Here's Leicester manager Nigel Pearson talking about tonight's game against Man City:

Do I expect any backlash or do I take encouragement from their recent results? I think a bit of both.

This is an example of how mistaken beliefs can cancel out. On the one hand, we have the hot hand fallacy (well, cold hand in this case); the idea that unusually bad performance persists*. On the other, there's the gambler's fallacy; the idea that a run of unusually bad outcomes increases the likelihood of a good one. In giving equal weight to both errors, Mr Pearson has come to a reasonable judgment.

This is an example of how cognitive biases can cancel out to produce an accurate opinion.

I suspect it's not an isolated instance. Bayesian conservatism - under-reaction - might sometimes offset over-reaction: behaviouralfinance.net has entries for both.

A new paper by Thomas Eisenbach and Martin Schmalz give us another example. Overconfidence, they say, might be used as a commitment device.

This is because many of us have time-inconsistent risk preferences: we don't worry about future risks until they are imminent, when we panic. For example, you might sign up for a charity parachute jump but then panic on the day. And actors and musicians feel stage fright just before they perform even though they chose to enter professions where they knew they'd have to go on stage. For retail investors, such preferences can be expensive. It can cause them to have heavy equity exposure in normal times, only to get cold feet when volatility increases, thus causing them to sell when prices are temporarily depressed.

Overconfidence, they say, can solve these problems. The investor who is overconfident about his abilities might think when shares fall "the market's being stupid; it'll come round to my way of thinking soon". This might be irrational overconfidence, but it saves him from the temptation to sell at the bottom. Similarly, the mediocre actor can overcome stage fright by telling himself that he's going to deliver a great performance.

I suspect that a lot of what we call rational behaviour is in fact the cancelling out of biases. This might occur within particular individuals, as in Mr Pearson's case. Or it might occur within groups. Maybe one reason why the stock market is (sometimes? often?) efficient isn't that all its participants are rational but rather that those who over-react offset those that under-react; experiments show that stupid traders can produce rational markets. There's (sometimes) wisdom in crowds and (often?) benefits to cognitive or ecological diversity because irrationalities can net out. This is why we often prefer committees to individual decision-making.

There's a political implication here. If we look only at individual cognitive biases, we'll be tempted to infer that stupidity is everywhere; after all, the list of such biases is a long one. This, though, is a mistake. We need the hard evidence of systematic error before we can infer that biases matter. And sometimes, this is lacking.

Take one example - savings. It's easy to infer from the ubiquity of hyperbolic discounting or akrasia that people are saving too little. If, however, we look at the wealth of those in or approaching retirement, things aren't so plain. And, indeed, it is possible to be overly prudent - to save too much and so deprive one's present self of happiness and one's future self of consumption capital and happy memories.

Perhaps, then, some of us overstate the amount of irrationality around us. Worse still, there might be unpleasant political motives for doing so.

* There's some doubt about whether the hot hand fallacy really is a fallacy. However, any Gooner seeing Olivier Giroud's performance against Monaco last week - when he missed a series of chances even though he went into the game in good "form" - will suspect it is indeed a fallacy. (This point might itself be an example of over-reaction).