Weighted Voting Systems - W.H. Freeman

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CHAPTER 11 Weighted Voting Systems 417assign weights and a quota to a two-voter system, there are only three ways thatthe power can be distributed: A as dictator, B as dictator, or consensus rule.Equivalent Voting SystemsTwo voting systems are equivalent if there is a way for all of the voters of thefirst system to exchange places with the voters of the second system and preserveall winning coalitions.The weighted voting systems [50 : 49, 1] and [4 : 3, 3], involving pairs ofvoters A, B, and C, D, respectively, are equivalent because in each system, unanimoussupport is required to pass a measure. We could have A exchange placeswith C, and B exchange places with D.Now consider two voting systems [2 : 2, 1] and [5 : 3, 6] involving the samepair of voters, A and B. In the first, [2 : 2, 1], A is a dictator, while in the second,[5 : 3, 6], B dictates. By having A and B exchange places with each other,we see that the two systems are equivalent. “Equivalent” does not mean “thesame.” Voter A would tell you that the system where he is the dictator is not thesame as the system where B is the dictator. The systems are equivalent becauseeach has a dictator.Every two-voter system is equivalent either to a system with a dictator or toone that requires consensus. As the number of voters increases, the number ofdifferent types of voting systems increases.Minimal Winning CoalitionsA minimal winning coalition is a winning coalition in which each voter is acritical voter.In a dictatorship, every coalition that includes the dictator is a winning coalition,but the only minimal winning coalition is the one that includes the dictatorand no other voters.Minimal Winning Coalitions: A Three-Voter SystemThe three-member committee from Example 5 uses the voting system [6 : 5, 3,1]. Let’s refer to its members as A, B, and C in order of decreasing weight. Thereare three winning coalitions. One, {A, B}, has weight 8, more than the quota, butit is minimal because both voters are critical. Another, {A, C}, with weight 6, isalso minimal. The third winning coalition, {A, B, C }, is not minimal becauseonly A is a critical voter. EXAMPLE 19
418 PART III Voting and Social ChoiceEXAMPLE 20The Four-Shareholder CorporationTable 11.4 lists the five winning coalitions in the corporation with the voting system[51 : 40, 30, 20, 10]. In each coalition, the critical voters have been identified.The minimal ones are those in which each voter is marked as critical: {A, B},{A, C }, and {B, C, D}. These minimal winning coalitions are displayed in Figure11.1. A voting system can be described completely by specifying its minimal winningcoalitions. If you want to make up a new voting system, instead of specifyingweights and a quota, you could make a list of the minimal winning coalitions.You would have to be careful that your list satisfies the following threerequirements:1. Your list can’t be empty. You have to name at least one coalition—otherwise, there would be no way to approve a motion.2. You can’t have one minimal winning coalition that contains another one—otherwise, the larger coalition wouldn’t be minimal.3. Every pair of coalitions in the list has to overlap—otherwise, two opposingmotions could pass.In the four-shareholder corporation (see Figure 11.1), you can see that theserequirements are satisfied. Now let’s construct some voting systems.ABCDFIGURE 11.1 Eachoval surrounds aminimal winningcoalition for the fourshareholdercorporation.{B,C,D} is a winningcoalition because ithas 60 votes, 9 morethan needed.
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Weighted Voting Systems - W.H. Freeman

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