Question

Suppose seven individuals constitute a society in which individuals cast votes for their most preferred social arrangement and that the arrangement with the greatest number of votes is always chosen. Devise an example of individual rankings of the three states \(A, B,\) and Csuch that state \(A\) is chosen when all three states are available but that state \(B\) is chosen if the "irrelevant" alternative Cis not available. (This amounts to showing that the constitution of this society does not obey Axiom 4 in Arrow's list.) How reasonable is your example? What does it indicate about the nature of Arrow's axiom?

Short answer

And what is the result when state C is removed? Answer: In the given example, when all three states (A, B, and C) are available for voting, state A is chosen as the societal preference. However, when state C is removed, state B becomes the preferred option.

Step-by-step solution

Define individual rankings

Let's assign rankings to the seven individuals for the three states \(A\), \(B,\) and \(C\): 1. \(ABC\) 2. \(BCA\) 3. \(BAC\) 4. \(AB.\ C\) 5. \(AB.\ C\) 6. \(CB.\ A\) 7. \(CBA\) We can now calculate their aggregate ranking and determine which state is preferred when all three states are available.

Calculate aggregate ranking when all states are available

To calculate the aggregate ranking, we will count how many times each state is ranked first, second, and third in the individual rankings. - \(A\) is ranked first in rankings 1, 4, and 5 (3 times). - \(B\) is ranked first in rankings 2, 3, and 6 (3 times). - \(C\) is ranked first in ranking 7 (1 time). Since state \(A\) has the greater number of first-place rankings (3) compared to state \(B\) (3) and state \(C\) (1), state \(A\) is chosen when all three states are available.

Calculate aggregate ranking when state C is not available

Now, let's remove state \(C\) from consideration and update the individual rankings accordingly: 1. \(AB\) 2. \(BA\) 3. \(BA\) 4. \(AB\) 5. \(AB\) 6. \(BA\) 7. \(BA\) Again, we will count how many times each state is ranked first and second in the updated individual rankings: - \(A\) is ranked first in rankings 1, 4, and 5 (3 times). - \(B\) is ranked first in rankings 2, 3, 6, and 7 (4 times). Since state \(B\) has the greater number of first-place rankings (4) compared to state \(A\) (3), state \(B\) is chosen when state \(C\) is not available.

Analyze the example and discuss the nature of Arrow's axiom

In this example, we have demonstrated that the constitution of this society does not obey Axiom 4 in Arrow's list. When all three states \(A\), \(B\), and \(C\) are available, state \(A\) is chosen as the societal preference. However, when state \(C\) is not available, state \(B\) is chosen over state \(A\). This indicates that the societal preference does not obey the independence of irrelevant alternatives, as removing the "irrelevant" alternative \(C\) changes the societal preference. This example demonstrates that the society's constitution does not always satisfy Arrow's Axiom 4 and can produce an unreasonable outcome. This indicates that the nature of Arrow's axiom may not always accurately represent societal preferences, as the removal of an alternative can change the outcome even though the preferences for the available options remain the same among the individuals. It also demonstrates the complexity of aggregating individual preferences into a well-defined group preference order.

Key concepts

Social Choice Theory

Social Choice Theory is a framework for analyzing collective decision-making processes, such as those involved in voting or policy selection. It attempts to understand how individual preferences can be aggregated to reach a collective decision that is representative of a society or group.

The theory explores various methods of voting and the conditions under which they will lead to fair and rational outcomes. Central to social choice theory is the idea that the preferences of individuals can be diverse and sometimes conflicting, yet a collective decision must be made that somehow reflects these preferences. Arrow's Impossibility Theorem, a fundamental result within social choice theory, shows the challenges inherent in trying to meet several fair criteria simultaneously when aggregating individual preferences into a group decision.

The aim of social choice theory is not only to study these mechanisms but also to understand the implications and limitations of different voting systems and how they impact group outcomes.

Voting Paradox

The Voting Paradox, also known as Condorcet's Paradox, highlights a situation in democratic processes where collective preferences cannot be determined consistently from individual rankings. This paradox occurs when the preferences of individuals within a group are cyclical, meaning there is no clear winner that can be defined.

For example, if a group prefers candidate A over B, B over C, and yet C over A, we have a cycle that does not lead to a transitive preference order for the group. This is problematic as it prevents a clear majority preference, and therefore, a fair and definitive decision cannot be reached.

The paradox is a fundamental problem in voting theory and demonstrates that even if each individual's preferences are rational and transitive, the group's collective preference can still be intransitive and therefore irrational. This challenges the idea that simply 'majority rules' can always lead to fair or rational social decisions.

Independence of Irrelevant Alternatives

The Independence of Irrelevant Alternatives (IIA) is a principle in social choice theory, particularly within the context of voting systems. It states that the collective preference between any two alternatives should not be affected by the introduction or removal of a third, unrelated alternative.

In essence, if society prefers option X over option Y, introducing a new option Z should not change this order of preference. Likewise, removing Z should not affect the established preference between X and Y. This principle is essential because it ensures that the group's choice reflects true preferences among the options, without being biased by other irrelevant factors.

Arrow's Impossibility Theorem highlights that no voting system can satisfy IIA together with other reasonable democratic conditions, such as unrestricted domain, non-dictatorship, and Pareto efficiency. The exercise involving states A, B, and C provides a clear example of how the introduction or exclusion of a choice (like state C) can influence the group's preference order, violating the IIA condition.