Here is a very insightful and worth studying set of plots showing the perceived range of probabilities under subjective measure scenarios. Source: https://github.com/zonination/perceptions
The charts above speak volumes about both, our (human) behavioural biases in assessing probabilities of events and the nature of subjective distributions.
First on the former. As our students (in all of my courses, from Introductory Statistics, to Business Economics, to advanced courses of Behavioural Finance and Economics, Investment Analysis and Risk & Resilience) would have learned (to a varying degree of insight and complexity), the world of Rational expectations relies (amongst other assumptions) on the assumption that we, as decision-makers, are capable of perfectly assessing true probabilities of uncertain outcomes. And as we all have learned in these classes, we are not capable of doing this, in part due to informational asymmetries, in part due to behavioural biases and so on.
The charts above clearly show this. There is a general trend in people assigning increasingly lower probabilities to less likely events, and increasingly larger probabilities to more likely ones. So far, good news for rationality. The range (spread) of assignments also becomes narrower as we move to the tails (lower and higher probabilities assigned), so the degree of confidence in assessment increases. Which is also good news for rationality.
But at that, evidence of rationality falls.
Firstly, note the S-shaped nature of distributions from higher assigned probabilities to lower. Clearly, our perceptions of probability are non-linear, with decline in the rate of likelihoods assignments being steeper in the middle of perceptions of probabilities than in the extremes. This is inconsistent with rationality, which implies linear trend.
Secondly, there is a notable kick-back in the Assigned Probability distribution for Highly Unlikely and Chances Are Slight types of perceptions. This can be due to ambiguity in wording of these perceptions (order can be viewed differently, with Highly Unlikely being precedent to Almost No Chance ordering and Chances Are Slight being precedent to Highly Unlikely. Still, there is a lot of oscillations in other ordering pairs (e.g. Unlikely —> Probably Not —> Little Chance; and We Believe —> Probably. This also consistent with ambiguity - which is a violation of rationality.
Thirdly, not a single distribution of assigned probabilities by perception follows a bell-shaped ‘normal’ curve. Not for a single category of perceptions. All distributions are skewed, almost all have extreme value ‘bubbles’, majority have multiple local modes etc. This is yet another piece of evidence against rational expectations.
There are severe outliers in all perceptions categories. Some (e.g. in the case of ‘Probably Not’ category appear to be largely due to errors that can be induced by ambiguous ranking of the category or due to judgement errors. Others, e.g. in the case of “We Doubt” category appear to be systemic and influential. Dispersion of assignments seems to be following the ambiguity pattern, with higher ambiguity (tails) categories inducing greater dispersion. But, interestingly, there also appears to be stronger ambiguity in the lower range of perceptions (from “We Doubt” to “Highly Unlikely”) than in the upper range. This can be ‘natural’ or ‘rational’ if we think that less likely event signifier is more ambiguous. But the same holds for more likely events too (see range from “We Believe” to “Likely” and “Highly Likely”).
There are many more points worth discussing in the context of this exercise. But on the net, the data suggests that the rational expectations view of our ability to assess true probabilities of uncertain outcomes is faulty not only at the level of the tail events that are patently identifiable as ‘unlikely’, but also in the range of tail events that should be ‘nearly certain’. In other words, ambiguity is tangible in our decision making.
Note: it is also worth noting that the above evidence suggests that we tend to treat inversely certainty (tails) and uncertainty (centre of perceptions and assignment choices) to what can be expected under rational expectations:
