Coalition DEFINITION

A coalition is a set of participants in a voting system that might vote in favor of a measure. The empty coalition is allowed, and it represents a situation when the voters unanimously oppose a motion.

Winning Coalition and Losing Coalition DEFINITION

A set of participants in a weighted voting system whose combined voting weight is equal to or greater than the quota is called a winning coalition. If the set of participants has a combined voting weight less than the quota, it is a losing coalition.

Critical Voter DEFINITION

Let W be a winning coalition, and let V be a voter who belongs to W. Then V is a critical voter in W if the coalition consisting of all the voters in W except V is a losing coalition.

EXAMPLE 10 A Criminal Trial

After the evidence is presented and the summations are complete, the jury deliberates and each juror votes “guilty” or “not guilty.” We will regard these as two separate motions. Thus there is a “guilty” motion and a “not guilty” motion for each count that is before the jury. The jury must reach a unanimous verdict—there is only one winning coalition, the entire jury, and every juror is a critical voter. if neither “guilty” nor “not guilty” is approved in this system, a mistrial is declared, and the prosecution may demand a new trial.

Refer to Table 11.1 on page 468 to determine if any state was a critical voter in the 2012 election. Notice that a participant’s location in the voting permutation does not matter. You only need to check if the state was in the winning coalition, and if it was, did it have sufficient electoral votes to make a difference?

Banzhaf Power Index DEFINITION

In a weighted voting system, the Banzhaf power index of a voter V is the number of winning coalitions in which V is a critical member.

EXAMPLE 11 A Dictator

Consider the five-voter system . We’ll name the voters and , where stands for dictator. A set of voters is a winning coalition if and only if belongs to it. We can find the number of subsets of by applying the multiplication principle. Construct a subset by deciding if is in or out, two possibilities, and do the same for , , and . The number of subsets is . All of the winning coalitions consist of combined with one of these 16 subsets. in each winning coalition, is the only critical voter. Thus, has a Banzhaf index of 16, and I each of the other participants has a Banzhaf index of 0.

Find the Banzhaf index of each voter in the system .

A voting combination is a list of the voters, recording how each voted. A voter is critical in a voting combination if a change in that voter’s vote, with no other voter switching, will change the outcome. The definition of the Banzhaf power index that we are using counts only a voter’s critical “yes” votes in a voting combination. However, Banzhaf’s original definition^a counted all voting combinations in which the voter was critical as a “yes” voter or as a “no” voter. Previous editions of this text used the original definition.

Let V be a voter, and suppose that V is a critical voter in b winning coalitions (so that according to the definition we are using, the Banzhaf power index of V is equal to b). Let W be one of the winning coalitions in which V is a critical voter, and let ҡ be the set formed by combining V with all of the voters who do not belong to W (see the figure).

We will say that ҡ is a blocking coalition because, by voting “no,” it can prevent a motion from passing. Voter V is simultaneously a critical “yes” voter in the winning coalition W and, if she should change her vote, a critical “no” voter in the blocking coalition ҡ. Therefore, for each of the b winning combinations where V is a critical “yes” voter, there is a corresponding blocking combination—where V but no one else switches their vote—in which she is a critical “no” voter; there are b of these voting combinations, too. With the original definition, the Banzhaf index would be . Both indices yield the same distribution of voting power, but the original Banzhaf power index of each voter is double the index according to the definition that we are using.

Extra Votes DEFINITION

The extra votes of a winning coalition indicate the number of votes by which its total weight exceeds the quota.

Grand Coalition DEFINITION

The grand coalition is the coalition that includes every voter.

EXAMPLE 12 The Banzhaf Model Applied to the Film Selection Committee

Let’s return to the committee of Example 4 on page 465. The members are Allen, Betty, and Cao, and they use the voting system. Let’s identify the critical voters in the grand coalition, {Allen, Betty, Cao}. if Allen leaves the coalition, what happens?

