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 its 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 this latter type of scenario inconsistent weighting. We investigate the conditions that give rise to such a phenomenon within the class of weighted voting games. We also study how the choice of the quota and the total weight can affect the probability of observing inconsistent weighting. Finally, we investigate various applications where inconsistent weighting is observed