Quantitative Finance on Philipp D. Dubach | Quantitative Finance & AI Strategy

The Anatomy of a Decentralized Prediction Market: Notes from the Polymarket Order Book

me@philippdubach.com (Philipp D. Dubach) — Sat, 02 May 2026 00:00:00 +0000

I spent the last two months running a Polymarket order-book collector. The collector runs on a small VM, subscribes to the public WebSocket feed, and writes one Parquet file per UTC hour. By 2026-04-15 the archive had grown to 1,262 hourly files, 30,287,264,368 events, 623.8 GB on disk, covering 52 calendar days and 385,198 distinct market ids. The first version of the paper is up on arXiv, the replication package is on GitHub and Zenodo (DOI 10.5281/zenodo.19811426), and the manuscript is under review at the Journal of Financial Markets.

This post walks through what’s in it.

Why bother

Prediction markets aggregate dispersed beliefs into a single price that, in equilibrium, behaves like a probability. The empirical literature has historically focused on price-level questions: forecast accuracy, calibration against realised outcomes, the longshot bias, and the extent to which informed and uninformed traders coexist on the same venue. Yet microstructure is what determines the trading cost of holding an informational position, and the wider the cost, the smaller the informational signal that survives to the price. A prediction market with noisy microstructure produces noisier prices than the headline-level aggregation literature implicitly assumes.

The microstructure of prediction markets has remained under-studied. Limit-order-book depth profiles, spread decompositions, and the behaviour of liquidity providers near resolution were largely unobservable on the early venues, which used scoring rules (IEM, Hanson’s logarithmic market scoring rule) or sparse parimutuel pools. Polymarket changed this. Since 2021 the venue has run a limit-order-book exchange on Polygon, settling in USDC against an on-chain conditional-token contract. The infrastructure is finally there. What’s missing is the data work to actually use it.

Two contributions

The paper has two empirical contributions, ordered by weight for the literature.

The first is a measurement result. Trade direction inferred from Polymarket’s public feed is noise-dominated: about 59% volume-weighted sign agreement with the on-chain ground truth on the top-100 stratum, well below the 80% Lee-Ready achieves on equities. Any Polymarket microstructure result that depends on trade direction has to source it from on-chain OrderFilled events.

The second is a set of eight cross-sectional stylized facts on a pre-registered 600-market panel, observed simultaneously over a single 28-day scrape window, with microstructure measures computed on the full event tape and a direct on-chain trade record. None of the eight facts requires the on-chain join, and each is reported on the full panel.

The two are complementary. The stylized facts characterise Polymarket’s microstructure on its own terms. The measurement result sets the conditions under which any trade-direction-dependent statement about Polymarket can be trusted.

The data

The primary input is a continuous tick-level archive of the public WebSocket feed from 2026-02-21 16:00 UTC through 2026-04-15 08:00 UTC. The schema is preserved verbatim from the WebSocket payload; eager Pydantic parsing was a multi-hour operation at this row count, so I parsed JSON only on the events that survived market-id and time-window filters.

Joining to on-chain trades took most of the engineering time. Polymarket’s CTF Exchange smart contract logs OrderFilled events whose payloads identify both counterparties and the aggressor side. I scraped 255,425,405 fills across a 28-day calibration window (2026-02-28 to 2026-03-27) using batched eth_getLogs calls against a Polygon RPC provider, with adaptive chunk sizing that respects provider-specific rate limits. The off-chain feed is keyed on market_id, the on-chain record on makerAssetId / takerAssetId. The bridge between the two is the (condition_id, yes_token_id, no_token_id) mapping pulled from the CLOB REST endpoint and cached locally. CLOB REST resolves all 385,198 archive market ids; the Gamma metadata API, which is sometimes used in the literature, indexes only 34,764 markets.

The 600-market panel selection rule (volume metric, random-stratum eligibility threshold, random seed, category scheme) was committed in a pre-registration document before computing the panel. A deterministic build script emits the panel parquet, whose SHA-256 is recorded back into the pre-registration document before any analysis runs. This goes beyond the empirical-microstructure norm; the cost is one document and one hash, the benefit is that a reader can verify no market was added or removed after the analysis ran.

Eight stylized facts

SF1 — Longshot spread premium

Quoted half-spread, binned by per-market mean mid price into ten deciles. Median spread is about 400 bps in the central [0.4, 0.6] range and climbs to 1,300-1,800 bps for markets trading below 0.10. The pattern is asymmetric: the low-probability side is wider than the high-probability side, which echoes the longshot bias documented in the racetrack and parimutuel literature (Snowberg and Wolfers, 2010; Thaler and Ziemba, 1988) and the prediction-market longshot evidence on Iowa Electronic Markets and TradeSports surveyed by Wolfers and Zitzewitz (2004). The direction is the same; the magnitude is not. The 1,300-1,800 bps full quoted spread on Polymarket’s lowest-probability decile (650-900 bps half-spread) is an order of magnitude wider than on a continuous-payoff sportsbook market, which reads less like the classical risk-love or misperception story and more like a liquidity-provision constraint.

SF2 — Depth concentration

For each market I summarise the L2 depth profile by the ratio $\text{depth}{L=1} / \text{depth}{L=10}$, the share of cumulative top-10 depth held at the top-of-book. A value of 1.0 means the entire top-10 depth sits at level 1 (a thin, top-heavy book); 0.1 matches a uniform grid where each level carries equal depth. For the 546 markets with non-null depth, the median ratio is 0.137, close to the uniform-grid benchmark, with $p_{10} = 0.033$ and $p_{90} = 0.428$. The folk view that prediction-market books are concentrated at top-of-book does not hold on Polymarket: depth is generally layered deeper into the book.

SF4 — Maker-wallet diversity

For each market I compute the volume-weighted Herfindahl index (HHI) of maker-address share across on-chain trades. Across 600 markets and 6.4M trades, the median HHI is 0.031 (about 32 effective makers). The distribution is right-skewed: $p_{90} = 0.119$ (about 8 effective makers) and a maximum of 0.40 (roughly 3 effective makers). Maker liquidity is decentralised on most markets in the panel, with a tail of thin or niche markets dominated by one to three wallets. This matters for any narrative that frames Polymarket as a venue dominated by a small number of professional liquidity providers. At least in the top-100 by volume, that’s not what the data show.

A trade is flagged as wash-suspect under a two-tier rule: (a) maker == taker (direct self-match), or (b) a flipped pair $(maker_a, taker_a) \leftrightarrow (taker_a, maker_a)$ within 128 blocks (Polygon finality buffer) on the same market. This is an explicit lower bound: the detector covers only direct and immediate-roundtrip self-counterparty patterns, not the extended-graph patterns that network-based classifiers like Cong et al. (2023) address on unregulated cryptocurrency token exchanges, where wash shares of 25-70% have been documented.

Across 600 markets and 6.4M trades, the median wash share is 0.97%, with p90 at 4.5%, p99 at 10.6%, and a maximum of 22.2%. The gap between this lower bound and the network-classifier estimates on token exchanges has two components: wash patterns that require multi-counterparty graph analysis to detect (which my detector does not cover), plus a venue-class difference in the underlying wash incentives. The first I can quantify only under a graph-classifier extension; the second is identification, not measurement.

SF8 — Depth decay near resolution

Do markets near resolution carry shallower books? I regress log mean depth at $L=10$ on log seconds-to-close at the panel window midpoint (2026-03-13), restricted to 322 markets with positive seconds-to-close and non-zero summary depth.

The bivariate slope is 0.818 (HC3 SE 0.113, $t = 7.2$, $R^2 = 0.13$). Category fixed effects (Crypto, Sports, Other, Geopolitics) attenuate the slope to 0.550 (HC3 SE 0.143, $t = 3.85$, $R^2 = 0.22$): roughly a third of the bivariate association is category-level confounding. Adding log panel-window volume on top of category attenuates further to 0.305 (HC3 SE 0.104, $t = 2.94$, $R^2 = 0.49$). The 0.31 slope works out to about 6% less mean depth per 10× reduction in seconds-to-close.

The category + log-volume specification is the conservative reading. Volume mediates the depth-time relationship because markets active longer accumulate more makers, and more makers means more depth, so a regression that omits volume attributes the makers-and-time channel to time alone. The 0.305 coefficient is the residual depth decay after that mediation is netted out; the 0.550 within-category slope reported in the abstract is the figure before mediation is netted out, and is the appropriate one to compare to a literature that does not condition on volume.

(SF3 covers Polygon block-clock alignment, SF5 the category-conditional spread, and SF6 the archive-ingestion latency. All three are in the paper. SF6 in particular is a property of the collector pipeline rather than of Polymarket; the median per-market p50 ingestion delay is 41.5 ms, which is a sanity check on the collector, not a microstructure result.)

The limits of orderbook-only inference

Six standard direction-dependent microstructure measures (effective spread, realized spread, Roll, Abdi-Ranaldo, Kyle’s $\lambda$, Amihud) need an aggressor sign for each trade. Empirical practice on equity venues sources that sign from a quote-driven feed via Lee-Ready or its variants. The Polymarket public feed does not expose enough information to do this reliably: the change_side field marks which side of the book moved, not which side initiated the trade.

I infer trades from the feed under a LOOSE rule (every resting-size decrement counts) and match the inferred buckets against on-chain OrderFilled events at 5-second and exact-price granularity, over four disjoint 7-day windows.

Panel mean (109 valid cells of 400, 55 markets): 0.615, market-clustered bootstrap 95% CI [0.579, 0.653]
Volume-weighted by matched-bucket count (total 125,080): 0.592, bootstrap 95% CI [0.542, 0.659]

A volume-weighted sign-agreement of about 59% sits just above the 50% chance baseline. Even when an inferred bucket matches an on-chain bucket in time and price, the inferred aggressor direction is wrong about two trades in five. The mechanism is the feed itself: price_change updates broadcast a post-match snapshot of the resting book without identifying the taker. The change_side field marks which side of the book moved, not which side initiated the trade, and using it as a sign proxy produces the about 59% agreement rate.

A noisy sign propagates to every measure that consumes it. On the comparable subset of the top-100 panel:

Effective half-spread changes sign on 67% of markets in the first 7-day window when feed-inferred direction is swapped for on-chain ground truth (50% in a second non-overlapping window).
Kyle’s $\lambda$ changes sign on 60% of markets in the first window (43% in the second).

Both windows sit in or below the chance band, well short of the about 80% Lee-Ready accuracy documented on Nasdaq.

