# 8.4: Weighted Voting

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$

In a corporate shareholders meeting, each shareholders’ vote counts proportional to the amount of shares they own. An individual with one share gets the equivalent of one vote, while someone with 100 shares gets the equivalent of 100 votes. This is called weighted voting, where each vote has some weight attached to it. Weighted voting is sometimes used to vote on candidates, but more commonly to decide “yes” or “no” on a proposal, sometimes called a motion. Weighted voting is applicable in corporate settings, as well as decision making in parliamentary governments and voting in the United Nations Security Council.

In weighted voting, we are most often interested in the power each voter has in influencing the outcome.

## Beginnings

We’ll begin with some basic vocabulary for weighted voting systems.

Vocabulary for Weighted Voting

Each individual or entity casting a vote is called a player in the election. They’re often notated as $$P_{1}, P_{2}, P_{3}, \ldots P_{N},$$ where $$N$$ is the total number of voters.

Each player is given a weight, which usually represents how many votes they get.

The quota is the minimum weight needed for the votes or weight needed for the proposal to be approved. A weighted voting system will often be represented in a shorthand form:$\left[q: w_{1}, w_{2}, w_{3}, \ldots, w_{n}\right] \nonumber$

In this form, $$q$$ is the quota, $$w_1$$is the weight for player 1, and so on.

Example $$\PageIndex{1}$$

In a small company, there are 4 shareholders. Mr. Smith has a 30% ownership stake in the company, Mr. Garcia has a 25% stake, Mrs. Hughes has a 25% stake, and Mrs. Lee has a 20% stake. They are trying to decide whether to open a new location. The company by-laws state that more than 50% of the ownership has to approve any decision like this. This could be represented by the weighted voting system:

$[51: 30, 25, 25, 20] \nonumber$

Here we have treated the percentage ownership as votes, so Mr. Smith gets the equivalent of 30 votes, having a 30% ownership stake. Since more than 50% is required to approve the decision, the quota is 51, the smallest whole number over 50.

In order to have a meaningful weighted voting system, it is necessary to put some limits on the quota.

Limits on the Quota

The quota must be more than ½ the total number of votes.

The quota can’t be larger than the total number of votes.

Why? Consider the voting system $$[q: 3, 2, 1]$$

Here there are 6 total votes. If the quota was set at only 3, then player 1 could vote yes, players 2 and 3 could vote no, and both would reach quota, which doesn’t lead to a decision being made. In order for only one decision to reach quota at a time, the quota must be at least half the total number of votes. If the quota was set to 7, then no group of voters could ever reach quota, and no decision can be made, so it doesn’t make sense for the quota to be larger than the total number of voters.

## Try it Now 1

In a committee there are four representatives from the management and three representatives from the workers’ union. For a proposal to pass, four of the members must support it, including at least one member of the union. Find a voting system that can represent this situation.

## A Look at Power

Consider the voting system [10: 11, 3, 2]. Notice that in this system, player 1 can reach quota without the support of any other player. When this happens, we say that player 1 is a dictator.

Definition: Dictator

A player will be a dictator if their weight is equal to or greater than the quota. The dictator can also block any proposal from passing; the other players cannot reach quota without the dictator.

In the voting system [8: 6, 3, 2], no player is a dictator. However, in this system, the quota can only be reached if player 1 is in support of the proposal; player 2 and 3 cannot reach quota without player 1’s support. In this case, player 1 is said to have veto power. Notice that player 1 is not a dictator, since player 1 would still need player 2 or 3’s support to reach quota.

Definition: Veto Power

A player has veto power if their support is necessary for the quota to be reached. It is possible for more than one player to have veto power, or for no player to have veto power.

With the system [10: 7, 6, 2], player 3 is said to be a dummy, meaning they have no influence in the outcome. The only way the quota can be met is with the support of both players 1 and 2 (both of which would have veto power here); the vote of player 3 cannot affect the outcome.

Definition: Dummy

A player is a dummy if their vote is never essential for a group to reach quota.

Example $$\PageIndex{2}$$

In the voting system [16: 7, 6, 3, 3, 2], are any players dictators? Do any have veto power? Are any dummies?

Solution

No player can reach quota alone, so there are no dictators.

Without player 1, the rest of the players’ weights add to 14, which doesn’t reach quota, so player 1 has veto power. Likewise, without player 2, the rest of the players’ weights add to 15, which doesn’t reach quota, so player 2 also has veto power.

Since player 1 and 2 can reach quota with either player 3 or player 4’s support, neither player 3 or player 4 have veto power. However they cannot reach quota with player 5’s support alone, so player 5 has no influence on the outcome and is a dummy.

## Try it Now 2

In the voting system $$[q: 10, 5, 3]$$, which players are dictators, have veto power, and are dummies if the quota is 10? 12? 16?

