[naniwaduni]

Perhaps surprisingly, no, approval voting is actually a score voting system, which provides more information than a strict preference ranking.

This isn't necessarily intuitive, and our immediate impulse might be to object that 2^n-1 is less than n!, but as a quick informal illustration, note that the voting system where each voter assigns each candidate a score in [0,1] and selects the candidate who earns the highest sum from all voters fairly straightforwardly violates Arrow's theorem. Now, consider an approval vote where each voter rolls a single random number in the range [0,1) and vote for each candidate whose score exceeds their random number, and we get asymptotically the same result (but with some error bars). It turns out that scoring, even on a 2-point scale, is just a better primitive operation than ranking!

(Also, it helps that 2^n-1>n! for n=2,3, which covers a surprisingly large proportion of interesting elections.)

[ClayShentrup]

Nailed it.