The Glosten-Harris spread decomposition makes this concrete. Restricted to the top-100 stratum and run on authoritative on-chain trades, the median effective half-spread on the top-100 panel is essentially zero (-0.0003 prob pp), as are the median transitory component (0.00001) and the median adverse-selection component (0.0). Once sign errors are removed, the dollar-weighted “adverse selection” that orderbook-only inference produces collapses, leaving the typical top-100 market with no detectable systematic spread component on either side.

The implication for anyone working on decentralised CLOB venues: the same constraint should bind on any venue where the off-chain matching layer broadcasts a post-match book state without exposing the taker identity (GMX v1, dYdX v3, Loopring’s historical CLOB, and similar hybrid architectures). The public feed exposes what cleared but not who initiated, and any direction-dependent microstructure measure on those venues will need an authoritative on-chain trade source.

What I’m not doing here

I had a long-ish exchange recently with someone scoping research on the same dataset, which is a useful proxy for what’s adjacent but out of scope.

Spread decomposition with bounded payoffs. Classical Glosten-Harris / Huang-Stoll on prediction markets is open. Prices are bounded in $(0,1)$, which breaks parts of the standard identification, so there is a real research question to do, not just a re-application.

Insider episodes and wallet-level patterns around resolution. Probably the most productive lane I am not working on. The JFM submission draws the privacy line at no deanonymisation, only aggregate wallet measures; where you draw it for your own work is your call.

Toxic flow. Some overlap with one of my planned follow-ups.

Latency arb (intra-venue). Doesn’t really work as a question on Polymarket. The two-clock gap in the WebSocket feed (SF6) is archive-ingestion delay, not trader latency, and without an exchange-side clock you can’t separate the two. Cross-venue is the workable framing.

Cross-venue arb. Open and product-relevant. Kalshi, PredictIt, sports-book mirrors. The data engineering is probably the harder part.

Replication

Everything is reproducible from the public on-chain record and your own WebSocket capture. I’m not redistributing the 624 GB raw archive (it is too large to move around practically), but the panel artifacts and the on-chain scrape pipeline are public:

Code: github.com/philippdubach/polymarket-microstructure
Replication package (DOI): 10.5281/zenodo.19811426
Paper: arXiv:2604.24366, under review at the Journal of Financial Markets
Collector: pmxt-dev/pmxt is a good starting point if you want to run your own. The Polymarket WebSocket is a public feed, so once you have a collector running you’ll have continuous coverage going forward, which for product work is probably more useful than a static historical slice anyway.

The CTF Exchange V1 → V2 cutover at the end of April 2026 closes my scrape window and opens a venue-evolution comparison. That, plus a per-market depth time series that would let SF8 move from a cross-sectional to a within-market depth-decay regression, plus a cross-venue analysis against Kalshi and sports-book mirrors that would address the price-discovery question this paper leaves open. Those are the obvious next things.

The Moral Philosophy of Investing in Ignorance

me@philippdubach.com (Philipp D. Dubach) — Wed, 22 Apr 2026 00:00:00 +0000

Investing at the Edge of Knowledge, Part 5 · Start with Part 1

“If in an unknowable world none of your bridges fall down, you are building them too strong.”

Zeckhauser’s version of this line refers to investments, not bridges, but the structural point is the same. A philosophy of investing that expects some failures is also a philosophy that accepts some losses will be borne by the people on the other side of the trade. Over four installments I’ve laid out a framework for thinking about what you can know (Part 1), why investors flee what they can’t (Part 2), how to assess what others know (Part 3), and how much to bet (Part 4). This final piece asks the question the framework doesn’t answer: when you profit from ignorance, is that a legitimate source of returns?

Zeckhauser opens his 2006 paper with a warning that is usually read as practical advice: “Do not read on if blame aversion is a prime concern.” I think there’s an ethical question underneath the practical one.

Three sources of profit

In the risk box, profit comes from superior calculation. Both sides had the same information. You ran the numbers better. This is the cleanest form of trading profit. The losing side made a computational error that was, in principle, avoidable.

In the uncertainty box, profit comes from superior estimation. You had a better model, better priors, or more data. The other side could have done the same analysis but didn’t. This is still relatively clean, though the boundary between “better estimation” and “inside information” requires constant policing, which is what securities regulation exists to do.

In the ignorance box, the source of profit shifts. The other side didn’t sell because they thought the price was fair. They sold because they couldn’t model the asset and their constraints forced a decision. The fund manager who sold the IGV at $80 during the SaaSpocalypse wasn’t wrong about the disruption risk from AI. They were unable to hold a position that couldn’t be defended to their risk committee, their clients, or their compliance team. The profit for the buyer comes from the gap between institutional rationality (the right decision for the manager’s career) and market rationality (the right price for the asset).

The distinction matters. Most alpha in UU situations is some form of constraint arbitrage: profiting from the gap between what an asset is worth and what institutions are able to pay for it. Time horizon arbitrage, where you can hold for five years and they can’t. Liquidity arbitrage, where you can accept illiquidity and they can’t. Career-risk arbitrage, where you can tolerate looking wrong and they can’t. All three produce genuine returns, and in none of them did the counterparty make an error. They made a rational decision given their constraints, and you profited from having different constraints.

One way to frame this is positive: the constraint-arbitrage investor is providing liquidity to a market that needs it. They’re buying when others are forced to sell, which improves price discovery and reduces the magnitude of mispricings. In this framing, the profit is compensation for bearing ambiguity that others can’t.

Another way to frame it is uncomfortable: the returns from ignorance flow to those who can afford to bear it, and that set of people is not random.

The sidecar problem, revisited

I discussed the sidecar concept in Part 3 as an information problem: how do you know the driver is skilled? Here I want to revisit it as an ethical problem: what kind of edge is the driver using?

Zeckhauser’s sidecar works cleanly when the driver has genuine capability. A real estate developer who can build and lease a building is creating value. A venture capitalist with operational expertise and a network of technical talent is creating value. The sidecar investor earns a share of that value creation. The profit comes from complementary skills combined with capital, and this is hard to object to on ethical grounds.

It gets murkier when the complementary asset is power rather than skill. Zeckhauser discusses a hypothetical Gazprom investment: “If you could comfortably determine that the Russian elite was investing on its own volition, and that foreigners would not be discriminated against…” The edge in that scenario isn’t analytical. It’s access to a political structure. The sidecar investor is riding alongside someone who can influence outcomes, not someone who can predict them. The distinction between capability and power as complementary assets is one the paper gestures at but doesn’t fully resolve.

Robb (2006) put his finger on a related problem. UU knowledge is “uncommunicable.” If a mechanism for generating excess returns could be expressed as a process, someone would have arbitraged it away. But if the driver can’t articulate their edge, the sidecar investor can’t distinguish between genuine insight, survivorship bias, and proximity to power. There’s an epistemological problem here, and an ethical one. You’re making a bet on someone whose advantage you can’t evaluate, which means you’re implicitly trusting that the advantage is legitimate.

Summers (2006) observed that identifying skilled UU managers may be no easier than picking investments directly. I suspect this is too generous. In many cases, identifying whether a sidecar driver has skill, power, or luck is harder than evaluating the underlying asset, because the asset at least has observable characteristics. The driver’s edge, by definition, doesn’t.

Blame, accountability, and the collective action problem

The Monday Morning Quarterback problem runs through Zeckhauser’s entire paper. Investors avoid Box F not because the expected value is negative but because a bad outcome will be judged harshly in retrospect. I’ve discussed this as a mechanism for mispricing (Parts 2 and 3). Here I want to name the distributional consequence.

If we want institutional investors to make UU bets, which would improve price discovery and reduce the mispricing that currently rewards unconstrained investors, we need governance structures that tolerate good decisions with bad outcomes. The current structure doesn’t. A pension fund CIO who buys the IGV at $80 and watches it fall to $70 will face questions that no amount of “the expected value was positive” can answer. The governance framework is built for the risk box, where decisions can be evaluated against a defined probability model. In the ignorance box, there is no model to evaluate against, which means there is no institutional language for “this was a good bet that happened to lose.”

The result is a collective action problem with distributional consequences. The returns from UU mispricing accrue disproportionately to wealthy individuals, family offices, and unconstrained investors like Buffett, precisely the people who can afford career risk, illiquidity, and blame. Pension funds, endowments, and retail investors in diversified vehicles are structurally excluded, not by regulation or by choice, but by governance frameworks that require the kind of probability estimates the ignorance box doesn’t produce.

I’m not sure this is a solvable problem. The fiduciary duty to beneficiaries is real, and “we invested in something we couldn’t model because the price seemed low” is not, and should not be, an acceptable fiduciary justification. But it’s worth naming the consequence: the epistemological structure of markets has equity implications. The returns from acting under ignorance flow to those who already have the most capacity to bear it. This is not a conspiracy. It’s a structural feature that emerges naturally from the interaction of ambiguity aversion, institutional constraints, and governance design.

Zeckhauser closes his paper by returning to David Ricardo at Waterloo. Ricardo wasn’t a military analyst. He didn’t have inside information about Wellington’s strategy. He just understood the structure of the situation: thin competition (most investors had fled), an eager seller (the British government needed capital), asymmetric payoffs (bounded downside, enormous upside), and a kind of not-knowing that was the same for everyone. He bought British government bonds on the eve of the battle and made a fortune.

The honest answer to “what’s your thesis?” in a UU investment is: “I don’t have one in the way you mean. I have a set of second-order inferences about what other people don’t know, what constraints they face, and why the price might be wrong even though I can’t tell you what the right price is.” That’s not a pitch deck. It’s a worldview. Most investment committees would reject it, which is, of course, part of why it works.

Charlie Munger, in his 1995 Harvard Law School speech on the psychology of human misjudgment, offered a compliment that I think is the best summary of everything this series has tried to say: “The right way to think is the way Zeckhauser plays bridge.” The compliment is precise. Bridge is a game of acting under uncertainty with imperfect information, where the quality of the decision is independent of the outcome, and where the best players are distinguished not by what they know but by how they reason about what they don’t know.

That might be the best definition of investing at the edge of knowledge I can offer.

Bet Sizing at the Frontier

me@philippdubach.com (Philipp D. Dubach) — Fri, 17 Apr 2026 00:00:00 +0000

Investing at the Edge of Knowledge, Part 4 · Start with Part 1

“He who acts in N plays to make his mean log of wealth as big as it can be made will, with odds that go to one as N soars, beat me who acts to meet my own tastes for risk.”

That’s Paul Samuelson, writing in one-syllable words. The title of his 1979 paper: “Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long.” Published in the Journal of Banking & Finance, a journal not typically known for its prose style. The playfulness of the writing masks the seriousness of the dispute underneath: a disagreement about the foundations of position sizing that remains unresolved half a century later.

