Theoretical models suggest that so much importance is placed on memorable highs and lows that people may end up overestimating rare events, even as new evidence accumulates.
In a recent study published in the journal Financial reviewresearchers have developed a theoretical model that shows how limited attention spans distort what people learn from experiences.
This model suggests that unusually positive or negative events may receive disproportionate psychological weight. As a result, rare outcomes can appear more likely than they actually are, even after a person observes large amounts of new information.
Why memorable events shape expectations
People often form expectations based on past experiences. However, behavioral research shows that vivid, unusual, or emotionally striking events are easier to remember than ordinary events.
For example, a market crash can have a greater impact on future expectations than an uneventful trading day. Similarly, sudden financial gains can encourage overly optimistic views about future outcomes.
Traditional economic models generally assume that people give appropriate weight to all available information. Instead, the new framework investigates what happens when attention is uneven and recent extreme events are more salient than everyday experiences.
How the model works
The researchers considered a model agent that continuously observes a series of outcomes and gradually forms beliefs about the likelihood of different outcomes.
Each new result is compared to a moving window of recent observations. The model ranks the results within that group and assigns attention weights based on that rank.
In the author’s main example, the largest and smallest results are weighted more heavily than observations near the middle. A broader framework may also represent people who ignore extremes, focus primarily on typical outcomes, or pay more attention to positive experiences than negative ones.
The researchers then derived mathematical equations that show what beliefs the agent ultimately forms and how quickly those beliefs approximate long-term patterns.
The numerical example used a window of 10 observations and a sequence of 1,000 results. The largest observation is given the highest weight, the smallest is given the second highest weight, and the remaining observations are given equal but lower weights.
Bias may persist even after repeated learning
In normal isoweight learning, beliefs should converge to the true distribution as more evidence accumulates. With newer models, this doesn’t necessarily happen.
As salient outcomes repeatedly receive special attention, the agent eventually learns a distorted version of the underlying distribution. Therefore, even if there are a large number of observations, the bias remains.
When both abnormally high and abnormally low results are emphasized, the model produces an inverted S-shaped distortion. This makes events at the tail of the distribution more likely and gives less weight to intermediate outcomes.
The opposite pattern occurs when agents primarily focus on representative or intermediate observations. In this case, the tail is underweighted, resulting in an S-shaped distortion.
This model can also generate optimistic or pessimistic beliefs. Paying more attention to big wins moves perceived results upwards, and emphasizing small wins moves perceived results downwards.
Why memory window size matters
The researchers also investigated how the number of recent observations used for comparison affected learning.
The ranks are relatively noisy due to the short window. An otherwise normal result may appear unusually high or low simply because it is compared to a few recent observations.
The longer the window, the more accurately the resulting rank reflects the position in the true distribution. For the weighting patterns considered in the paper, this resulted in even more pronounced probability distortions.
As the total number of observations increased, the estimates became more accurate. However, learning accuracy remained low in parts of the distribution where small changes in objective probability caused large changes in perceived probability.
Overreaction and underreaction can coexist
This model also helps explain why people overreact to some information but underreact to other information.
Very rare new observations carry more weight than traditional learning and lead to stronger changes in beliefs. More common observations receive less weight and produce weaker responses.
This means that the same learning process can produce overreactions to vivid events and underreactions to mundane evidence. This result is consistent with broader psychological research showing that memorable information can have more influence than statistically representative information.
Potential impact on financial behavior
In the simplified version of the model, emphasizing abnormally high outcomes increased perceived average returns, whereas emphasizing abnormally low outcomes decreased average returns. The direction and magnitude of the bias also depend on whether the underlying distribution is positively or negatively skewed.
Under some conditions, distorted learning also reduces the perceived difference between investments with high and low Sharpe ratios. This framework thus provides a possible explanation for why investors pursue positively biased returns without necessarily inherently prioritizing risk or distortion.
conclusion
This study provides a mathematical explanation of how limited attentional capacity can lead to persistent errors in probability judgments. When salient experiences repeatedly carry more weight in the mind than ordinary experiences, the accumulation of evidence can strengthen rather than correct distorted beliefs.
This framework links attention and memory to several well-known behavioral patterns, such as overestimation of rare events, holding optimistic or pessimistic expectations, and coexistence of overreactions and underreactions.
However, this paper is theoretical and did not test the model with human participants or real investors. Also, its main result assumes that the observations are independent and generally come from a continuous distribution.
Further empirical research is needed to determine the extent to which the proposed mechanisms reflect real-world learning and decision-making.
