Robot Tug-of-War

August 2021 : Puzzle

The Robot Weightlifting World Championship was such a huge success that the organizers have hired you to help design its sequel: a Robot Tug-of-War Competition!

In each one-on-one matchup, two robots are tied together with a rope. The center of the rope has a marker that begins above position 0 on the ground. The robots then alternate pulling on the rope. The first robot pulls in the positive direction towards 1; the second robot pulls in the negative direction towards -1. Each pull moves the marker a uniformly random draw from [0,1] towards the pulling robot. If the marker first leaves the interval [‑½,½] past ½, the first robot wins. If instead it first leaves the interval past -½, then the second robot wins.

However, the organizers quickly noticed that the robot going second is at a disadvantage. They want to handicap the first robot by changing the initial position of the marker on the rope to be at some negative real number. Your job is to compute the position of the marker that makes each matchup a 50-50 competition between the robots. Find this position to seven significant digits—the integrity of the Robot Tug-of-War Competition hangs in the balance!