In Part 3 I described how to assess whether the other side of a trade knows something you don’t. But even if you’re confident you’re in Box D or Box F (shared uncertainty or shared ignorance, where neither side has an information edge), you still need to decide how much capital to commit. And in a UU world, the most famous formula for answering that question stops working.

What Kelly actually says

J.L. Kelly Jr. (1956) was a physicist at Bell Labs, not a finance researcher. His paper, “A New Interpretation of Information Rate,” was about communication channels, not portfolios. The insight was a connection between Shannon’s information theory and gambling: the maximum exponential growth rate of a gambler’s capital equals the rate of information transmission over a noisy channel.

The formula itself is simple. For a binary bet with probability p of winning and odds of b to 1, the optimal fraction of your bankroll to wager is f = (bp - q) / b, where q = 1 - p. If you have a 60% chance of winning an even-money bet, Kelly says invest 20% of your capital. The appeal is a mathematical proof: given sufficient repetitions, a Kelly bettor will, with probability approaching one, end up wealthier than anyone using any other fixed-fraction strategy. It maximizes the geometric growth rate of the portfolio, which is the growth rate that actually compounds over time.

Ed Thorp was the first to take this seriously as an investment principle. He used it to beat blackjack (documented in Beat the Dealer, 1962), then applied it to warrant pricing and convertible arbitrage through his hedge fund, Princeton Newport Partners, which returned roughly 15% annually with minimal drawdowns over two decades. Elwyn Berlekamp, Kelly’s research assistant at Bell Labs, later became the key figure who restructured Renaissance Technologies’ Medallion Fund in 1989, applying Kelly-based position sizing to thousands of short-duration trades. Medallion returned roughly 66% annually before fees from 1988 through 2021. Bill Gross used Kelly-adjacent thinking at PIMCO. The framework has serious practitioners with serious track records.

What Samuelson actually objected to

Samuelson’s critique is often misunderstood as “Kelly doesn’t work.” That’s not what he said. What he said is more precise and more interesting.

Kelly maximizes the expected logarithm of wealth. This is only optimal if your utility function is logarithmic. Log utility means you are exactly indifferent between the status quo and an even-money bet that would either double or halve your total wealth. Most people would not take that bet. If you are more risk-averse than log utility implies (and most human beings are), Kelly systematically overbets, exposing you to drawdowns that are mathematically acceptable but psychologically devastating. If you are less risk-averse (say, risk-neutral), Kelly underbets: a risk-neutral investor should go all-in on every positive expected value opportunity.

Underneath the math, Kelly and Samuelson are arguing about something more basic. Kelly treats position sizing as a mathematical optimization problem with a unique solution. Samuelson insists it is a preference problem with as many valid solutions as there are utility functions. Both are correct within their own frameworks. The dispute is about which framework applies. And because the answer depends on the investor’s risk preferences, which are not observable from the outside and may not even be stable over time, the dispute is in principle unresolvable.

Practitioners tend to resolve it pragmatically. Most Kelly users bet “half Kelly” or “quarter Kelly,” sacrificing some expected growth for lower variance. There’s also a second reason for fractional Kelly that has nothing to do with utility: estimation error. The formula is exquisitely sensitive to how accurately you’ve estimated p, and overestimating your edge by even a few percentage points pushes you past optimal into a regime that destroys capital over time. Half Kelly is partly a hedge against your own overconfidence in the inputs. This works in practice, but it concedes Samuelson’s point: the “optimal” fraction depends on your tolerance for pain and the precision of your estimates, not just on the mathematics of compound growth. The formula provides a ceiling, not a prescription.

Why the debate is beside the point in UU

Here is where both Kelly and Samuelson run into the same wall.

Kelly requires p (the probability of winning) and b (the payoff ratio). Samuelson’s alternative utility-maximization frameworks require the same inputs, or richer ones: full probability distributions over outcomes. Every formal system for position sizing assumes you can parameterize your uncertainty. In the ignorance box, as defined in Part 1, you can’t. The parameters themselves are objects of ignorance. p is not an imprecise estimate waiting for better data. It is undefined, because the state space over which p would be calculated hasn’t been enumerated. This is the territory Knight called true uncertainty, distinguishing it from measurable risk a century ago, and it is where every formal position-sizing tool stops working at the same time.

Zeckhauser walks through five reasons UU money management resists formal modelling, and each one breaks a different standard assumption.

Most UU investments are illiquid for unknown periods. You can’t rebalance, which means sequential portfolio optimization models don’t apply. Worse, markets charge enormous premiums to cash out illiquid assets, so your exit price is not your mark-to-market. Even the toy models of optimal sequential investment assume the hard problems away: known probabilities, known time horizons, known liquidity. There’s also the embarrassing fact that smart people disagree about position sizing even on problems where probabilities are known, which gives you no reason to expect convergence when they aren’t. And when genuinely unknowable events arrive (the 1987 crash, the 1997 Asian crisis, the 2020 pandemic), the money-management problems that emerge are precisely the ones no model anticipated.

None of this means do nothing. It means formula-based precision is unavailable and pretending otherwise is dangerous. Plugging estimated probabilities into Kelly when the estimates are themselves wild guesses doesn’t give you a rigorous answer with a known error bar. It gives you false precision, which is worse than an honest admission of ignorance because it generates unwarranted confidence in the position size.

Diversification versus concentration

Standard portfolio theory says: diversify. Spread capital across uncorrelated assets to minimize idiosyncratic risk. In the risk box (Box 1 from Part 1), where expected returns and covariance matrices are estimable, this is correct. The marginal expected return on each position is similar, and diversification is free insurance.

In the ignorance box, this logic inverts. If you’ve identified one or two opportunities in Zeckhauser’s Box D or Box F, the quadrants from Part 3 where neither side has an information edge, and you have a genuine advantage from complementary skills, constraint arbitrage, or structural access, spreading capital evenly across a dozen positions dilutes the few bets where your advantage is real. Zeckhauser makes this point sharply: investors routinely allocate almost the same percentage to an investment where they expect a 30% return as to one where they expect 10%. Consider the SaaSpocalypse: the IGV at $80 with an RSI of 18 and 17% sector earnings growth was, on the framework, a Box F opportunity with a large absolute advantage for anyone with a multi-year time horizon. The Maxim B response would be to concentrate, not diversify. Most institutional investors did the opposite.

Buffett’s practice reflects this. Berkshire Hathaway’s top five holdings routinely exceed 70% of its equity portfolio. This is Kelly-adjacent thinking: bet big when your edge is big. But it operates without the false precision of computing a Kelly fraction, because Buffett doesn’t pretend to know his probability of winning to two decimal places. He knows the price is low relative to his assessment of value, his time horizon exceeds the market’s, and the business is durable enough to survive things he can’t foresee. The probability distribution of outcomes is not something he claims to have, and he doesn’t pretend otherwise.

Zeckhauser uses a bridge analogy that I think captures this well. A bridge player makes hundreds of decisions in a single session, balancing expected gains and losses on every hand. But no serious bridge player computes Kelly fractions mid-hand. They develop judgment over thousands of hands about when to bid aggressively and when to pass. The judgment is trained by feedback, but the individual decision is not a calculation. It’s pattern recognition combined with temperament, a sense of when the odds are tilted enough to justify the risk.

This is where Maxim B does its work: “The greater is your expected return, the larger your advantage, the greater the percentage of your capital you should put at risk.” It sounds obvious. Read it again. In a world where position sizing formulas require inputs you don’t have, Maxim B is the honest replacement: a heuristic that says “bet proportionally to your edge,” without pretending to quantify the edge precisely. Combine it with Zeckhauser’s diagnostic, and you get the closest thing to a position sizing framework that works in UU: “If in an unknowable world none of your investments looks foolish after the fact, you are staying too far away from the unknowable.”

I find that diagnostic unsettling in the right way. It says that some of your bets should look bad. Not because you were wrong, but because the honest price of participation in UU markets is a portfolio that occasionally embarrasses you. If every position is defensible in hindsight, you’ve been too cautious. You’ve been optimizing for looking smart rather than for capturing the returns that only come to those willing to look foolish.

The first four parts of this series have addressed what you can know, what you can’t, how to assess what others know, and how much to bet. The final question is different, and less comfortable: when you profit from ignorance, from others’ institutional inability to act in UU situations, who exactly are you profiting from? That’s Part 5.

The Geometry of Who Knows What

me@philippdubach.com (Philipp D. Dubach) — Mon, 13 Apr 2026 00:00:00 +0000

Investing at the Edge of Knowledge, Part 3 · Start with Part 1

“One of these days in your travels, a guy is going to show you a brand-new deck of cards on which the seal is not yet broken. Then this guy is going to offer to bet you that he can make the jack of spades jump out of this brand-new deck of cards and squirt cider in your ear. But, son, you do not accept this bet, because as sure as you stand there, you’re going to wind up with an ear full of cider.”

Zeckhauser opens his 2006 paper with this advice from Sky Masterson’s father in Guys and Dolls. The lesson is as old as markets: if someone offers you a bet where they seem to know something you don’t, they probably do. Don’t take that bet.

But Zeckhauser’s point isn’t the lesson. It’s the exception. What happens when nobody has the marked deck? When the ambiguity is shared, when neither side can enumerate the states of the world, the Sky Masterson rule stops applying, and the investors who keep following it anyway leave money on the table.

In Part 1 I laid out Zeckhauser’s taxonomy: risk, uncertainty, and ignorance as three distinct problems. In Part 2 I examined why investors flee the third box, the mechanism of ambiguity aversion. This piece asks the question that follows: when you’re facing someone on the other side of a trade, how do you figure out whether they know something you don’t?

The two matrices

Zeckhauser draws two matrices that I think are the most underappreciated diagrams in the paper.

The first covers investing under uncertainty, where the possible states are known but probabilities are hard. It’s a 2x2: Easy or Hard for You to Estimate Value crossed with Easy or Hard for Others. Box A (easy for both) is the standard competitive market: lots of participants, tight spreads, no edge for anyone. Box B (easy for you, hard for others) is where you’re the informed party: think a biotech scientist evaluating a drug trial readout. Box C (hard for you, easy for others) is the danger zone, the other side has the marked deck, and the Sky Masterson rule applies in full. Box D (hard for both) is where it gets interesting. Neither side has an information advantage. Both are operating under genuine uncertainty. Buffett’s earthquake reinsurance sits here.

The second matrix covers investing under ignorance, where even the possible states are unknown. It’s simpler: a 2x1. Unknown to You and Known to Others (Box E) versus Unknown to You and Unknown to Others (Box F). Box E is dangerous. Box F is opportunity.

