Excursions in Modern Mathematics, 7e: 1.Conclusion - 2Copyright © 2010 Pearson Education, Inc. 1 The Mathematics of Voting CONCLUSION Elections, Fairness, and Arrow’s Impossibility Theorem

Excursions in Modern Mathematics, 7e: 1.Conclusion - 3Copyright © 2010 Pearson Education, Inc. “small” elections of various kinds play a pervasive and important role in all our lives, from getting a job to deciding where to go to dinner there are many different formal methods that can be used to decide the outcome of an election we focused on four basic methods- mostly because they are simple and commonly used - but there are many others, some quite elaborate and exotic Themes of Chapter 1

Excursions in Modern Mathematics, 7e: 1.Conclusion - 4Copyright © 2010 Pearson Education, Inc. we saw that the results of an election can change if we change the voting method Themes of Chapter 1 The Math Club example dramatically illustrated this point - each of the four voting methods produced a different winner. Since there were four candidates, we can say that each of them won the election - just pick the “right” voting method.

Excursions in Modern Mathematics, 7e: 1.Conclusion - 5Copyright © 2010 Pearson Education, Inc. Given an abundance of voting methods, how do we determine which are “good” voting methods and which are not so good? In a democratic society the most important quality we seek in a voting method is that of fairness. This leads us to the last, but probably most important, theme of this chapter - the notion of fairness criteria. Fairness criteria set the basic standards that a fair election should satisfy. We discussed four of them in this chapter - let’s review what they were: Themes of Chapter 1

Excursions in Modern Mathematics, 7e: 1.Conclusion - 6Copyright © 2010 Pearson Education, Inc. The Majority Criterion A majority candidate should always win the election. After all, it seems clearly unfair when a candidate with a majority of the first-place votes does not win. Fairness Criteria

Excursions in Modern Mathematics, 7e: 1.Conclusion - 7Copyright © 2010 Pearson Education, Inc. The Condorcet Criterion A Condorcet candidate should always win the election. When the candidates are compared two at a time, the Condorcet candidate beats each of the other candidates. How could it be fair to declare a different candidate as the winner? Fairness Criteria

Excursions in Modern Mathematics, 7e: 1.Conclusion - 8Copyright © 2010 Pearson Education, Inc. The Monotonicity Criterion Suppose candidate X is a winner of the election, but for one reason or another there is a new election. If the only changes in the ballots are changes in favor of candidate X (and only X), then X should win the new election. Fairness Criteria

Excursions in Modern Mathematics, 7e: 1.Conclusion - 9Copyright © 2010 Pearson Education, Inc. The Independence-of-Irrelevant- Alternatives Criterion (IIA) Suppose candidate X is a winner of the election, but for one reason or another there is a new election. If the only changes are that one of the other candidates withdraws or is disqualified, then X should win the new election. The flip side of this criterion is that a winner of the election should not be penalized by the introduction of irrelevant new candidates who have no chance of winning. Fairness Criteria

Excursions in Modern Mathematics, 7e: 1.Conclusion - 10Copyright © 2010 Pearson Education, Inc. Other fairness criteria beyond the ones listed above have been proposed, but these four are sufficient to establish a minimum standard of fairness - a fair voting method should clearly satisfy all four of the above. Surprisingly, none of the four voting methods we discussed in this chapter meets this minimum standard - far from it! Fairness Criteria

Excursions in Modern Mathematics, 7e: 1.Conclusion - 11Copyright © 2010 Pearson Education, Inc. Borda count method violates the Majority criterion, the Condorcet criterion, and the IIA criterion. Plurality-with-elimination method violates the Condorcet criterion, the monotonicity criterion, and the IIA criterion. Pair-wise comparisons method violates the IIA criterion. Fairness Criteria

Excursions in Modern Mathematics, 7e: 1.Conclusion - 12Copyright © 2010 Pearson Education, Inc. What voting method satisfies the minimum standards of fairness set forth by the above four criteria? For democratic elections involving three or more candidates, the answer is that there is no such voting method. Given the obvious importance of having fair elections in a democracy, how is it possible that no one has come up with such a voting method? Fair Voting Method

Excursions in Modern Mathematics, 7e: 1.Conclusion - 13Copyright © 2010 Pearson Education, Inc. In 1949 mathematical economist Kenneth Arrow demonstrated that this is indeed an impossible task. Arrow’s impossibility theorem essentially says (his exact formulation of the theorem is slightly different from the one given here) that it is mathematically impossible for a democratic voting method to satisfy all of the fairness criteria. Arrow’s impossibility theorem is surprising and paradoxical. It tells us that making decisions in a consistently fair way is inherently impossible in a democratic society. Arrow’s Impossibility Theorem