To better define power, we need to introduce the idea of a coalition. A coalition is a group of players voting the same way. In the example above, {P1, P2, P4} would represent the coalition of players 1, 2 and 4. This coalition has a combined weight of 7+6+3 = 16, which meets quota, so this would be a winning coalition.

A player is said to be critical in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition. In the coalition {P1, P2, P4}, every player is critical. In the coalition {P3, P4, P5}, no player is critical, since it wasn’t a winning coalition to begin with. In the coalition {P1, P2, P3, P4, P5}, only players 1 and 2 are critical; any other player could leave the coalition and it would still meet quota.

Definition: Coalition

A coalition is any group of players voting the same way. A coalition is a winning coalition if the coalition has enough weight to meet quota.

Definition: Critical Player

A player is critical in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition.

Example $$\PageIndex{3}$$

In the Scottish Parliament in 2009 there were 5 political parties: 47 representatives for the Scottish National Party, 46 for the Labour Party, 17 for the Conservative Party, 16 for the Liberal Democrats, and 2 for the Scottish Green Party. Typically all representatives from a party vote as a block, so the parliament can be treated like the weighted voting system:

[65: 47, 46, 17, 16, 2]

Solution

Consider the coalition {P1, P3, P4}. No two players alone could meet the quota, so all three players are critical in this coalition.

In the coalition {P1, P3, P4, P5}, any player except P1 could leave the coalition and it would still meet quota, so only P1 is critical in this coalition.

Notice that a player with veto power will be critical in every winning coalition, since removing their support would prevent a proposal from passing.

Likewise, a dummy will never be critical, since their support will never change a losing coalition to a winning one.

Dictators,veto, and Dummies and Critical Players

• A player is a dictator if the single-player coalition containing them is a winning coalition.
• A player has veto power if they are critical in every winning coalition.
• A player is a dummy if they are not critical in any winning coalition.

## Calculating Power: Banzhaf Power Index

The Banzhaf power index was originally created in 1946 by Lionel Penrose, but was reintroduced by John Banzhaf in 1965. The power index is a numerical way of looking at power in a weighted voting situation.

Calculating the Banzhaf Power Index

To calculate the Banzhaf power index:

1. List all winning coalitions
2. In each coalition, identify the players who are critical
3. Count up how many times each player is critical
4. Convert these counts to fractions or decimals by dividing by the total times any player is critical

Example $$\PageIndex{4}$$

Find the Banzhaf power index for the voting system [8: 6, 3, 2].

Solution

We start by listing all winning coalitions. If you aren’t sure how to do this, you can list all coalitions, then eliminate the non-winning coalitions. No player is a dictator, so we’ll only consider two and three player coalitions.

$$\left\{P_{1}, P_{2}\right\}$$ Total weight: 9. Meets quota.

$$\left\{P_{1}, P_{3}\right\}$$ Total weight: 8. Meets quota.

$$\left\{P_{2}, P_{3}\right\}$$ Total weight: 5. Does not meet quota.

$$\left\{P_{1}, P_{2}, P_{3}\right\}$$Total weight: 11. Meets quota.

Next we determine which players are critical in each winning coalition. In the winning two-player coalitions, both players are critical since no player can meet quota alone. Underlining the critical players to make it easier to count:

$$\left\{\underline{P}_{1}, \underline{P}_{2}\right\}$$

$$\left\{\underline{P}_{1}, \underline{P}_{3}\right\}$$

In the three-person coalition, either P2 or P3 could leave the coalition and the remaining players could still meet quota, so neither is critical. If P1 were to leave, the remaining players could not reach quota, so P1 is critical.

$$\left\{P_{1}, P_{2}, P_{3}\right\}$$

Altogether, P1 is critical 3 times, P2 is critical 1 time, and P3 is critical 1 time.

Converting to percents:

$$\begin{array}{l} P_{1}=3 / 5=60 \% \\ P_{2}=1 / 5=20 \% \\ P_{3}=1 / 5=20 \% \end{array}$$

Example $$\PageIndex{5}$$

Consider the voting system [16: 7, 6, 3, 3, 2]. Find the Banzhaf power index.

Solution

The winning coalitions are listed below, with the critical players underlined.

$$\begin{array}{l} \left\{P_{1}, P_{2}, P_{3}\right\} \\ \left\{P_{1}, P_{2}, P_{4}\right\} \\ \left\{P_{1}, P_{2}, P_{3}, P_{4}\right\} \\ \left\{P_{1}, P_{2}, P_{3}, P_{5}\right\} \\ \left\{P_{1}, P_{2}, P_{4}, P_{5}\right\} \\ \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\} \end{array}$$

Counting up times that each player is critical:

• $$P_1 = 6$$
• $$P_2 = 6$$
• $$P_3 = 2$$
• $$P_4 = 2$$
• $$P_5 = 0$$

Total of all: 16

Divide each player’s count by 16 to convert to fractions or percents:

$$\begin{array}{l} P_{1}=6 / 16=3 / 8=37.5 \% \\ P_{2}=6 / 16=3 / 8=37.5 \% \\ P_{3}=2 / 16=1 / 8=12.5 \% \\ P_{4}=2 / 16=1 / 8=12.5 \% \end{array}$$

The Banzhaf power index measures a player’s ability to influence the outcome of the vote. Notice that player 5 has a power index of 0, indicating that there is no coalition in which they would be critical power and could influence the outcome. This means player 5 is a dummy, as we noted earlier.