The point most people miss is about misidentification. Most investors assume they’re in Box C or Box E: the other side knows more. This assumption is the legacy of standard information asymmetry models in finance, where Akerlof’s lemons problem (1970) and the Glosten-Milgrom bid-ask spread model (1985) trained a generation to worry about adverse selection. Those worries are justified in Boxes A through C and in Box E. But in Box D and Box F, you’re not facing an informed counterparty. You’re facing someone equally confused, or someone who has left the market entirely because they can’t tolerate the confusion.

Bazerman and Samuelson (1983) showed that even in clean experimental settings, people are terrible at accounting for why the other side is willing to trade. Their winner’s curse experiments found that bidders consistently failed to discount for the fact that winning an auction is bad news about your estimate’s accuracy. In a UU world, this failure compounds. You can’t compute the conditional expectation of the asset’s value given that the other side is selling, because neither of you can define the state space over which that expectation would be calculated.

The practical question is always: am I in Box C or Box D? Am I in Box E or Box F? And the answer is almost never available from the data. It’s a judgment call, informed by what you know about the seller’s constraints, their institutional context, and whether the source of the ambiguity is private or shared.

Institutional blindness as structural opportunity

The California Earthquake Authority story is Zeckhauser’s best illustration, and it deserves the full telling.

In the late 1990s, California needed reinsurance for earthquake risk. The authority offered a $1 billion slice at premiums that worked out to roughly five times actuarial value. Wall Street said no. Not because investment banks thought the Earthquake Authority possessed secret seismological knowledge. Nobody has an informational edge over the reinsurer when it comes to tectonic plate movement. The ambiguity was shared: Box F.

Wall Street said no because their internal processes required probability estimates that didn’t exist. Compliance teams required distributional assumptions about tail risk that nobody could provide. Risk models required defined scenarios, and “catastrophic earthquake in the next 12 months” didn’t fit neatly into any existing framework. The honest assessment, “we have no idea about the probability, but the price is very high,” didn’t fit the form. Buffett took the entire slice.

This is Zeckhauser’s Maxim H: “Do not engage in the heuristic reasoning that just because you do not know the risk, others do.” The Wall Street banks weren’t outcompeted by someone with better information. They were outcompeted by someone with fewer institutional constraints. Buffett could hold a position that was impossible to model because he answered to shareholders who trusted his judgment, not to compliance officers who required his models.

Generalize this, and you get a structural feature of UU markets that doesn’t go away. Fiduciary duty requires estimable risk. Compliance models require defined scenarios. Career risk creates what Zeckhauser calls Monday Morning Quarterback (MMQ) risk: the danger that a bad outcome on a good decision destroys your reputation. Professional investors face a permanent bias toward the risk box (known probabilities) and away from the ignorance box (unknown states). This isn’t a market inefficiency waiting to be arbitraged. It’s an institutional constant. And it creates a permanent supply of mispriced assets for those without the same constraints.

Aumann (1976) proved that rational agents with common priors who share their posterior beliefs must converge: they cannot “agree to disagree.” The theorem is elegant and, in UU markets, irrelevant. Aumann assumes common priors and known state spaces. In Box 3, both assumptions fail. The state space is undefined, so there are no common priors to start from. Disagreement in UU isn’t a puzzle to be resolved by more information exchange. It’s the default condition. Two equally rational investors can look at the same situation and reach opposite conclusions without either one being wrong, because they’re not disagreeing about probabilities. They’re disagreeing about what world they’re in.

The advantage-versus-selection formula

Zeckhauser offers a framework for deciding when to invest despite potential adverse selection. Your expected return depends on three things: your absolute advantage (a), the probability the other side is better informed (p), and the selection factor (s), how much their information hurts you. Invest when the combination exceeds the cost of entry.

The formula matters less than the logic behind it. A large absolute advantage provides insurance against adverse selection. Zeckhauser’s Maxim E: “A significant absolute advantage offers some protection against potential selection.”

What counts as absolute advantage? Complementary skills are the classic answer: the real estate developer who creates value a passive investor cannot, the venture capitalist whose operational expertise and network make the company worth more than the sum of its capital. Their return isn’t compensation for bearing risk. It’s a share of value they helped create. Sidecar investors, Zeckhauser’s term for those who invest alongside skilled operators, earn excess returns because access to these deals is limited and the value creation is real.

But complementary skills aren’t the only form of advantage. In the SaaSpocalypse, the “absolute advantage” for a buyer at IGV $80 was time horizon. If you could hold for three to five years, tolerate the MMQ risk of further drawdowns, and ignore the career consequences of looking wrong for a few quarters, you had a structural edge over institutional sellers who couldn’t do the same. That’s not analytical skill. It’s constraint arbitrage. And constraint arbitrage is a legitimate form of absolute advantage, because fiduciary requirements and career incentives are structural features that won’t disappear next quarter.

Larry Summers (2006) raises the obvious objection to the sidecar concept: “identifying skilled UU managers may be no easier than picking market-beating investments directly.” The sidecar doesn’t solve the epistemological problem. It relocates it from asset selection to manager selection. How do you know the driver is skilled rather than lucky?

Richard Robb (2006) pushes further. He argues that UU knowledge is “uncommunicable.” If a mechanism for generating excess returns could be expressed as a process, someone would have arbitraged it away. Ricardo, on the eve of Waterloo, might have said “British Government bonds offer a high reward for the risk.” But what would it look like for that statement to be proven false? The claim is unfalsifiable because it lives in the ignorance box where probability statements don’t have clear empirical content. If the sidecar driver can’t explain their edge in terms you can evaluate, how do you distinguish skill from survivorship bias?

I think both objections are correct and both miss something. They’re correct that sidecar investing doesn’t eliminate the evaluation problem. But they miss that the evaluation problem has different difficulty levels depending on context. Evaluating whether a real estate developer can build and lease a building is easier than evaluating whether a macro hedge fund can predict interest rates. Evaluating whether Buffett’s insurance math is sound is easier than evaluating whether a biotech startup’s drug candidate works. The sidecar concept isn’t “trust someone blindly.” It’s “invest alongside someone whose edge you can partly verify, in situations where your own analytical advantage is zero.”

Knowing the geometry of who-knows-what is necessary but not sufficient. You’ve identified a Box D or Box F opportunity. You’ve assessed your absolute advantage. You’ve decided the other side isn’t better informed. Now you need to decide how much to bet. In a UU world, the most famous formula for position sizing, the Kelly Criterion, breaks down in the ways you’d expect. That’s Part 4.

Ambiguity by Design

me@philippdubach.com (Philipp D. Dubach) — Wed, 08 Apr 2026 00:00:00 +0000

Investing at the Edge of Knowledge, Part 2 · Start with Part 1

Ellsberg’s urn experiment is one of the cleanest results in decision theory. Daniel Ellsberg (1961) put two urns in front of subjects. Urn A: 50 red balls, 50 black. Urn B: 100 balls, red and black, ratio unknown. Pay $100 if you draw the right color. Most people chose Urn A, the known 50/50 bet. Fine so far. But here’s the problem: they chose Urn A regardless of which color they were betting on. Bet on red? Prefer Urn A. Bet on black? Still prefer Urn A. This is incoherent. If you think Urn B has fewer red balls (making you avoid it for a red bet), you should prefer it for a black bet. The subjects weren’t estimating probabilities at all. They were fleeing the feeling of not knowing the probability. Ellsberg proved that people make systematically different choices when probabilities are unknown versus known, even when the unknown probabilities carry no actual informational disadvantage. Richard Zeckhauser’s contribution was to ask

What happens to prices when an entire market makes this choice simultaneously?

The experimental evidence

Ellsberg’s result spawned a body of work that, six decades later, has only strengthened the original finding. Fox and Tversky (1995) added a twist that matters enormously for financial markets. Their “comparative ignorance hypothesis” showed that ambiguity aversion intensifies when people can compare themselves to someone who appears more knowledgeable. In a non-comparative setting, where subjects evaluated an ambiguous bet in isolation, ambiguity aversion largely disappeared. But the moment subjects could compare their knowledge to someone else’s, the aversion came roaring back.

In markets, there is always someone who appears more confident. Every sell-side note, every CNBC segment, every hedge fund manager interviewed at Davos projects certainty that you don’t feel. The comparative ignorance effect is permanently activated in financial markets. You don’t just feel uncertain. You feel uncertain relative to someone who seems to know, and the gap between their apparent confidence and your honest confusion is what drives the exit decision.

Zeckhauser’s own experimental evidence in the 2006 paper extends this further. He ran lottery choice experiments comparing willingness to bet on standard probabilistic gambles versus events with unknown and unknowable (UU) outcomes (to familiarize yourself with this framework start with Part 1.) People refused to distinguish between small probabilities of UU events even when the expected value difference was large. The feeling of not-knowing overwhelmed the arithmetic of expected value. Separately, he documented that individuals explicitly warned about overconfidence are still surprised 35% of the time on quantities where they should be surprised only 2% of the time. We simultaneously know less than we think (overconfidence) and refuse to act on what we do know when probabilities are ambiguous (ambiguity aversion).

Is ambiguity aversion rational?

This turns out to be a harder question than it looks, and the answer matters for how you think about the mispricing mechanism. The case for “yes, it’s rational” is surprisingly strong. Gilboa and Schmeidler (1989) proved that a decision maker who evaluates bets by the worst-case probability in their set of plausible priors is behaving in a way that satisfies all the standard axioms of rational choice except one: the Sure-Thing Principle that Ellsberg’s experiment violates. Their maxmin expected utility model says: if you don’t know the probability, evaluate the bet as if the probability is the worst one consistent with your information. This is formally coherent. It’s also roughly what a good risk manager does when facing an uncertain tail risk. Bewley (2002), as I discussed in Part 1, showed that dropping the completeness axiom produces a framework where inertia, refusing to act, is the rational response when you cannot rank the alternatives. If you can’t tell which bet is better, sticking with the status quo isn’t lazy. It’s defensible.

The case for “no, it’s a bias” rests on the Ellsberg experiment itself. The subjects preferred a known 50% chance over an unknown chance that they could bet on either side of. There is no informational disadvantage. The probability they’re fleeing might be 50%, might be 30%, might be 70%, but since they can bet on either color, the expected value is the same regardless. The aversion is to the experience of not-knowing, not to any actual asymmetry in the bet. That looks more like a bug than a feature.

