Quantitative Finance FAQ

An order book lists resting buy and sell orders by price. Level 2 (L2) data show the aggregate size at each price. Polymarket exposes two event types. book_snapshot gives a complete L2 snapshot of one market side at subscription and at irregular intervals. price_change gives the new resting size at one level. change_side identifies the side that moved, not the side that initiated the trade. The feed never identifies the taker. An algorithm that uses only the feed therefore cannot identify whether a buyer or seller initiated the trade reliably.

Equity researchers often use Lee-Ready or a variant to distinguish buyer-initiated from seller-initiated trades. These methods assume that the feed contains enough information. Polymarket's feed does not. change_side identifies the side that moved, not the side that initiated the trade. Across the 100 highest-volume markets and four 7-day windows, volume-weighted sign agreement is about 59%. This is just above the 50% chance baseline. It is well below the approximately 80% Lee-Ready accuracy on Nasdaq.

A quoted spread is the gap between the best buy and sell prices. The mid-price is their average. One basis point is 0.01 percentage point. The quoted half-spread is half the quoted spread relative to the mid-price. On Polymarket's central price decile, it is around 200 basis points. It reaches 650-900 basis points in the lowest-probability decile. The effective half-spread measures a trade's execution cost relative to the mid-price. On liquid US equities after decimalization, it is in the single-digit basis-point range. Polymarket is roughly an order of magnitude wider. This is consistent with longer prediction-market holding periods and substantially less market-maker capital.

This study flags direct self-matches and one-step round trips as wash-suspect trades. Across the 600-market panel, the median self-counterparty wash share is 0.97% per market. The 90th percentile is 4.5%, the 99th is 10.6%, and the maximum is 22.2%. These results are lower bounds. The detector does not cover multi-counterparty graph patterns. Cong et al. (2023) report 25-70% wash shares on unregulated cryptocurrency token exchanges. Polymarket is well below that range, although the venue-class incentives differ.

The longshot spread premium is the extra spread charged on low-probability contracts. The mid-price is the average of the best buy and sell prices. One basis point is 0.01 percentage point. The quoted half-spread is about 200 basis points in the central [0.4, 0.6] mid-price range. The full quoted spread reaches 1,300-1,800 basis points in the lowest-probability decile. Its half-spread is 650-900 basis points. The low-probability side is wider than the high-probability side. This magnitude looks less like a behavioral longshot bias and more like a liquidity-provision constraint. Low-probability binary contracts give market makers bounded upside and asymmetric downside. Market makers therefore charge a larger inventory-risk premium than on a continuous-payoff sportsbook market.

Researchers should source trade direction from on-chain OrderFilled events on the Conditional Token Framework (CTF) Exchange smart contract. They should not infer it from the public WebSocket feed. In OrderFilled, makerAssetId and takerAssetId show which side held US Dollar Coin (USDC). This shows directly whether a buyer or seller initiated the trade. The replication package at github.com/philippdubach/polymarket-microstructure joins off-chain and on-chain records. It also provides patches that let existing measurement code accept those on-chain records as the trade-direction source.

The question has no clean answer. In the risk and uncertainty boxes, profit comes from being right: you estimated better. In the ignorance box, profit often comes from willingness to act when others can't, due to fiduciary requirements, career risk, or compliance models that require probability estimates. The sellers weren't wrong about the asset. They were unable to hold it. Whether profiting from that gap is legitimate depends on whether you view it as constraint arbitrage (providing liquidity to a market that needs it) or as exploiting structural advantages that accrue mainly to the already-wealthy.

The people who can profit from UU mispricing tend to be those who can afford career risk, illiquidity, and blame: wealthy individuals, family offices, and unconstrained investors like Buffett. The people who cannot participate are pension funds, endowments, and retail investors in diversified vehicles, precisely because their governance structures require estimable risk. The epistemological structure of markets has equity implications: the returns from ignorance flow disproportionately to those who already have the most.

In his 1995 Harvard Law School speech on the psychology of human misjudgment, Munger argued that the right way to think about probabilistic decisions is the way Zeckhauser plays bridge: making hundreds of decisions under uncertainty, balancing expected gains and losses, and accepting that good decisions sometimes lead to bad outcomes. The compliment is precise because bridge rewards reasoning about what you don't know, not just what you do.

