Why should it depend on how much you stand to lose if Maxine wins? No matter the value (as long as it doesn’t approach your total wealth—or is this what you meant?) to you of Maxine’s election, you’ll gain more if you bet following the odds more closely.
Yes, if the people selling you insurance are rational, then you would gain more on average by not buying insurance, putting edge cases aside. That is true both of ordinary insurance and bets taken for insurance.
But the point of insurance is to reduce risk, not maximize gain.
For example, suppose Maxine winning would cost my business $200, and I cannot lose more than $50 and stay in business. Then I see a bet that pays $200 if Maxine wins, which costs me $50 to buy. It would be worth taking that bet regardless of the probability of Maxine winning, if I’m very risk-averse regarding losing my business. It turns a possible loss of $200 into a guaranteed loss of $50.
If the actual probability of Maxine winning is 10%, then the expected value of not betting is $-20, while the expected value of betting is $-50, so if I want to maximize gain I should not take the bet. However, taking the bet has a maximum loss of $50, while not taking the bet has a maximum loss of $200 (costing me my business), so in taking the bet I’ve gone from a 10% chance of losing my business to a 0% chance of losing my business. So if I want to minimize the probability of losing my business (all else equal) I should take the bet.
Why should it depend on how much you stand to lose if Maxine wins? No matter the value (as long as it doesn’t approach your total wealth—or is this what you meant?) to you of Maxine’s election, you’ll gain more if you bet following the odds more closely.
Yes, if the people selling you insurance are rational, then you would gain more on average by not buying insurance, putting edge cases aside. That is true both of ordinary insurance and bets taken for insurance.
But the point of insurance is to reduce risk, not maximize gain.
For example, suppose Maxine winning would cost my business $200, and I cannot lose more than $50 and stay in business. Then I see a bet that pays $200 if Maxine wins, which costs me $50 to buy. It would be worth taking that bet regardless of the probability of Maxine winning, if I’m very risk-averse regarding losing my business. It turns a possible loss of $200 into a guaranteed loss of $50.
If the actual probability of Maxine winning is 10%, then the expected value of not betting is $-20, while the expected value of betting is $-50, so if I want to maximize gain I should not take the bet. However, taking the bet has a maximum loss of $50, while not taking the bet has a maximum loss of $200 (costing me my business), so in taking the bet I’ve gone from a 10% chance of losing my business to a 0% chance of losing my business. So if I want to minimize the probability of losing my business (all else equal) I should take the bet.