I think the answer is “it depends,” and the distinction matters. Ambiguity aversion is rational when there might be a better-informed party on the other side of the trade. If you’re buying a stock and you suspect the seller knows something you don’t, demanding a discount for your ignorance is not a bias. It’s adverse selection protection. But ambiguity aversion is irrational when you can establish that nobody knows more than you do. When the ambiguity is universal, when the entire market is confused because the state space itself is new, the discount demanded by ambiguity-averse investors is a pricing error, not a risk premium.

This is where I land: ambiguity aversion is a sensible default that gets systematically overweighted in specific situations. The skill is the distinction. And the distinction is judgment, not math.

Discomfort as information

Zeckhauser’s most counterintuitive move in the paper is turning ambiguity aversion from a problem into a signal. His Speculation 1 states it directly: “UUU investments drive off speculators, which creates the potential for an attractive low price.”

The logic is recursive. Your discomfort when facing an ambiguous situation tells you something, but not about the asset. It tells you about the competitive field. If you’re uncomfortable, most other potential buyers have already left. The very thing that makes you want to sell, the feeling of not-knowing, is the same thing that has thinned the competition and compressed the price. David Ricardo buying British government bonds on the eve of Waterloo was uncomfortable. Warren Buffett writing earthquake reinsurance for the California Earthquake Authority at roughly five times actuarial value was comfortable only because he had done this inference before: the discomfort of everyone else was the opportunity itself. Zeckhauser’s Maxim G puts it memorably

Discounting for ambiguity is a natural tendency that should be overcome, just as should be overeating.

Both ambiguity aversion and overeating are evolved heuristics that served us well in ancestral environments and poorly in modern ones. In a small tribal group where the unknown reliably correlated with danger, fleeing ambiguity kept you alive. In a financial market where ambiguity-averse institutional capital mechanically exits positions it can’t model, the same instinct creates a systematic transfer of wealth from the ambiguity-averse to the ambiguity-tolerant.

Bohnet and Zeckhauser (2004) identified a related mechanism they called “betrayal aversion.” People demand stronger odds when a betraying human rather than indifferent nature determines the outcome. In markets, this manifests as an extra discount demanded when the ambiguity involves a counterparty who might be exploiting your ignorance. The mere possibility that someone on the other side knows more amplifies the ambiguity premium beyond what the uncertainty alone would justify.

Now apply all of this to a more recent example, the SaaSpocalypse. I wrote about the details elsewhere, but the relevant point here is the mechanism. When Anthropic released the Claude Cowork plugins in late January, institutional investors didn’t sit down and estimate the probability that AI would replace CRM. They faced something worse: they couldn’t define what “replacing CRM” would even mean. The state space was undefined, as I argued in Part 1. And when the state space is undefined, the entire institutional machinery for processing uncertainty breaks down simultaneously.

Fiduciary duty requires estimable risk. Compliance models require defined scenarios. Portfolio managers face career risk: losing money on a position you can’t explain is a firing offense; missing a rally in something you sold is merely embarrassing. The institutional constraints compound the ambiguity aversion. Each layer of oversight demands a model, and the model requires defined states, and the states don’t exist yet. The rational response for any individual institutional actor was to sell. The collective result was an IGV drawdown of 32% while sector earnings grew 17%, an RSI of 18, and $2 trillion in evaporated market cap.

The sellers weren’t acting on information. They were acting on ambiguity aversion, amplified by comparative ignorance (everyone else seemed to be selling too), amplified by career risk (nobody gets fired for selling software before the AI disruption), amplified by betrayal aversion (maybe the AI insiders knew something the market didn’t). Stack these amplifiers on top of Ellsberg’s basic finding, and you get a price that reflects the intensity of collective discomfort rather than any assessment of fundamentals.

Zeckhauser describes the investor’s challenge with a bridge analogy: you have to make peace with good decisions that lead to bad outcomes. Buying the IGV at $80 with an 18 RSI and 17% earnings growth is, on the framework, a good decision. If it drops to $70 first, that doesn’t make it a bad decision. But making that distinction under ambiguity is not an analytical skill. It’s a temperamental one. It requires accepting that “I don’t know” is not disqualifying and that the discomfort you feel is shared, priced in, and possibly overpriced. That’s harder than any calculation.

Knowing that ambiguity aversion creates mispricing is the easy part. The hard part is what comes next: when you’re facing someone on the other side of a trade in a UU world, how do you figure out whether they know something you don’t, or whether they’re just less uncomfortable than you are? That’s the domain of sidecar investing and strategic inference. That’s Part 3.

Three Kinds of Not-Knowing

me@philippdubach.com (Philipp D. Dubach) — Sat, 04 Apr 2026 00:00:00 +0000

Investing at the Edge of Knowledge, Part 1

David Ricardo made a fortune buying British government bonds four days before the Battle of Waterloo. He was not a military analyst. He had no basis to compute the odds of Napoleon’s defeat, or victory, or any of the ambiguous outcomes in between. But he understood something that most of his contemporaries did not: the nature of his own ignorance was the same as everyone else’s, the seller was desperate, competition was thin, and the pounds he’d gain if Wellington won were worth far more than the pounds he’d lose if Wellington fell.

Ricardo’s edge was not information. It was a correct assessment of what kind of not-knowing he was facing.

That distinction, between different kinds of not-knowing, is mostly absent from finance. Richard Zeckhauser, the Frank P. Ramsey Professor of Political Economy at Harvard, made it the foundation of his 2006 paper “Investing in the Unknown and Unknowable,” published in Capitalism and Society. The paper takes no derivatives and runs no regressions. What it does instead is more valuable: it provides a taxonomy of not-knowing, and then shows why the category that finance theory handles worst is the one where the biggest fortunes have been made.

This is Part 1 of a five-part series that works through Zeckhauser’s framework and extends it. The goal is not a literature review. It’s an attempt to build a working vocabulary for the kind of investing that modern portfolio theory was never designed to address.

The taxonomy

Zeckhauser presents three categories of not-knowing. Each demands different skills. Each rewards a different kind of investor. And the jump between them is not a smooth gradient. It’s a cliff.

The first box is risk. Probabilities are known, distributions of returns are known, and the challenge is optimization. This is the world of the capital asset pricing model, of mean-variance portfolios, of the efficient frontier. You hold a 60/40 stock-bond portfolio and rebalance quarterly. The math is clean. The Nobel Prizes were awarded. Finance education lives here.

The second box is uncertainty. You can identify the possible states of the world, but you can’t assign reliable probabilities. A corporate bond analyst looking at incomplete financials knows the company might default or might not, knows the recovery rate might be 40 cents or 60 cents, but can’t compute a precise probability for either. The skill that pays here is Bayesian estimation: forming the best prior you can from limited data, updating as information arrives, and having the temperament to act on imperfect beliefs. This is harder than Box 1, but it’s still recognizable territory. Decision theory was built for it.

The third box is ignorance. Zeckhauser abbreviates it UU: unknown and unknowable. Here, even the identity of possible future states is undefined. You don’t have a distribution to estimate because you can’t enumerate what you’re estimating over. The question isn’t “what’s the probability of outcome X?” It’s “what even is X?” This is where Ricardo was standing at Waterloo. This is where Warren Buffett was standing in 1996 when he wrote a $1.5 billion reinsurance policy for the California Earthquake Authority at a premium far above actuarial estimates, coverage that the capital markets had failed to place. The New York financial community couldn’t model the risk. Buffett’s insight was that nobody could, that the Authority was not better informed about seismic activity than he was, and that the price was absurdly favorable given the symmetry of ignorance.

The boxes are not a spectrum. You don’t get from Box 2 to Box 3 by adding more uncertainty. You get there when the state space itself is undefined. In Box 2, you might not know whether a company will default, but you know that “default” and “no default” are the relevant categories. In Box 3, you don’t even know the categories. That’s a qualitative difference, not a quantitative one.

Why finance forgot the third box

The strange thing is that the third box was identified a century ago. Twice, independently, in the same year.

Frank Knight published Risk, Uncertainty and Profit in 1921. His central argument, the origin of what economists now call Knightian uncertainty, was that entrepreneurial profit is compensation for bearing true uncertainty: situations where probabilities cannot be meaningfully calculated. Risk, in Knight’s framework, is insurable. Uncertainty is not. The distinction is not about the degree of confidence in your estimate. It’s about whether the concept of a probability estimate even applies.

John Maynard Keynes published A Treatise on Probability the same year. His angle was different but convergent. Keynes introduced the idea of the “weight of evidence”: a thin body of evidence yields low weight even when the point estimate looks reasonable. In his 1937 Quarterly Journal of Economics article, he made the distinction explicit: “By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty.” Roulette is risky. The future of interest rates, the price of copper twenty years out, the obsolescence of a technology: these are uncertain in the deeper sense. The distinction mattered to Keynes, and it should matter to anyone building a portfolio.

Both arguments lost. The discipline moved toward formalization, and formalization required calculable probabilities. The efficient markets hypothesis, rational expectations, CAPM, Black-Scholes: all of these live in Box 1 or assume that Box 2 can be reduced to Box 1 with sufficient data and computing power. This isn’t a criticism of these models within their domain. They’re brilliant engineering for the problems they were designed to solve. It’s a claim about the boundaries of that domain, and about how much of real-world investing sits outside it.

LeRoy and Singell (1987) offered a provocative reinterpretation in the Journal of Political Economy: Knight’s real distinction, they argued, was about insurability, not probability. Uncertainty describes situations where insurance markets collapse because of moral hazard and adverse selection, not simply because probabilities are subjective. This reading is more radical than the standard one. It says the breakdown isn’t epistemic (we don’t know enough) but structural (the market itself can’t price the risk). That structural breakdown is precisely what happened in 1996 when Wall Street couldn’t write the California earthquake policy, and again in 2025 when insurance markets in parts of the American Southeast and West simply stopped functioning.

Kay and King picked up this thread in their 2020 book Radical Uncertainty, arguing that the conflation of risk and uncertainty has caused systematic mismanagement across finance and policy. Their prescription is “narrative reasoning” rather than probabilistic optimization for decisions facing genuine uncertainty. I’m not sure narrative reasoning is sufficient, but I’m confident that probabilistic optimization is insufficient. The honest position is somewhere in between, and Zeckhauser’s framework gives you the vocabulary to think about where.

Bewley (2002) formalized the problem differently. Working from a 1986 Cowles Foundation paper, he dropped the completeness axiom from expected utility theory. In standard theory, you can always rank alternatives: you prefer A to B, or B to A, or you’re indifferent. Bewley said: sometimes you simply can’t rank them. When alternatives are incomparable, sticking with the status quo is rational, not a bias. This gives mathematical expression to something practitioners know in their bones: there’s a difference between “I’m going to hold because I think the price will go up” and “I’m going to hold because I have no coherent basis for predicting what will happen and the cost of acting without a basis is higher than the cost of staying put.”

