Arrow’s Impossibility Theorem: What is it?

A social choice dilemma known as Arrow’s impossibility theorem highlights the drawbacks of ranked voting systems. It claims that following the necessary rules of fair voting procedures will prevent establishing a clear order of preferences. The broad impossibility theorem is another term for Arrow’s impossibility theorem, which bears Kenneth J. Arrow’s name as an economist.

Realizing the Impossibility of Arrow Theorem

To have a democratic society, voices must be heard. For instance, when a new government has to be created, elections are held, and voters turn out to cast their ballots. The winner of the election and the person who will serve as the next elected official are then determined by tallying millions of ballots.

In all situations where preferences are ranked, it is impossible to create a social ordering without disobeying one of the following rules, according to Arrow’s impossibility theorem:

• Non-dictatorship: Voters’ preferences should be taken into account.
• Pareto Efficiency: Individual choices must be honored even when they are unanimous. If candidate A is preferred by all voters over candidate B, candidate A should prevail.
• Independence of Irrelevant Alternatives: If one option is eliminated, the ranking of the remaining options shouldn’t change. For example, if Candidate A is ranked higher than Candidate B, candidate A should remain in that position even if Candidate C, a third contender, is eliminated.
• Unconstrained Domain: Voting must take into consideration each voter’s preferences.
• Social ordering: People should be allowed to rank the options however they like and mark any ties.

It was hailed as a significant advancement when Arrow’s impossibility theorem, a component of social choice theory that examines whether a society can be organized to represent individual preferences, was first proposed. Later, it was frequently utilized to analyze issues with welfare economics.

Arrow’s Impossibility Theorem illustration

Let’s have a look at an illustration of the kind of issues that Arrow’s impossibility theorem points out. Consider the following scenario: voters are asked to select their preferences for A, B, and C among three projects that the nation’s annual tax revenue may fund. It is up to the 99 voters in this nation to decide which of the three projects shall get annual financing in the following order: best to worst.

• One-third of the voters (33%) prefer A to B and B to C.
  One-third of the voters (33%) prefer B to C and prefer C to A.
  33 votes go C > A > B, with 1/3 choosing C over A and B over C.

Therefore,

• 66 people favor A over B.
  66 people favor B over C.
  66 people favor C over A.

Consequently, a two-thirds majority of voters choose A over B, B over C, and C over A, which is a contradictory outcome given that voters were required to rank their preferences for the three options.

According to Arrow’s Theorem, it is impossible to formulate a social ordering on a problem like the one mentioned above without breaking one of the following conditions: non-dictatorship, Pareto efficiency, independence of irrelevant alternatives, unrestricted domain. If the conditions cited above in this article, i.e., non-dictatorship, Pareto efficiency, independence of irrelevant alternatives, and social ordering, are to be part of the decision-making criteria.

The origins of Arrow’s Improbable Theorem

In honor of economist Kenneth J. Arrow, the theorem bears his name. The theory was first presented in Arrow’s Ph.D. thesis and later made public in his 1951 book Social Choice and Individual Values. Arrow had a lengthy career as a professor at Harvard University and Stanford University. He won the Nobel Memorial Prize in Economic Sciences in 1972 for his original article, A Difficulty in the Concept of Social Welfare.

In addition to these themes, Arrow’s research has examined the economics of racial discrimination, endogenous growth theory, collective decision-making, and social choice theory.

Conclusion

• Arrow’s impossibility theorem, a social-choice dilemma, demonstrates the difficulty of having the optimal voting system.
• It claims that following the necessary rules of fair voting procedures will prevent a clear order of preferences from being established.
• Kenneth J. Arrow was awarded the Nobel Memorial Prize in Economic Sciences for his discoveries.