risking it all
A. dependence
Right, so you’re building a bridge across a river. You’ve been asked by the government to build it and they’ve given you $1000.
With such a tiny budget you need to keep the cost down and the bridge up and you find that your local builders are the cheapest. They quote $1000 for the bridge and say it will take a day to build1. You say “wow what a great deal” and the next day you’re $1000 lighter and a bridge heavier.
But you live in Japan. Every day there is a 50% chance of a monsoon. A monsoon would damage the bridge and these damages would cost $22 to repair. However, every day there is also a 1% chance of an earthquake. The builders weren’t following building standards particularly closely2, so if there is an earthquake the bridge will collapse.
These monsoon happenings and the earthquake happenings are independent of each other.
You can either eliminate the risk of the monsoon or the risk of the earthquake3 with the resources you have. But you can’t do both. You need to estimate the cost of each of these risks and devote your money to tackling one of them.
The expected daily cost of the monsoon is
\[\begin{align} & = P_\text{monsoon} c_\text{monsoon} \\ & = 0.5 \times $ 22 \\ & = $ 11 \end{align}\]
and the expected daily cost of the earthquake is
\[\begin{align} & = P_\text{earthquake} c_\text{earthquake} \\ & = 0.01 \times $ 1000 \\ & = $ 10 \end{align}\]
where \(c_\text{earthquake}\) is assumed to be the cost of rebuilding the bridge only. We’ve done the maths, so we get a cookie, and we choose to mitigate the monsoon.
B. independent strut
The difficulty isn’t a trick within the problem, it is that even though the events are independent the risks are dependent. The probability of the earthquake happening does not affect the probability of the monsoon happening. However, if the earthquake happens and the bridge collapses then there is no more bridge. Although the typhoon could happen the very next day there will be no bridge for it to damage.
The expected daily cost of the monsoon is actually
\[\begin{align} & = P_\text{monsoon}(\bar{P}_\text{earthquake}^{(d-1)}) c_\text{monsoon}\\ & = 0.5 \times 0.99^{(d-1)} \times $22 \\ & = $ 11 \times 0.99^{(d-1)} \end{align}\]
where \(d\) is the number of days into the future we’re looking and \(\bar{P}\) is the probability of an event not happening.
At the start of the period the expected cost of the monsoon is really similar to what we previously calculated. The expected cost decreases as a function of time - as the risk that an earthquake has occurred increases. By the eleventh day the expected cost of the monsoon is less than the cost of the earthquake and it keeps getting smaller in comparison.
What did we learn from this? Well we learnt to not trust local builders for a start, buy Carillion! We also learnt that events can be independent but there can be dependence between their risks. As we know the probabilites aren’t dependent, this dependence is actually carried in the “impact” term - \(c_\text{monsoon} \sim f(P_\text{earthquake})\).
We learnt that if we don’t consider this dependence we make the wrong decision. These interdependencies get complex real quick. Even large organisations, like Carillion and other bridge builders, can’t adequately model these dependencies.
C. it gets bigger
We’ve been so down in the bridge struts that we’ve neglected other risks. The pandemic comes as a disease that is almost invisible, highly contagious, and sterilises the host. “How could we have planned for this? How could we have known?”
Now, the problem is that this pandemic affects the value function of our bridge. When the generation turns over like tilled soil, and the last person on Earth breathes their last, the bridge is almost useless but for the deer and the foxes.
The government allocated these funds. They had a few pots of £1000 for bridges. They had a bit of money left over they could allocate to risk mitigations. They analyzed all the best locations and chose contractors to build and maintain the bridges. “These are valuable bridges,” they said as they did it by the book.
Looking at each of the bridges they were able to allocate risk mitigation funds. £100 per day here to stop earthquakes from damaging a bridge. £20 per day there to stop beavers stealing the wood. The problem is that they didn’t look wholistically across the suite of risks they are faced with. Where there were clearly dependent risks (where the dependence is carried in the probabilities like earthquakes increasing the risk of tsunamis) the builders are smart enough to account for dependence. But where we are looking at an alteration of the impact term of a risk event then the contingency can become difficult to model. For example, what would the bridge be worth if a pandemic hits and everyone works from home? How about if the bridge is right next to a hospital? How does this affect the impact of an earthquake destroying the bridge?
A pandemic doesn’t occur on any bridge’s risk register. Yet, if it occurs it can seriously affect the value of the bridge - perhaps even eliminating its value. There is a systematic tendency to overvalue localised risk against global risk. We then mitigate really poorly against these risks, as we do not distribute appropriate resources.