Why Knightian uncertainty is growing

This hundred-year-old taxonomy feels more relevant in 2026 than it did in 2006. Technological change creates entirely new categories of outcomes faster than models can absorb them. The state space itself is expanding.

I wrote recently about the SaaSpocalypse paradox: the market simultaneously pricing AI capex failure and AI destroying all enterprise software, when both cannot be true. That sell-off is a textbook example of Zeckhauser’s third box. The problem wasn’t that investors struggled to estimate the probability of known outcomes. The problem was that the outcomes themselves were undefined. What does “CRM” mean when AI agents replace human users? What does “per-seat licensing” mean when the number of seats might go to zero or might multiply by ten as agents proliferate? What does “enterprise software moat” mean when the moat was always the trained-user interface and the interface is now natural language? These aren’t questions with difficult probability estimates. They’re questions where the categories haven’t been invented yet.

Nobody in January 2026 could enumerate the states of the world for enterprise software post-Claude Cowork plugins. Not “the probabilities were hard to estimate.” The states themselves were undefined. That’s not Box 2. That’s Box 3.

And Box 3 is where the IGV software ETF fell 32% from its September peak, where hedge funds made $24 billion shorting the sector, where the RSI hit 18 (the most oversold reading in the ETF’s history), and where earnings growth continued at 17%. The disconnect between operating results and market prices is exactly what Zeckhauser’s framework predicts: when the state space is undefined, investors who require defined state spaces to make decisions leave the market. Their departure compresses prices beyond what any fundamental analysis would justify. The mispricing lives in the gap between what the asset is worth and what institutions are able to hold.

This pattern will recur. AI is not the last technology that will generate new categories of outcomes that nobody anticipated. Every time it happens, the same sequence plays out: Box 3 conditions emerge, institutions flee because their models require Box 1 or Box 2 inputs, prices overshoot, and unconstrained investors who understand the nature of their own ignorance pick up the pieces. Zeckhauser wrote his paper two decades ago. The mechanism he described has, if anything, accelerated.

The taxonomy tells you what kind of problem you’re facing. It doesn’t tell you what to do about it. That requires understanding why most investors run from Box 3, and whether running is rational. That’s Part 2.

Long Volatility Premium

me@philippdubach.com (Philipp D. Dubach) — Sat, 14 Feb 2026 00:00:00 +0000

The real value of tail hedging is not in the hedge itself. It’s in what the hedge enables.

In The Variance Tax I wrote about the ½σ² formula: compound returns equal arithmetic returns minus half the variance, and because the penalty is quadratic, large drawdowns destroy wealth in ways that are hard to recover from. A portfolio that falls 50% needs 100% just to break even. That piece was about the problem. This one is about a potential solution, and about whether paying for crash protection can actually improve total returns rather than drag them.

There is a chart circulating in quantitative finance circles that should not exist. It shows a strategy that buys put options on the S&P 500 and, when layered on top of a stock portfolio, improves total returns while simultaneously reducing volatility and maximum drawdown. The chart comes from Patrick Causley at One River Asset Management in a paper called “Heretical Thinking: The Long Volatility Premium” and it makes a specific claim: that long volatility, properly constructed, is not a cost center but a compensated factor that deserves to sit alongside value, momentum, and trend in institutional portfolios.

The conventional wisdom says buying puts is a losing game. The dominant empirical finding is that a volatility risk premium (VRP) exists: from 1990 to 2018, the average VIX level was 19.3% while average realized S&P 500 volatility was just 15.1%, a persistent gap of 4.2 percentage points. Options are, on average, overpriced relative to what materializes. The CBOE S&P 500 PutWrite Index, which systematically sells S&P 500 puts against cash collateral, rose 1,835% from 1986 to 2018. The CBOE 5% Put Protection Index, which buys puts as a hedge, rose only 708%. As Bondarenko (2019) documented, the PUT Index achieved 9.54% annualized versus 9.80% for the S&P 500 but with far lower volatility (9.95% vs. 14.93%), yielding a Sharpe ratio of 0.65 versus 0.33 for put buyers.

So selling options earns money. Buying them bleeds money. That is the consensus. This article is about why that framing, while technically correct, misses something important.

I. Separating beta from convexity

The key insight from the One River paper is that the raw return of a put option conflates two partially independent components, and conflating them has led to a categorical error in how most allocators think about tail hedging.

When you buy a put, your P&L is driven by delta (directional exposure to the underlying), gamma (the acceleration of that exposure as the market moves), and vega (sensitivity to implied volatility). The problem with naively holding puts is that delta embeds a massive short-beta position. Since the equity risk premium is one of the most reliable premia in equity markets, you are fighting a powerful headwind. Your puts bleed value every day the market does not crash, and that bleed overwhelms the occasional windfall when it does.

Causley’s move is straightforward. Neutralize the short-beta by adding enough long equity exposure to offset the embedded delta. What remains is a beta-neutral “long volatility factor” that isolates gamma and vega. Stack this on top of an equity program and the historical results over approximately 40 years are striking: the beta-1 portfolio with long volatility outperformed a portfolio without it while producing lower volatility and a shallower maximum drawdown.

The persistence of this phenomenon, even with a simplistic implementation using monthly 5% OTM puts from the CBOE’s PPUT index, is what makes the paper interesting rather than dismissible. A more sophisticated execution (better strike selection, dynamic sizing, multi-tenor rolls) would likely improve results further. But the baseline already makes the case.

II. Why would a long volatility premium exist?

If markets are efficient, a beta-adjusted long volatility position should not deliver a positive premium. Three mechanisms suggest why it might.

The first is the rebalancing premium. When you hold negatively correlated assets and rebalance systematically, you extract what the literature calls a “rebalancing bonus” where the geometric return exceeds the weighted average of individual arithmetic returns. Recent work in the Investment Analysts Journal formalizes this for tail hedging specifically. A long volatility position that delivers explosive gains during crashes and modest losses during calm markets, rebalanced against equities, creates a structural tailwind. You systematically sell the hedge at high prices after crashes and buy it back cheaply during calm, monetizing mean reversion.

The second is that stock-volatility correlation intensifies dramatically during crashes. When equities fall sharply, implied volatility does not just rise proportionally, it spikes exponentially. The hedge’s payoff is largest precisely when the portfolio most needs it. This convexity, once beta-adjusted, can more than compensate for the ongoing cost of the position.

The third is a supply-demand imbalance. Institutional investors are structurally short volatility in numerous ways: through equity ownership itself, through structured products with embedded short option positions, and through strategies that implicitly sell insurance (risk parity, short vol ETFs, pension de-risking). Meanwhile, the supply of long volatility is limited by behavioral challenges. As Jody Deio of Aearon Risk Advisors explains: “People don’t have the patience to wear these exposures for any long period of time. You’re happy being basically a wasting asset. And a wasting asset is nothing that any investment committee or client meeting wants to deal with.” This behavioral gap between the demand for protection and the willingness to supply it may create a structural premium for those who can withstand the psychology.

All three mechanisms connect back to the variance tax. The ½σ² drag on compound returns means that reducing drawdown severity has a nonlinear effect on terminal wealth. By truncating left-tail outcomes, even a costly hedge can increase compound wealth through the compounding channel alone.

III. What the research actually says

The claim that long volatility is a compensated factor runs against a large body of literature documenting the short volatility premium. But the two are not necessarily contradictory. They operate at different levels of analysis.

Research from Barclays found that while the VRP has positive equity market beta, it also has excess alpha above that beta exposure. A linear regression of the VRP against S&P 500 returns found a significant positive intercept of roughly 3.48 volatility points independent of equity market direction. This suggests that both buying and selling volatility can capture distinct premia depending on how the trade is structured and what exposures are isolated.

Goldman Sachs Asset Management’s 2025 analysis has the clearest framing I have seen. Their key finding: even an idealized, 99% reliable tail-risk hedging strategy provides a standalone annual return boost of only about 0.8 basis points. Trivial. But that is not the point. The real value comes from what the hedge enables. Because tail-risk hedges reduce the impact of severe drops, they allow a portfolio to take on more equity risk, to increase beta. The gains from this “risk budget reallocation” can be substantial, especially for institutional investors with fixed drawdown constraints. In Goldman’s framing, tail-risk hedging is not a standalone return generator. It is an offensive weapon that enables more aggressive positioning in core assets. This is philosophically closer to how Formula 1 teams think about pit stops: they cost time, but soft tires allow faster laps, resulting in a faster overall race.

The numbers from Universa’s live track record add color. During Q1 2020, as the COVID pandemic triggered a 34% crash in the S&P 500, Universa delivered a 4,144% return. But Spitznagel himself downplays these headline figures, noting that “any punter can devise a trade that does well in a crash. The key is how you do in a crash relative to the rest of the time.” The Wall Street Journal reported that a strategy consisting of just a 3.3% allocation to Universa with the rest in the S&P 500 had a compound annual return of 12.3% over 10 years through February 2018, beating the S&P 500 itself. A 3.3% tail position improving total portfolio returns over a decade is not intuitive. But it follows directly from the variance tax arithmetic.

IV. Puts vs. trend: the tortoise and the hare

AQR’s research on tail hedging, published in the Journal of Systematic Investing, complicates the picture in a way I find genuinely useful for portfolio construction. They compare two fundamental approaches: buying out-of-the-money puts and multi-asset trend-following.

Puts act as the hare. They deliver spectacular returns in sudden crashes like COVID, when put-buying strategies returned over +42% in a single month. But they are expensive to maintain and their long-term expected return is negative. Trend-following acts as the tortoise. It cannot provide the same reliable downside protection as index puts, but has delivered surprisingly consistent safe-haven performance when most needed while earning positive long-run returns.

AQR’s follow-up paper examined the five largest 60/40 drawdowns since 2000 and found that options-based strategies outperformed in shorter drawdowns while trend-following posted its most impressive returns during protracted bear markets. Since longer drawdowns are arguably more damaging to long-term wealth (they impair compounding for extended periods, which brings us back to the variance tax), AQR leans toward trend-following as the more practical hedge for most investors.

But the strategies are genuinely complementary. Recent academic work combining both approaches via a portable alpha framework found statistically significant alpha of 0.25% per month after controlling for traditional equity factors, with the strongest outperformance during periods of market turmoil. Puts for the fast crash, trend for the slow bleed.

