Evaluations: Clones, Teaming and Independence Criteria 2

Arrow's Impossibility Theorem

The search for a perfectly fair voting system ended abruptly when Kenneth Arrow established his famous Impossibility Theorem. Given the four reasonable requirements for fairness stated below, he showed that no voting system could satisfy all four criteria simultaneously. Any particular electoral method with two or more voters choosing between three or more options would not meet one or more of these four demands.

• Non-dictatorship: The election outcome rankings must not always reflect the preferences of one particular voter but should treat all voters equally.
• Unrestricted domain: Each voter may strictly rank all the candidates in any order they wish and the resultant election outcome must yield a unique and complete ranking of all those candidates.
• Pareto condition: This condition now replaces the earlier monotonicity and non-imposition criteria. See the earlier Evaluations: General Criteria section for both the Pareto and monotonicity requirements. Non-imposition means that any possible permutation of election outcome rankings is permissible.
• Independence of Irrelevant Alternatives (IIA): See the preceding page.

As GV must fail at least one of these four requirements, how does it measure up? See below.

• Non-dictatorship: GV satisfies this criterion as it treats all voters equally.
• Unrestricted domain: GV satisfies this criterion too. Every voter is free to rank the candidates in any preferred order. The election outcome rankings will include every candidate and, as the GV algorithm is deterministic, the same unique outcome will be produced every time the same election and profile is rerun.
• Pareto condition: GV satisfies this condition and the monotonicity criterion; see the earlier Evaluations: General Criteria section. GV also satisfies the non-imposition requirement as any possible permutation of election outcome rankings is achievable.
• Independence of Irrelevant Alternatives (IIA): GV fails this criterion; see the preceding page.

GV only fails one of the impossible set of four requirements for a fair voting system. Like all other positional voting systems, it easily passes the first three criteria but fails the IIA criterion.

Problems with Irrelevant Alternatives

Suppose that three candidates A, B and C are each considering whether to stand in a forthcoming election. In any scenario the relative preference of every voter for one candidate over another is unchanged. Provided two or more candidates contest this election, the four possible scenarios in terms of candidate nominations are stated below. The resultant outcome of each scenario is also stated.

• Candidates A, B & C all stand and A wins.
• Only candidates A & B stand and B wins.
• Only candidates A & C stand and A wins.
• Only candidates B & C stand and B wins.

Candidates A and B are potential winners but C faces defeat in any scenario. Candidates A and B are most likely to stand while C, as an irrelevant alternative, might decide not to compete. If C does compete against A and B, A will win. If C does not, B will win. The outcome is clearly dependent on the decision of the supposedly irrelevant alternative C. Who is the rightful victor here, A or B? By failing the IIA criterion, it remains uncertain as to which result is the fairer one.

If the decision of candidate C to stand or not was based solely on their own perceived chances of winning or losing, then either outcome is legitimate. If however C conspired with B to ensure that A was defeated (by C not standing) then the election was adversely affected by this nomination strategy. Alternatively, C could have conspired with A to defeat B (by C standing). In practice, it is generally impossible to know whether any nomination is sincere or is a strategic attempt to manipulate the election outcome. Also, in the real world, all candidates (including - with hindsight - supposedly irrelevant ones) are likely to have some impact on the election campaign and hence also on its outcome.

Some pairwise comparison (Condorcet) methods do comply with the IIA criterion. Unfortunately, voters here can rank candidates in intransitive cycles; for example, they might prefer A to B, B to C and C to A! Understandably, this leads to a wide variety of paradoxes that Condorcet methods are well-known for. Compounding this loss of transitivity, no information about the strength of the preference for one candidate over another is permissible. For example, one voter may prefer candidate A over B by a large margin while another voter may slightly prefer B to A.

Perversely, it is this lack of intensity information and the loss of transitivity that makes it easier to satisfy the IIA requirement. So paradoxically, by allowing nonsensical cycles and by ignoring the strengths of voter preferences a voting system can then be judged to be fair! Positional voting systems such as GV and CHPV only allow transitive preferences to be expressed. Also, the strengths of voter preferences are explicitly embedded in the positional weightings employed. It is therefore arguable that provided these weightings adequately reflect the views of the voters, then any election outcome is indeed fair and reasonable; regardless of whether any irrelevant alternative intervenes or not.

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