Professor Stephen Unger (Columbia University, NY)
explains why range voting has, and approval voting can have,
greater expressive power than ranked voting methods

Assume 5 voters and 3 candidates A, B, C.
Suppose the ranked votes are:

#voters

Their Vote

2

A>B>C

1

B>A>C

1

C>B>A

1

C>A>B

Then the "Condorcet winner" (beats all rivals in man-to-man two-choice elections)
is A.

Suppose we interpret the first and third voting patterns to mean that
the sincere evaluations of the 3 candidates are "excellent," "very good,"
"terrible," and assign corresponding weights 9, 8, 0, and that we
interpret the second and fourth patterns to mean "excellent," "terrible,"
"worst," then the weights might be 9, 1, 0.

The corresponding range votes would be:

#voters

Their Vote

2

A=9, B=8, C=0

1

A=1, B=9, C=0

1

A=0, B=8, C=9

1

A=1, B=0, C=9

Then the Range Voting
winner is B (summed-score 33=2×8+9+8+0).

For Approval Voting
it would be reasonable to consider 8 as approval and 1 as not
approved, so the Approval Vote would look like:

#voters

Their Vote

2

A=1, B=1, C=0

1

A=0, B=1, C=1

1

A=0, B=0, C=1

The Approval Voting winner is B.

But if we took the same ranked votes and, for lines 1 and 3 considered
the rankings to be "excellent," "terrible," "worse," and line 2 to mean
"excellent," "very good," and "terrible," we would find, when the weights
are changed appropriately, that A would win both the Range and Approval
elections.

This illustrates how both Range and Approval can have more discriminating power
then any ranked system. Or we could say that Approval and Range both allow the voter
to be more precise.

For ranking systems where equal ranking is permitted (at present, most do not permit that),
then we can think of Approval as the simplest case of such systems and
therefore such ranking systems are more powerful than Approval (but remain
weaker than Range).
But then we have to look at complexity.
For both
voters, and, more important, for the tallying process, Range and Approval both are
much simpler.