Chapter 2: The Mathematics of Power 2.1 An Introduction to Weighted Voting

Chapter 2:The Mathematics of Power2.1 An Introduction to Weighted Voting

Excursions in Modern Mathematics, 7e: 1.1 - 2Copyright © 2010 Pearson Education, Inc.Copyright © 2014 Pearson Education. All rights reserved. 2.1-2

In a democracy we take many things for granted, not the least of which is the idea that we are all equal. When it comes to voting rights, the democratic ideal of equality translates into the principle of one person-one vote. But is the principle of one person-one vote always fair? Should one person-one vote apply when the voters are institutions or governments, rather than individuals?

Weighted Voting


What we are talking about here is the exact opposite of the principle of one voter–one vote, a principle best described as one voter–x votes and formally called weighted voting. Weighted voting is not uncommon; we see examples of weighted voting in shareholder votes in corporations, in business partnerships, in legislatures, in the UnitedNations, and, most infamously, in the way we elect the President of the United States.

Weighted Voting


The Electoral College consists of 51 “voters” (eachof the 50 states plus the District of Columbia), eachwith a weight determined by the size of its Congressional delegation (number of Representatives and Senators). At one end of the spectrum is heavyweight California (with 55 electoral votes); at the other end of the spectrum are lightweights like Wyoming, Montana, North Dakota, and the District of Columbia (with a paltry 3 electoral votes). The other states fall somewherein between.

Electoral College


The 2000 and 2004 presidential elections broughtto the surface, in a very dramatic way, the vagaries and complexities of the Electoral College system, and in particular the pivotal role that a single state (Florida in 2000, Ohio in 2004) can have in the final outcome of a presidentialelection.

In this chapter we will look at the mathematicsbehind weighted voting, with a particular focus on the question of power.

Electoral College


We will use the term weighted voting system to describe any formal voting arrangement in which voters are not necessarily equal in terms of the number of votes they control. We will only consider voting on yes–no votes, known as motions. Note that any vote between two choices (say A or B) can be rephrased as a yes–no vote (a Yes is a votefor A, a No is a vote for B). Every weighted voting system is characterized by three elements:

Weighted Voting System


The players.

We will refer to the voters in a weighted voting system as players. Note that in a weighted voting system the players may be individuals, but they may also be corporations, governmental agencies, states, municipalities, or countries. We will use N to denote the number of players in a weighted voting system, and typically (unless there is a good reason not to) we will call the players P1, P2,…, PN .



The weights.

The hallmark of a weighted voting system is that each player controls a certain number of votes, called the weight of the player. We will assume that the weights are all positive integers, and we will use w1, w2,…, wN to denote the weights of P1,

P2,…, PN, respectively. We will use V = w1 + w2 +…+wN to denote the total number of votes in the

system.



The quota.

In addition to the player’s weights, every weighted voting system has a quota. The quota is the minimum number of votes required to pass a motion, and is denoted by the letter q. While the most common standard for the quota is a simple majority of the votes, the quota may very well be something else.



In the U.S. Senate, for example, it takes a simple majority to pass an ordinary law, but it takes a minimum of 60 votes to stop a filibuster, and it takes a minimum of two-thirds of the votes to override a presidential veto.

In other weighted voting systems the rules may stipulate a quota of three-fourths of the votes, or four-fifths, or even unanimity (100% of the votes).



The standard notation used to describe a weighted voting system is to use square bracketsand inside the square brackets to write the quota q first (followed by a colon) and then the respective weights of the individual players separated by commas. It is convenient and customary to list the weights in numerical order, starting with the highest, and we will adhere to this convention throughout the chapter.

Notation


Thus, a generic weighted voting system with N players can be written as:

Notation

[q: w1, w2,…, wN]

(with w1 ≥ w2 ≥ … ≥ wN)

GENERIC WEIGHTED VOTING SYSTEM WITH N PLAYERS


Four partners (P1, P2, P3, and P4) decide to start a

new business venture. In order to raise the $200,000 venture capital needed for startup money, they issue 20 shares worth $10,000 each. Suppose that P1 buys 8 shares, P2 buys 7 shares, P3

buys 3 shares, and P4 buys 2 shares, with the usual agreement that one share equals one vote in the partnership. Suppose that the quota is set to be two-thirds of the total number of votes.

