# Free market insurance is incompatible with knowledge

Yes, it’s an extreme title, but it’s true. The idea of insurance is to average risk over a large group of people. If advance information exists about the outcomes of individuals, it’s impossible for a fully competitive free market to provide insurance.

In particular, free markets cannot provide health insurance.

To see this, consider a function $u : S \to R$ which assigns a utility value to each point of a state space $S$. For example, one of the elements of $S$ could be “you will have cancer in 23 years”. This outcome is bad, so the corresponding $u(s)$ would be a large, negative number.

We also have a probability distribution $p : S \to R$ over $S$. Without insurance, the expected value of $u$ is $E(u) = \sum_{s \in S} p(s) u(s)$. With insurance, we can average over a large number of people to change the utility function to be closer to the average. For simplicity, we’ll consider only the case of perfect insurance, where the new utility function is exactly the average. In the perfect insurance model, we pay an insurance company $E(u)+o$, and in return they agree to pay us $-u(s)$ depending on the particular outcome $s$. $o$ is an extra amount to cover administrative costs, risks due to lack of independence and finite numbers of customers, and profit (in the case of imperfect competition).

Assuming no one has any prior knowledge of the state $s$, the only way for different insurers to compete in the perfect insurance model is to reduce overhead. Everyone looks the same, so there’s no advantage in charging different amounts to different people. The insurers profit from anyone with $u(s) \gt E(s)$ and lose money from anyone with $u(s) \lt E(s)$, but there’s nothing they can do about it if they can’t tell the difference in advance.

Now assume there’s some prior knowledge about the state, say $S = K \times U$ where $K$ is known in advance and $U$ is unknown. In the absence of regulation, it becomes possible for an insurance company to charge different amounts based on the different $k \in K$. In particular, it’s possible for an insurer to sell policies only to people with a favorable value of $k$, and charge $E(u \vert k) \gt E(u)$. In a free market, anyone with a favorable value of $k$ will flock to these cheaper policies. Insurers offering policies to those with unfavorable values of $k$ will have to raise rates in order to stay in business, since they will have lost the customers from which they make money. Assuming a sufficient level of competition, the price of all insurance policies will converge on $E(u \vert k) + o(k)$.

The result is that we’re now insuring only over the uncertainty contained in $U$, not $K$. In the worst case, if $K = S$, $E(u \vert k) = u(s)$ and insurance vanishes completely.

Whether this is good or bad policy-wise depends on what $K$ and $U$ look like. For car insurance, $K$ includes whether the driver was considered at fault in accidents in the past, whether they’ve driven drunk, whether they drive a muscle car or a Honda Civic, etc. Charging different amounts depending on these factors seems fair, since intuitively these factors can be considered the “fault” of the individual. Similarly, charging more for home owners insurance if you live in the path of a hurricane is also (arguably) reasonable.

In the case of car insurance, even with these known factors out of a way, the space of uncertainty $U$ is still quite large. It includes the actions of other drivers, random equipment failure, invisible road conditions, etc. It is impossible for insurers to predict these factors, which means that private, free market insurance can efficiently insure against them.

For health insurance,the space of known factors includes all past medical history and preexisting conditions, public genetic information including gender and race, healthy or unhealthy lifestyle, etc. In many cases, it includes information about the current medical problem, since insurers have significant control over what kind of treatment people can receive once they are diagnosed. Now, we can argue about whether it’s fair to blame people for unhealthy lifestyles, but I highly doubt anyone will argue that black men should be held responsible for their higher rates of prostate cancer.

If we accept that the space of known factors $K$ is too large, the only way to reduce it is to apply some type of regulation to reduce the effective size of $K$. A fair amount of subtlety is required to make such regulation effective. For example, let’s say we ban insurers from discriminating based on race, but still allow them to collect information about healthy lifestyle. It’s healthy to play sports, so the insurer might ask whether the person plays basketball. People who play basketball are more likely to be black than those who don’t (caveat: I’m just guessing here), and therefore it’s quite possible that they have higher risks of prostate cancer. Unless the government is smarter than the insurers (impossible, since the insurers have access to the text of laws), the only reliable way to solve this is to ban knowledge of $K$ entirely.

However, banning insurers from using knowledge of $K$ is dangerous unless you also ban customers from using knowledge of $K$. In an extreme case, it would be very bad to allow people to buy insurance policies in response to accidents of unexpected diagnoses. Everyone would wait until they needed medical coverage to buy insurance, and all insurers would rapidly go out of business.

In general, if individuals are allowed to use any information prohibited to insurers, and the space of available policies is large enough, sufficiently diligent individuals with favorable $k$ values can use this information to lower their insurance premiums without raising their risk. Insurers will have to raise their premiums in response, which results in an increase in cost for those with unfavorable $k$ values.

In fact, assuming sufficient options and perfect competition, the result of this individual choice would be exactly the same as if the insurers were allowed to use knowledge of $K$! Wow. I didn’t fully understand that point before writing this post.

The conclusion is that if we believe true health insurance is a good thing, and that health insurance means insuring over factors which can be known in advantage, free markets don’t work either for insurers or for individuals. We can’t allow insurers to base prices on prior knowledge, and we can’t even allow individuals to choose which policy they buy based on their knowledge of their own medical history.

Hmm. The individual side of this is somewhat unfortunate, but I don’t see any way around this argument.