# Formal Matters on the Topic of Risk Mitigation: A Mathematical Perspective
## 1 Introduction
### Risk Management in Formulas  
The current commonly (at least in the field of natural hazards) accepted informal expression of risk as the product of hazard, exposure (or value), and vulnerability is due to [2]:  
\[\text{Risk}=\text{Hazard}\times\text{Exposure}\times\text{Vulnerability} \tag{1}\]  
where Hazard indicates the probability of occurrence of the event (e.g., lava flow inundation, earthquake, etc. ), Exposure is a quantification of (the value of) the people, systems, and property potentially subject to the hazardous phenomenon (in fact, Ref. [2] explicitly uses the term Value rather than Exposure), and Vulnerability is a quantification of the effective relative impact of the event, expressed as a percentage, with \(V=1\) indicating total loss (100%) and \(V=0\) indicating total resilience. Contextually, the author also presents some key elements for risk management (in the context of volcanic hazard, but of general applicability), such as land-use planning (\"zoning\") to reduce exposure, and preparedness (including monitoring, early warning systems and response planning) to reduce vulnerability. We note that Equation (1) is essentially _qualitative_ in nature, rather than quantitative, as pointed out e.g., by [3]. Other functional relationships are available in the literature, especially in engineering contexts. For example, Ref. [4], and more recently [5], prefer an even more informal  
\[\text{Risk}=\text{Uncertainty}+\text{Damage}. \tag{2}\]When relating risk to hazard, Ref. [4] provides a different, but still informal relationship:  
\[\text{Risk}=\frac{\text{Hazard}}{\text{Safeguards}} \tag{3}\]  
that provides an explicit indication of the possibility of intervention to reduce risk. This is in contrast to [2] whose formula (1) provides no explicit mention of the possible quantification of mitigation efforts, even though the paper presents several approaches to risk management. In the field of natural hazards, an extension of (1) that includes an explicit dependency on mitigation is given by [3]:  
\[\text{Risk}=\text{Hazard}\times\text{Exposure}\times(\text{Vulnerability}- \text{Risk mitigation efforts}), \tag{4}\]  
and more recently in [6] that provides a different formulation for mitigated risk as:  
\[\text{Risk}=\frac{\text{Hazard}\times\text{Exposure}\times\text{Vulnerability} }{\text{Mitigation measures}}. \tag{5}\]  
Equations (4) and (5) are also intended to be informal and qualitative rather than quantitative, although, like (1), they _can_ be used in a more quantitative sense (see also Section 2.4). For example, (5) could be used to compute an a posteriori value for the _efficiency_ of the given mitigation measures as \(M=R_{U}/R_{M}\), where \(M\) represents the efficiency of the mitigation measures, \(R_{U}\) the unmitigated risk, and \(R_{M}\) the mitigated risk. Arguably, (4) and (5) illustrate a philosophical difference in the approach to risk mitigation: while [3] focuses exclusively on reducing vulnerability, [6] applies mitigation to risk _as a whole_, and it is thus closer to the arguments brought forth in [2], which include the exposure-related land-use planning as a risk management feature. In relation to the other formulas seen so far, it may also be considered a more detailed version of (3), and a more general version of (4), even though the latter is not clear, due to the different choice of mathematical operators used to indicate the influence of mitigation measures. Considering their qualitative, informal nature, the functional difference between (4) and (5) is actually largely inessential. It is only when aiming at a more rigorous and quantitative assessment of risk and its mitigation that the specifics of the mathematical formulation become relevant--an aspect that so far has received more attention in engineering [4; 5; 7; 8] than in natural hazards [9; 10; 11]. In engineering, this quantifying effort is required to manage the multi-objective problem of minimizing both the risk and the costs associated with the mitigation [8].