Arrow's impossibility theorem: Difference between revisions
→Systems which evade Arrow's criteria
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</ref><ref>{{Cite web| title = Podcast 2012-10-06: Interview with Nobel Laureate Dr. Kenneth Arrow| work = The Center for Election Science| accessdate = 2020-03-20| date = 2015-05-25| url = https://www.electionscience.org/commentary-analysis/voting-theory-podcast-2012-10-06-interview-with-nobel-laureate-dr-kenneth-arrow/|quote=CES: Now, you mention that your theorem applies to preferential systems or ranking systems. Dr. Arrow: Yes CES: But the system that you’re just referring to, Approval Voting, falls within a class called cardinal systems. So not within ranking systems. Dr. Arrow: And as I said, that in effect implies more information.}}</ref> The typical example is [[score voting]] but there are also several [[Multi-Member System |multi-winner systems]]{{clarify}} which purport to pass all three of Arrow's original criteria. Additionally, there are cardinal systems which do not pass all criteria, but this is not due to Arrow's theorem; for example [[Ebert's Method]] fails [[Monotonicity]].
=== Benefits ===
However, note that one of the assumptions used in Arrow's Theorem is that voters do not change their preferences on a given set of candidates regardless of whether candidates not in the set are running or not running. If even a single voter [[Normalization|normalizes]] their rated ballot or otherwise deviates from this assumption, it fails. Example:<blockquote>1: A:10 B:6 C:0▼
There are two main "benefits" that come from evading Arrow's theorem: when candidates enter or drop out of the race, this doesn't impact the choice between the remaining candidates, and when voters are trying to impact the race between a certain set of candidates, they need only alter the portions of their ballot that show their preferences among that set of candidates.
▲However, note that to obtain the first benefit, one of the assumptions used in Arrow's Theorem is that voters do not change their preferences on a given set of candidates regardless of whether candidates not in the set are running or not running. If even a single voter [[Normalization|normalizes]] their rated ballot or otherwise deviates from this assumption, it fails. Example:<blockquote>1: A:10 B:6 C:0
1: B:10 C:4 A:0
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1: A:10 B:0</blockquote>Scores are A 20, B 10, and now A wins in Score voting. This example uses the standard [[Condorcet paradox]] but presented in rated form.
=== Caveats ===
However, subsequent social choice theorists have expanded on Arrow's central insight, and applied his ideas more broadly. For example, the [[Gibbard-Satterthwaite theorem]] (published in 1973) holds that any deterministic process of collective decision making with multiple options will have some level of [[strategic voting]]. As a result of this much of the work of social choice theorists is to find out what types of [[strategic voting]] a system is susceptible to and the level of susceptibility for each. For example [[Single Member system | Single Member systems]] are not susceptible to [[Free riding]].
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