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Arrow’s Impossibility Theorem: Definition – Explanation – Example

What is Arrow’s Impossibility Theorem?

Arrow's Impossibility TheoremArrow’s Impossibility Theorem asserts that a clear order of preferences is impossible to determine while adhering to mandatory fair voting procedures. In other words, the Theorem states that converting individuals’ preferences from a fair ranked-voting election system cannot yield clear community-wide ranked preferences. The theorem is sometimes known as “The General Possibility Theorem” or “Arrow’s Paradox.” The research was initially based on economics, but has strong social choice applications. Arrow’s impossibility theorem is a social-choice conundrum that highlights the shortcomings of ranked voting systems. Arrow’s impossibility theorem is also known as the universal impossibility theorem, after economist Kenneth J. Arrow.

According to the impossibility theory, it is impossible for a ranked-voting system to attain a community-wide order of preferences.  Specifically,  by collecting ranking preferences while meeting a set of constraints when there are more than two possibilities. However, those same prerequisites are the requirements for a reasonably fair voting procedure.

Arrow’s Impossibility Theorem – A Closer Look

Arrow’s impossibility theorem is a social choice theory.  It investigates the combination of individual preferences, welfares, and opinions to attain asocial welfare or community-wide decisions. Moreover, it examines the shortcomings of a ranked-voting election system. The theory suggests it is impossible for a ranked-voting system to attain a community-wide order of preferences by collecting and converting individuals’ preference orders while meeting a set of constraints. Particularly, when there are more than two possibilities.  However, these prerequisites are the requirements for a reasonably fair voting procedure.

People’s voices must be heard for democracy to function. When it comes time to form a new government, an election is held.  Of course, that’s when people go to the polls to vote and millions of ballots are then counted.  The goal is to decide who the most popular candidate is and who will be the next elected politician. However, according to Arrow’s impossibility theorem, it is impossible to establish a social ordering in all circumstances when preferences are ranked without breaking one of the conditions.  The theorem asserts that no rank-order electoral system can be created that meets all fairness criteria all the time.

Fairness Conditions in Arrow’s Impossibility Theorem

There is a set of conditions (criteria) for a reasonably fair electoral procedure. It includes non-dictatorship, unrestricted domain, independence of irrelevant alternatives, social ordering, and Pareto efficiency.

Non-Dictatorship

Non-dictatorship implies that a single voter and his or her preferences cannot represent the entire society. The social welfare function must take into account the wishes of multiple voters. As a result, there is no single voter with the ability to always determine the group’s preference.

Unrestricted Domain

Unrestricted domain necessitates counting all of the preferences of every voter.  The result is a complete ranking of social preferences and voting should account for all individual preferences.

Independence of Irrelevant Alternatives (IIA)

The independence of irrelevant alternatives condition states that when individuals’ rankings of a subset’s irrelevant alternatives change, the subset’s social ranking should not change. If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged.  This should hold true even if voters’ preferences between other pairs like X and Z, Y and Z, or Z and W change. In other words, if a choice is removed, then the others’ order should not change. If candidate A ranks ahead of candidate B, candidate A should still be ahead of candidate B, even if a third candidate, candidate C, is removed from participation.

Social Ordering

The social ordering condition necessitates that voters be able to order their options in a connected and transitive manner.  For example, from better to worse. Each person should be allowed to organize the options any way they like and identify ties.

Pareto Efficiency

Individuals’ unanimous preferences must be respected for Pareto efficiency. If every voter strictly favors one of the alternatives over another, the order of social preferences must correspond with the order of individual preferences. The outcome should be unaffected by the preference profile and individual choices must be respected.  Therefore, if every voter chooses candidate A over candidate B, candidate A should be elected. (Source: corporatefinanceinstitute.com)

Critique of Arrow’s Impossibility Theorem

Arrow’s impossibility theorem was initially put forth as an economic theory.  It explores whether a society can be organized in a way that represents individual preferences. However, it was quickly hailed as a major development in social choice theory. The theory is still widely utilized to analyze problems in welfare economics.

