Thanks for this! It applies to a lot of different kinds of insurance. Car insurance, for instance, isn't financially great (except liability umbrella in many cases), but having the insurance company set standards and negotiate with the other driver (or THEIR insurance company) is much simpler than having to do it yourself, potentially in court.
For some kinds of insurance, there's also tax treatment advantages. Because it's usually framed as responsible risk reduction (and because insurance bundles some lobbying into the fees), premiums are sometimes untaxed, and payouts are almost always untaxed. This part only affects the financial considerations, but can be significant.
Yes, insurance for your own car's value is usually not great - it's bounded and in most cases cars are relatively easily replaceable with something functionally almost as good for relatively low capital expense.
Insurance for liability to third parties is worthwhile for almost everyone, since the scale of damages in the upper tail exceeds almost everyone's accessible wealth.
since the scale of damages in the upper tail exceeds almost everyone's accessible wealth
Car insurance is [edit: in the US] bounded: a standard policy will cover you up to some cap (ex: $50k). I think maybe your comment is a better argument for umbrella insurance, though that is also not infinite.
An incredibly productive way of working with the world is to reduce a complex question to something that can be modeled mathematically and then do the math. The most common way this can fail, however, is when your model is missing important properties of the real world.
Consider insurance: there's some event with probability X% under which you'd be out $Y, you want to maximize the logarithm of your wealth, and your current wealth is $Z. Under this model, you can calculate (more) the most you should be willing to pay to insure against this.
This is a nice application of the Kelly criterion, though whether maximizing log wealth is a good goal is arguable (ex: bankruptcy is not infinitely bad, the definition of 'wealth' for this purpose is tricky). But another one thing it misses is that many things we call "insurance" have important properties that diverge from this model:
There can be a collective bargaining component. For example, health insurance generally includes a network of providers who have agreed to lower rates. Even if your bankroll were as large as the insurance company's, this could still make taking insurance worth it for access to their negotiated rates.
An insurance company is often better suited to learn about how to avoid risks than individuals. My homeowner's insurance company requires various things to reduce their risk: maybe I don't know whether to check for Federal Pacific breaker panels, but my insurance company does. Title insurance companies maintain databases. Specialty insurers develop expertise in rare risks.
Insurance can surface cases where people don't agree on how high the risk is, and force them to explicitly account for it on balance sheets.
Insurance can be a scapegoat, allowing people to set limits on otherwise very high expenses. Society (though less LW, which I think is eroding a net-positive arrangement) generally agrees that if a parent buys health insurance for their child then if the insurance company says no to some treatment we should perhaps blame the insurance company for being uncaring but not blame the parent for not paying out of pocket. This lets the insurance company put downward pressure on costs without individuals needing to make this kind of painful decision.
Relatedly, agreeing in advance how to handle a wide range of scenarios is difficult, and you can offload this to insurance. Maybe two people would find it challenging to agree in the moment under which circumstances it's worth spending money on a shared pet's health, but can agree to split the payment for pet health insurance. You can use insurance requirements instead of questioning someone else's judgement, or as a way to turn down a risky proposition.
There are still cases where the model is useful: none of these benefits would apply to insuring my mandolin, computer, or a flight, and none of these are a large enough portion of my wealth for the calculator to say I should get the insurance. But if you apply the model without thinking about how well it applies in a particular case it will often tell you not to buy insurance in cases where insurance would actually help.