I know that a GPS receiver requires at least 4 satellites to locate its position in 3D, and I have read elsewhere that having more than 4 satellites would increase its location accuracy. My question is is there a way to calculate how much of a reduction in error is gained due to the increased number of satellites?
Accuracy is about being close to the "real" value of measure. The 4 satellites are the minimum to solve the x,y,z,t equations to get you location, so you have a single observation of your location with 4 satellite.
With more satellites you can combine them to get more than one set of solutions or location and (probably) you can select the observations got from the satellites with better or stronger sginal, but it is just guessing.
The more observations of the measure you want to have, in your case your location, the better.