The bias persists even after observing thousands of new events.
A new mathematical model published in the Review of Finance offers a formal account of something ancient in human experience: that we do not weigh all moments equally, and that this imbalance quietly reshapes our understanding of what is likely and what is not. Researchers show that when dramatic or extreme events command disproportionate attention, the resulting distortion of probability beliefs persists stubbornly across time — not as a correctable error, but as a structural feature of how selective minds learn. The finding places a rigorous framework around a familiar intuition: that a single vivid catastrophe can outweigh a thousand ordinary days, and that markets, like memories, are haunted by what stands out rather than what is true.
- A mathematical model now demonstrates that attention bias does not merely nudge probability estimates — it permanently warps them, even as new evidence accumulates by the thousands.
- The same mechanism that makes a market crash unforgettable renders a decade of steady gains nearly invisible, creating a mind that overreacts to the dramatic and underreacts to the routine simultaneously.
- Investors may be chasing positively skewed returns not from a conscious appetite for risk, but because selective memory has quietly convinced them that exceptional gains are more common than they are.
- The model's distortions grow more pronounced — not less — with longer memory windows, challenging the assumption that more experience naturally produces better-calibrated judgment.
- No human participants have yet been tested against the theory, leaving the framework as a compelling mathematical portrait of bias that still awaits its confrontation with actual minds.
A market crashes, and the memory lingers for years. A thousand quiet trading days follow, but they dissolve into background noise. Researchers have now built a mathematical model to show precisely how this asymmetry warps our sense of probability — and why even vast quantities of new evidence may fail to correct it.
Published in the Review of Finance, the study begins from a simple premise: not all experiences weigh equally in memory. Dramatic outcomes — a sudden windfall, a sharp plunge — dominate our mental landscape far longer than the gradual, unremarkable movements that surround them. Traditional economic theory assumes we absorb information evenly and update beliefs accordingly. This model asks what happens when attention is structurally uneven.
The researchers constructed a model agent observing a continuous stream of outcomes. Each new observation is ranked within a recent window of events, and that rank determines how much weight it receives in shaping future beliefs. When extreme outcomes — both high and low — receive the most attention, the agent learns a distorted reality: rare events appear more probable than they are, moderate outcomes less so. This inverse-S-shaped distortion persists even after thousands of new observations. Longer memory windows improve precision in some respects but amplify distortion in others.
The model also explains a paradox long noted by psychologists: people can overreact to vivid information and underreact to routine evidence at the same time. The same learning mechanism produces both responses, depending only on how memorable a given observation is.
For financial markets, the implications are pointed. Investors may pursue positively skewed returns not from an explicit preference for risk, but because selective attention has quietly convinced them that exceptional gains are more common than the data warrants. The researchers are careful to note that the model remains theoretical and untested with real participants. But it offers a rigorous account of why people overestimate rare events, hold stubbornly skewed expectations, and seem unable to learn their way out of distortions that experience alone should correct.
A market crashes. You remember it vividly for years. A thousand quiet trading days pass, but they blur together into background noise. This asymmetry—the way our minds cling to the dramatic and forget the ordinary—shapes how we understand the world in ways we rarely notice. Researchers have now built a mathematical model to show exactly how this selective attention warps our sense of probability, and why even mountains of new evidence may fail to correct the distortion.
The study, published in the Review of Finance, begins with a simple observation: people learn from experience, but not all experiences weigh equally in memory. A sudden financial windfall sticks with us more than a dozen days of modest, steady returns. A stock market plunge haunts our expectations far longer than the gradual climbs that bookend it. Traditional economic theory assumes we absorb all available information fairly and adjust our beliefs accordingly. The new framework asks a different question: what happens when attention is uneven, when extremes dominate the mental landscape?
To answer this, the researchers constructed a model agent observing a continuous stream of outcomes. Each new piece of data gets ranked within a recent window of observations—say, the last ten events. That rank then determines how much weight the outcome receives in shaping the agent's beliefs about future probability. In their main scenario, the largest and smallest observations get the most attention, while everything in the middle gets less. The framework is flexible enough to represent other patterns too: people who ignore extremes, those who focus on typical outcomes, or those who weight positive surprises more heavily than negative ones.
