How to dispell the myth that Arrows theorem applies to Cardinal systems

There is a weird argument that basically goes like this.

“I know Arrow’s theorem is not related to Cardinal systems but I like Ordinal systems so I need to defend them from Arrow’s theorem. Gibbard-Satterthwaite theorem applies to both Cardinal and Ordinal systems and it is somewhat similar to Arrow’s. Gibbard-Satterthwaite says all systems have strategic voting and Arrow’s says Ordinal systems suffer from specific strategic related flaws. Therefore, Arrows theorem applies to Cardinal systems”

Its wrong and illogical but the FairVote people have managed to get a ton of people to buy this. Is there a way we can get very clear statements publicly available?

1 Like
1. GS is about strategic voting. Cardinal systems have some amount of strategy, but crucially (especially in approval), most of it is deciding WHICH of several semi honest votes to cast, and you rarely have to totally flip a preference.
2. Arrow’s theorem is about IIA, which in real elections manifests as the spoiler effect and blocks 3rd parties from getting votes. Meanwhile, with Approval and Score, voting for your favorite NEVER hurts your compromise unless, of course, your favorite wins.

Its about Independence of Irrelevant Alternatives and Pareto Criterion. In the worst cases Pareto fails by failing monotonicity. But anyway, we all know this. How do we make it clear to others that this is the case?

If we take the following preferences

1 A>B>C
1 B>C>A
1 C>A>B

and assume voters in a cardinal method are following the polls and strategically exaggerate their preferences, then you’ll have an iterated Condorcet cycle in the polling (A is leading the polls, then suddenly C, then suddenly B, then repeat) where it isn’t clear who will win on Election Day. But now if one of the candidates drops out, you have a clear majority preference for one of the two remaining candidates, and under strategic voting, that majority-preferred candidate gets a boost in probability of winning. So while you can’t have a failure of IIA with cardinal methods when nobody changes their ballot to adjust to someone dropping out or entering the race, you can have them when voters do that, which it seems clear enough voters will do to make cardinal methods fail IIA at least once in practice.

Another obvious one:
26 A10 B9
25 A9 B10
49 C10
C wins here, but if one of A or B drops out, then under strategic voting, the remaining one of them may have a higher chance of winning (if the majority faction maximally supports the remaining candidate).
Or if the example is modified to have C-top voters originally give some marginal support to B (because they prefer B>A) then B would win, but if B drops out that might make C win.

Also, by this definition of Pareto

if every individual prefers a certain option to another, then so must the resulting societal preference order.

isn’t it at least theoretically possible for cardinal methods to fail Pareto if, say, between two terrible candidates, everyone just rates both of them a 0, since they’ll be perceived as equal in the final result, rather than one being better than the other?

It is possible for poorly designed Cardinal systems to fail every criteria. The proofs are about which criteria a well designed system must fail. Arrow’s theorem does not mean that Cardinal systems are great. Arrow’s theorem is not about Cardinal systems.

I think you guys have made my point. There is general confusion about these theorems.