Example $$\PageIndex{6}$$

Revisiting the Scottish Parliament, with voting system [65: 47, 46, 17, 16, 2], the winning coalitions are listed, with the critical players underlined.

$$\begin{array}{l} \left\{\underline{P}_{1,} \underline{P}_{2}\right\} \\ \left\{\underline{P}_{1,} \underline{P}_{2}, P_{3}\right\} \quad \left\{\underline{P}_{1}, \underline{P}_{2}, P_{4}\right\} \\ \left\{\underline{P}_{1}, \underline{P}_{2}, P_{5}\right\} \quad \left\{\underline{P}_{1}, \underline{P}_{3}, \underline{P}_{4}\right\} \\ \left\{\underline{P}_{1}, \underline{P}_{3}, \underline{P}_{5}\right\} \quad \left\{\underline{P}_{1}, \underline{P}_{4}, \underline{P}_{5}\right\}\\ \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{4}\right\} \quad \left\{\underline{P}_{2}, \underline{P}_{3}, \underline{P}_{5}\right\}\\ \left\{P_{1}, P_{2}, P_{3}, P_{4}\right\} \quad \left\{P_{1}, P_{2}, P_{3}, P_{5}\right\} \\ \left\{\underline{P}_{1}, P_{2}, P_{4}, P_{5}\right\} \quad \left\{\underline{P}_{1}, P_{3}, P_{4}, P_{5}\right\}\\ \left\{\underline{P}_{2}, P_{3}, P_{4}, P_{5}\right\} \\ \left\{P_{1}, P_{2}, P_{3}, P_{4}, P_{5}\right\} \end{array}$$

Counting up times that each player is critical:

District

Times critical

Power index

$$P_1$$ (Scottish National Party)

9

9/27 = 33.3%

$$P_2$$ (Labour Party)

7

7/27 = 25.9%

$$P_3$$ (Conservative Party)

5

5/27 = 18.5%

$$P_4$$ (Liberal Democrats Party)

3

3/27 = 11.1%

$$P_5$$ (Scottish Green Party)

3

3/27 = 11.1%

Interestingly, even though the Liberal Democrats party has only one less representative than the Conservative Party, and 14 more than the Scottish Green Party, their Banzhaf power index is the same as the Scottish Green Party’s. In parliamentary governments, forming coalitions is an essential part of getting results, and a party’s ability to help a coalition reach quota defines its influence.

## Try it Now 3

Find the Banzhaf power index for the weighted voting system [36: 20, 17, 16, 3].

Example $$\PageIndex{7}$$

Banzhaf used this index to argue that the weighted voting system used in the Nassau County Board of Supervisors in New York was unfair. The county was divided up into 6 districts, each getting voting weight proportional to the population in the district, as shown below. Calculate the power index for each district.

 District Weight Hempstead #1 31 Hempstead #2 31 Oyster Bay 28 North Hempstead 21 Long Beach 2 Glen Cove 2

Solution

Translated into a weighted voting system, assuming a simple majority is needed for a proposal to pass:

[58: 31, 31, 28, 21, 2, 2]

Listing the winning coalitions and marking critical players:

 {H1, H2} {H1, OB} {H2, OB} {H1, H2, NH} {H1, H2, LB} {H1, H2, GC} {H1, H2, NH, LB} {H1, H2, NH, GC} {H1, H2, LB, GC} {H1, H2, NH, LB. GC} {H1, OB, NH} {H1, OB, LB} {H1, OB, GC} {H1, OB, NH, LB} {H1, OB, NH, GC} {H1, OB, LB, GC} {H1, OB, NH, LB. GC} {H2, OB, NH} {H2, OB, LB} {H2, OB, GC} {H2, OB, NH, LB} {H2, OB, NH, GC} {H2, OB, LB, GC} {H2, OB, NH, LB, GC} {H1, H2, OB} {H1, H2, OB, NH} {H1, H2, OB, LB} {H1, H2, OB, GC} {H1, H2, OB, NH, LB} {H1, H2, OB, NH, GC} {H1, H2, OB, NH, LB, GC}

There are a lot of them! Counting up how many times each player is critical,

 District Times critical Power index Hempstead #1 16 16/48 = 1/3 = 33% Hempstead #2 16 16/48 = 1/3 = 33% Oyster Bay 16 16/48 = 1/3 = 33% North Hempstead 0 0/48 = 0% Long Beach 0 0/48 = 0% Glen Cove 0 0/48 = 0%

It turns out that the three smaller districts are dummies. Any winning coalition requires two of the larger districts.

