It's a minor point, but Arrow's work first appeared in the early 1950s, not the 1960s.
More significant, there are voting systems that ask voters to rate candidates, not rank them, and thereby escape the consequences of Arrow's theorem. These include approval voting, of which I am an advocate, that ask voters simply to approve or not approve of candidates. But range or score voting, and a new system proposed by Balinski and Laraki, majority judgment voting, allow finer ratings, differing on whether they elect the candidate with the highest mean or median rating. Note that ratings are not equivalent to rankings--for example, a voter can rate all except one candidate tops--so Arrow's framework that assumes preference rankings is not applicable to such systems.
Beginning in the 1980s, approval voting was adopted by a number of science and engineering societies in multicandidate elections (i.e., with more than two candidates). Pleasingly, it almost always elects a consensus choice, who is generally a Condorcet winner (a candidate who can beat all other candidates in pairwise contests).
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