Of course, we could be overreacting to the magnitude of the problem, as when we panic about a scenario that has a 1 in a trillion chance of occurring but that sounds disastrous if it did happen. It's not clear, though, what the "appropriate" level of concern should be for a disaster of a given magnitude and chance of happening. So the charge of irrationality is harder to level using this argument about a single event.
But we also get comparisons of risk between events wrong, for example when we fear traveling by airplane more than traveling by car, even though planes are safer. Here the case for irrationality is straightforward: for a given level of disaster (say, breaking your arm, dying, or whatever), we should panic more about the more probable ways that it can occur. The plane vs. car example makes us look irrational.
Still, there's another way we could measure how sensible our response is, only instead of comparing two sources of danger at the same point in time, comparing the same source of danger at different points in time. That is, for a given level of disaster, any change in the probability of it happening over time should cause us to adjust our level of concern accordingly. If dying in a plane crash becomes less and less likely over time, people should become less and less afraid of flying. When I looked into this before, I found that the NYT's coverage of murder and rape has become increasing out-of-touch with reality: while the crime statistics show the murder and rape rates falling after the early 1990s, the NYT devoted more and more of its articles to these crimes. So at least at the Newspaper of Record, they were responding irrationally to danger.
But what about the average American -- maybe the NYT responds in the opposite way that we might expect because when violent crime is high, people see and hear about plenty of awful things outside of the media, so that writing tons of articles about murder and rape wouldn't draw in lots more readers. In contrast, when the society becomes safer and safer, an article about murder or rape is suddenly shocking -- just when you thought things were safe! -- and so draws more readers, who start to doubt their declining concern about violence.
The General Social Survey asks people whether there's any area within a mile of their house where they're afraid to go out at night. Here is a plot of the percent of people who say that they are afraid to go out at night, along with the homicide rate for that year:
Clearly there is a tight fit between people's perception of danger and the reality underlying that fear. The Spearman rank correlation between the two within a given year is +0.74. That's assuming that people respond very quickly to changes in violence; it might be even higher because there appears to be somewhat of a lag between a change in the homicide rate and an appropriate change in the level of fear. For example, the homicide rate starts to decline steadily after a peak in 1991, but people's fear doesn't peak until two years later, when it too steadily declines. That makes sense: even if you read the crime statistics, those don't come out until two years later. To respond right away, you'd have to be involved in the collection and analysis of those data. The delay is more likely due to people hearing through word-of-mouth that things are getting better -- or not getting negative word-of-mouth reports -- and that it takes awhile for this information to spread throughout people's social network.
Putting all of the data together presents a mixed picture on how rational or irrational our response to the risk of danger is. But here is one solid piece that average people -- not those with an incentive to misrepresent reality, either in a more negative or more positive way -- do respond rationally to risk.
GSS variables used: fear, year
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