Determining overall risk

Abstract

Abstract The risk journal literature lacks a clear and simple account of the conceptual issues involved in determining the overall risk of an action, and in explaining how risk is additive. This article attempts to bring a measure of clarity to these issues in as basic and non‐technical a way as possible. First of all, the view that risk is 'expected harm' is explained. The view that risk is a quantitative concept is then defended. The distinction between the risk run by doing action A in respect of possible outcome x, and the overall risk run by doing action A in general is explained, as is the position that the overall risk of A is determined by summing the risks of each possible harm that A could give rise to. The article then explains how risks can be summed over time, as long as the probabilities involved are determined according to probability theory. Finally, the article explains that in a doing a risk‐benefit analysis of A, positive aspects of a possible outcome x, where x is harmful on balance, must be incorporated into x's level of harm rather than incorporated into the benefit side of the risk‐benefit analysis of A. Keywords: riskrisk analysisprobabilityharmexpected utilityrisk‐benefit analysis Acknowledgements I wish to thank the Leverhulme Trust for funding my research position at the Institute for the Study of Genetics, Biorisks and Society at the University of Nottingham, during which time this paper was written. Thanks also to Nicholas Shackel and Andrew Woodfield for helpful comments and suggestions. Notes 1. For a similar view, see Rescher Citation1983, p. 6. 2. In "What is the Definition of Risk?", (unpublished). 3. The terms 'good' and 'bad' are deliberately used here in a very broad sense – 'good', for example, doesn't just refer to moral goodness, but any sort of goodness. There will, of course, be disagreements about what things count as good and bad, but such disagreements are not relevant here. 4. This is not to claim that we will always be right about whether x is, on balance, good or bad or neutral. Nor is it to say that we always can determine whether x is good or bad – it may be that in some cases we have no idea of whether x is good, bad or neutral. 5. I do not assume here that x's probability can always be determined. Not only may we be unable to assign a precise quantity to the probability of x, we may even be unable in some case to say anything more general about x's probability, such as whether it is high or low. 6. There is a complication that arises here. Expected utility theory usually is understood in economics and decision theory to involve subjective probabilities – understood as degrees of belief – rather than objective probabilities (see, for example, Bengt Hansson Citation1975, p. 175). But risk analysis traditionally involves, as Sven Ove Hansson says, objective probabilities (1993, p. 17). I shall take 'expected harm' in a more neutral sense and allow it to include either subjective and/or objective probabilities. (In actual fact, an increasing number of risk analysts are incorporating subjective probabilities – or at least subjective probability methods, such as Bayesianism – into risk analysis. See, for example, Kaplan Citation1988; Bier and Mosleh Citation1988; Martz and Waller Citation1988; Apostolakis Citation1990; Siu and Kelly Citation1998.). 7. Strictly speaking, even crossing the bridge could cause negative outcomes other than falling into the ravine. One could slip and break one's legs. One could get a bad splinter from the handrail. In practise we usually ignore these possibilities because the risk level for them is so low. 8. See Tversky Citation1967 for some empirical work on this matter. 10. The odds of x happening without y will not be the same as they were for x in Table 1, because every 25 days x occurs on average five times, but only four of these don't involve y (and 4/25 = 0.16). 9. For those who are new to probability, the probability of events x and y happening together is expressed more formally as p(x & y). If x and y are independent – that is, if the occurrence of either does not make it more likely that the other happens (or has happened) – then p(x & y) = p(x)×p(y). That is, the probability of x and y together equals the product of the probability of each. If x and y are dependent, though – that is, if x makes y more likely and vice versa – then p(x & y) = p(x) × p(y|x) (or p(y) ×p(x|y)). That is, the probability of x and y together equals the probability of x multiplied by the probability of y given that x has occurred (read '|' as 'given that'). That will differ from p(x)×p(y), because p(y|x) by definition is different in dependent cases from p(y). For example, suppose p(x) is 0.06, and p(y) is 0.2. If x and y are independent then p(x & y) = 0.012. But suppose that if x occurs then y is twice as likely to occur as otherwise would be the case. The probability of x and y in this dependent case is therefore given by p(x)×p(y|x), which in this case is 0.06×0.4 = 0.024. 11. So there is a difference between the probability, before you start, of getting to the third year, and the probability of getting to the third year given that you have made it to the second year. In the latter case, you have new information (namely, that you survived the first roll) which is not available in the former case, and the original probabilities must be adjusted to accommodate this new evidence. 12. This can be formalized as p(e1 v e2 v e3), where 'v' stands for 'or'. 13. Of course, it may be that you have other information about the firm that tells you that Fred's chances should not be revised downwards, or perhaps you have reason to think that Fred's persistence eventually will be recognized by the firm as an admirable quality in itself, which will eventually gain him the job. But these are further ways in which Fred's situation is not analogous to the dice case. 14. For a recent general overview of some of the issues involved in the wighing of risks against benefits, see Hansson (Citation2004) 15. I presume here that there is no chance that the broken leg will involve complications not covered for by the compensation. 16. It might be said that I should not talk of the risk of x when I have tabled not the risk of x, but the risk (or benefit) of x without z, and x with z. But even so, on this representation there is less risk in doing A in Situation 3 than in Situation 1. 17. It should be noted that my analysis does not tell us anything about whether we should be risk‐seeking or risk averse. This is because my view concerns the meaning of the term 'risk'. It is not a theory of rationality. A theory of rationality should of course be informed by a correct view of what 'risk' means, but the two are nonetheless distinct. For this reason the complaint of Kaplan and Garrick (Citation1981, pp. 13–14) that the 'probability times harm' view of risk does not allow us to distinguish between low harm‐high probability and low probability‐high harm events is misguided. The fact that the 'probability times harm' view may assign the same risk level to these two types of event does not mean that we cannot distinguish between them on the basis that one is low harm‐high probability and the other is the opposite. Whether we should do so depends on what the correct theory of rationality says.