And apparently insurance companies can make money because the expected utility of buying insurance is lower than it’s price.
No, the expected monetary value of insurance is lower than its price. (Assuming that the insurance company’s assessment of your risk level is accurate.) You’re equivocating between money and utility, which is the source of your confusion.
Suppose I offered a simple wager: we flip a coin, and if it comes up heads, I give you a million dollars. But if it comes up tails, you owe me a million dollars, and I get every cent you earn until that debt is paid. Is this bet fair?
Monetarily, yes. But even if I skew the odds in your favor a little, maybe 60⁄40, I’ll bet you still don’t want to take it. Why not? Isn’t an expected return of $200,000 wildly in your favor?
Yeah, but that doesn’t matter. An extra million dollars would make your life somewhat better; spending the next twenty years flat broke would make your life drastically worse. Expected utility is very negative.
The utility of money is sometimes claimed to be logarithmic. For small amounts of money you can use a linear approximation, but if the outcome can shift you to a totally different region of the curve, the concavity becomes very important.
The utility of money is sometimes claimed to be logarithmic.
More generally it’s that the marginal utility of money decreases, making buying insurance potentially an expected gain in utils even when the expectation of financial gain is negative.
Not if the marginal utility of money decreases as you have more.
If your utility function has convex parts then it might be possible, though. If you have $1000 but owe the Mafia $2000 and they’re coming to collect, betting it all on black might be a good idea.
No, the expected monetary value of insurance is lower than its price. (Assuming that the insurance company’s assessment of your risk level is accurate.) You’re equivocating between money and utility, which is the source of your confusion.
Suppose I offered a simple wager: we flip a coin, and if it comes up heads, I give you a million dollars. But if it comes up tails, you owe me a million dollars, and I get every cent you earn until that debt is paid. Is this bet fair?
Monetarily, yes. But even if I skew the odds in your favor a little, maybe 60⁄40, I’ll bet you still don’t want to take it. Why not? Isn’t an expected return of $200,000 wildly in your favor?
Yeah, but that doesn’t matter. An extra million dollars would make your life somewhat better; spending the next twenty years flat broke would make your life drastically worse. Expected utility is very negative.
The utility of money is sometimes claimed to be logarithmic. For small amounts of money you can use a linear approximation, but if the outcome can shift you to a totally different region of the curve, the concavity becomes very important.
More generally it’s that the marginal utility of money decreases, making buying insurance potentially an expected gain in utils even when the expectation of financial gain is negative.
Having this in mind, could it be possible to construct such roulette betting system, which have positive expected utility value?
Not if the marginal utility of money decreases as you have more.
If your utility function has convex parts then it might be possible, though. If you have $1000 but owe the Mafia $2000 and they’re coming to collect, betting it all on black might be a good idea.