Question

Suppose there are three individuals in society trying to rank three social states \((A, B,\) and C). For each of the methods of social choice indicated, develop an example to show how (at least) one of the Arrow axioms will be violated. a. Majority rule without vote trading. b. Majority rule with vote trading. c. Point voting where each voter can give \(1,2,\) or 3 points to each alternative and the al ternative with the highest point total is selected.

Short answer

Question: Give an example that demonstrates each of the following social choice methods failing to satisfy one of Arrow's axioms: (a) Majority rule without vote trading, (b) Majority rule with vote trading, and (c) Point voting. Answer: (a) Majority rule without vote trading violates the Independence of Irrelevant Alternatives axiom, as shown in the scenario where removal of alternative C changes the ranking between A and B. (b) Majority rule with vote trading also violates the Independence of Irrelevant Alternatives axiom, as the same scenario from example a applies even with vote trading. (c) Point voting, with individuals allocating 1, 2, or 3 points per alternative, violates the Independence of Irrelevant Alternatives axiom when the removal of alternative B changes the ranking between alternatives A and C.

Step-by-step solution

Example a: Majority rule without vote trading

Consider these preferences for three individuals: Individual 1: A > B > C Individual 2: B > A > C Individual 3: B > C > A Using majority rule without vote trading, preferences are as follows: - A vs B: B wins (2-1) - B vs C: B wins (2-1) - C vs A: A wins (2-1) The alternative with the most wins is B, but in this scenario, the Independence of Irrelevant Alternatives axiom is violated. If we were to remove alternative C, the preferences would be: Individual 1: A > B Individual 2: B > A Individual 3: B > A B still wins (2-1), but the ranking between A and B has shifted.

Example b: Majority rule with vote trading

Consider these preferences for three individuals: Individual 1: A > B > C Individual 2: B > A > C Individual 3: C > B > A Suppose Individual 3 trades their vote with Individual 2, for B as the winner. This would result in: Individual 1: A > B > C Individual 2: B > A > C Individual 3: B > C > A Using majority rule with vote trading, the same results occur as in Example a, violating the Independence of Irrelevant Alternatives axiom.

Example c: Point voting (1, 2, or 3 points per alternative)

Consider these preferences and point allocations for three individuals: Individual 1: A (3 points) > B (2 points) > C (1 point) Individual 2: B (3 points) > A (2 points) > C (1 point) Individual 3: B (3 points) > C (2 points) > A (1 point) The total points for each alternative: A: 3+2+1 = 6 points B: 2+3+3 = 8 points C: 1+1+2 = 4 points Alternative B is selected as it has the most points. However, if alternative B is removed, the scenario will now look like this: Individual 1: A (2 points) > C (1 point) Individual 2: A (2 points) > C (1 point) Individual 3: C (2 points) > A (1 point) The total points for A and C with B removed: A: 2+2+1 = 5 points C: 1+1+2 = 4 points Alternative A is selected, but the ranking between A and C shifted. Thus, this method violates the Independence of Irrelevant Alternatives axiom.

Key concepts

Social Choice Theory

Social choice theory is a fascinating field that explores how collective decision-making can represent individual preferences within a society. This theory examines various methods used to aggregate individual preferences into a single societal choice, which can include voting, auctions, and other decision-making processes.

One of the main objectives of social choice theory is to design fair and efficient systems that consider every individual's preference. These systems aim to produce outcomes that reflect the collective will of the people. However, reaching a consensus that satisfies everyone's preferences is often challenging, and certain paradoxes can arise.

Arrow's Impossibility Theorem is a fundamental insight within social choice theory. It proves that no voting system can perfectly convert individual preferences into a collective decision without violating at least one fairness criterion when there are three or more options. This shows the inherent difficulties in creating a fair voting system that can truly capture the societal preference as a whole.

Independence of Irrelevant Alternatives

The Independence of Irrelevant Alternatives (IIA) is a crucial condition in Arrow's Impossibility Theorem. It states that if a group prefers option X over option Y, then introducing a new irrelevant option Z should not change the relative preference of X and Y. In everyday language, adding an alternative that wasn't initially considered should not affect the decision outcome between two known options.

However, as seen in the examples, this axiom is often violated in practical voting systems. For instance, when you remove or introduce a third option in some voting methods, like point voting or majority rule, it can shift the ranking of the original choices. In these cases, the decision between X and Y is influenced by the presence or absence of Z, which ends up being contrary to the IIA condition.

This violation highlights one of the key limitations of various voting methods, as it demonstrates that the results can be inconsistent when alternatives change. A fair and consistent decision-making process should ideally meet the IIA condition to ensure the reliability of collective choices.

Voting Systems

Voting systems are the mechanisms used to aggregate individual preferences into a collective decision. They play a central role in social choice theory and come in various types, such as majority rule, point voting, and ranked choice voting, each with its unique strengths and challenges.

The examples provided showcase different voting systems and how they can violate certain axioms, such as the Independence of Irrelevant Alternatives. With majority rule, individuals simply choose the option they prefer most, often leading to a winner based on who garners the most votes. However, this system can fail when votes are traded or when unexpected options appear, impacting the final decision.

Point voting allows individuals to distribute points to each choice according to their preference level. While it provides some flexibility, it struggles with the IIA condition, as changes in the options can influence the outcome unexpectedly. These issues underscore the difficulties in designing a perfect voting system that accurately reflects the collective will without inherent biases or paradoxes.