This question is from the September/October issue of the Actuarial Review magazine.
Kim and Ann start with an equal amount of money, but they value money very differently. Kim acts to maximize the expected squared amount of her total wealth. For example, Kim is indifferent between a 100 percent chance of her wealth being 1 dollar and a risk for her total wealth of a 75 percent chance of 0 and a 25 percent chance of 2 dollars. In contrast, Ann tries to maximize the expected square root of the amount of her wealth. For Ann, a 100 percent chance of 1 dollar total wealth is the same as a risk for total wealth of a 75 percent chance of 0 and a 25 percent chance of 16 dollars.
Kim and Ann can each choose a percentage, from 0 percent to 100 percent, of their initial wealth to gamble on a fair coin flip, with the winner claiming the total amount that both bet.
The coin flip is voluntarily negotiated beforehand so that the percentage each of them bets — the percentage need not be equal — is acceptable to the other one. What combinations of betting fractions would be mutually acceptable to both of them? You might want to draw a graph. Do you have an opinion about what specific combination of betting percentage they might settle on before flipping?