The weighted voting system that Americans are most familiar with is the Electoral College system used to elect the President. In the Electoral College, states are given a number of votes equal to the number of their congressional representatives (house + senate). Most states give all their electoral votes to the candidate that wins a majority in their state, turning the Electoral College into a weighted voting system, in which the states are the players. As I’m sure you can imagine, there are billions of possible winning coalitions, so the power index for the Electoral College has to be computed by a computer using approximation techniques.

## Calculating Power: Shapley-Shubik Power Index

The Shapley-Shubik power index was introduced in 1954 by economists Lloyd Shapley and Martin Shubik, and provides a different approach for calculating power.

In situations like political alliances, the order in which players join an alliance could be considered the most important consideration. In particular, if a proposal is introduced, the player that joins the coalition and allows it to reach quota might be considered the most essential. The Shapley-Shubik power index counts how likely a player is to be pivotal. What does it mean for a player to be pivotal?

First, we need to change our approach to coalitions. Previously, the coalition $$\left\{P_{1}, P_{2}\right\}$$ and $$\left\{P_{2}, P_{1}\right\}$$ would be considered equivalent, since they contain the same players. We now need to consider the order in which players join the coalition. For that, we will consider sequential coalitions – coalitions that contain all the players in which the order players are listed reflect the order they joined the coalition. For example, the sequential coalition

<P2, P1, P3> would mean that P2 joined the coalition first, then P1, and finally P3. The angle brackets < > are used instead of curly brackets to distinguish sequential coalitions.

Definition: Pivotal Player

A sequential coalition lists the players in the order in which they joined the coalition.

A pivotal player is the player in a sequential coalition that changes a coalition from a losing coalition to a winning one. Notice there can only be one pivotal player in any sequential coalition.

Example $$\PageIndex{8}$$

In the weighted voting system [8: 6, 4, 3, 2], which player is pivotal in the sequential coalition <P3, P2, P4, P1>?

Solution

The sequential coalition shows the order in which players joined the coalition. Consider the running totals as each player joins:

$$P_3 \quad \text { Total weight: 3 } \quad \text { Not winning}$$

$$P_3, P_2 \quad \text { Total weight: 3+4 = 7 } \quad \text { Not winning}$$

$$P_3, P_2, P_4 \quad \text { Total weight: 3+4+2 = 9 } \quad \text { Winning}$$

$$P_3, P_2, P_4, P_1 \quad \text { Total weight: 3+4+2+6 = 15 } \quad \text { Winning}$$

Since the coalition becomes winning when $$P_4$$ joins, $$P_4$$ is the pivotal player in this coalition.

Calculating Shapley-Shubik Power Index

To calculate the Shapley-Shubik Power Index:

1. List all sequential coalitions
2. In each sequential coalition, determine the pivotal player
3. Count up how many times each player is pivotal
4. Convert these counts to fractions or decimals by dividing by the total number of sequential coalitions

How many sequential coalitions should we expect to have? If there are $$N$$ players in the voting system, then there are $$N$$ possibilities for the first player in the coalition, $$N – 1$$ possibilities for the second player in the coalition, and so on. Combining these possibilities, the total number of coalitions would be:$N(N-1)(N-2)(3-N) \ldots(3)(2)(1)\nonumber$This calculation is called a factorial, and is notated $$N !$$ The number of sequential coalitions with $$N$$ players is $$N !$$

Example $$\PageIndex{9}$$

How many sequential coalitions will there be in a voting system with 7 players?

Solution

There will be 7! sequential coalitions.

As you can see, computing the Shapley-Shubik power index by hand would be very difficult for voting systems that are not very small.

Example $$\PageIndex{10}$$

Consider the weighted voting system [6: 4, 3, 2]. We will list all the sequential coalitions and identify the pivotal player. We will have 3! = 6 sequential coalitions. The coalitions are listed, and the pivotal player is underlined.

$$< P_{1}, \underline{P}_{2}, P_{3} > \quad < P_{1}, \underline{P}_{3}, P_{2} > \quad< P_{2}, \underline{P}_{1_{2}} P_{3} >$$

$$<P_{2}, P_{3}, \underline{P}_{1} > \quad <P_{3}, P_{2}, \underline{P}_{1} > \quad <P_{3}, \underline{P}_{1}, P_{2} >$$

Calculate the Shapley-Shubik Power Index.

Solution

$$P_1$$ is pivotal 4 times, $$P_2$$ is pivotal 1 time, and $$P_3$$ is pivotal 1 time.

 Player Times pivotal Power index P1 4 4/6 = 66.7% P2 1 1/6 = 16.7% P3 1 1/6 = 16.7%

For comparison, the Banzhaf power index for the same weighted voting system would be P1: 60%, P2: 20%, P3: 20%. While the Banzhaf power index and Shapley-Shubik power index are usually not terribly different, the two different approaches usually produce somewhat different results.