Zeckhauser's sidecar concept works cleanly when the driver has genuine capability: a real estate developer who creates value, a venture capitalist with operational expertise. The profit is a share of value creation. It becomes ethically murkier when the complementary asset is power rather than skill, as in Zeckhauser's Gazprom example where the edge comes from access to political elites rather than from analytical or operational superiority.

Developed by J.L. Kelly at Bell Labs in 1956, the Kelly Criterion is a formula for optimal bet sizing that maximizes the long-run geometric growth rate of wealth. For a binary bet, it says to invest f = (bp - q) / b of your capital, where p is the probability of winning, q is the probability of losing, and b is the odds. Given sufficient time, a Kelly bettor will almost certainly end up wealthier than anyone using a different strategy. The catch: it requires knowing p and b precisely.

Paul Samuelson argued in a famous 1979 paper (written entirely in one-syllable words) that the Kelly Criterion only maximizes expected utility for investors with logarithmic utility functions. Log utility implies you'd accept an even-money bet to double or halve your wealth, which most people wouldn't. For more risk-averse investors, Kelly overbets. The dispute is whether position sizing is a mathematical optimization problem with one answer (Kelly) or a preference problem with many answers (Samuelson).

Both Kelly and Samuelson assume you know your probability of winning (p) and the payoff ratio (b). In UU situations, you know neither. The parameters themselves are objects of ignorance. This is not a fixable data problem where better research could estimate p more precisely. It is a category problem: the state space over which p would be defined is itself undefined. Kelly gives no answer when its inputs are unavailable.

Half Kelly means betting half the fraction the formula prescribes. Practitioners use it for two reasons. First, the Kelly fraction is exquisitely sensitive to estimation error in p and b: overestimating your edge by even a few percentage points can push the formula past optimal into a regime that destroys capital over time. Second, full Kelly accepts drawdowns that are mathematically tolerable but psychologically devastating, around 12 percent on a typical bet versus around 3.5 percent for half Kelly. The trade-off: half Kelly cuts compounded return by roughly 25 percent in exchange for substantially lower variance.

Yes, indirectly. Elwyn Berlekamp, who served as Kelly's research assistant at Bell Labs, restructured the Medallion Fund in 1989 around Kelly-based position sizing applied to thousands of short-duration trades. The fund returned roughly 66 percent annually before fees from 1988 through 2021. Medallion's success is one of the strongest empirical arguments for Kelly, but it works because the trades have estimable probabilities and very short holding periods. That is the opposite of the UU situations where Kelly's inputs are undefined.

Zeckhauser's Maxim B: 'The greater is your expected return, the larger your advantage, the greater the percentage of your capital you should put at risk.' This sounds obvious but is actually radical because it replaces formula-based precision with judgment-based heuristics. He also offers a diagnostic: 'If in an unknowable world none of your investments looks foolish after the fact, you are staying too far away from the unknowable.'

Sidecar investing means putting your money alongside someone who has complementary skills you lack, such as a real estate developer or biotech venture capitalist. The investor rides in a sidecar pulled by a driver with genuine operational expertise. The more confident you are in the driver's integrity and ability, the more attractive the sidecar investment, since its price reflects the scarcity of access rather than public information.

In standard markets, the concern is that the other side knows the value of the asset better than you do. Akerlof's lemons problem and Glosten-Milgrom's bid-ask spread model both assume someone has superior information about a defined quantity. In UU (unknown and unknowable) markets, neither side can enumerate the possible states of the world. The fear of adverse selection persists, but it is often unfounded because the ambiguity is shared. The mispricing lives in the gap between assumed and actual information asymmetry.

Zeckhauser's framework says your return depends on your absolute advantage (complementary skills), the probability the other side is better informed, and the selection factor (how much their information hurts you). A large absolute advantage, such as a longer time horizon or operational expertise, provides insurance against adverse selection. You don't need to know more than the other side. You need an edge they can't replicate.

Not because they thought the Earthquake Authority had inside information about seismic risk. Because their internal processes required probability estimates, their compliance teams required distributional assumptions, and the honest assessment of 'we have no idea, but the price is very high' did not fit their institutional decision-making forms. The opportunity existed because institutional constraints prevented sophisticated capital from acting on it.