Cooperation, Insurance and The Law of Large Numbers

Mathematics, risk and society. How a theorem in probability shows that pro-social behaviour (insurance) mitigates individual risk and supports long-term survival.

Contents

Introduction — A simple model of savings
Part I — The benefits of an insurance scheme
Part II — The law of large numbers
Part III — When insurance schemes fail
Appendix — Math for the simulation

Inspired by Ole Peters (lecture) and Nassim Nicholas Taleb.

This article strings together a surprising series of thoughts which I’ve had over the past year. Expected values, ensemble averages, time averages, empirical vs theoretical mean. Insurance, cooperation, sharing, culture, tradition, conservatism and politics. Risk, correlation, contagion and catastrophe. Portfolio diversification and market delusion. Survival, elimination, and evolution. And a lot more, if you have the imagination.

The detailed math is in part II and the appendix.

Introduction: A Simple Model of Savings

Suppose that you have a stable job. You get paid some fixed amount periodically. You also have to pay bills and other expenses, but they are generally stable and predictable. (These expenses are already added into the income figure, so income is net income.)

This is how your savings might look like over time.

Not all bills are predictable. What if you have a medical emergency or a fine? These bills come occasionally, and can be very costly. This might be how your savings looks over time.

What happens if an unexpected bill is too costly? Can you pull yourself out from less than 0?

Life is random.

Some days are good, and you make progress. Everything goes as expected.

Then there are other days. Something unexpected happens and it’s not good. You tell yourself that you will be fine. After all, battling adversity is a normal part of life, right? But then you check your savings. You have nothing left.

What do you do? Get a loan? Declare bankruptcy?

What if you had… insurance?

Part I: The Benefits of an Insurance Scheme

Consider an arrangement where individuals agree to help each other sustain losses which would otherwise be fatal. We create a simplified model for this, but in real life, insurance-like arrangements can appear in many forms.

  • Relatives, friends, or even members of the local church, contributing to a shared pool of wealth.
  • The general public paying premiums to an insurance company, which then agrees to cover large, unexpected expenses.

I would not be surprised if these sorts of behaviors were present in other social animals, or even microbes. Evolutionary biologists might have something to say about this.

The two trajectories show that the two individuals go below zero at certain points in time. Blue goes below zero at around the 15th time step, while Green goes below zero at around the 200th time step.

Without any sort of external support (debt, insurance, mercy), the trajectories would end early and the two individuals would be eliminated.

Feels a bit unfair.

(* Expected individual savings assuming no claims: see appendix for explanation.)

The only difference in the simulation is that a contribution is now subtracted from individual savings and added to the shared savings. Note the key differences between the trajectories:

  • Insured, the two individuals do not get eliminated early. They survive over the entire time span.
  • The individual savings growth under insurance is reduced. You can see that the green trajectory goes below 0 earlier (~160) under insurance, instead of later (~200), and the final savings is also less. But of course, there is no final savings with an early elimination.

Right now, insurance seems to be a trade-off between survival and maximizing individual savings growth. But this is just one example.

Does cooperation prevent elimination in general?

We consider insurance arrangements of N people.

For communities, survival is defined as the shared savings remaining non-negative. The elimination time is interpreted as the time when shared savings goes negative. This leads to the death of the individual who could not claim from the collective pool, but note that this does not eliminate everyone. (A limitation of this simulation).

There are 1000–3000 simulations, and the proportion of simulations still active (not yet eliminated) is plotted against time.

Observations

  • Individuals and communities are most vulnerable to elimination in the early stages. This makes sense, as initial savings is low.
  • Elimination is less likely for the communities which have survived longer. This makes sense. Given that these communities survived early and have positive expected savings growth, we expect a relatively large pool of savings to have been built up, which makes these communities robust to unexpected expenses.
  • The survival probability increases as the number of people in the insurance arrangement increases.