## Try it Now 4

Find the Shapley-Shubik power index for the weighted voting system [36: 20, 17, 15].

## Exercises

In exercises 1-8, determine the apportionment using

1. Hamilton’s Method
2. Jefferson’s Method
3. Webster’s Method
4. Huntington-Hill Method
5. Lowndes’ method
1. A college offers tutoring in Math, English, Chemistry, and Biology. The number of students enrolled in each subject is listed below. If the college can only afford to hire 15 tutors, determine how many tutors should be assigned to each subject.

Math: 330 English: 265 Chemistry: 130 Biology: 70

1. Reapportion the previous problem if the college can hire 20 tutors.
2. The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion 20 salespeople given the information below.
 Shift Morning Midday Afternoon Evening Average Number of Customers 95 305 435 515
1. Reapportion the previous problem if the store has 25 salespeople.
2. Three people invest in a treasure dive, each investing the amount listed below. The dive results in 36 gold coins. Apportion those coins to the investors.

Alice: $7,600 Ben:$5,900 Carlos: $1,400 1. Reapportion the previous problem if 37 gold coins are recovered. 2. A small country consists of five states, whose populations are listed below. If the legislature has 119 seats, apportion the seats. A: 810,000 B: 473,000 C: 292,000 D: 594,000 E: 211,000 1. A small country consists of six states, whose populations are listed below. If the legislature has 200 seats, apportion the seats. A: 3,411 B: 2,421 C: 11,586 D: 4,494 E: 3,126 F: 4,962 1. A small country consists of three states, whose populations are listed below. A: 6,000 B: 6,000 C: 2,000 1. If the legislature has 10 seats, use Hamilton’s method to apportion the seats. 2. If the legislature grows to 11 seats, use Hamilton’s method to apportion the seats. 3. Which apportionment paradox does this illustrate? 1. A state with five counties has 50 seats in their legislature. Using Hamilton’s method, apportion the seats based on the 2000 census, then again using the 2010 census. Which apportionment paradox does this illustrate? County 2000 Population 2010 Population Jefferson 60,000 60,000 Clay 31,200 31,200 Madison 69,200 72,400 Jackson 81,600 81,600 Franklin 118,000 118,400 1. A school district has two high schools: Lowell, serving 1715 students, and Fairview, serving 7364. The district could only afford to hire 13 guidance counselors. 1. Determine how many counselors should be assigned to each school using Hamilton's method. 2. The following year, the district expands to include a third school, serving 2989 students. Based on the divisor from above, how many additional counselors should be hired for the new school? 3. After hiring that many new counselors, the district recalculates the reapportion using Hamilton's method. Determine the outcome. 4. Does this situation illustrate any apportionment issues? 2. A small country consists of four states, whose populations are listed below. If the legislature has 116 seats, apportion the seats using Hamilton’s method. Does this illustrate any apportionment issues? A: 33,700 B: 559,500 C: 141,300 D: 89,100 1. Explore and describe the similarities, differences, and interplay between weighted voting, fair division (if you’ve studied it yet), and apportionment. 2. In the methods discussed in the text, it was assumed that the number of seats being apportioned was fixed. Suppose instead that the number of seats could be adjusted slightly, perhaps 10% up or down. Create a method for apportioning that incorporates this additional freedom, and describe why you feel it is the best approach. Apply your method to the apportionment in Exercise 7. 3. Lowndes felt that small states deserved additional seats more than larger states. Suppose you were a legislator from a larger state, and write an argument refuting Lowndes. 4. Research how apportionment of legislative seats is done in other countries around the world. What are the similarities and differences compared to how the United States apportions congress? 5. Adams’s method is similar to Jefferson’s method, but rounds quotas up rather than down. This means we usually need a modified divisor that is smaller than the standard divisor. Rework problems 1-8 using Adam’s method. Which other method are the results most similar to? 6. Consider the weighted voting system [47: 10,9,9,5,4,4,3,2,2] 1. How many players are there? 2. What is the total number (weight) of votes? 3. What is the quota in this system? 7. Consider the weighted voting system [31: 10,10,8,7,6,4,1,1] 1. How many players are there? 2. What is the total number (weight) of votes? 3. What is the quota in this system? 8. Consider the weighted voting system [q: 7,5,3,1,1] 1. What is the smallest value that the quota q can take? 2. What is the largest value that the quota q can take? 3. What is the value of the quota if at least two-thirds of the votes are required to pass a motion? 9. Consider the weighted voting system [q: 10,9,8,8,8,6] 1. What is the smallest value that the quota q can take? 2. What is the largest value that the quota q can take? 3. What is the value of the quota if at least two-thirds of the votes are required to pass a motion? 10. Consider the weighted voting system [13: 13, 6, 4, 2] 1. Identify the dictators, if any. 2. Identify players with veto power, if any 3. Identify dummies, if any. 11. Consider the weighted voting system [11: 9, 6, 3, 1] 1. Identify the dictators, if any. 2. Identify players with veto power, if any 3. Identify dummies, if any. 12. Consider the weighted voting system [19: 13, 6, 4, 2] 1. Identify the dictators, if any. 2. Identify players with veto power, if any 3. Identify dummies, if any. 13. Consider the weighted voting system [17: 9, 6, 3, 1] 1. Identify the dictators, if any. 2. Identify players with veto power, if any 3. Identify dummies, if any. 14. Consider the weighted voting system [15: 11, 7, 5, 2] 1. What is the weight of the coalition {P1,P2,P4} 2. In the coalition {P1,P2,P4} which players are critical? 15. Consider the weighted voting system [17: 13, 9, 5, 2] 1. What is the weight of the coalition {P1,P2,P3} 2. In the coalition {P1,P2,P3} which players are critical? 