Single payer still acts as an insurer and needs actuaries (even if they're not called that) to figure out which procedures make sense with their limited resources.

Not really. There is no need to price the cost of insuring a 30 year old white male living in Croydon. If he is ill, he gets treatment. If he is not ill, he doesn't.

The only thing close to actuarial science that the NHS actually needs is QALYS calculations. This is only really used for calculating how worthwhile very expensive treatments that target the very old or very disabled.

The number of people needed to do that is very small, compared to the number needed to calculate prices in a non-single payer context.

Then they need to estimate how many procedures of each kind to contract, or how large facilities to maintain, in Croydon. Which in part depends on the number of 30 year old white males living there.

If they get it wrong, he doesn't magically get treatment anyway. He either has to wait, or travel elsewhere (and someone still has to figure out where and when). Otherwise, they waste money on unused capacity.

Perhaps, but you don't need an army of sophisticated actuaries to figure this year's number for Croydon is not too different from the last 20 years of data. There are some prediction and budgeting problems to solve of course but they are of an entirely different character when you're not dealing with a for profit or marketplace solution.

All you need to know is that you 'maydie' if you go to Croydon's hospitals ;)

Australia had/has state provided third party vehicle insurance as part of the required yearly vehicle tax.

Aside from economies of scale and protection from risk this was also incredibly useful in that one could (if qualified) legally drive any vehicle displaying a valid windscreen ticket.

er so what does he think the NI Stamp (social security) pays for

It's supposed to be for Works & Pensions (unemployment and pensions basically) but I think it now goes into the general tax pot. It's not health insurance though.