# Risk Appetites - betting percentages

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?

Let's suppose (this is just a matter of choice of units) that both start off with 1 unit of money. If Kim Square bets an amount $$a$$ while Ann Root bets an amount $$b$$ then twice Kim's expected utility is $$(1-a)^2+(1+b)^2$$, which will be $$>2$$ iff the point $$(a,b)$$ lies outside the circle with centre $$(1,-1)$$ and radius $$\sqrt2$$. (So this circle passes through the origin, which should not be surprising considering what the coordinates mean.) Twice Ann's expected utility is $$\sqrt{1+a}+\sqrt{1-b}$$; the region where this is $$>2$$ is (after a little algebraic manipulation) the "exterior" of a parabola whose axis is along the line $$a+b=0$$ which, again, passes through the origin.

The intersection of these two regions is

as it happens not empty: it's a sort of curvilinear wedge extending northeastward from (0,0) and broadening and curving downward as it does so. At the origin it has zero thickness; at $$a=1$$ it extends from about $$b=0.42$$ to $$b=0.66$$.

There's no single obvious way for K&A to pick a specific bet; which they decide on may depend on questions like which of them may / must make the first concrete proposal, which of them feels like they have more leverage in the negotiations, etc. But if we suppose that an impartial third party is going to make the decision then

one "simple" possibility which is well inside both participants' comfort zones is (1,1/2), which in terms of the "utilities" I described above gives Kim an expected utility of 1.125 units and Ann an expected utility of 1.061ish units; if we take the view that these utilities should be equal (which there's no particular reason to do, really) or, equivalently, try to maximize the minimum of the two, we get $$a=1$$ and $$b$$ is the root of a certain 6th-degree equation and is approximately 0.465. Both participants' "utilities" are then about 1.073.

For a start, we can say that

Kim would risk a $$\sim0.9961$$ ($$1-\frac1{256}$$) chance of $$\0$$ in order to win $$\16$$, whereas the much more sensible Ann would accept $$\2$$ against a $$\sim0.2929$$ ($$1-\frac1{\sqrt{2}}$$) chance of nothing.

This seems drastically unfair, and so I say:

no bets were placed.