16. Find the Banzhaf power distribution of the weighted voting system [27: 16, 12, 11, 3] 17. Find the Banzhaf power distribution of the weighted voting system [33: 18, 16, 15, 2] 18. Consider the weighted voting system [q: 15, 8, 3, 1] Find the Banzhaf power distribution of this weighted voting system, 1. When the quota is 15 2. When the quota is 16 3. When the quota is 18 19. Consider the weighted voting system [q: 15, 8, 3, 1] Find the Banzhaf power distribution of this weighted voting system, 1. When the quota is 19 2. When the quota is 23 3. When the quota is 26 20. Consider the weighted voting system [17: 13, 9, 5, 2]. In the sequential coalition <P3,P2,P1,P4> which player is pivotal? 21. Consider the weighted voting system [15: 13, 9, 5, 2]. In the sequential coalition <P1,P4,P2,P3> which player is pivotal? 22. Find the Shapley-Shubik power distribution for the system [24: 17, 13, 11] 23. Find the Shapley-Shubik power distribution for the system [25: 17, 13, 11] 24. Consider the weighted voting system [q: 7, 3, 1] 1. Which values of q result in a dictator (list all possible values) 2. What is the smallest value for q that results in exactly one player with veto power but no dictators? 3. What is the smallest value for q that results in exactly two players with veto power? 25. Consider the weighted voting system [q: 9, 4, 2] 1. Which values of q result in a dictator (list all possible values) 2. What is the smallest value for q that results in exactly one player with veto power? 3. What is the smallest value for q that results in exactly two players with veto power? 26. Using the Shapley-Shubik method, is it possible for a dummy to be pivotal? 27. If a specific weighted voting system requires a unanimous vote for a motion to pass: 1. Which player will be pivotal in any sequential coalition? 2. How many winning coalitions will there be? 28. Consider a weighted voting system with three players. If Player 1 is the only player with veto power, there are no dictators, and there are no dummies: 1. Find the Banzhof power distribution. 2. Find the Shapley-Shubik power distribution 29. Consider a weighted voting system with three players. If Players 1 and 2 have veto power but are not dictators, and Player 3 is a dummy: 1. Find the Banzhof power distribution. 2. Find the Shapley-Shubik power distribution 30. An executive board consists of a president (P) and three vice-presidents (V1,V2,V3). For a motion to pass it must have three yes votes, one of which must be the president's. Find a weighted voting system to represent this situation. 31. On a college’s basketball team, the decision of whether a student is allowed to play is made by four people: the head coach and the three assistant coaches. To be allowed to play, the student needs approval from the head coach and at least one assistant coach. Find a weighted voting system to represent this situation. 32. In a corporation, the shareholders receive 1 vote for each share of stock they hold, which is usually based on the amount of money the invested in the company. Suppose a small corporation has two people who invested$30,000 each, two people who invested $20,000 each, and one person who invested$10,000. If they receive one share of stock for each \$1000 invested, and any decisions require a majority vote, set up a weighted voting system to represent this corporation’s shareholder votes.
33. A contract negotiations group consists of 4 workers and 3 managers. For a proposal to be accepted, a majority of workers and a majority of managers must approve of it. Calculate the Banzhaf power distribution for this situation. Who has more power: a worker or a manager?
34. The United Nations Security Council consists of 15 members, 10 of which are elected, and 5 of which are permanent members. For a resolution to pass, 9 members must support it, which must include all 5 of the permanent members. Set up a weighted voting system to represent the UN Security Council and calculate the Banzhaf power distribution.
35. In the U.S., the Electoral College is used in presidential elections. Each state is awarded a number of electors equal to the number of representatives (based on population) and senators (2 per state) they have in congress. Since most states award the winner of the popular vote in their state all their state’s electoral votes, the Electoral College acts as a weighted voting system. To explore how the Electoral College works, we’ll look at a mini-country with only 4 states. Here is the outcome of a hypothetical election:
 State Smalota Medigan Bigonia Hugodo Population 50,000 70,000 100,000 240,000 Votes for A 40,000 50,000 80,000 50,000 Votes for B 10,000 20,000 20,000 190,000
1. If this country did not use an Electoral College, which candidate would win the election?
2. Suppose that each state gets 1 electoral vote for every 10,000 people. Set up a weighted voting system for this scenario, calculate the Banzhaf power index for each state, then calculate the winner if each state awards all their electoral votes to the winner of the election in their state.
3. Suppose that each state gets 1 electoral vote for every 10,000 people, plus an additional 2 votes. Set up a weighted voting system for this scenario, calculate the Banzhaf power index for each state, then calculate the winner if each state awards all their electoral votes to the winner of the election in their state.
4. Suppose that each state gets 1 electoral vote for every 10,000 people, and awards them based on the number of people who voted for each candidate. Additionally, they get 2 votes that are awarded to the majority winner in the state. Calculate the winner under these conditions.
5. Does it seem like an individual state has more power in the Electoral College under the vote distribution from part c or from part d?
6. Research the history behind the Electoral College to explore why the system was introduced instead of using a popular vote. Based on your research and experiences, state and defend your opinion on whether the Electoral College system is or is not fair.
1. The value of the Electoral College (see previous problem for an overview) in modern elections is often debated. Find an article or paper providing an argument for or against the Electoral College. Evaluate the source and summarize the article, then give your opinion of why you agree or disagree with the writer’s point of view. If done in class, form groups and hold a debate.
2. To decide on a new website design, the designer asks people to rank three designs that have been created (labeled A, B, and C). The individual ballots are shown below. Create a preference table.

