Inconsistent weighting in weighted voting games

In a weighted voting game, each voter has a given weight, and a coalition of voters is successful if the sum of their individual weights exceeds a given quota. Such voting systems translate the idea that voters are not all equal by assigning them different weights. In such a situation, two voters are symmetric in a game if interchanging the two voters leaves the outcome of the game unchanged. Two voters with the same weight are naturally symmetric in every weighted voting game, but the converse statement is not necessarily true. We call the latter scenario inconsistent weighting. We investigate herein the conditions that give rise to such a phenomenon within the class of weighted voting games, and we study how the choice of the quota, the total weight, and the number of voters can affect the probability of observing inconsistent weighting.