The standard case: decision making under risk

Let us compare two random experiments: fair coin flipping and the draw of a multidimensional random variable, with a possibly complicated joint probability density. The two cases may look quite different. The first one can be represented by a quite simple Bernoulli random variable, and calculating expected values of whatever function of the outcome is pretty straightforward. On the contrary, calculating an expectation in the second case may involve a possibly awkward multidimensional integral. However, the difference is more technical and computational than conceptual, as in both cases we assume that we have a full picture of uncertainty. The important point is that the risk we are facing is linked to the realization of a random variable, which is perfectly known from a probabilistic perspective. With reference to Fig. 14.1, we are pretty sure about:

1. The possible future scenarios, since we know exactly what might possibly happen, and there is no possibility unaccounted for.
2. The probabilities of these scenarios, whatever meaning we attach to them.

Hence, we have a full picture of the scenario tree. This standard case has been labeled as decision under risk, to draw the line between this and more radical views about uncertainty. Risk may be substantial; to see this, consider a one-shot decision when there is a very dangerous, but very unlikely scenario. Which decision model is appropriate to cope with such a case? There is no easy answer, but, at least, we have no uncertainty about our description of uncertainty itself.