The average shortest length of a random sequence of tones from a scale which is needed to identify the key.
If the scale size is N then there are N! (N factorial) different sequences of all tones. To calculate the efficiency, for all sequences the shortest necessary length for key identification is summed and this is divided by N! and again divided by N to get the relative length, a number between 0 and 1.
So efficiency measures how many tones help you to identify the key. A low number means there are many tones that don't help to do this. It may be interpreted as a measure of the assymmetry of a scale with respect to all rotations and translations of itself.
If a listener hears random samples of a pitch set tuned to a given scale, then the Rothenberg efficiency of the scale is the time it takes her, on average, to determine which scale position each pitch represents.