Allen is a critical voter because the remaining voters form a losing coalition when he changes his vote. This is not surprising: He has veto power.

If Betty changes her vote, the remaining voters still form a winning coalition:

Betty is not a critical voter in this coalition, and you can verify for yourself that Cao is also not a critical voter.

To determine the Banzhaf power indices of Allen, Betty, and Cao, refer to Table 11.4. The table lists all winning coalitions and the extra votes in each—that’s the total weight of the coalition minus the quota, 6. To obtain the list, the noncritical voters were removed, one by one, from the grand coalition. This yielded two winning coalitions, {Allen, Betty} and {Allen, Cao}. We cannot remove voters from these two coalitions without making losing coalitions, so we stop. The voters in each winning coalition with weight greater than the extra votes are the critical voters. We find Allen is critical in all three coalitions, while Betty and Cao are each critical in just one. Hence the Banzhaf index is (3, 1, 1).

478

In Example 12, Allen was a critical voter in the grand coalition and he had veto power. Is it possible for a voter in a weighted voting system to be a critical voter in the grand coalition and not have veto power?

Minimal Winning Coalition DEFINITION

A winning coalition is minimal if every voter who belongs to the coalition is critical in that coalition.

Can the number of extra votes of a minimal winning coalition be greater than 0?

EXAMPLE 13 A Five-Voter System

use the weighted voting system . List the winning coalitions and find their critical voters.

We will start with the grand coalition. it has a total weight of 19, so there are 7 extra votes. Since no voter has more than 7 votes, the grand coalition has no critical voters.

479

Step 1 is to remove members of the grand coalition one at a time and obtain five winning coalitions: (0 extra votes, a minimal winning coalition); and (3 extra votes each; all voters except and are critical); (4 extra votes, only is critical); and (5 extra votes, only is critical).

Step 2 is to remove noncritical voters from the winning coalitions found in Step 1. We obtain , a minimal winning coalition with 0 extra votes, by removing from or by removing from . Only count this once! By removing from or from , one gets . In the same way, there are two ways to get the minimal winning coalitions and from and one other four-member coalition listed above. Finally, the minimal winning coalition can be obtained from or by removing the last voter from either one.

Table 11.5 summarizes the results and determines the Banzhaf power index of each participant. There are 11 winning coalitions in all. When compared with 120 permutations for the Shapley–Shubik index, it is manageable.

Show the other ways to get the minimal winning coalitions and .

Combinations

The U.S. Supreme Court, in its 1962 decision of the case Baker v. Carr, instituted a principle that legislative districts within states should be equal in population. The decision was based on the “Equal Protection” clause of the Fourteenth Amendment to the U.S. Constitution. Dissenting, Justice Frankfurter declared that the process of equalizing populations would lead to a “mathematical quagmire.”

The decision had repercussions for state and county legislative districts throughout the United States. Some district boundaries were redrawn to equalize populations. To avoid redrawing district boundaries the New Jersey legislature, the New York City Board of Estimate, Nassau County, and several other county boards of supervisors in New York State resorted to weighted voting, where the voting weight of a legislator (or supervisor) would be roughly proportional to the population being represented.

In a 1965 law review article, John F. Banzhaf iii pointed out that three of the six supervisors of Nassau County, New York, were dummy voters (see Example 3 on page 464). in the same article he argued that the weighted voting system employed by the New Jersey legislature was unfair. The article inspired legal action against several elected bodies that employed weighted voting systems.

A lawsuit invalidated the voting system of the Board of Supervisors of Washington County, New York, in 1967. in its decision, the New York State Court of Appeals provided a way to fix a weighted voting system: Each supervisor’s Banzhaf power index, rather than his or her voting weight, should be proportional to the population of the district that he or she represents. Borrowing Justice Frankfurter’s phrase, the Court predicted that its remedy would lead to a “mathematical quagmire.”

