Now suppose that Pivotal Voter moves B above A , but keeps C in the same position and imagine that any number (or all!) of the other voters change their ballots to move C above B , without changing the position of A . Then aside from a repositioning of C this is the same as Profile k from
Part One and hence the societal outcome ranks B above A . Furthermore, by IIA the societal outcome must rank A above C , as in the previous case. In particular, the societal outcome ranks B above C , even though Pivotal Voter may have been the only voter to rank B above C . By IIA this conclusion holds independently of how A is positioned on the ballots, so Pivotal Voter is a dictator for B over C . Part Three: There can be at most one dictator [ edit ] Part Three: Since voter k is the dictator for B over C , the pivotal voter for B over C must appear among the first k voters. That is, outside of Segment Two. Likewise, the pivotal voter for C over B must appear among voters k through N . That is, outside of Segment One. In this part of the argument we refer back to the original ordering of voters, and compare the positions of the different pivotal voters (identified by applying Parts One and Two to the other pairs of candidates). First, the pivotal voter for B over C must appear earlier (or at the same position) in the line than the dictator for B over C : As we consider the argument of Part One applied to B and C , successively moving B to the top of voters' ballots, the pivot point where society ranks B above C must come at or before we reach the dictator for B over C . Likewise, reversing the roles of B and C , the pivotal voter for C over B must at or later in line than the dictator for B over C . In short, if k X/Y denotes the position of the pivotal voter for X over Y (for any two candidates X and Y ), then we have shown k B/C ≤ k B/A ≤ k C/B . Now repeating the entire argument above with B and C switched, we also have k C/B ≤ k B/C . Therefore we have k B/C = k B/A = k C/B and the same argument for other pairs shows that all the pivotal voters (and hence all the dictators) occur at the same position in the list of voters. This voter is the dictator for the whole election. Interpretations of the theorem [ edit ] Although Arrow's theorem is a mathematical result, it is often expressed in a non-mathematical way with a statement such as "No voting method is fair," "Every ranked voting method is flawed," or "The only voting method that isn't flawed is a dictatorship" . These statements are simplifications of Arrow's result which are not universally considered to be true. What Arrow's theorem does state is that a
deterministic preferential voting mechanism - that is, one where a preference order is the only information in a vote, and any possible set of votes gives a unique result - cannot comply with all of the conditions given above simultaneously.
- Spring '17
- Economics, Social Choice and Individual Values, Voting system, Social choice theory, Arrow's impossibility theorem, aggregate production, pivotal voter