What is Arrow’s Impossibility Theorem?

Arrow’s Impossibility Theorem states that no ranked voting system can simultaneously satisfy a set of reasonable criteria when there are three or more distinct options. Here are the key criteria that any fair voting system would ideally meet:
– Unrestricted Domain: All voters’ preferences should be used to determine the outcome. This means that every possible combination of voter preferences should be considered.
– Non-Dictatorship: No single voter’s preference should control every possible outcome. In other words, no one person should have the power to dictate the result.
– Pareto Efficiency: If all voters prefer one option over another, the system should rank the preferred option higher. This ensures that if there is a unanimous preference, it is reflected in the outcome.
– Independence of Irrelevant Alternatives: The outcome should not be changed by adding or removing options that are not contenders. For example, introducing a new candidate who has no chance of winning should not affect the ranking between other candidates.
These criteria seem reasonable and intuitive, but Arrow’s Theorem shows that it is impossible to satisfy all of them simultaneously in a ranked voting system.

The Axioms and Assumptions

Let’s explore each of these axioms in more detail:

Unrestricted Domain

This axiom ensures that all possible voter preferences are considered. It means that voters can have any combination of preferences without restriction. For instance, in an election with three candidates (A, B, and C), voters can rank them in any order (e.g., A > B > C or C > B > A).

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Non-Dictatorship

This prevents any single voter from dictating the outcome. In a democratic system, this is crucial because it ensures that no individual has absolute power over the decision-making process.

Pareto Efficiency

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This ensures that if all voters prefer one option over another, the system ranks the preferred option higher. For example, if every voter prefers candidate A over candidate B, then candidate A should be ranked higher than candidate B in the final outcome.

Independence of Irrelevant Alternatives

This axiom is often violated in practice. For instance, consider an election where two candidates (A and B) are strong contenders and a third candidate (C) has no chance of winning but splits some votes from candidate A. The presence of candidate C could change the ranking between A and B even though C is not a serious contender.

Implications and Examples

The practical implications of Arrow’s Impossibility Theorem are significant:
– Plurality Voting: In a simple plurality voting system (where voters choose one candidate), introducing a new candidate can split votes and change the outcome even if the new candidate has no real chance of winning.
– Plurality with Runoff: Even systems that use runoff elections can fail to meet these criteria because adding or removing candidates can affect who makes it to the runoff.
– Democratic Processes: The theorem affects democratic processes by highlighting that no voting system can perfectly reflect collective preferences without violating at least one of these reasonable criteria.

Limitations and Exceptions

While Arrow’s Impossibility Theorem is powerful, it does have limitations:
– Straight Plurality Methods: The theorem does not apply to all voting methods; for example, straight plurality methods or votes between only two candidates do not face these issues.
– Cardinal Voting: Cardinal voting systems (where voters assign numerical scores to candidates) can provide more information and potentially avoid some of the issues highlighted by Arrow’s Theorem. However, these systems also have their own set of challenges.

Impact on Social Choice Theory and Collective Decision-Making

The broader impact of Arrow’s Impossibility Theorem on social choice theory is substantial:
– Mechanism Design: The theorem influences the design of mechanisms for collective decision-making. It forces designers to consider trade-offs between different criteria and to be aware that no system is perfect.
– Beyond Electoral Voting: The implications extend beyond electoral voting to other areas where collective decisions are made, such as committee decisions or policy-making processes.

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