So we can see that cooperation does indeed support the survival of individuals in a community, and the larger the cooperation scheme, the better.

We now consider how insurance arrangements are affected by different levels of commitment.

In previous scenarios, individuals increased their income by an average of 5, with a contribution of 0.8. We now change the amount contributed by each individual. In particular, we want to consider:

  • Maximum greed: contribution = 0
  • Maximum self-sacrifice: contribution = 5

The average unexpected loss in each time period was 4, so the expected individual savings growth is:

  • Positive when the contribution is between 0 and 1 (keeping an income of 4 to 5 on average)
  • Negative when the contribution is greater than 1 (keeping an income of less than 4 on average)

We now simulate these scenarios and find the probability of survival, with the definition being the same as above.

Dashed line — 0.8 contribution (from past simulations). Increasing contributions makes elimination less likely. What does this say about the benefits of self-sacrifice?

Unsurprisingly, when individuals contribute less, it hurts the community’s ability to protect its members.

Something to note is the non-linearly increasing benefit of contributions. For example, the difference between a contribution of 0 and 1 is about 30% at time-step 60, but the difference between a contribution of 1 and 2 is closer to 10%. It looks like additional contributions have diminishing benefits.

What do these scenarios with large contributions look like? We look at a scenario where 3 people contribute 2, so that their expected individual savings growth is -1.

The individual trajectories are constantly returning to 0 and they are claiming from the shared savings. But ignoring who gets what resources, the total shared income is 3*5=15 and the unexpected losses cost 3*4=12. This is a net positive for the community, so the growth of the shared savings makes sense. We can clearly see that the individuals have sacrificed their own individual growth but the collective thrives.

  • Individuals face a risk of early elimination due to random, unexpected expenses.
  • By entering into an insurance arrangement, individuals can form a community and protect each other from early elimination.
  • Larger communities are better at preventing first deaths.
  • Generous communities are better at preventing first deaths.

Part II: Society and The Law of Large Numbers

What does insurance and cooperation have to do with mathematics and the law of large numbers? First, we look at the Law of Large Numbers (LLN) as it is commonly known.

The Law of Large Numbers: Sum of observed savings growth over n time-steps approaches the quantity n multiplied by the theoretical one-step savings growth.

The LLN basically states that random phenomena will eventually behave predictably in the long-run, and this can be predicted via the expected value. But what does it say about the short-run?

The Law of Large Numbers, by omission, states that the observed behaviour is not necessarily going to follow the expected behaviour in the short term.

We demonstrate this with a simulation over 1 million time steps.

The plots show the trajectory over the different time scales. What is notable is that the trajectory is far more volatile over shorter time spans, which means that the trajectory is not always moving as expected. However, the graph on the right shows very little difference between the simulated and expected trajectory.

This plot shows the mean savings growth from the beginning, calculated up to each point in time. Notice how extreme the fluctuations are during the beginning. It can take a long time until the savings growth you experience matches what you are theoretically expecting.

In theory (LLN), you can achieve the mean growth rate over the long term. But in the real world, your long term survival is not guaranteed. We have already seen that early fluctuations can be deadly. If you are eliminated early, then what will be your long term expected earnings? Zero. The law of large numbers won’t help you.

If insurance protects you from being eliminated, then you get to stay in the game. You get to play for the long run. You get to see your long-run average growth.

In other words — from your individual perspective, you get the benefits of the law of large numbers.

Green would have been eliminated early, but because of insurance, Green survives and experiences the LLN growth.

So far, the law of large numbers has showed that from the individual perspective, the average trumps volatility over long time spans. But we can also consider an average over many people for the same point in time (ensemble average).

For large n, the quantity which is redistributed (after collection from n people) approaches the theoretical one-step savings growth for one person.

In terms of the simulations, we are taking all of the savings growth and redistributing it equally amongst the N community members.

Plot of average savings at each point in time.

