# What is Arrow’s impossibility theorem [Explained with example]

Arrow’s impossibility theorem is very important in welfare economics. The theorem proves that it is impossible to develop a fully fair and transitive system of voting. Kenneth Arrow tried to develop a fair system but failed to do so. In this article, we shall discuss this theorem in detail and will also discuss examples to have a better understanding of how it is very important in today's time.

## History

The theorem is named after a very economist Kenneth J. Arrow who introduced this theorem in his PhD thesis. Kenneth has a long respectable career at Harvard and Stanford University. The theorem was highlighted and recognized as important in later years around 1951 when Kenneth published his book named “social choice and individual values”. Later another very popular research in welfare economics earned him the Nobel prize in economic sciences in the year 1972. The theory of Arrow’s impossibility is still very applicable in welfare economics.

## What is Arrow’s impossibility theorem [Explained with example]

Arrow’s impossibility theorem is a very interesting and intriguing phenomenon that focuses on the paradox of social choices explaining all the flaws and problems of ranked systems of voting. The theorem states that any clear or accurate set of preferences cannot be predicted while sticking to the compulsory principles of fair voting methods. Arrow’s impossible theorem was first introduced by a famous economist named Kenneth J. Arrow. The other name for this theorem is the general impossibility theorem.

The modern world is based on the principles of democracy, most countries around the world follow democracy for electing their representatives and leaders. Democracy is strong when the voice of the majority is heard, and a fair system is developed for electing the most preferred individual. The most common example is general elections, when one regime completes its reign and it's time for a new leader to be elected, elections are called, and people go to specific voting polls to cast their votes. All the voting slips are then counted by officials to determine the chosen leader, and the person with the most votes wins.

## Understanding Arrow’s impossibility theorem

According to Arrow’s theorem whichever situation in which preferences are to be ranked, it becomes impossible to formalize a general social ordering without breaking one of the following rules.

• Non-dictatorship: The choice and preference of the majority must be considered the decision must be open and fair. All people must be able to express their preferences and make their own decisions.
• Pareto efficiency: Unanimous decisions must be respected if all the individuals are voting for participant A over B. Then participant A will win.
• Independence of alternative choices: If one choice is removed from the category the rest of the order must stay the same. For example, if participant A gets more votes than participant B and C. Then even if participant C withdraws, the winner must be A regardless.
• Unrestricted Domain: All individuals being impacted by the decision must be allowed to vote.
• Social ordering: this condition requires that the voters must be able to exercise their choices to vote even if their choices are interrelated and are in order from best to best.

Arrow concluded that while choosing between more than two options, it becomes impossible to maintain all the mentioned conditions. An example is of Plurality method, which allows individuals to vote for their favorite candidate and under these conditions the person with less than 50% can also become the winner.

Arrow’s impossibility theorem plays a very crucial role in welfare economics. The theory is a part of social choice theory, a very important theory in economics that focuses on the fact that can a society be ordered in a way that reflects the choices of most individuals. There are two types of arrangements described in this theorem transitive and intransitive. Transitive means that the preferences are arranged in a sensible order.

For example, if Apples are preferred over mangoes and mangoes are preferred over strawberries then the order is said to be transitive. On the other hand, if the order of preferences is such that Apples are preferred over mangoes, Mangoes are preferred over strawberries, but strawberries are preferred over apples. Then this order does not make sense and hence it is intransitive.

## Example

To better understand the theorem, it is very important to consider an example. Let’s assume that three friends go to McDonald’s for lunch. To make this example easier let’s say that there are only three different types of burgers available at McDonald’s. The three friends will have to declare their preferences for the variety of burgers available. The three friends are named John, Brady, and Carol. All three friends have to rank their choices according to their tastes, in order from best to worst. After carefully thinking and making their choices, all three friends rank their choices as follows:-

John: ABC

Carol: CAB

The results can be explained as follows:-

John prefers A over B and B over C.

Brady prefers B over C and C over A.

Carol prefers C over A and A over B.

So, we can say that:-

2/3 prefer A over B

2/3 prefer B over C

2/3 prefer C over A

A paradox is occurring as 2/3 of each of the majority prefers A over B, B over C and C and A. hence, the Kenneth impossibility theorem gets validated. To consider a practical real-life example the US elections of 1992 can be studied. Bill Clinton won the elections with only 43% of votes while his competitors George W. Bush and Ross Perot got 38% and 19% votes respectively.

## Conclusion

To conclude, Arrow’s impossibility theorem explains that it is impossible to construct a perfect voting system for voting out a single winner, who also satisfies all the criteria. No one individual can pick out the winner, the preference of the majority must be considered. Voters can arrange their preferences in whatever way they deem suitable. If all the voters select one candidate over the other then the other candidate cannot win. The presence or absence of a candidate who did not win must not impact the overall results of the elections.

The Arrow’s impossibility is most used in welfare economics. There have been many other theories that have been inspired by this theorem. Voting systems must be transparent and allow the voice of the majority to come through. The absence of a fair voting system can lead to non-democratic decisions. Through his research Arrow aimed to set the ground for the development of a fair and transparent voting system, but while doing this he proved that it is impossible to develop a truly transitive system.