Example 2.1 Venture Capitalism


Since the total number of votes in the partnership is V = 20, and two-thirds of 20 is 13 1/3, we set the quota to q = 14. Using the weighted voting system notation we just introduced, the partnership can be described mathematically as [14: 8, 7, 3, 2].

Example 2.1 Venture Capitalism


Imagine the same partnership discussed in Example 2.1, with the only difference being that the quota is changed to 10 votes. We might be tempted to think of this partnership as the weighted voting system [10:8,7,3,2], but there is aproblem here: the quota is too small, making it possible for both Yes’s and No’s to have enough votes to carry a particular motion.

Example 2.2 Anarchy


(Imagine, for example, that an important decision needs to be made and P1 and P4 vote yes and P2

and P3 vote no. Now we have a stalemate, since

both the Yes’s and the No’s have enough votes to meet the quota.)

In general when the quota requirement is less than simple majority (q ≤ V/2) we have the potential for both sides of an issue to win–a mathematical version of anarchy.

Example 2.2 Anarchy


Once again, let’s look at the partnership introducedin Example 2.1, but suppose now that the quota is set to q = 21, more than the total number of votes inthe system. This would not make much sense. Underthese conditions no motion would ever pass and nothing could ever get done–a mathematical version of gridlock.

Example 2.3 Gridlock


Given that we expect our weighted voting systemsto operate without anarchy or gridlock, from hereon we will assume that the quota will always fall somewhere between simple majority and unanimity of votes. Symbolically, we can express these requirements by two inequalities: q > V/2 andq ≤ V. These two restrictions define the range of possible values of the quota:

The Range of Values of the Quota


The Range of Values of the Quota

V/2 < q ≤ V

(where V = w1 + w2 + … + wN)

RANGE OF VALUES OF THE QUOTA


Let’s consider the partnership introduced in Example 2.1 one final time. This time the quota is set to be q = 19. Here we can describe the partnership as the weighted voting system [19: 8, 7,3, 2]. What’s interesting about this weighted voting system is that the only way a motion can pass is by the unanimous support of all the players. (Note that P1, P2, and P3 together have 18 votes–they still need

P4’s votes to pass a motion.)

Example 2.4 One Partner–One Vote?


In a practical sense this weighted voting system is no different from a weighted voting system in which each partner has 1 vote and it takes the unanimous agreement of the four partners to pass a motion (i.e., [4: 1, 1, 1, 1]).

Example 2.4 One Partner–One Vote?


The surprising conclusion of Example 2.4 is that theweighted voting system [19: 8, 7, 3, 2] describes a one person–one vote situation in disguise. This seems like a contradiction only if we think of one person–one vote as implying that all players have an equal number of votes rather than an equal say in the outcome of the election. Apparently, these two things are not the same! As Example 2.4 makesabundantly clear, just looking at the number of votes a player owns can be very deceptive.

Equal Say versus Equal Number of Votes


Consider the weighted voting system [11: 12, 5, 4]. Here one of the players owns enough votes to carry a motion single handedly. In this situation P1 is

in complete control–if P1 is for the motion, then the

motion will pass; if P1 is against it, then the motion will fail. Clearly, in terms of the power to influence decisions, P1 has all of it. Not surprisingly, we will say

that P1 is a dictator.

Example 2.5 Dictators


In general a player is a dictator if the player’s weight is bigger than or equal to the quota. Since the player with the highest weight is P1, we can conclude that if there is a dictator, then it must beP1. When P1 is a dictator, all the other players, regardless of their weights, have absolutely no say on the outcome of the voting–there is never a timewhen their votes really count. A player who neverhas a say in the outcome of the voting is a player who has no power, and we will call such players dummies.

Dictator and Dummies


Four college friends (P1, P2, P3, and P4) decide to go

into business together. Three of the four (P1, P2, and

P3) invest $10,000 each, and each gets 10 shares in

the partnership. The fourth partner (P4) is a little short on cash, so he invests only $9000 and gets 9 shares. As usual, one share equals one vote. The quota is set at 75%, which here means q = 30 out of a total of V = 39 votes. Mathematically (i.e., stripped of all the irrelevant details of the story), thispartnership is just the weighted voting system[30: 10, 10, 10, 9].

