In a robot tug-of-war matchup, two robots are tied together with a rope. The center of the rope has a marker that is above some position 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, from wherever it currently is, a distance drawn uniformly at random from [0,1] (independently for each pull, and the pull length is not affected by the marker's position) towards the pulling robot; the marker's position carries over from one pull to the next. 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.
Starting from position 0, the robot going second is at a disadvantage. To make the game fair, the marker's starting position is moved to some negative real number c (with −½ < c < 0); the first robot still pulls first.
Question. Find the starting position c that gives each robot exactly a 50% chance of winning. Give c to seven significant digits.