ABC, ABC, ACB, BAC, BCA, BCA, ACB, CAB, CAB, BCA, ACB, ABC

1. To decide on a movie to watch, a group of friends all vote for one of the choices (labeled A, B, and C). The individual ballots are shown below. Create a preference table.

CAB, CBA, BAC, BCA, CBA, ABC, ABC, CBA, BCA, CAB, CAB, BAC

1. The planning committee for a renewable energy trade show is trying to decide what city to hold their next show in. The votes are shown below.
 Number of voters 9 19 11 8 1st choice Buffalo Atlanta Chicago Buffalo 2nd choice Atlanta Buffalo Buffalo Chicago 3rd choice Chicago Chicago Atlanta Atlanta
1. How many voters voted in this election?
2. How many votes are needed for a majority? A plurality?
3. Find the winner under the plurality method.
4. Find the winner under the Borda Count Method.
5. Find the winner under the Instant Runoff Voting method.
6. Find the winner under Copeland’s method.
1. A non-profit agency is electing a new chair of the board. The votes are shown below.
 Number of voters 11 5 10 3 1st choice Atkins Cortez Burke Atkins 2nd choice Cortez Burke Cortez Burke 3rd choice Burke Atkins Atkins Cortez
1. The student government is holding elections for president. There are four candidates (labeled A, B, C, and D for convenience). The preference schedule for the election is:
 Number of voters 120 50 40 90 60 100 1st choice C B D A A D 2nd choice D C A C D B 3rd choice B A B B C A 4th choice A D C D B C
1. The homeowners association is deciding a new set of neighborhood standards for architecture, yard maintenance, etc. Four options have been proposed. The votes are:
 Number of voters 8 9 11 7 7 5 1st choice B A D A B C 2nd choice C D B B A D 3rd choice A C C D C A 4th choice D B A C D B
1. How many voters voted in this election?
2. How many votes are needed for a majority? A plurality?
3. Find the winner under the plurality method.
4. Find the winner under the Borda Count Method.
5. Find the winner under the Instant Runoff Voting method.
6. Find the winner under Copeland’s method.
1. Consider an election with 129 votes.
1. If there are 4 candidates, what is the smallest number of votes that a plurality candidate could have?
2. If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have?
2. Consider an election with 953 votes.
1. If there are 7 candidates, what is the smallest number of votes that a plurality candidate could have?
2. If there are 8 candidates, what is the smallest number of votes that a plurality candidate could have?
3. Does this voting system having a Condorcet Candidate? If so, find it.
 Number of voters 14 15 2 1st choice A C B 2nd choice B B C 3rd choice C A A
1. Does this voting system having a Condorcet Candidate? If so, find it.
 Number of voters 8 7 6 1st choice A C B 2nd choice B B C 3rd choice C A A
1. The marketing committee at a company decides to vote on a new company logo. They decide to use approval voting. Their results are tallied below. Each column shows the number of voters with the particular approval vote. Which logo wins under approval voting?
 Number of voters 8 7 6 3 A X X B X X X C X X X
1. The downtown business association is electing a new chairperson, and decides to use approval voting. The tally is below, where each column shows the number of voters with the particular approval vote. Which candidate wins under approval voting?
 Number of voters 8 7 6 3 4 2 5 A X X X X B X X X X C X X X X D X X X X
1. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and C decided to drop out of the election, which caused B to become the winner, which is the primary fairness criterion violated in this election?
2. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If for some reason the election had to be held again and many people who had voted for C switched their preferences to favor A, which caused B to become the winner, which is the primary fairness criterion violated in this election?
3. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If in a head-to-head comparison a majority of people prefer B to A or C, which is the primary fairness criterion violated in this election?
4. An election resulted in Candidate A winning, with Candidate B coming in a close second, and candidate C being a distant third. If B had received a majority of first place votes, which is the primary fairness criterion violated in this election?
5. In the election shown below under the Plurality method, explain why voters in the third column might be inclined to vote insincerely. How could it affect the outcome of the election?
 Number of voters 96 90 10 1st choice A B C 2nd choice B A B 3rd choice C C A
1. In the election shown below under the Borda Count method, explain why voters in the second column might be inclined to vote insincerely. How could it affect the outcome of the election?
 Number of voters 20 18 1st choice A B 2nd choice B A 3rd choice C C
1. Compare and contrast the motives of the insincere voters in the two questions above.
2. Consider a two party election with preferences shown below. Suppose a third candidate, C, entered the race, and a segment of voters sincerely voted for that third candidate, producing the preference schedule from #17 above. Explain how other voters might perceive candidate C.
 Number of voters 96 100 1st choice A B 2nd choice B A
1. In question 18, we showed that the outcome of Borda Count can be manipulated if a group of individuals change their vote.