In the left plot, there are only 5 people. The average savings growth still fluctuates, but the average savings is visibly less volatile than the individual trajectories. As the number of people increases, the average savings growth fluctuates even less. Eventually, the volatility in savings growth is eliminated, as the average savings growth is exactly the theoretical mean, even though we only have 30 time-steps.

By using redistribution to reduce individual exposure to volatility, individuals are protected from unexpected expenses which might eliminate them. Thus, we get a theoretical basis for why increased community size and contributions (redistribution/sharing) might led to better survival.

  • The law of large numbers states that in the long-term, the observed growth will match the expected growth.
  • The law of large numbers does not state anything about the short-term. In real life, short-term volatility can eliminate individuals.
  • An insurance scheme protects individuals from short term volatility, allowing them to experience the long-term growth which is predicted by the law of large numbers.
  • The law of large numbers predicts that redistribution in large, cooperative communities will allow individuals to reduce volatility in their savings growth.

Part III: Why Insurance Fails

We have now seen the survival benefits of insurance and “large numbers”. However, constructed models are not real life. What are the key differences between these constructions and the real world? And why does insurance fail in the real world?

We assumed that everyone had a consistently positive savings growth before insurance, but this might not be the case in real life. A more realistic picture might include inconsistent income (switching jobs, getting fired), or even negative savings growth. And if the individuals are losing more than they gain in the long term, then it is simply not possible to maintain a shared pool of resources to protect the community.

It also makes sense that many people would choose to make large purchases (discretionary spending) after accumulating a large pool of savings. Thus, the long-term pool of individual savings might not grow as large as we simulated, so volatility could be more harmful to people in the real world.

The volatility was simulated from an exponential distribution, mostly out of convenience. In the real world, risk events do not necessarily come from neat probability distributions. Thus, the way risk affected individuals in these simulations may not reflect all the ways that risk can affect people in the real world. For example, a house fully burnt down can be an elimination of nearly 100% of wealth.

We saw that the larger communities are better at protecting members from elimination. However, in the real world, it is not easy to ask people to cooperate on such a large scale. In general, trust is limited to local communities — friends, families and neighbours. The size of a social network places a cap on the number of people who would be willing to insure each other (assuming the ability to do so).

In real life, we can’t simply increase the required contribution to the collective pool. Most people want individual savings growth (profit motive), which places a cap on the contribution which can be collected.

Illusion of safety → increased risk → negative savings growth
Suppose that a group of people start off net positive in savings growth, and they later agree to insure each other. Knowing that they are protected, they might then feel open to risky behaviours. This could put them into net negative savings growth. And if everyone did this, then the insurance scheme would no longer work.

Restricting the freedom to take risks
This justifies a need to restrict the freedom to take risks. Insurance companies charge higher premiums and refuse to cover individuals from certain types of risk events. And human culture has historically been very conservative and restrictive on freedom — perhaps these societies were better at survival, compared to other societies which failed to restrict risky behaviours.

Entrepreneurship and beneficial risks.

Suppose that there are a class of people who can perform great feats, but with a low probability of success. The gain from their success far exceeds the loss from their failure, so the theoretically expected savings growth is extremely high. However, they do not want to risk their life, so they choose to take a different path, which is less dependent on chance. By putting these people into insurance arrangements, society can give them the safety and confidence to take these improbable but high payoff risks.

The law of large numbers does not work if individual risks are correlated.

Consider this illustrative example. We flip a coin. If it lands H, we get +1, and if T, then -1.

One coin
H = +1 (50%)
T = -1 (50%)

Two independent coins
HH=+2 (25%)
HT/TH = 0 (50%)
TT = -2 (25%)

But now imagine the two coins are perfectly correlated. If one lands H, then the other must land H.

Two correlated coins
HH = +2 (50%)
TT = -2 (50%)

Suddenly, there is no difference between this series of two correlated tosses vs. a single toss with double the payoff.