Where most tail hedges fail, and the benchmark problem

Here is where most investors get burned. A CAIA Association paper compared multiple tail-risk strategies against a deliberately boring benchmark: holding cash. Cash achieved a reduction of 80% of portfolio tail risk and 81% of portfolio standard deviation compared to an S&P 500-only portfolio, with an information ratio of 0.67. Several popular tail-risk strategies, particularly those involving short-dated VIX futures and 1-month variance swaps, actually failed to beat this cash benchmark, with performance drags of 355 and 203 basis points respectively.

This is the finding that should make any allocator uncomfortable. If your sophisticated tail hedge cannot beat holding Treasury bills, you are paying for complexity that destroys value. The specific implementation matters a lot, and many “obvious” approaches (VIX futures being the most popular) are structurally flawed because of contango decay in the VIX term structure that steadily erodes returns during calm periods.

A 2025 paper in the Journal of Futures Markets adds a related finding: naïve hedging strategies outperformed more complex models for tail-risk hedging, consistent with earlier findings on variance-minimizing hedges. The explanation lies in model risk. Sophisticated approaches require more assumptions about market dynamics, and when those assumptions are wrong (as they inevitably are during the very tail events you are trying to hedge) the resulting model misspecification can leave hedging portfolios with higher-than-expected risk. This is a familiar problem in ML: the more parameters you fit on in-sample data, the worse your out-of-sample performance when the regime changes. Tail events are precisely when regimes change.

There is also a benchmark problem that poisons the conversation around tail hedging. A portfolio of stocks plus put options gets compared against a portfolio of just stocks. When the market rises steadily for years, the hedged portfolio naturally underperforms and the hedge looks like a waste of money. This comparison is intellectually dishonest. A portfolio with puts has less risk than a portfolio without them. Comparing them as equivalent is like comparing a levered equity portfolio to an unlevered one and concluding that leverage “works” because it outperformed during a bull market. The appropriate comparison for a hedged portfolio is against a portfolio with similar risk, whether achieved through lower equity allocation, higher cash balances, or other risk-reducing measures. When Bhansali and Davis (2010) conducted this more appropriate comparison, they found that offensive tail hedging, using the freed-up risk budget to increase equity exposure, resulted in superior risk-adjusted performance. The hedge was not a drag. It was an enabler.

What I take away

Most of the interesting questions in finance are not about individual positions but about what positions enable. Tail hedging is boring in isolation. What it does to the rest of the portfolio, the willingness to stay invested during drawdowns, the capacity to hold concentrated positions, the ability to rebalance into cheap assets after crashes rather than capitulating, that is where the return comes from. Spitznagel and Goldman agree on this even if they agree on little else.

The optimal tail hedge allocation is a psychological question, not a mathematical one. Most practitioners suggest 1 to 5% of portfolio value, sized to offset a meaningful portion of equity losses during a severe 30 to 50% drawdown. But the right number is the one that allows you to stay invested in your core portfolio through the worst of times without abandoning the strategy. If you cannot stomach three years of negative carry on a put overlay, the correct allocation for you is zero, not five percent.

The framing I find most useful is Goldman’s. Do not evaluate the hedge in isolation. Evaluate what it enables. A 3% tail hedge allocation that reduces max drawdown from 50% to 25% frees up enough risk budget to increase equity exposure by 10 to 15 percentage points. The incremental return from that higher equity allocation over a full market cycle will, in most scenarios, more than compensate for the cost of the hedge. The hedge is the enabler, not the alpha.

Whether you implement this with puts, trend-following, or both depends on your time horizon and what kind of drawdown keeps you up at night. Fast crashes favor puts. Slow bleeds favor trend. If you do not know which one is coming (you do not), blend them.

The caveats are real. All backtests benefit from hindsight. Transaction costs and bid-ask spreads in options markets are material and not fully captured in the CBOE indices used as benchmarks. The behavioral challenge of holding a position that bleeds money most of the time is hard to overstate, especially for allocators who report to investment committees that look at monthly returns.

But the turkey metaphor from One River’s presentation is apt. A statistician turkey who, right up until Thanksgiving, can prove with perfect p-values that the farmer is benevolent. The turkey’s model is flawless within the distribution of observed data. The problem is that the data does not contain the event that matters most. Tail hedging is the strategy of the paranoid turkey. The empirical evidence suggests this paranoia can be not just protective but profitable, provided you implement it with discipline and use it not as a way to avoid risk but as a foundation for taking more of it.

Variance Tax

me@philippdubach.com (Philipp D. Dubach) — Fri, 06 Feb 2026 00:00:00 +0000

Let’s say your portfolio returned +60% in 2024, then fell 40% in 2025. That’s an annualized average return of +10%. Actual return after two years: minus 4% (i.e $100 * 1.6 * 0.6 = $96).

That 14-point gap is what we call the variance tax aka variance drain or volatility drag and it’s one of the least intuitive forces in investing.

Take any series of returns with arithmetic mean μ and volatility σ. The compound growth rate, the one that actually determines your wealth, is approximately:

$$G ≈ μ − ½σ²$$

This comes from a second-order Taylor expansion of ln(1+r). Take expectations, and the mean log return equals the arithmetic mean minus half the variance. Everything else drops out. Half the variance. That is the tax. The same correction term appears when you solve geometric Brownian motion via Itô’s lemma (the drift of log(S) is μ − σ²/2, not μ) so whether you come at it from discrete compounding or continuous-time stochastic calculus, you land in the same place. And because it is quadratic, doubling volatility does not double the cost. It quadruples it. And what we learned during covid, if anything at all, is that we generally have a hard time to mentally abstract exponential growth rates.

Treasury bonds at 5% vol pay about 0.1% per year in variance drain. Barely noticeable. The S&P 500 at 16% vol pays 1.3%. A 3x leveraged ETF at 48% vol pays 11.5%. Jacquier, Kane, and Marcus (2003) studied S&P 500 returns from 1926 to 2001: arithmetic mean 12.49%, geometric mean 10.51%. The gap is 1.98 percentage points. The formula predicts ½ × 0.203² = 2.06%.

Looking at the last row, we see that tripling leverage triples the arithmetic return but delivers nearly the same compound return as 2x. The linear gain gets eaten by the quadratic penalty.

Same 10% arithmetic return, different volatility. After 30 years, the zero-volatility path reaches $1,745. At 15% vol, $1,280. At 30%, $498. At 50% vol you have lost more than half your money despite averaging +10% per year.

Now apply leverage. If you lever an asset by factor L, the arithmetic return scales linearly (Lμ) but the variance drain scales quadratically (½L²σ²). The compound return becomes:

$$G(L) ≈ r + L(μ − r) − ½L²σ²$$

Take the derivative, set to zero. The leverage that maximizes compound wealth:

$$L^{\ast} = (μ − r) / σ²$$

For the S&P 500 with roughly 7% excess return and 16% vol, L* comes out to about 2.7x.

This is the Kelly criterion (which you might know from utility theory or gambling heuristics but in fact, as we see here, it falls straight out of the variance tax formula.) Beyond Kelly, every dollar of additional leverage costs more in variance drain than it earns in expected return. The curve bends over and eventually goes negative. In practice, most practitioners use “half-Kelly” — sizing positions at L*/2 — because the formula assumes you know μ and σ precisely, and you don’t. Estimation error in either parameter can push you past the peak and onto the losing side of the curve. Half-Kelly sacrifices roughly 25% of the theoretical growth rate but dramatically reduces drawdown risk.

You can see this play out in practice. ProShares UPRO, the 3x S&P 500 ETF, has returned roughly 28% annualized over the past decade during one of the strongest bull markets in history. The S&P 500 compounded at about 10% over the same period. Linear 3x leverage would imply roughly 30%. Variance drain accounts for the gap, and that was in a favorable environment. In 2022, when the S&P fell about 19%, UPRO dropped 70%. The effect is even starker in higher-volatility underlyings: ProShares TQQQ, the 3x Nasdaq-100 ETF, sat roughly flat from its 2021 highs through early 2025 while the unlevered QQQ had long since recovered — a textbook case of variance drain overwhelming the leverage premium in a choppy market.

The same half-sigma-squared shows up across finance. It is why stock prices follow log-normal distributions, not normal ones. Why put options cost more than equidistant calls. Why the Black-Scholes d₁ and d₂ terms carry a ½σ²t adjustment. Why a $100 stock’s true geometric midpoint between $150 up and $50 down is not $100 but $86.60, because ln(150/100) = ln(100/66.67). Wherever returns compound and volatility is nonzero, the variance tax is being collected.

Is Private Equity Just Beta With a Lockup?

me@philippdubach.com (Philipp D. Dubach) — Thu, 29 Jan 2026 00:00:00 +0000

The pitch used to be simple: accept illiquidity, get rewarded. Lock up your capital for seven years, tolerate capital calls and J-curves, and in exchange you’d earn returns that public markets couldn’t touch. It was the defining bargain of institutional investing for two decades.

AQR’s latest capital market assumptions make for uncomfortable reading if you’re an allocator to private markets. Their expected real return for U.S. buyouts over the next 5-10 years is 4.2%. For U.S. large cap public equities, it’s 3.9%. That’s a 30 basis point premium for accepting years of lockup, unpredictable capital calls, limited transparency, and the very real risk of picking the wrong manager.

Private credit looks even worse. Expected returns dropped 0.5 percentage points year over year as spreads narrowed and base rates came down. The asset class that was supposed to be the sensible alternative to stretched equity valuations now offers less compensation than it did twelve months ago.

This isn’t a temporary dislocation. It’s the logical endpoint of too much capital chasing the same opportunities. When every pension fund, endowment, and sovereign wealth fund decides they need 20-30% allocation to alternatives, the returns that made alternatives attractive get arbitraged away. The money didn’t find alpha. It became beta (with a lockup).

I read more reports and the a16z State of the Markets 2026 isn’t less interesting. The dispersion numbers tell an interesting story. In venture capital, top decile managers generate 31.7% IRR while bottom decile managers return negative 7%. The spread between winners and losers is enormous. But that spread is precisely why average returns have compressed. Access to top-tier funds has always been limited, and everyone else is fighting over what’s left.

AQR’s framework suggests something that few allocators want to hear: the illiquidity premium might be negative for most investors. If you’re not in the top quartile of manager selection, you’re accepting lockup risk for returns you could approximate in public markets with better liquidity and lower fees.

The counterargument, and it’s a reasonable one, is that private markets offer exposure to companies you simply can’t access in public markets anymore. This part is true. 87% of U.S. companies with more than $100 million in revenue are now private. The top 10 private companies represent 38% of total unicorn valuation, and that share has nearly doubled since 2020. SpaceX, OpenAI, Anthropic, Databricks, Stripe: these are category-defining businesses, and they’re not on any exchange.

