# Arrow’s Impossibility Theorem Definition

### What is Arrow’s Impossibility Theorem?

Arrow’s impossibility theorem is a paradox of social choice illustrating the shortcomings of classified voting systems. It indicates that a clear order of preference cannot be determined while respecting the mandatory principles of fair voting procedures. Arrow’s impossibility theorem, named after economist Kenneth J. Arrow, is also known as the general impossibility theorem.

### Key points to remember

• Arrow’s impossibility theorem is a paradox of social choice illustrating the impossibility of having an ideal voting structure.
• It indicates that a clear order of preference cannot be determined while respecting the mandatory principles of fair voting procedures.
• Kenneth J. Arrow won a Nobel Prize in economics for his discoveries.

### Understanding Arrow’s Impossibility Theorem

Democracy depends on the voice of people being heard. For example, when it is time to form a new government, an election is called and people go to the polls to vote. Millions of ballots are then counted to determine who is the most popular candidate and the next elected one.

According to Arrow’s impossibility theorem, in all cases where preferences are classified, it is impossible to formulate a social order without violating one of the following conditions:

• Non-dictatorship: The wishes of several voters must be taken into account.
• Pareto efficiency: Unanimous individual preferences must be respected: if each voter prefers candidate A to candidate B, candidate A should prevail.
• Independence of irrelevant alternatives: If a choice is withdrawn, the order of the others should not change: if candidate A ranks ahead of candidate B, candidate A must always be ahead of candidate B, even if a third candidate, candidate C, is withdrawn from participation.
• Unlimited domain: Voting must take into account all individual preferences.
• Social order: Each individual should be able to order the choices in any way and indicate the links.

Arrow’s impossibility theorem, which is part of social choice theory, an economic theory that considers whether a society can be ordered in a way that reflects individual preferences, has been hailed as a major breakthrough. It was then widely used to analyze the problems of the welfare economy.

### Example of Arrow’s impossibility theorem

Let’s look at an example illustrating the type of problems highlighted by Arrow’s theorem of impossibility. Consider the following example, where electors are asked to rank their preference over candidates A, B and C:

• 45 votes A> B> C (45 people prefer A to B and prefer B to C)
• 40 votes B> C> A (40 people prefer B to C and prefer C to A)
• 30 votes C> A> B (30 people prefer C to A and prefer A to B)

Candidate A has the most votes, so he / she will be the winner. However, if B was not in the running, C would be the winner, because more people prefer C than A. (A would have 45 votes and C would have 70). This result is a demonstration of Arrow’s theorem.

### Special considerations

The Arrow Impossibility Theorem is applicable when voters are asked to rank all candidates. However, there are other popular voting methods, such as approval voting or plurality voting, which do not use this framework.

### History of Arrow’s Impossibility Theorem

The theorem is named after economist Kenneth J. Arrow. Arrow, who had a long teaching career at Harvard University and Stanford University, introduced the theorem into his doctoral thesis and then popularized it in his 1951 book, Social Choice and Individual Values. The original document, titled A Difficulty in the Concept of Social Welfare, won him the Nobel Prize in Economics in 1972.

Arrow’s research has also explored social choice theory, endogenous growth theory, collective decision-making, the information economy and the economics of racial discrimination, among other topics.