# A framework for thinking about risk

How long do you want to live? 80 years? 120 years? 200 years?  1000 years?

The longer you want to live, the more risk averse you should be, but many things in our normal lives involve a non-trivial amount of risk.

We can break our wanted life expectancy into a statistical life and compute the corresponding risk budgets. Below, I have computed it for a range of wanted life expectancies.

Total life Maximum activity risk daily risk budget
$80$y $1.5\cdot10^{-6}$ $3\cdot 10^{-5}$
$120$y $10^{-6}$ $2\cdot 10^{-5}$
$200$y $6\cdot10^{-7}$ $1\cdot 10^{-5}$
$1000$y $10^{-7}$ $3\cdot 10^{-6}$

For the maximum activity risk, we assume that each activity on average takes one hour. The daily risk budget is the maximum daily risk you can tolerate.

However, just being alive in the US counts $2.2\cdot 10^{-5}$ towards your daily risk budget which doesn't leave much for added risk, or much longer lives. Individual activities that are not counted in the average above, also adds to the risk budget. For example, scuba-diving carries a $10^{-5}$ risk of dying per dive. Travelling is also a risky behavior. For example, there's about $10^{-6}$ risk of dying from travelling $370$km by car, or $1600$km by jet, or $10000$km by train. This also exposes that humans are bad at risk assessment. Many people are scared of flying while they're okay with driving the same distance even though it's more than $4$ times as dangerous. It also shows that by far the safest way of travelling is by train.

However, most activities we regularly engange in carry only a relatively small risk of death, so for the framework to be useful, we need a way of comparing small terminal and larger non-terminal risks.

## How to deal with non-terminal risks?

Some risks are not as severe as other risks. For example, most people would agree that they would rather break an arm than completely lose the arm. And they would rather lose an arm than completely die.

One way of accouting for different risk-severitys is by weigting them. This way the total risk of an activity becomes the expectation, or the weighted sum of the individual risks

$R = \sum_i^N w_i p_i$

There are some limitations of this model however. For example, how many times do you want to break an arm before you would rather lose it? If you weigthed breaking your arm as $0.2$, and losing an arm as $1$, then you would be indifferent between breaking an arm $5$ times and losing it one time. Moreover, you'd rather completely lose your arm than you would break it $6$ times.

However, there's still a hiarchy of risk-severitys. Most people would agree that they would rather break an arm ($B$) than lose it $(L)$, and would rather lose the arm than die ($D$). There's a associative property playing in here that is since

$L$ is worse than $B$ and $D$ is worse than $L$ then $D$ is also worse than $B$, so we have a certain order $B,L,D$ from least severe to most severe. potentially, we would be able to place any risk-category somewhere in the relative order. We might also reason since there are many things worse than breaking a arm but less bad than death, dying is much worse than breaking an arm. Using this approach we can also get a distance metric which gives us a sense of degree of risk similar to the weighting from before but without running into the same problems. We do, however, lose some quantifibility since we are only using a heuristic distance function.

This becomes a problem if you have to choose between two situations one situation where there is a very low chance of a severe outcome happening, and another situation where there is a high probability of a low-severity outcome happening. Somewhat constructed you might imagine that you are in a burning building and have to jump out of the window. Not jumping will lead to certain death which is undesirable, but when you jump you have the choice of landing on your legs which comes with a high probability of breaking the leg, or you can land on your stomach which will distribute the force more evenly but comes with a low chance of banging your head into the ground killing you - which choice would you make?

Most people would probably gravitate towards breaking their legs over a low risk of dying. This would suggest that perhaps you would accept an arbitrarily high probability of a smaller-severity risk over an arbitraily small probability of a higher-severity risk. But what about the extreme case? Most people would probably rather have a 0% probability of dying over a 100% probability of breaking a leg.

Likewise, people would probably accept $0+\epsilon$ probability of dying over a $1-\epsilon$ probability of breaking a leg for sufficiently small $\epsilon > 0$. So where is the break? What's the maximum $\epsilon$? I don't know how to estimate maximum $\epsilon$. I would think that people implicitly estimate it in their daily lives, but aren't consistent. It would probably also depend on the distance between the risk categories.

Let us assume we have a risk model with $N$ risk factors, and we have two possible actions with associated risk severity profiles $A,B$ and their associated probabilities $p_A,p_B$. To make it simpler let's assume that $A and that there are $n$ risk factors between $A,B$. We can then find the optimal action by

$\min ( p_A \frac{N-n}{2N} , p_B (\frac{1}{2}+\frac{n}{2N}) )$

If we have a not-very severe action let's say the least severe risk compared to the most severe risk then $n=N$ and we find that

\begin{aligned} &\min ( p_A \frac{N-N}{2N} , p_B (\frac{1}{2}+\frac{N}{2N}) )\\ &\min (0 , p_B) \end{aligned}

We find that we will always choose the least severe risk action unless there is no probability for the high severity risk. For risks that are closer where $n$ is smaller, the probabilities of the risks happening account for more.

We see that if the severities of the risks are very close to each other, then we would choose the action with the lower probability, but if there is a large difference between the severity levels, then the relative probabilities matter less, and we are more likely to choose an action with a higher probability of a less severe outcome.

This improves the weigthed action selection algorithm since we will always choose to break an arm over dying. Furthermore, we can extend it to encode individual preferences as a function $f(p,w)$ that depends on the probability and severity

$\min ( f(p_A, \frac{N-n}{2N}) , f(p_B, (\frac{1}{2}+\frac{n}{2N})) )$

which is unique to each individual.

We see here for several probabilities of $B$ that as $B$ becomes more severe relative to $A$ we are more likely to accept the consequences of $A$ even at a higher probability.

In fact, as long as the probability of $A$ is below the curve, we would rather choose action $A$ over $B$.

At $n/N=0$ that is when $A$, $B$ have the same severity, the probabilities alone determine which of the actions we choose  If $p_A < p_B$ then we choose $A$. However, as the relative severity of $B$ compared to $A$ increases, we become more willing to accept $A$ even if the probability of the consequence being realized is greater than that of $B$.

At some point, the severeity of $B$ is so bad compared to $A$ that we always will prefer $A$ to $B$ even if there is a very low (but greater than zero) probability $B$ happening and a $100\%$ probability of $A$ happening. The higher the probability of $B$ the sooner we will prefer $A$ no matter what.

Finally, this gives us a way of converting between risks of different severities and find their equivalent $p_B$ which we can set to the equivalent probability of death.

We see that if $n/N=0$ that is the severity is in the same class as $B$ (in this case death), then we do not have any not have any preference between $A$ and $B$ at the same probability. However, as $B$ becomes relatively more severe than $A$ (indicated by higher $n/N$) then the equivalent $p_B$ becomes lower for a given $p_A$. This lets us effectively deal with non-terminal risks using just a single risk-tolerance threshold.

While this model is easier to compute, the heuristic distance metric assumes that risky activities are evenly distributed in severity which is not guranteed.

## Conclusion

We have introduced a framework for thinking about risk in terms of risk budgets, and relating terminal and non-terminal risks to a single consistent risk tolerance level.

We have seen that a given life expectancy gives us an associated risk budget, and that society has calibrated the risk tolerance to the mean life expectancy, so to dramatically increase life expectancy, we need to rethink our risk tolerance, and become more risk adverse.