This \(f\) satisfies SO because \(g\) does. It satisfies WP because there are more real voters than phantoms (\(2n\) to \(1\); we could have used any odd number of phantoms smaller than \(2n\)). \(f\) satisfies I because the phantom ordering is the same in all profiles of the expanded domain. If the domain includes sufficient variety among profiles then \(f\) also satisfies D and is Arrow consistent. This nice idea of phantom voters was introduced by Moulin (1980), who used it to characterize a class of voting schemes that are non-manipulable, in that they do not provide opportunities for strategic voting. 