# Understanding Arrow's Impossibility Theorem: Definition, History, and Example

## What Is Arrow's Impossibility Theorem?

Arrow's Impossibility Theorem reveals the inherent challenges in ranked voting systems, demonstrating why a fair preference order is unattainable without breaking key voting principles. Named after [economist](https://www.investopedia.com/terms/e/economist.asp) [Kenneth J. Arrow](https://www.investopedia.com/terms/k/kenneth-arrow.asp), this foundational theory highlights the paradoxes in achieving a perfect voting system.

### Key Takeaways

*   Arrow's impossibility theorem demonstrates the flaws in ranked voting systems, making it impossible to establish a socially ideal order of preferences while meeting fair voting principles.  
    
*   The key conditions challenged by Arrow's theorem include nondictatorship, Pareto efficiency, independence of irrelevant alternatives, unrestricted domain, and social ordering.  
    
*   An example of Arrow's theorem is the voting paradox, where a group’s multiple rankings lead to inconsistent collective preferences.  
    
*   Economist Kenneth J. Arrow introduced the theorem, earning a Nobel Memorial Prize for his significant contributions to social choice theory.  
    
*   Despite the theorem's implications, alternative voting methods like approval or plurality voting are not constrained by these same assumptions.

## Dive Deeper into Arrow's Impossibility Theorem

Democracy relies on everyone's voice being heard. In elections, people vote to choose the next leader. Votes are counted to find the most popular candidate.

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

*   **Nondictatorship**: The wishes of multiple voters should be taken into consideration.
*   [**Pareto Efficiency**](https://www.investopedia.com/terms/p/pareto-efficiency.asp): Unanimous individual preferences must be respected; if every voter prefers candidate A over candidate B, candidate A should win.
*   **Independence of Irrelevant Alternatives**: 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 third candidate C is removed from participation. 
*   **Unrestricted Domain**: Voting must account for all individual preferences.
*   **Social Ordering:** Each individual should be able to order the choices in any way and indicate ties.

Arrow's impossibility theorem is part of [social choice theory](https://www.investopedia.com/terms/s/social-choice-theory.asp), which studies how a society can reflect individual preferences. It became an important tool for analyzing issues in [welfare economics](https://www.investopedia.com/terms/w/welfare_economics.asp).

## Illustrating Arrow's Impossibility Theorem With a Practical Example

Let’s look at an example illustrating the type of problems highlighted by Arrow's impossibility theorem. Consider the following example, where voters are asked to rank their preference among three projects: A, B, and C. This country has 99 voters who are each asked to rank the order, from best to worst, for which of the three projects should receive the annual funding.

*   33 vote A > B > C (1/3 prefer A over B and prefer B over C)
*   33 vote B > C > A (1/3 prefer B over C and prefer C over A)
*   33 vote C > A > B (1/3 prefer C over A and prefer A over B)

In this scenario, 66 voters prefer A over B, 66 voters prefer B over C, and 66 voters prefer C over A.

This represents a paradoxical result, as a two-thirds majority of voters prefer A over B, B over C, and C over A.

Arrow’s theorem posits that it is impossible to formulate a social ordering on a problem such as indicated in this example without violating one of the above conditions.

> **Fast Fact**
>
> Arrow’s impossibility theorem is also applicable when voters are asked to rank political candidates. However, there are other popular voting methods, such as approval voting or plurality voting, that do not use this framework.

## Unveiling the Legacy and History of Arrow's Impossibility Theorem

Named after economist Kenneth J. Arrow, the theorem was introduced in his doctoral thesis and later popularized in his 1951 book, "Social Choice and Individual Values." Arrow taught at Harvard and Stanford during his long career. His original paper, "A Difficulty in the Concept of Social Welfare," won him the [Nobel Memorial Prize in Economic Sciences](https://www.investopedia.com/terms/n/nobel-memorial-prize-in-economic-sciences.asp) in 1972.

Arrow's research has also explored the social choice theory, [endogenous growth theory](https://www.investopedia.com/terms/e/endogenousgrowththeory.asp), collective decision-making, the economics of information, and the economics of racial discrimination, among other topics.

## What Is Social Choice Theory?

Social choice theory is a field of study focused on the mechanisms of collective decision-making. The central questions of the field include voting systems and group choices. It has ramifications in economics, politics, behavioral science, and beyond.

## Who Was Kenneth J. Arrow?

Kenneth J. Arrow was an American economist well-known for his contributions to general equilibrium theory and welfare theory. In 1972, Arrow became the youngest person to be awarded the Nobel Memorial Prize in Economic Sciences. Arrow taught at numerous institutions, including Stanford University, Harvard University, and the University of Chicago, and at least five of his students have gone on to win the Nobel Prize, as well.

## What Is Independence of Irrelevant Alternatives?

Independence of irrelevant alternatives is an assumed condition within decision theory that states that preferences among options should be unchanged if additional but irrelevant alternatives are introduced. In the context of an election, consider a ballot with multiple candidates. If a losing candidate were to be eliminated from the ballot, the result of the election should not change under the condition of independence of irrelevant alternatives. If it does change, then the condition has been violated.

## The Bottom Line

Arrow's impossibility theorem demonstrates the inherent limitations of ranked voting systems by showing that no voting method can satisfy all key fairness conditions simultaneously: nondictatorship, Pareto efficiency, independence of irrelevant alternatives, unrestricted domain, and social ordering.

This theorem, a crucial concept within social choice theory, underscores the complexity and challenges in collective decision-making. It highlights the work of Kenneth J. Arrow, whose insights revolutionized the understanding of voting systems and earned him a Nobel Prize in Economic Sciences.

> **Article Sources**
>
> 1.  Stanford University, Stanford Encyclopedia of Philosophy. "[Arrow's Theorem](https://plato.stanford.edu/entries/arrows-theorem/)."
>     
> 2.  The Nobel Prize. "[Kenneth J. Arrow: Facts](https://www.nobelprize.org/prizes/economic-sciences/1972/arrow/facts/)."
>     
> 3.  Stanford University. "[Nobel Prize-Winner Kenneth Arrow Dies](https://news.stanford.edu/stories/2017/02/nobel-prize-winner-kenneth-arrow-dies)."
>     
> 4.  Stanford University, Stanford Magazine. "[Nobel Laureate and Economist](https://stanfordmag.org/contents/nobel-laureate-and-economist#:~:text=At%20age%2051%2C%20Kenneth%20Arrow,Palo%20Alto%20on%20February%2021.)."
>     
> 5.  University of Nebraska-Lincoln. "[Independence of Irrelevant Alternatives and Arrow's Theorem](https://mathbooks.unl.edu/Contemporary/sec-5-5-arrow.html)."