Note that usually it’s a scale of 0–10 or 0–5, so there wouldn’t be scores of 53, etc.

But if you had asked them to rank only candidates 3 and 4, you might have got scores like 30, 10 so candidate 3 scores3 times as high as candidate 4.

Ranking has the same problem, though. If there are three candidates, and you would rank them A>B>E, but then C and D enter the race, and you would rank those A>B>C>D>E, then C and D have widened the gap between B and E, making the B>E preference appear three times as strong as it was with only 3 candidates.

The paradoxes of social choice arise from the fact that the ordinal scales that are taken as the starting point are also context dependent. On such a scale, the distance between two alternatives is given by the number of intermediate alternatives and changes as these are added or subtracted. This is the reason for the violation of Arrow’s axiom of independence of irrelevant alternatives. Both Fleming (1952) and Harsanyi 1955, 1977) have shown that social choice procedures that satisfy reasonable conditions are possible, as long as preferences are expressed by means of an independent cardinal scale.

Voting and the Cardinal Aggregation of Judgments

Constructing a consistent cardinal ranking is probably way more difficult and time consuming than constructing a consistent ordinal ranking.

That doesn’t make sense. Constructing consistent cardinal ratings just requires you to put candidates into a few categories. Constructing a consistent ordinal ranking requires you to make a bunch of pairwise comparisons, including between items that you don’t have a strong preference between.

Have you participated in ranked elections with many candidates? It’s not easy to produce a full ranking.