Greedy

Setting your level of risk

Choose your risk tolerance x and your payoff each round will be f(x) whose graph is draw at the right.

So far it is clear that to get a high payoff, you want a high x.  

But there’s another factor.

After each round a random number z is chosen between 0 and 1. If  z is greater than x, you can stay in the game for the next round, whereas if  z is less than x, you need to leave the game—no more rounds. In that case, you are able to take your winnings so far with you.

The problem is to find the optimal x, the level of risk that will maximize your expected take home pay.

This game is a bit sophisticated but it has an elegant analysis. To handle the random nature of z we will need to revisit the Dartboard Theorem, but that’s an interesting enough resut itself. 

A surprising finding is that there is a solution in all three standard mathematical branches: graphical, algebraic, and analytical (calculus).