Example 2.6 Unsuspecting Dummies


Everything seems fine with the partnership until one day P4 wakes up to the realization that with the quota set at q = 30 he is completely out of the decision-making loop: For a motion to pass P1, P2,

and P3 all must vote Yes, and at that point it makes no difference how P4 votes. Thus, there is never

going to be a time when P4’s votes are going to make a difference in the final outcome of a vote. Surprisingly, P4–with almost as many votes as the other partners–is just a dummy!

Example 2.6 Unsuspecting Dummies


Consider the weighted voting system [12: 9, 5, 4, 2]. Here P1 plays the role of a “spoiler”–while not havingenough votes to be a dictator, the player has enough votes to prevent a motion from passing. This happens because if we remove P1’s 9 votes the sumof the remaining votes (5 + 4 + 2 = 11) is less than the quota q = 12. Thus, even if all the other players voted Yes, without P1 the motion would not pass. In a situation like this we say that has veto power.

Example 2.7 Veto Power


A player who is not a dictator has veto power if a motion cannot pass unless the player votes in favor of the motion. In other words, a player withveto power cannot force a motion to pass (the player is not a dictator), but can force a motion tofail. If we let w denote the weight of a player withveto power, then the two conditions can be expressed mathematically by the inequalities w < q (the player is not a dictator), and V – w < q (the remaining votes in the system are not enough to pass a motion).

Veto Power


Veto Power

A player with weight w has veto power if and

only if w < q and V – w < q (where V = w1 + w2

+ … + wN).

VETO POWER


Chapter 2:The Mathematics of Power2.2 Banzhaf Power


2.2 Banzhaf Power

We can already draw an important lesson:

In weighted voting the players’ weights can be deceiving. Sometimes a player can have a fewvotes and yet have as much power as a player with many more votes; sometimes two players have almost an equal number of votes, and yet one player has a lot of power and the other one has none.


2.2 Banzhaf Power

To pursue these ideas further we will need a formal definition of what “power” means and how it can be measured. In this section we will introduce a mathematical method for measuring the power of the players in a weighted voting system. This method was first proposed in 1965 by a law professor named John Banzhaf III.


2.2 Banzhaf Power

We start our discussion of Banzhaf’s method with an application to the U.S. Senate.

The circumstances are fictitious, but given party politics these days, not very far-fetched.


Example: The US Senate

The U.S. Senate has 100 members, and a simple majority of 51 votes is required to pass a bill. Suppose the Senate is composed of 49 Republicans, 48 Democrats, and 3 Independents, and that Senators vote strictly along party lines—Republicans all vote the same way, so do Democrats, and even the Independents are sticking together.



Under this scenario the Senate behaves as the weighted voting system351: 49, 48, 34. The three players are the Republican party (49 votes), the Democraticparty (48 votes), and the Independents (3 votes).

There are only four ways that a bill can pass:

•All players vote Yes. The bill passes unanimously, 100 to 0.

•Republicans and Democrats vote Yes; Independents vote No. The bill passes 97 to 3.



•Republicans and Independents vote Yes; Democrats vote No. The bill passes 52 to 48.

•Democrats and Independents vote Yes; Republicans vote No. The bill passes 51 to 49.

That’s it! There is no other way that a bill can pass this Senate. The key observation now is that the Independents have as much influence on the outcome of the vote as the Republicans or the Democrats (as long as they stick together): To pass, a bill needs the support of any two out of the three parties.



•Republicans and Independents vote Yes; Democrats vote No. The bill passes 52 to 48.

•Democrats and Independents vote Yes; Republicans vote No. The bill passes 51 to 49.

That’s it! There is no other way that a bill can pass this Senate. The key observation now is that the Independents have as much influence on the outcome of the vote as the Republicans or the Democrats (as long as they stick together): To pass, a bill needs the support of any two out of the three parties.


The legislative body for the EU (called the EU Council of Ministers) operates as a weighted voting system where the different member nations have weights that are roughly proportional to their respective populations (but with some tweaks that favor the small countries). The second column shows each nation’s weight in the Council of Ministers. The total number of votes is V = 345, withthe quota set at = 255 votes.

The European Union


The third column gives the relative weights (weight/345) expressed as a percent. The last column shows the Banzhaf power index of each member nation, also expressed as a percent. Wecan see from the last two columns that there is a very close match between Banzhaf power and weights (when we express the weights as a percentage of the total number of votes).

