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.
We can approximate user accuracy as:
Accuracy = URE * DOP
The DOP is purely determined by the relative position of the satellites, while URE includes all other possible sources of error. I'm not sure how much URE depends on the number of satellites, but we could assume it is constant. Then, we just need to calculate the DOP which is given by some matrices which you can find in Wikipedia or any GNSS textbook. The problem is that the DOP depends on the number of satellites and their relative position (for example, two satellites on the same position won't lead to better accuracy). Thus, to get some numbers we need to use specific satellite positions. An option would be to choose some location, say New York and get the GPS orbits. Then we could compute how the DOP changes considering X satellites. Guess this could be done analytically or computationaly.