The critical finding emerges when you run the numbers forward. Under normal learning—where every observation counts equally—beliefs should converge on the true distribution as evidence accumulates. More data should correct errors. But in this model, that correction never fully arrives. When extreme outcomes repeatedly grab attention, the agent eventually learns a distorted version of reality, and that bias persists even after observing thousands of new events. If both unusually high and unusually low outcomes receive extra weight, the model produces what researchers call an inverse-S-shaped distortion: events at the extremes appear more probable than they are, while moderate outcomes seem less likely. The opposite happens when people focus mainly on middle-ranked observations—the tails get underweighted, and the distortion takes an S-shape.
The size of the memory window matters too. With a short window, an ordinary outcome can look extreme simply because it's being compared with only a handful of recent events. With a longer window, ranks more accurately reflect where an outcome truly sits in the distribution. Yet even with longer windows, the weighting pattern examined in the paper led to more pronounced distortions, not fewer. Precision improved as total observations increased, but learning remained imprecise in parts of the distribution where small changes in actual probability produced large changes in perceived probability.
Perhaps most intriguingly, the model explains why people can simultaneously overreact to some information and underreact to other information. A vivid, unusual observation carries disproportionate weight and triggers a strong shift in beliefs. An ordinary observation gets less weight and produces a weaker response. The same learning mechanism generates both overreaction to memorable events and underreaction to routine evidence—a pattern that matches what psychologists have long observed about how memorable information exerts more influence than statistically representative information.
For investors and financial markets, the implications are suggestive. When people emphasize unusually high returns, they perceive average returns as higher; when they emphasize low returns, they perceive them as lower. Under some conditions, distorted learning also reduces perceived differences between investments with high and low Sharpe ratios—a possible explanation for why investors pursue positively skewed returns without necessarily having an explicit preference for risk or skewness. They may simply be learning from a distorted version of their own experience.
The researchers are careful to note the limits of their work. The model is theoretical; it has not been tested with human participants or real investors. It assumes observations are independent and drawn from a continuous distribution. Further empirical research will be needed to determine whether this mechanism actually explains how people learn and decide in the real world. But the framework offers a mathematical explanation for several well-documented behavioral patterns: why people overestimate rare events, why they hold stubbornly optimistic or pessimistic expectations, and why they can overreact and underreact in the same breath. The next question is whether the theory holds up when it meets actual human minds.
Notable Quotes
When remarkable outcomes repeatedly receive extra attention, the agent eventually learns a distorted version of the underlying distribution— The research model's findings
The Hearth Conversation Another angle on the story
So the model is saying that vivid events don't just feel more important—they actually prevent us from learning the true odds, no matter how much time passes?
Exactly. If a market crash gets extra mental weight every time you recall it, and you keep comparing new events to that memory, the crash remains overweighted in your beliefs about future crashes. More data doesn't fix it; it just gets filtered through the same distorted lens.
But surely after seeing a thousand uneventful trading days, wouldn't the crash fade into proper perspective?
Not in this model. The mechanism itself—ranking each new outcome and weighting by rank—keeps extremes prominent. The crash doesn't fade; it stays in the comparison window, and its rank keeps it influential.
What about people who are naturally pessimistic versus optimistic? Does the model account for that?
It does. If you weight positive surprises more heavily, you learn an optimistic distortion. If you weight negative surprises more, you learn a pessimistic one. Same mechanism, different emotional emphasis.
And this could explain why investors chase skewed returns without consciously wanting to take on risk?
Yes. They're not necessarily risk-seeking. They're learning from their own experience, but their experience has been filtered through selective attention. They've noticed the big wins more than the steady days, so the distribution they've learned looks different from the actual one.
What would it take to test this in the real world?
You'd need to track how actual investors update their beliefs as they observe returns, and see if the distortion pattern matches the model's predictions. The researchers acknowledge that's the next step.