1. Show that it is possible for a single voter to change the outcome under Borda Count if there are four candidates.

2. Show that it is not possible for a single voter to change the outcome under Borda Count if there are three candidates.

2. Show that when there is a Condorcet winner in an election, it is impossible for a single voter to manipulate the vote to help a different candidate become a Condorcet winner.

3. The Pareto criterion is another fairness criterion that states: If every voter prefers choice A to choice B, then B should not be the winner. Explain why plurality, instant runoff, Borda count, and Copeland’s method all satisfy the Pareto condition.

4. Sequential Pairwise voting is a method not commonly used for political elections, but sometimes used for shopping and games of pool. In this method, the choices are assigned an order of comparison, called an agenda. The first two choices are compared. The winner is then compared to the next choice on the agenda, and this continues until all choices have been compared against the winner of the previous comparison.

1. Using the preference schedule below, apply Sequential Pairwise voting to determine the winner, using the agenda: A, B, C, D.
 Number of voters 10 15 12 1st choice C A B 2nd choice A B D 3rd choice B D C 4th choice D C A
1. Show that Sequential Pairwise voting can violate the Pareto criterion.
2. Show that Sequential Pairwise voting can violate the Majority criterion.
1. The Coombs method is a variation of instant runoff voting. In Coombs method, the choice with the most last place votes is eliminated. Apply Coombs method to the preference schedules from questions 5 and 6.
2. Copeland’s Method is designed to identify a Condorcet Candidate if there is one, and is considered a Condorcet Method. There are many Condorcet Methods, which vary primarily in how they deal with ties, which are very common when a Condorcet winner does not exist. Copeland’s method does not have a tie-breaking procedure built-in. Research the Schulze method, another Condorcet method that is used by the Wikimedia foundation that runs Wikipedia, and give some examples of how it works.
3. The plurality method is used in most U.S. elections. Some people feel that Ross Perot in 1992 and Ralph Nader in 2000 changed what the outcome of the election would have been if they had not run. Research the outcomes of these elections and explain how each candidate could have affected the outcome of the elections (for the 2000 election, you may wish to focus on the count in Florida). Describe how an alternative voting method could have avoided this issue.
4. Instant Runoff Voting and Approval voting have supporters advocating that they be adopted in the United States and elsewhere to decide elections. Research comparisons between the two methods describing the advantages and disadvantages of each in practice. Summarize the comparisons, and form your own opinion about whether either method should be adopted.
5. In a primary system, a first vote is held with multiple candidates. In some states, each political party has its own primary. In Washington State, there is a "top two" primary, where all candidates are on the ballot and the top two candidates advance to the general election, regardless of party. Compare and contrast the top two primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc.
6. In a primary system, a first vote is held with multiple candidates. In some many states, where voters must declare a party to vote in the primary election, and they are only able to choose between candidates for their declared party. The top candidate from each party then advances to the general election. Compare and contrast this primary with general election system to instant runoff voting, considering both differences in the methods, and practical differences like cost, campaigning, fairness, etc.
7. Sometimes in a voting scenario it is desirable to rank the candidates, either to establish preference order between a set of choices, or because the election requires multiple winners. For example, a hiring committee may have 30 candidates apply, and need to select 6 to interview, so the voting by the committee would need to produce the top 6 candidates. Describe how Plurality, Instant Runoff Voting, Borda Count, and Copeland’s Method could be extended to produce a ranked list of candidates.

## Reference

1. References (21)