I'm well aware that there are many different factors to be considered, such as ease of installation/maintenance, conversion from AC to DC, environment, distance, speed, frequency of turns, and so forth.
Setting all of those aside for now, suppose we build a 10-mile long demonstration track with some hills and turns along the way and featuring both a third rail and an overhead cable. Suppose further that we have two trains (one equipped with a pantograph, the other with contact shoes). They have the exact same weight and are programmed to accelerate and decelerate at the same rate, and run at the same speed.
Which one will have consumed more electricity at the end of the ten-mile run?
Bonus Question: A third rail is a lot wider and thicker than an overhead wire. Is more electricity wasted that way?