The European Union


This is an indication that, unlike in the Nassau CountyBoard of Supervisors, in the EU votes and power go hand in hand and that this weighted voting system works pretty much the way it was intended to work.

The European Union


Chapter 2:The Mathematics of Power2.3 Shapley to Shubik Power


A different approach to measuring power, first proposed by American mathematician Lloyd Shapley and economist Martin Shubik in 1954. The key difference between the Shapley-Shubik measure of power and the Banzhaf measure of power centers on the concept of a sequential coalition. In the Shapley-Shubik method the assumption is that coalitions are formed sequentially:

Shapley-Shubik Measure of Power


Players join the coalition and cast their votes in anorderly sequence (there is a first player, then comes a second player, then a third, etc.).

Thus, to an already complicated situation we areadding one more wrinkle–the question of the order in which the players join the coalition.

Shapley-Shubik Measure of Power


When we think of the coalition involving three players P1, P2, and P3, we think in terms of a set. For convenience we write the set as {P1, P2, P3}, but

we can also write in other ways such as {P3, P2, P1},{P2, P3, P1}, and so on. In a coalition the only thing that matters is who are the members–the order in which we list them is irrelevant.

Example 2.13Three-Player Sequential Coalitions


In a sequential coalition, the order of the players does matter–there is a “first” player, a “second” player, and so on. For three players P1, P2, and P3, we can form six

different sequential coalitions:

P is the first player, the P joined in, and last came P3;

as shown on the next slide.


P1,P

2,P

3,

P1,P

3,P

2; P

2,P

1,P

3; P

2,P

3,P

1;

P3,P

1,P

2; P

3,P

2,P

1;




Pivotal player. In every sequential coalition there is a player who contributes the votes that turn what was a losing coalition into a winning coalition–we call such a player the pivotal player of the sequential coalition. Every sequential coalition has one and only one pivotalplayer. To find the pivotal player we just add the players’ weights from left to right, one at a time, until the tally is bigger or equal to the quota q. The player whose votes tip the scales is the pivotal player.

Shapley-Shubik Approach


The Shapley-Shubik power index. For a given player P, the Shapley-Shubik power index of P is obtained by counting the number of times P is a pivotal player and dividing this number by the total number of times all players are pivotal. The Shapley-Shubik power index of a player can be thought of as a measure of the size of that player’s “slice” of the “power pie” and can be expressed either as a fraction between 0 and 1 or as a percent between 0 and 100%.



The Shapley-Shubik power index.

We will use 1 (“sigma-one”) to denote the Shapley-

Shubik power index of P1, 2 to denote the Shapley-

Shubik power index of P2, and so on.



The Shapley-Shubik power distribution. The complete listing of the Shapley- Shubik power indexes of all the players is called the Shapley-Shubik power distribution of the weighted voting system. It tells the complete story of how the (Shapley-Shubik) power pie is divided among the players.



Summary of Steps

Step 1. Make a list of all possible sequential coalitions of the N players. Let T be the number of such coalitions. (We will have a lot more to say about T soon!)

COMPUTING A SHAPLEY-SHUBIK POWER DISTRIBUTION

Step 2. In each sequential coalition determine the pivotal player. (For bookkeeping purposes underline the pivotal players.)


Summary of Steps

Step 3. Count the total number of times that P1 is pivotal. This gives SS1, the pivotal count for P1. Repeat for each of the other players to find SS2, SS3, . . .,SSN .

Step 4. Find the ratio 1 = SS1/T. This gives the Shapley-Shubik power index of P1. Repeat for each of the other players to find 2, 3, . . . , N . The complete list of ’s gives the Shapley-Shubik power distribution of the weighted voting system.


The Multiplication Rule is one of the most useful rules of basic mathematics.

Multiplication Rule

If there are m different ways to do X and n different ways to do Y, then X and Y together can be done in m n different ways.

THE MULTIPLICATION RULE


A bigger ice cream shop offers 5 different choices of cones, 31 different flavors of ice cream (this is not a boutique ice cream shop!), and 8 different choices of topping. The question is, If you are goingto choose a cone and a single scoop of ice cream but then add a topping for good measure, how many orders are possible?

Example 2.15 Cones and Flavors Plus Toppings


The number of choices is too large to list individually, but we can find it by using the multiplication rule twice: First, there are5 31 = 155 different cone/flavor combinations, and each of these can be combined with one of the 8 toppings into a grand total of 155 8 = 1240 different cone/flavor/topping combinations. The implications – as well as the calories – are staggering!

