Sunday, 25 April 2010


Humans are notoriously bad at judging risks. I’m talking about the decision making process behind actions. We all make hundreds of decisions every day. When do we decide to cross the road? What do we decide to put in our sandwich for lunch? What religion do we choose to follow? Not all these sound like risks, but all decisions to take risks are nonetheless decisions and follow the same decision making process as any other decision. We don’t consider a decision to be a risk when the cost of making the wrong decision isn’t very great. That’s the only distinction between a conventional decision and a risk.

The decision making process is actually pretty straightforward, as I shall demonstrate with my Risk MatrixTM later. However, there is a substantial amount of judgment to be done, and this is where our human nature allows us to screw up and make the wrong decisions and put ourselves at risk.

Every decision starts with a question: Should I do action X?

Action X might be to cross the road, for instance. Then there are two factors to consider: probability and cost/benefit, both of which can be either positive or negative. These are shown in the Risk MatrixTM below.

In our road crossing example, there might be few cars on the road so the probability of successfully crossing the road (good outcome; P) will be rather high compared to the probability of being involved in an accident (bad outcome; p). Let’s say you decide that you have a 99% chance of successfully crossing the road. So P = 0.99 and p = 0.01.

Next we need to do some cost-be

nefit analysis. We can assume that there are more benefits of reaching the other pavement than

there are costs. It could be that the supermarket is on the

other side of the street. The benefit is that we can get some bread and milk before going home. The cost is that it’ll take us longer to get home since we’ll have to do the shopping and cross back over the road. But we decide that overall the benefits are greater than the costs, so we assign a net benefit of +1.

On the other hand, there are considerable costs associated with unsuccessfully crossing the road and ending up in an accident. These include injury resulting in time spent at the hospital, preventing you from doing things you would need or want to do. There could be some benefits, such as not having to go to school and hand in your unfinished homework, but overall the costs will outweigh the benefits. Since there is some uncertainty in exactly how bad the accident could be, we’ll assign it an average cost of -100 (it would be worth about 100 shopping trips).

Now in order to determine whether to perform the action of crossing the road, we multiply the cost/benefit by the probability:

(P x b) + (p x c)

If the result is a positive number, then the action is worth making. If the result is negative, then the action should not be taken; it is too risky!

In our example,

(P x b) + (p x c)

= (0.99 x 1) + (0.01 x -100)

= 0.99 - 1

= -0.01

The result is negative and we conclude that in these conditions crossing the road would not be a smart move. Perhaps wait until some cars have gone past and re-evaluate.

However, remember at the start I said that humans are notoriously bad at judging risks? Well this hasn’t changed; the issue is in the values we assign to each part of the equation. If we were drunk we might misjudge the likelihood of succeeding; the road might be busy and we ignore a high probability of collision. Maybe we’re young and innocent and don’t realise the implications of being involved in an accident and misjudge the net cost of a bad outcome.

Sometimes we’re blinded by one impressive value in the Risk MatrixTM and do not notice the importance of the other values. Here are two examples.

The first is that of the gambler. The question is should he put all his money on red? The benefit of winning is obviously massive. He could walk away with thousands of pounds. So excited is he by this prospect that he ignores the underwhelming probability of him actually succeeding. Coupled with a high cost to failure, he shouldn’t make the gamble, but it can be easy to be blinded by the spectacular benefit of success.

The other example is that of the agnostic. He comes across a religious group that tells him that their god punishes non-belief by eternal suffering in the afterlife. The agnostic may at first question the existence of this god, but the religious group tells him that even if he’s not sure whether or not that their god exists, at least he can avoid eternal suffering by becoming a follower. The question is should I ignore this god? The cost of a bad outcome (that after all this god does exist) is so great that he might ignore to consider the probability of the god existing, which could turn out to be so slight as to mitigate the impressive costs.

If there were to be a moral to this story, it would be to consider all corners of the Risk MatrixTM when making an important life-changing decision. Always make sure you have a good understanding of both the probabilities and costs/benefits involved.


  1. Your maths is fine. In fact, it's very easy to generalise your "risk matrix" to lotteries with multiple outcomes, rather than just two. You can also be a little more delicate than just "positive is good, negative is bad," too - with the way your risk matrix is set up, it's the case that *higher* is good and lower is bad. This allows you to compare different actions - do I cross road A or road B to get to the shop, perhaps. Both might give negative scores (dying is bad), but you still need to cross one of them or you'll starve..

    More on the mathematical side of things here: Warning: rather technical, but the St. Petersburg paradox is fun. Of course, as you go on to say in the post, all of this theory assumes *rational* behaviour in making choices, which you don't always get..

    (You write science very well indeed, btw. Your style is most compelling.)

  2. Thanks for the kind words. Comments like these are really inspiring, actually :)