Arrow’s Impossibility Theorem: Can We Hit a Societal Bullseye?

Many students have expressed discontent with our current voting system, especially as it relates to the ideas of “choosing between two bad options” or “wasting your vote” on a third candidate. Many articles, bills, and petitions have called for electoral reform, along with celebrities such as Lady Gaga. In this article I will go over the economic idea of a social choice rule, or mechanism which describes how we can move from individual preferences to group preferences, or social preferences. For a primer on the economic idea of preferences, see last week’s post, Accounting for Taste: Ice Cream Preferences.

One of the best articles I saw reflected back to the Republican primaries, in which the remaining two candidates, Trump and Cruz, had some of the highest unfavorability ratings in the country, and didn’t seem to represent moderate Republicans well. The economist Justin Wolfers described this outcome as related to the Condorcet Paradox, which shows how social preferences can appear cyclical, even if the individual preferences are not. To quote Wolfers,

Think about the Republican Party as an alliance of moderates, conservatives and populists. A coalition of populists and moderates will vote to ensure a populist beats a conservative. Conservatives and populists will vote together to ensure a conservative beats a moderate. And moderates and conservatives will join forces to help a moderate beat a populist.

In preference notation, this would be expressed as Populist ≻ Conservative ≻ Moderate ≻ Populist, which creates circular preferences violating the axiom of transitivity, such that a moderate candidate can be both better and worse than a populist candidate.

In attempting to create a true social preference, the economist Kenneth Arrow created several axioms of social preferences:
1. Unrestricted Domain: for any logically possible set of rational individual preferences, there is a social ordering R. For example, if someone A ≻ B, and B ≻ C, then transitivity would imply that  A ≻ C.
2. Independence of Irrelevant Alternatives: If an additional choice is added to the choices, for example adding choice C to the original choice set A and B, then adding C shouldn’t change the original preferences between A and B. An example of where this failed was in the 2000 presidential election between George W. Bush, Al Gore, and Ralph Nader. If the initial preferences were Gore ≻ Bush, then adding Nader shouldn’t change the outcome, but that obviously wasn’t the case.
3. Pareto Principle: If everyone prefers A to B, then B must not be chosen if A is available to be chosen.
4. Non-Dictatorship: There is no single voter i with the individual preference P, such that P is the winning social preference, unless all voters have the same P.

As Arrow looked at applying these axioms, he found that when voters have three or more options, a ranked voting electoral system cannot convert individual ranked preferences into a community-wide rankings while satisfying his 4 axioms. This result became known as Arrow’s Impossibility Theorem, and is applied to issues with different voting systems.

Lastly, I’ll end with a quote by Arrow himself, “Most systems are not going to work badly all of the time. All I proved is that all can work badly at times.”

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