Five lawsuits filed over a period of 25 years challenged weighted voting in the Nassau County Board of Supervisors. Thus, the mathematical quagmire that the appeals court had feared actually materialized. Although Nassau County made a sincere attempt to comply with the Washington County decision, every voting system that it devised faced a new lawsuit. With conflicting expert testimony, the U.S. District Court finally ruled in 1993 that weighted voting was inherently unfair.

Banzhaf’s article was aptly titled “Weighted Voting Doesn’t Work.” (it, and the Washington County decision, are included in the Suggested Readings at the end of this chapter. The Nassau County quagmire is the subject of Exercise 5 on page 496. You can explore what happened to the Board of Estimate in Exercises 38 and 39 on page 499).

Combinations and Combinations Symbol DEFINITION

The combinations symbol stands for the number of ways to choose members from a set of elements—that is, the number of combinations of objects, taken at a time.

Formula 1: Basic Combinations Formula FORMULA

EXAMPLE 14 Calculate

By formula 1, . Using ; , and ; we get .

This is not how we should calculate combinations, because it is more efficient to cancel before multiplying. The first cancellation is

Then

Use Formula 1 to verify that .

Calculate and without using a calculator.

EXAMPLE 15 Banzhaf Power in Alice’s Committee

Alice’s Banzhaf index has already been determined: It is .

Bill, a weight-1 participant, will be a critical voter in any winning coalition with exactly 6 votes. He will need, in addition to his own vote, Alice’s vote and the votes of two other participants, or, without Alice, he will need the votes of five other participants. He must get the two or the five votes from the seven other weight-1 participants—he can’t vote twice!

There are winning coalitions in which Bill casts a critical vote and Alice is included, and winning coalitions where Bill casts a critical vote and Alice is not included. Bill’s Banzhaf power index is . The other participants have the same index.

By the Banzhaf model, Alice is times as powerful as Bill. This is reasonably close to the Shapley–Shubik model’s assessment that Alice is 4 times as powerful as Bill.

Formula 2: Symmetry of Combinations FORMULA

Use Formula 1 to verify Formula 2.

EXAMPLE 16 The Revolt: Banzhaf Model

In Example 9, Alice’s committee was transformed to a weighted voting system by a pact in which three weight-1 voters agreed to let Bill have their votes. Alice will be a critical voter in any winning coalition when she is joined by a losing coalition with weight 3, 4, or 5. This losing coalition could be Bill alone or Bill joined by one weight-1 voter, or it could be three or all four weight-1 voters. There is coalition consisting of Bill alone, and coalitions with Bill and one weight-1 voter. There are coalitions of three or four weight-1 voters, so Alice’s Banzhaf index is .

Bill will be a critical voter in any winning coalition when the other voters have a total weight of 2, 3, 4, or 5. These other voters could be Alice plus up to two weight-1 voters; there are of these coalitions. Without Alice, Bill would need two or more weight-1 voters; there are such coalitions, so Bill’s Banzhaf index, 22, is more than twice Alice’s.

A weight-1 voter, Zoë, must be joined by other voters with total weight 5 to form a winning coalition in which she is critical. That could be Alice and two of the three other weight-1 voters; there are of these coalitions. if Zoë is joined by Bill, exactly one of the three other weight-1 voters is needed. There are three of these coalitions too, so Zoë’s Banzhaf index is 6.

The table below displays the Shapley-Shubik (SSPI) and Banzhaf (BPI) power indices of the voters in the Electoral College, as compared with the voter’s weight as a percent of 538 (PCT), the total weight of all the voters. It shows that for the most part both measures of power agree closely with the actual share of power that a participant in the college has by virtue of its voting weight. There is an exception, though. California, whose voting weight is slightly more than 10% of 538, has more than its share of power by either measure. The power indices shown were calculated with the online Power Index Calculator (see page 502).

