When drawing the arcs in order to bisect a line segment why must the width of the compass be more than half of the segment?

It's done to make sure the arcs intersect to form the points needed to make the perpendicular bisector.

Check out the diagram below. I have segment AB that I'm trying to bisect (ie cut in half). If I plot circles centered at A and B, such that each radius is half of AB, then the circles will not intersect. This is shown in figure 1. I used full circles instead of arcs, but it's the same basic idea.

Figure 2 has the circles touch at one point, and this happens when both have a radius of half of AB. That one point is the midpoint of AB.

It's only when we go over the halfway point is when the two circles cross at two spots, call them points C and D. Line CD is the perpendicular bisector of segment AB. This is shown in figure 3.

In my opinion, it's preferable to have the radius equal to AB itself so that we're guaranteed to be larger than AB/2. To me it's easier because A and B are already present as points.