Limited attention can make rare events seem more likely over time

In a recent study published in the journal Review of Finance, researchers developed a theoretical model showing how limited attention can distort what people learn from experience.

The model suggests that unusually positive or negative events can receive disproportionate mental weight. As a result, rare outcomes may appear more likely than they really are, even after a person has observed a large amount of new information.

Why memorable events shape expectations

People often form expectations by drawing on previous experiences. However, behavioral research suggests that vivid, unusual, or emotionally striking events are easier to recall than ordinary ones.

A market crash, for example, may have a greater influence on future expectations than many uneventful trading days. Similarly, a sudden financial gain may encourage an overly optimistic view of future outcomes.

Traditional economic models generally assume that people give appropriate weight to all available information. The new framework instead examines what happens when attention is uneven and recent extremes stand out more than routine experiences.

How the model works

The researchers considered a model agent that observes a continuous sequence of outcomes and gradually forms beliefs about the likelihood of different outcomes.

Each new outcome is compared with a moving window of recent observations. The model ranks the outcome within that group and assigns it an attention weight based on its rank.

In the authors’ main examples, the largest and smallest outcomes receive more weight than observations near the middle. The broader framework can also represent people who ignore extremes, focus mainly on typical outcomes, or pay more attention to positive than negative experiences.

The researchers then derived mathematical expressions showing what beliefs the agent would eventually form and how quickly those beliefs would approach their long-term pattern.

A numerical example used a window of 10 observations and a sequence of 1,000 outcomes. The largest observation received the greatest weight, the smallest received the second-greatest weight, and the remaining observations received equal but lower weights.

Bias can persist despite repeated learning

Under ordinary equal-weight learning, beliefs should converge on the true distribution as more evidence accumulates. In the new model, this does not necessarily happen.

When remarkable outcomes repeatedly receive extra attention, the agent eventually learns a distorted version of the underlying distribution. The bias, therefore, persists even with an extremely large number of observations.

When both unusually high and unusually low outcomes are emphasized, the model produces an inverse-S-shaped distortion. This makes events in the tails of the distribution appear more likely and gives less weight to moderate outcomes.

The opposite pattern arises when the agent focuses mainly on representative or middle-ranked observations. In that case, the tails are underweighted, and the resulting distortion has an S-shape.

The model can also generate optimistic or pessimistic beliefs. Giving more attention to large outcomes shifts perceived results upward, while emphasizing small outcomes shifts them downward.

Why the size of the memory window matters

The researchers also examined how the number of recent observations used for comparison affects learning.

With a short window, ranks are relatively noisy. An otherwise ordinary outcome can appear unusually high or low simply because it is being compared with only a few recent observations.

With a longer window, an outcome’s rank more closely reflects its position in the true distribution. Under the weighting pattern examined in the paper, this led to more pronounced probability distortions.

The estimates became more precise as the total number of observations increased. However, learning remained less precise in parts of the distribution where small changes in objective probabilities produced large changes in perceived probabilities.

Overreaction and underreaction can coexist

The model also helps explain why people may overreact to some information but underreact to other information.

A highly unusual new observation carries greater weight than it would under conventional learning, leading to a stronger change in beliefs. A more ordinary observation receives less weight and produces a weaker response.

This means that the same learning process can generate overreaction to vivid events and underreaction to routine evidence. The result is consistent with wider psychological research showing that memorable information can exert more influence than statistically representative information.

Possible implications for financial behavior

In a simplified version of the model, emphasizing unusually high outcomes raised perceived average returns, while emphasizing unusually low outcomes reduced them. The direction and size of the bias also depended on whether the underlying distribution was positively or negatively skewed.

Under some conditions, distorted learning also reduced perceived differences between investments with high and low Sharpe ratios. The framework, therefore, offers a possible explanation for why investors may pursue positively skewed returns without necessarily having an inherent preference for risk or skewness.

Conclusion

The study presents a mathematical explanation of how limited attention can lead to lasting errors in probability judgments. When standout experiences repeatedly carry more weight in the mind than ordinary ones, accumulating evidence may reinforce distorted beliefs rather than correct them.

The framework links attention and memory to several well-known behavioral patterns, including overestimating rare events, holding optimistic or pessimistic expectations, and coexisting overreaction and underreaction.

However, the paper is theoretical and does not test the model in human participants or real investors. Its main results also assume that observations are independent and generally drawn from a continuous distribution.

Further empirical research is needed to determine how closely the proposed mechanism reflects real-world learning and decision-making.