additional contract that would most reduce current uncertainty.

Behind the interface would sit an event graph. A contract could be represented as

Here, φi is a logical proposition, τi its temporal structure, Si its designated information source, and Ri its resolution function. Typed relationships among these objects would generate probability constraints. Violations would produce explicit coherence certificates. Missing relationships would remain visible as uncertainty, preventing an analyst’s preferred model from silently filling them.

Language models could help extract candidate relations from natural-language contracts. Recent work on semantic clustering shows why this is useful. But a system intended to support financial or scientific inference should distinguish a suggested relationship from a proved one. “These contracts sound related” is not equivalent to a machine-checkable derivation that one event implies another under their actual settlement rules.

The final output may be a set of reconstructed distributions, together with bounds on what all of them imply.

Return to the two contracts at the beginning. A conventional screen says:

Rate cut by year-end: 60 percent
Recession by year-end: 60 percent

An honest world-model adds:

Both events: 20 to 60 percent
Direction of dependence: not identified
Additional contract needed: the conjunction, or an equivalent conditional market
Settlement basis: verify the source, deadline, and revision rules for each contract

The second display looks less decisive. It contains more information.

Prediction markets remain valuable because they force claims into tradable form, update them in public, and expose confidence to financial consequence. They turn disagreement into a continuously revisable public object and make changes in collective expectation visible with unusual speed. None of that requires pretending that a vector of prices is a complete theory of the future.

A price vector is a projection of possible worlds through the contracts a market happened to list. Some distinctions survive that projection. Others collapse. Reading the visible coordinates also requires determining what the projection has erased.

Markets are valuable within those limits. Trouble begins when precision in their answers is mistaken for completeness in their questions.

References

  1. Oriol Saguillo, Vahid Ghafouri, Lucianna Kiffer, and Guillermo Suarez-Tangil, “Unravelling the Probabilistic Forest: Arbitrage in Prediction Markets”, arXiv preprint, 2025. ↩
  2. Agostino Capponi, Alfio Gliozzo, and Brian Zhu, “Semantic Trading: Agentic AI for Clustering and Relationship Discovery in Prediction Markets”, arXiv preprint, 2025. ↩
  3. Polymarket, “Resolution”, platform documentation. ↩
  4. Kalshi, “Weather Markets”, platform documentation. ↩
  5. Kathryn Blackmond Laskey et al., “Graphical Model Market Maker for Combinatorial Prediction Markets”, Journal of Artificial Intelligence Research 63, 2018. ↩
  6. Juan Pablo Madrigal-Cianci, Camilo Monsalve Maya, and Lachlan Breakey, “Prediction Markets as Bayesian Inverse Problems: Uncertainty Quantification, Identifiability, and Information Gain from Price-Volume Histories under Latent Types”, arXiv preprint, 2026. ↩