EXAMPLE 17 The Jury

A jury of 12 members must reach a unanimous verdict. This means that the only winning coalition is the grand coalition. The Banzhaf power index of each member of the jury is 1. Thus the probability that a juror will be a critical voter is . One might think that most criminal trials would end as mistrials, since it is so unlikely that the jurors would agree. But the Banzhaf model only touches on the voting system. it does not consider all of the evidence that the jurors see before they vote.

EXAMPLE 18 Majority Rule

A committee uses majority rule. What is the probability that a member of the committee will cast a critical vote in a winning coalition? Since every voter has weight 1, no one is a critical voter in a winning coalition unless the coalition has 0 extra votes; it must be a minimal winning coalition.

If there are three members, , , and , then the minimal winning coalitions with are and . The Banzhaf power index of is 2, so the probability that will be critical is

With four voters, the minimal winning coalitions will have three members. would have to be joined by two of the other three members, so ’s Banzhaf power index is . The probability that will be critical is .

If the number of voters is even, say , then a minimal winning coalition has members. Voter would be joined by of the other members to form a minimal winning coalition. The Banzhaf power index of would be and the probability that will be a critical voter is

With an odd number of voters a minimal winning coalition would still have members. Voter would be joined by of the other voters to form a minimal winning coalition, so the Banzhaf power index of would be , and the probability that is critical would be

Table 11.6 shows a few of these probabilities.

Formula 3: Sum Formula for Combinations FORMULA

Formula 4: An Approximation FORMULA

is approximately equal to

Use Formula 4 to approximate the probabilities for the even values of in the first two columns of Table 11.6. Compare your results with the probabilities given in the table (which were calculated using Formula 1, not the approximation). Don’t bother with the third column because Formula 4 was used to compute those probabilities.

In Self Check 9 (page 472) you calculated the Shapley- Shubik index of a 90-voter system in which Alice had 30 votes and the other 89 voters each had 1 vote apiece, where the quota was a simple majority, 60. You found that Alice had a Shapley-Shubik index of , just as in the committee with voting system . According to the Banzhaf model Alice had a slightly larger share of the power (about 35%) in the nine-voter system. You are not expected to calculate the Banzhaf index of the 90-member committee, because—while it is simple to do so in principle— the arithmetic is hard to manage. Alice will be a critical voter in any winning coalition when joined by between 30 and 59 of the other voters. There are

of these coalitions, and this is her Banzhaf index. Expressed in scientific notation, Alice’s Banzhaf index is approximately . A weight-1 voter will be critical when joined by Alice and 29 other weight-1 voters or by 59 weight-1 voters. This voter’s Banzhaf index is . This is a pretty big number too, approximately , or about the number of molecules in 9 milliliters of water (nearly 2 teaspoons). Alice’s Banzhaf index is 2043 times as large, comparable to the number of molecules in 18.5 liters (about 5 gallons) of water. in the 90-member committee, the models differ radically. Let’s use the water analogy to measure this. Alice has 18.5 liters, each of the 89 weight-1 voters has 0.009 liters. The sum of all the Banzhaf indices is liters. Alice’s share is of the power according to the Banzhaf model, but 33.3% of the power in the Shapley-Shubik model.

How can we explain this difference? With the Shapley-Shubik model, Alice is a pivot if she is in one of 30 places in a permutation that has 90 places. Hence she has one-third of the power. For the Banzhaf index, suppose Alice votes “yes” and each of 89 weight-1 voters tosses a coin. What is the probability that between 29 and 59 of the tosses is heads? The average number of heads is 44.5, and it’s very unlikely that the total number of heads will be more than 15 away from that value. A weight-1 voter will be critical if 29 of the other 88 weight-1 voters, and Alice, throw heads, or if 59 of the other weight-1 voters throw heads and Alice throws tails. This is improbable, because on the average there will be 44 heads when 88 coins are thrown, and it will not be often that the number of heads will be exactly 15 more or less than 44.