To be risk neutral on the money themselves you need to be extremely rich or betting on very tiny amounts (and at very least you need to have $500 000 to bet).
If you are risk neutral on logarithm of amount of money (which is more realistic), the pay off is a>0 and you start off with n>0 in assets, then your expected pay-off after betting x is 0.5 ln(n-x)+0.5 ln(a+n-x) , the change of utility is 0.5 ln(n-x)+0.5 ln(1000000+n-x) - log(n) and there’s zero change in utility when it equals zero when 0.5 log(n-x)+0.5 log(a+n-x) = log(n) , which has solution x=0.5 a + n − 0.5 sqrt(a^2 + 4 * n^2) .
Plugging in 1 million in payoff and $100 000 in assets, you should bet up to about 90 000 $ , which is somewhat surprising. Usually, the utility function is not logarithmic though, as if you were to lose a lot of money you would have to deal with all the logistics of moving to cheaper apartment or the like, so people would be willing to bet less. Actually, (rational) people just compare two imagined outcomes directly to each other rather than convert each to a real number first, this is better when you only partially imagine an outcome, so that you can do both equally partially. Estimating 2 utilities and then comparing runs into problems when estimations are necessarily approximate.
edit: reddit is designed for people who use italics more often than algebra.
To be risk neutral on the money themselves you need to be extremely rich or betting on very tiny amounts (and at very least you need to have $500 000 to bet).
If you are risk neutral on logarithm of amount of money (which is more realistic), the pay off is a>0 and you start off with n>0 in assets, then your expected pay-off after betting x is 0.5 ln(n-x)+0.5 ln(a+n-x) , the change of utility is 0.5 ln(n-x)+0.5 ln(1000000+n-x) - log(n) and there’s zero change in utility when it equals zero when 0.5 log(n-x)+0.5 log(a+n-x) = log(n) , which has solution x=0.5 a + n − 0.5 sqrt(a^2 + 4 * n^2) .
Plugging in 1 million in payoff and $100 000 in assets, you should bet up to about 90 000 $ , which is somewhat surprising. Usually, the utility function is not logarithmic though, as if you were to lose a lot of money you would have to deal with all the logistics of moving to cheaper apartment or the like, so people would be willing to bet less. Actually, (rational) people just compare two imagined outcomes directly to each other rather than convert each to a real number first, this is better when you only partially imagine an outcome, so that you can do both equally partially. Estimating 2 utilities and then comparing runs into problems when estimations are necessarily approximate.
edit: reddit is designed for people who use italics more often than algebra.