But access isn’t the same as returns. You can have exposure to the most exciting companies in the world and still underperform a boring index fund if you pay too much or pick the wrong vintage. The S&P 500 minimum market cap eligibility has tripled since 2019 to $22.7 billion. Companies are staying private longer, which means more value creation happens before public investors get a chance. It also means private investors are paying up for that privilege.

Value creation has moved earlier in the company lifecycle. For IPOs between 2014-2019, only 12% of median value was created in private markets. For 2020-2023 IPOs, that number jumped to 55%. If you want to capture returns from the next generation of important companies, you probably need private market exposure.

The question really is what you’re paying for it.At 4.2% expected returns versus 3.9% for public equities, you’re paying in liquidity and flexibility for almost nothing in expected return. The premium that justified the allocation model has been competed away. If you’re in the top 5% of venture funds earning 60%+ IRR, none of this applies. For everyone else, the world has moved on.

Against All Odds: The Mathematics of 'Provably Fair' Casino Games

me@philippdubach.com (Philipp D. Dubach) — Sun, 25 Jan 2026 00:00:00 +0000

Gambling can be harmful and lead to significant losses. Participation is subject to local laws and age restrictions. Always gamble responsibly. Need help? Visit BeGambleAware.org

Crash games represent a category of online gambling where players place bets on an increasing multiplier that can ‘crash’ at any moment. The fundamental mechanic requires players to cash out before the crash occurs; successful cash-outs yield the bet amount multiplied by the current multiplier, while failure results in total loss of the wager.

The specific game I came across is a variant that employs an aircraft flight metaphor. Let’s call it Plane Game. What intrigued me wasn’t the game itself but that it said “provably fair” on the startup screen, which I assumed to be a typo at first. I stand corrected:

A provably fair gambling system uses cryptography to let players verify that each outcome was generated from fixed inputs, rather than chosen or altered by the operator after a bet is placed. The casino commits to a hidden “server seed” via a public hash, combines it with a player-controlled “client seed” and a per-bet nonce, and later reveals the server seed so anyone can recompute and confirm the result.

The stated Return-to-Player (RTP) of that specific game is 97%, implying a 3% house edge. After watching a few rounds, the perceived probability felt off. And if there’s something that gets my attention, it’s the combination of games and statistics. So I did what any reasonable person would do: I watched another 20,000 rounds over six days (112 hours total) and wrote a paper about it.

The distribution below shows the classic heavy tail: most rounds crash quickly at low multipliers, while rare events produce 100x or even 1000x payouts. The maximum I observed was 10,000x. This extreme variance creates the illusion of big wins just around the corner while the house edge operates relentlessly over time.

For a crash game with RTP = r (where 0 < r < 1), the crash multiplier M follows a specific probability distribution. The survival function is particularly relevant:

$$P(M \geq m) = \frac{r}{m}$$

This means the probability of reaching at least multiplier m before crashing equals r/m. For any cash-out target, the expected value of a unit bet works out to:

$$E[\text{Profit}] = P(M \geq m) \times m - 1 = \frac{r}{m} \times m - 1 = r - 1 = -0.03$$

This mathematical property makes crash games theoretically “strategy-proof” in expectation. No cash-out timing strategy should yield better long-term results than another.

The empirical data matches theory almost perfectly. A 2x target succeeds about 48.5% of the time. Aiming for 10x? That works only 9.7% of rounds. The close fit between my observations and the theoretical line confirms the stated 97% RTP.

So is the game fair? My analysis says yes. Using three different statistical methods (log-log regression, maximum likelihood, and the Hill estimator), I estimated the probability density function exponent at α ≈ 1.98, within 2.2% of the theoretical value of 2.0. This contrasts with Wang and Pleimling’s 2019 research that found exponents of 1.4 to 1.9 for player cashout distributions. The key distinction: their deviations reflect player behavioral biases (probability weighting), not game manipulation. The random number generator produces fair outcomes.

I then ran Monte Carlo simulations of 10,000 betting sessions under four different strategies: conservative 1.5x cashouts, moderate 2.0x, aggressive 3.0x, and high-risk 5.0x targets.

Every single strategy produces negative expected returns. The conservative approach has lower variance but still loses. The aggressive strategies lose faster with higher variance.

The consumer protection angle is what concerns me most. My data revealed 179 rounds per hour with 16-second median intervals. At that pace, with a 3% house edge per round, players face expected losses exceeding 500% of amounts wagered per hour of play. The manual cashout mechanic creates an illusion of control, masking the deterministic nature of losses.

The game is provably fair in the cryptographic sense. The mathematics check out. But mathematical fairness doesn’t ensure consumer safety. The house always wins, and it wins fast.

The only winning strategy is not to play

The full paper preprint with methodology and statistical details is available on SSRN. Code and data are on GitHub.

It Just Ain’t So

me@philippdubach.com (Philipp D. Dubach) — Sun, 15 Jun 2025 00:00:00 +0000

It ain’t what you don’t know that gets you into trouble. It’s what you know for sure that just ain’t so.

This (not actually) Mark Twain quote from The Big Short captures the sentiment of realizing that some foundational assumptions might be empirically wrong.

A recent article by Anton Vorobets that I came across in Justina Lee’s Quant Newsletter presents compelling evidence that challenges one of the field’s fundamental statistical assumptions, that asset returns follow normal distributions. Using 26 years of data from 10 US equity indices, he ran formal normality tests (Shapiro-Wilk, D’Agostino’s K², Anderson-Darling) and found that the normal distribution hypothesis gets rejected in most cases. The supposed “Aggregational Gaussianity” that academics invoke through Central Limit Theorem arguments? It’s mostly wishful thinking enabled by small sample sizes. As Vorobets observes:

Finance and economics academia is unfortunately driven by several convenient myths, i.e., claims that are taken for granted and spread among university academics despite their poor empirical support.

The article highlights significant practical consequences for portfolio management and risk assessment. Portfolio optimization based on normal distribution assumptions ignores fat left tails—exactly the kind of extreme downside events that can wipe out portfolios. This misspecification can lead to inadequate risk management and suboptimal asset allocation decisions. Vorobets suggests alternative approaches, including Monte Carlo simulations combined with Conditional Value-at-Risk (CVaR) optimization, which better accommodate the complex distributional properties observed in financial data. While computationally more demanding, these methods offer improved alignment with empirical reality.

Reading this piece gave me a few ideas for extensions I might want to explore in an upcoming personal project: (1) While Vorobets focuses on US equity indices, similar analysis across fixed income, commodities, currencies, and alternative assets would provide a more comprehensive view of distributional properties across financial markets. Each asset class exhibits distinct market microstructure characteristics that may influence distributional behavior. (2) Global Market Coverage: Extending the geographic scope to include developed, emerging, and frontier markets would illuminate whether the documented deviations from normality represent universal phenomena or are specific to US market structures. Cross-regional analysis could reveal important insights about market development, regulatory frameworks, and institutional differences. (3) Building upon Vorobets’ foundation, there are opportunities to incorporate multivariate normality testing, regime-dependent analysis, and time-varying parameter models. Additionally, investigating the power and robustness of different statistical tests across various market conditions would strengthen the methodological contribution. (4) Examining different time horizons, market regimes (pre- and post-financial crisis, COVID period), and potentially higher-frequency data could provide deeper insights into when and why distributional assumptions break down.

Beyond Monte Carlo: Tensor-Based Market Modeling

me@philippdubach.com (Philipp D. Dubach) — Sun, 11 May 2025 00:00:00 +0000

A fascinating new paper from Stefano Iabichino at UBS Investment Bank explores what happens when you take the attention mechanisms powering modern AI and apply them to Wall Street’s most fundamental pricing problems, tackling what might be quantitative finance’s most intractable challenge.

The problem is elegantly simple yet profound: machine learning models are great at finding patterns in historical data, but financial theory demands that arbitrage-free prices be independent of past information. As the authors put it:

We contend that a fundamental tension exists between the usage of ML methodologies in risk and pricing and the First Fundamental Theorem of Finance (FFTF). While ML models rely on historical data to identify recurring patterns, the FFTF posits that arbitrage-free market prices are independent of past information.

Their solution? Transition Probability Tensors (TPTs) that function like attention mechanisms in neural networks, dynamically weighting relationships between risk factors while maintaining mathematical rigor. Instead of learning from history, these tensors capture “dynamic, context-aware relationships across dimensions” in real-time.

The practical results are impressive: simulating 210 quantitative investment strategies across 100,000 market scenarios in just 70 seconds, while identifying optimal hedging strategies and stress-testing future market conditions. The framework even adapts to different volatility regimes, shifting focus toward tail events during high-volatility periods—exactly like attention mechanisms focusing on relevant context. Whether it scales beyond this impressive proof-of-concept remains to be seen, but it’s seems to be a genuine attempt to resolve the fundamental tension between AI’s pattern-seeking nature and finance’s requirement for arbitrage-free pricing.

Quantitative Finance on Philipp D. Dubach | Quantitative Finance & AI Strategy

The Anatomy of a Decentralized Prediction Market: Notes from the Polymarket Order Book

Why bother

Two contributions

The data

Eight stylized facts

SF1 — Longshot spread premium

SF2 — Depth concentration

SF4 — Maker-wallet diversity

SF7 — Self-counterparty wash share

SF8 — Depth decay near resolution

The limits of orderbook-only inference

What I’m not doing here

Replication

The Moral Philosophy of Investing in Ignorance

Three sources of profit

The sidecar problem, revisited

Blame, accountability, and the collective action problem

Bet Sizing at the Frontier

What Kelly actually says

What Samuelson actually objected to

Why the debate is beside the point in UU

Diversification versus concentration

The Geometry of Who Knows What

The two matrices

Institutional blindness as structural opportunity

The advantage-versus-selection formula

Ambiguity by Design

The experimental evidence

Is ambiguity aversion rational?

Discomfort as information

Three Kinds of Not-Knowing

The taxonomy

Why finance forgot the third box

Why Knightian uncertainty is growing

Long Volatility Premium

I. Separating beta from convexity

II. Why would a long volatility premium exist?

III. What the research actually says

IV. Puts vs. trend: the tortoise and the hare

Where most tail hedges fail, and the benchmark problem

What I take away

Variance Tax

Is Private Equity Just Beta With a Lockup?

Against All Odds: The Mathematics of 'Provably Fair' Casino Games

It Just Ain’t So

Beyond Monte Carlo: Tensor-Based Market Modeling