This is what correlation does. If risk events occur at the same time, then the unexpected expenses arrive at the frequency that an individual experiences, but with a scaled-up severity. The community becomes a scaled up version of the individual. And it faces the same short-term volatility risk.

The ideal insurance scheme must have claims from the collective pool at different times. This makes claims more frequent, but smaller in size — but this reduces extreme fluctuations! Uncorrelated risk (or even negatively correlated risk) ensures that these claims are likely to occur at different times.

Sharing risk is bad, but hard to stop.

It is interesting to think about some real world examples.

Financial bubbles — correlated risk.
Properly diversified asset portfolio — negatively correlated risk.

To stay safe from correlations, people need to avoid taking the same sorts of risk. But this idea is easier said than done, as similar behaviour is often the very thing which holds a social network together, while deviating from the crowd can make one look foolish. And when everyone takes the same sorts of risk, then these people become uninsurable from these risks.

Insurance At Larger Scales

Sierpinski Carper — Fractal

The grid is a 3x3 collection of grids. But which grid am I talking about?

Correlation is a very serious issue that insurance companies face. For example, an insurance company might protect homeowners in a region from house fires and random damage. But if there is a natural disaster that destroys every house in the region, then the insurance company faces a very severe expense that might eliminate their cash reserves.

What if we could move up a scale?
Individuals face their own risk of their house burning down, and usually, it only happens to one house at a time. Similarly, a region face its own risk of a natural disaster, and usually, it only happens to one region at a time (uncorrelated). Thus, we can reconceptualise a region as an individual, and create a community of regions. Then we can create an insurance scheme for regions.

This is the idea behind re-insurance. Insurance companies provide their services to individuals and businesses. However, these companies tend to operate in a certain region, and in the event of a widespread catastrophe the insurance company may face simultaneous claims from all of its clients. When the insurer can’t cope with such an event, it can claim funds from a re-insurer, which collects funds from multiple insurance companies operating in different countries and facing different (and hopefully uncorrelated) risks.

This solves the issue of correlation and simultaneous claims. Simply create a larger insurance scheme, so that the insurers are insured!

If insurance can protect individuals, and reinsurers can protect insurers, then (assuming everyone contributes to their insurance schemes) can we be completely safe from unexpected events?

No. Consider this.

How many people are there in a community?
How many local communities are there in a state/province?
How many states/provinces are there in a country?
How many countries are there on Earth?
How many planets are there with life (and can we communicate and share resources with them)?

For the first half of the list, there are a sufficient number of entities at the individual level which are likely to cooperate. Thus, by the law of large numbers, these entities will be able to experience long-term growth and they will also be able to reduce short-term savings volatility with resource sharing.

But as we consider larger scales, there are far fewer entities ready and willing to assist each other — which means no reduction of volatility. These entities will experience the full effects of unexpected and extreme events.

And now consider the Earth, affected by a global pandemic or a change in the climate/environment. The world economy is no longer functioning. Countries cannot ask other countries for help. We are all powerless to help each other. What are our options — can we ask other planets for help? There is no life outside of Earth which can help us.

This has major implications. Lower-level entities (people, local communities) are able to take (non-contagious) risks safely, knowing that they can be readily insured. But higher-level entities (nations, the Earth) do not readily receive assistance from other higher-level entities. They are uninsurable from systemic threats.

Thus, in matters of risk, the typical behaviours of individuals cannot be scaled up to higher-levels. The only option is to avoid global systemic risks.

What are the systemic threats, and where do they come from?

Contagion. Multiplicative risk.

  • Global warming
  • Pandemics
  • Engineered genes spreading in the wild.
  • Massive debt and the possibility of a series of defaults.

Are we prepared? Are we insured? I don’t think so.

Appendix: the math behind the simulations in Part I

Thanks for your patient reading. Help your neighbours, and let your neighbours help you.

Math, stats, data. Influenced by the complex systems perspective. I prefer to take the critical view.

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