Friday, March 18, 2011

mutual underwriting

A particularly weird idea that's popped up in my head in the last month is on some level an extension of the old idea of a mutual insurance company, wherein policyholders are also residual owners of the company; in caricature, everyone pays in somewhat more at the beginning of say a six month period than they expect to lose and everyone gets back a portion of what's left after paying losses incurred during the period. If the overall risk level was higher than initially estimated, people may not get back the rebate they were hoping for, but if the overall risk level is lower, people end up effectively paying a lower premium for the period in which they were covered. They thereby insure their risk by spreading it among their fellow policyholders, remaining exposed to unexpected levels of overall risk, but they don't face the problem of perhaps believing that the insurance company is being overly conservative in setting its premiums — if that's true, the policyholders will ultimately get back the difference.

These require, in some sense, less absolute underwriting, but still require relative underwriting; if 100 homeowners in identical homes are buying insurance, that's easy, but if 12 of them have propane tanks next to their houses and the other 88 don't, charging everyone the same premium doesn't seem as fair. One solution here is to subdivide the groups: let the 88 buy insurance from each other, and the 12 buy insurance from each other, without the cross-subsidy. Each time we sub-divide, though, the insurance becomes less useful — I don't have the law of large numbers working for me terribly effectively when there are only 12 of us, since the whole point of buying insurance was not to be exposed to a risk of large loss, and one twelfth of a house is a large loss — and, since any two houses are different, at the very least in location (e.g. distance from fire stations), this creates a problem in which someone has to decide which houses are similar enough that it is better to throw them in the same pool, and which differences are sufficiently salient that they should be in different pools, even at the cost of a higher variance of outcomes for the policyholders.*

The idea that popped into my head is essentially that we let groups decide on their own which groups to join. Obviously simply saying, "Here are two pools: the safe pool, and the risky pool. Which do you want to join?" isn't going to work — the safe people need to be able to exclude the risky people in some fashion. One idea is to cap the number of people in each group, and let members of oversubscribed groups vote on which of the other (attempted) subscribers to keep; this gets close to the self-underwriting flavor that I was looking for when I thought about it. The same sort of arrangement could apply to health insurance, or even to mortgage lending (in something like a credit union), though that introduces other complications.

The details of the voting would be interesting, though; does everyone in the pool list their favorite 99 co-poolers and the top 100 get to be in the pool? Or perhaps everyone gets to vote for as many as they like, and the top 100 are in. Or maybe rank applicants in order, and give a certain number of points for first-place votes, etc. Or perhaps we should do something recursive; what if the group of the top 100 applicants, as measured by the votes of all applicants, differs from the group of the top 100 applicants as those 100 applicants themselves vote (i.e. excluding the votes of the rejected applicants). What we'd really like is some sort of stable outcome in which everyone is in a group, and nobody would prefer to be in a different group that would be eager to swap that person in for some other current member of the group. Can we get that?

Well, the answer turns out to be "no". Imagine 4 people: Alice, Bob, Carol, and Doug. They are to be divided into two groups of two people. Alice prefers to be with Bob, Bob with Carol, Carol with Alice; Doug is the last choice of all three of them. Now consider a prospective grouping; the person who is grouped with Doug is the first choice of one of the other two, and can go to them and say, "hey, let me join your group." No matter how the four people are divided into groups, there is always someone from each group who would rather be with each other than with their current group; every possible grouping is, in this sense, unstable.

For large numbers of people, with large groups, and with highly correlated preferences — that is to say, if people largely agree on who are the safest risks — then the probability of this being a problem in actual practice get very small very quickly. You could probably use just about any system you want to get groups down to 105, let them whittle it down to 100, and you would almost always have a stable alignment. This theoretical curiosity, then, isn't the biggest problem with the idea, though it is, I think, one of the more interesting.

The bigger practical problem is that it requires, at least in its most naive formulations, that everyone have an opinion about everyone else's level of riskiness, and an easy means of conveying it. I can imagine ways of getting around it, but on some level underwriting is a service provided by the insurance companies, who are presumably more or less expert at it; I would rather let Geico figure out which of my neighbors are the better risks, and allow me to put my efforts toward blogging about interesting but largely impractical ideas.

* In practice, I imagine everyone is thrown in the same pool with some policyholders asked to pay e.g. 1.5 times as much as others; that clearly still leaves an underwriting problem, and leads less naturally to the idea I'm trying to present.

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