# Formal Matters on the Topic of Risk Mitigation: A Mathematical Perspective
## 3 Formalizing Risk Mitigation
### Cost Functions  
In practice, risk mitigation has a cost: levees must be raised, ditches must be dug, buildings must be reinforced, and infrastructure needs to be changed to accommodate for the redistribution of population, etc.  
Mathematically, this can be taken into account by associating a _cost function_\(\kappa\) to each pair of configurations \(C_{0},C_{1}\), with \(\kappa(C_{0},C_{1})\geq 0\) modeling the cost of migration from configuration \(C_{0}\) to configuration \(C_{1}\).  
In risk mitigation, it is therefore in general appropriate to look for new configurations such that the cost of migration from the previous to the new configuration is less than the difference in risk, i.e., for configurations in the set  
\[\mathcal{C}_{<0}^{\kappa}=\left\{C\in\mathcal{C}:R(\Omega,C)<R(\Omega,C_{0})+ \kappa(C_{0},C)\right\}\]  
for some cost function \(\kappa\). This is a way to express mathematically the idea that the cost of reducing risk should not be higher than the value saved by reducing the risk.  
In this case, one could consider the optimality of a lower risk configuration \(C\in\mathcal{C}_{<0}^{\kappa}\) for example as the \"most bang for the buck\", i.e., a configuration that minimizes both \(R(\Omega,\cdot)\) and \(\kappa(C_{0},\cdot)\). This translates to a multi-objective optimization problem and the study of Pareto-optimal configurations [8; 17].  
The previous formulation is formally complete if the cost of migration from \(C_{0}\) to \(C\) is unique. This, however, is not the case in general: for example, the same change in single-building resilience (from \(Q_{0}\) to \(Q\)) may be achieved with different engineering efforts, each with a different cost.  
One possible approach to simplify this is to consider as \(\kappa(C_{0},C)\) the _minimum_ cost necessary to enact the change in configuration (or at least the infimum of the costs, if the minimum does not exist). This is sufficient to make the cost function unique but may result in unrealistic results in the estimation of \(C_{<0}^{\kappa}\), since, in practice, the costs will have a probability of being higher than the estimated lower bound \(\kappa\), possibly resulting in a risk mitigation process that is in practice more expensive than the expected savings in value loss.  
A more complete way to approach the multiplicity of the cost function for each pair of configurations is to take inspiration from homotopies. Mathematically, we define a _transition_ from configuration \(C_{0}\) to configuration \(C_{1}\) as a function \(M_{C_{0},C_{1}}:[0,1]\to\mathcal{C}\) such that \(M_{C_{0},C_{1}}(0)=C_{0}\) and \(M_{C_{0},C_{1}}(1)=C_{1}\). The mitigation cost is then associated not with the _endpoints_ of the transition \(C_{0},C_{1}\), but with the specific transition, i.e., not \(\kappa(C_{0},C_{1})\), but \(\kappa(M_{C_{0},C_{1}})\).  
If we indicate by \(\mathcal{M}(C_{0},C_{1})\) the set of possible transitions from \(C_{0}\) to \(C_{1}\), risk mitigation as an optimization problem translates then to the problem of finding \(C\in\mathcal{C}\) such that there exists \(M_{C_{0},C}\in\mathcal{M}(C_{0},C)\) such that \(R(\Omega,C)<R(\Omega,C_{0})+\kappa(M_{C_{0},C})\). Of course, while this description is more accurate and complete, it significantly increases the search space of the problem.