Question

Exercise 3: Voting

Three politician are holding a vote to see which policy they should prompt in their next joint campaign. Here are their preferences:

John Mary                 Will

1st choice     Education      Job creation       Healthcare

2nd choice   Job creation    Healthcare           Education

3rd choice    Healthcare    Education          Job creation

(a)The politicians vote by majority rule. If the vote is Education vs Job creation, which policy will win? Job creation vs Healthcare? And Healthcare vs Education?

(b)Define or explain transitive preferences. In the current example, is the aggregation of preferences by pairwise voting transitive? Explain.

(c)They decide to vote in a single-elimination rule: two votes and the winner of the first round proceeds on to the second (final) round. Now, suppose John is in charge of deciding which order to hold the votes. He wants to make sure that his favorite policy is selected. How should John stack the order of voting?

(d)Consider Mary is not a sincere voter while the other students are; that is, Mary votes strategically while the others vote in their most preferable choice. Would the result in (c)change? Explain.

Answer 1

Arrow's impossibility theorem is a social-choice paradox illustrating the flaws of ranked voting systems. It states that a clear order of preferences cannot be determined while adhering to 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 TAKEAWAYS

• Arrow's impossibility theorem is a social-choice paradox illustrating the impossibility of having an ideal voting structure.
• It states that a clear order of preferences cannot be determined while adhering to mandatory principles of fair voting procedures.
• Kenneth J. Arrow won a Nobel Memorial Prize in Economic Sciences for his findings.

Understanding Arrow's Impossibility Theorem

Democracy depends on people's voices being heard. For example, when it is time for a new government to be formed, an election is called, and people head to the polls to vote. Millions of voting slips are then counted to determine who is the most popular candidate and the next elected official.

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: 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 a third candidate, 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, part of social choice theory, an economic theory that considers whether a society can be ordered in a way that reflects individual preferences, was lauded as a major breakthrough. It went on to be widely used for analyzing problems in welfare economics.

Example of Arrow's Impossibility Theorem

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 of candidates A, B and C:

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

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

Special Considerations

Arrow’s 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, that don’t 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 in his doctoral thesis and later popularized it in his 1951 book Social Choice and Individual Values. The original paper, titled A Difficulty in the Concept of Social Welfare, earned him the Nobel Memorial Prize in Economic Sciences in 1972.

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

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