Robot Long Jump

March 2023 : Solution

For March’s puzzle, you had to determine the Nash equilibrium of a tricky probabilistic model of Robot Long Jump. It turned out the optimal play involved waiting until a robot’s position was at least some threshold x and then jumping, where x satisfies the equation

(x3 - 3x + 2)ex = 3x.

This threshold comes to ~0.416195355. Given that, the chance of any given round scoring a positive number is (1-x)ex, and so the final answer is (1-(1-x)ex) ~ 0.114845886

Congrats to the solvers that successfully computed the strategy and this probability!