Disadvantages

While Arrow’s theorem has been widely regarded as a seminal piece of work in the field of social choice, it does have a few issues.

  • Preferences – Arrow accounts for an individual voter’s order of preference.  However, he does not account for the intensity of each preference. On a scale of 1 to 5, how much more does a voter prefer their first choice over their second? Sometimes the difference may be marginal (1) and other times, the difference may be great (5).
  • Interpersonal comparison – The theorem also omits interpersonal comparison between voters. Essentially, voter A might have more to gain by Candidate X winning the election as compared to voter B. B may be significantly worse-off than A if Candidate X wins the election for whatever reason.
  • Rational assumption – Arrow’s theorem makes the assumption that voter preferences are rational. Throughout time, society does not always behave rationally.

Are All Elections Unfair?

People frequently claim that Arrow demonstrated that “there are no good or fair election processes. This is not correct because many election procedures are not covered by the hypotheses of Arrow’s theorem. Arrow’s result, in particular, applies only to procedures in which voters rank all candidates.  These criteria are not met by many popular voting techniques, such as approval voting or plurality voting. Furthermore, in any particular setting, one may question whether the “reasonable” requirements are genuinely reasonable. And Arrow’s Theorem does not apply if there are only two candidates.  In that common scenario, it is clear that plurality voting is a social welfare function that fulfills fair election criteria. Plurality voting is expressing a preference for one candidate over the other.  There are no further conditions as there are no other candidates.

Example of Arrow’s Impossibility Theorem

Consider an example of the type of difficulties highlighted by Arrow’s impossibility theorem. Imagine a scenario where voters are asked to rate their preference for three potential council members: X, Y, and Z. As a result, the voters rate the three individuals in order of preference, from best to worst.

  • 1/3 of voters prefer Candidate X over Y, and prefer Y over Z
  • 1/3 of voters prefer Candidate Y over Z, and prefer Z over X)
  • 1/3 of voters prefer Candidate Z over X, and prefer X over Y)

Therefore,

  • 2/3 of voters prefer X over Y
  • 2/3 of voters prefer Y over Z
  • 2/3 of voters prefer Z over X

So a two-thirds majority of voters prefer X over Y and Y over Z and Z over X.  This is a perplexing result based on the obligation to rank order the three choices. However, there are other popular voting techniques that do not require order ranking of the candidates.  For example approval voting or plurality voting.

Nevertheless, according to Arrow’s theorem, it is impossible to formulate a social ordering on the election outlined above.  Not without violating one of the conditions designated as part of the decision-making criteria. Specifically, non-dictatorship, Pareto efficiency, independence of irrelevant alternatives, unrestricted domain, and social ordering,

History of Arrow’s Impossibility Theorem

The theorem bears the name of economist Kenneth J. Arrow who taught at Harvard and Stanford for many years. He introduced the theory in his doctoral thesis and popularised it in his 1951 book Social Choice and Individual Values. In 1972, Arrow was awarded the Nobel Memorial Prize in Economic Sciences for his original paper, A Difficulty in the Concept of Social Welfare. Arrow’s research has also explored social choice theory, endogenous growth theory, collective decision making, information economics, and racial discrimination economics, among other things.

John R. Hicks and Kenneth J. Arrow “for their pioneering contributions to general economic equilibrium theory and welfare theory” (Source: nobelprize.org)

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Tax Credit ApartmentsTax credit apartments are housing projects or apartment complexes that let landlords or owners claim tax credits in exchange for offering units to low-income tenants with restricted rents. In general, tax credit apartments are complexes owned by a developer or landlord who participates in the Federal Low-income Housing Tax Credit (LIHTC) Program. These owners can claim tax credits for eligible buildings in return for renting some or all of the apartments to low-income tenants at a restricted rent.

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