Every social ranking function corresponds to a social choice function, and vice-versa, which is why they’re equivalent. The Ranking→Choice direction is trivial.
The opposite direction starts by identifying the social choice for a given ranking. Then, you delete the winner and run the same algorithm again, which gives you a runner-up (who is ranked 2nd); and so on.
Social ranking is often cleaner than working with an election algorithm because those have the annoying edge-case of tied votes, so your output is technically a set of candidates (who may be tied).
Every social ranking function corresponds to a social choice function, and vice-versa, which is why they’re equivalent. The Ranking→Choice direction is trivial.
The opposite direction starts by identifying the social choice for a given ranking. Then, you delete the winner and run the same algorithm again, which gives you a runner-up (who is ranked 2nd); and so on.
Social ranking is often cleaner than working with an election algorithm because those have the annoying edge-case of tied votes, so your output is technically a set of candidates (who may be tied).