Arguably, the first principle of fairness for a voting method is that each vote should count equally in the final outcome. In this chapter, we study voting systems that violate this basic principle. These are voting systems in which differences between voters are purposely introduced to reflect differences in the rank of the player or, in the case of regions, the size of the population. Examples of such systems include the U.S. Electoral College, the U.N. Security Council, and countless corporate and legislative committees. As is customary, we study such voting systems under the simplifying assumption that there are only two possible outcomes of the election, “yes” and “no,” a yes-no voting system as in Definition 7.4. In this setting, the relative power of the individual voters is the central issue. We apply the counting techniques, notably the Combinations Formula (Theorem 11.12), developed in Chapter 11 to study the power of voters in weighted systems. We also consider the question of assigning weights to voters in a yes-no voting system. This problem leads us to the notion of a mathematical invariant. Finally, we show that Question 1.6 regarding the U.S. Senate introduced in Chapter 1 leads to a combinatorial identity called the recursive law for combinations.

Yes-No Voting Systems and Power

Much of the controversy over voting methods stems from issues arising from the presence of multiple candidates. Even the most democratic methods can appear “unfair” in the attempt to untangle social preferences when their are several competing alternatives. We remove this issue here and focus on elections with only two candidates.