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ike comments on What makes buying insurance rational? - Less Wrong Discussion
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The expected monetary value of insurance is negative (or rather, negative in real dollars. It can be positive in nominal dollars but underperform inflation.)
But the utility is not linear in money. Losing e.g. $10,000 might be 20 times as bad as losing $1,000. If so, you should pay $1,000 100% of the time to avoid paying $10,000 8% of the time.
The insurance company averages out over many buyers, so their utility is roughly linear.
Insurance is just trading against different utility scales.
Yep. Also note that if you had $1M in the bank, you would then not prefer to buy insurance for something on the order of $10k.