Example 2.15 Cones and Flavors Plus Toppings


How do we count the number of sequential coalitions with four players? Using the multiplication rule, we can argue as follows: We can choose anyone of the four players to go first, then choose anyone of the remaining three players to go second, then choose any one of the remaining two playersto go third, and finally the one player left goes last. Using the multiplication rule we get a total of4 3 2 1 = 24 sequential coalitions with four players.

Example 2.16Counting Sequential Coalitions


If we have five players, following up on our previous argument, we can count on a total of

5 4 3 2 1 = 120 sequential coalitions, and with N players the number of sequential coalitions is

N (N – 1) … 3 2 1. The number

N (N – 1) … 3 2 1 is called the factorial of N and is written in the shorthand form N!.

Factorial


Sequential Coalitions

N! = N (N – 1) … 3 2 1 gives the number of sequential coalitions with N players.

THE NUMBER OF SEQUENTIAL COALITIONS


We will now revisit Example 2.10, the NBA draft example. The weighted voting system in this example is [6: 4, 3, 2, 1], and we will now find its Shapley-Shubik power distribution.

Example 2.18Shapley-Shubik Power and the NBA Draft

Step 1 and 2. Table 2-10 shows the 24 sequentialcoalitions of P1, P2, P3, and P4. In each sequential

coalition the pivotal player is underlined. (Each column corresponds to the sequential coalitions with a given first player.)


Step 3. The pivotal counts are SS1 = 10, SS2 = 6, SS3 = 6 and SS4 = 2.



Step 4. The Shapley-Shubik power distribution is given by


1

10

2441

2

3%

2

6

2425%

3

6

2425%

4

2

248

1

3%


If you compare this result with the Banzhaf power distribution obtained in Example 2.10, you will notice that here the two power distributions are the same. If nothing else, this shows that it is not impossible for the Banzhaf and Shapley- Shubik power distributions to agree. In general, however, for randomly chosen real-life situations it is very unlikely that the Banzhaf and Shapley-Shubik methods will give the same answer.



Chapter 2:The Mathematics of Power2.4 Subsets and Permutations


Subsets and Permutations

Subsets and Coalitions

Coalitions are essentially sets of players that joinforces to vote on a motion. (This is the reason we used set notation in Section 2.2 to work withcoalitions.) By definition a subset of a set is any combination of elements from the set. This includes the set with nothing in it (called the empty set and denoted by 5 6) as well as theset with all the elements (the original set itself).


Subsets of {P1, P2, P3} and {P1, P2, P3, P4}

The following table shows the eight subsets of the set {P1, P2, P3} and the 16 subsets ofthe set {P1, P2, P3, P4}.

The subsets of {P1, P2, P3, P4} are organized into two groups—the “no P4” group and the “add P4” group. The subsets in the “no P4”group are exactly the 8 subsets of {P1, P2, P3}; the subsets of the “add P4” groupare obtained by adding P4 to each of the 8 subsets of {P1, P2, P3}.





The key observation in Example 2.20 is that by adding one more element to a set we double the number of subsets, and this principle applies to sets of any size. Since a set with 4 elements has 16 subsets, a set with 5 elements must have 32 subsets and a set with 6 elements must have 64 subsets. This leads to the following key fact:



Number of subsets.

A set with N elements has 2N subsets.Now that we can count subsets, we can also count coalitions. The only difference between coalitions and subsets is that we don’t consider the empty set a coalition (the purpose of a coalition is to cast votes, so you need at least one player to have a coalition).



Number of coalitions.A weighted voting system with N players has 2N – 1 coalitions.

Finally, when computing critical counts and Banzhaf power, we are interested in listing just the winning coalitions. If we assume that there are no dictators, then winning coalitions have to have at least two players and we can, therefore, rule out all the single-player coalitions from our list.



Number of coalitions.A weighted voting system with N players has 2N – 1 coalitions.

Finally, when computing critical counts and Banzhaf power, we are interestedin listing just the winning coalitions. If we assume that there are no dictators, thenwinning coalitions have to have at least two players and we can, therefore, rule outall the single-player coalitions from our list.



Number of coalitions of two or more players. A weighted voting system with Nplayers has 2N - N - 1 coalitions of two or more players.

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Chapter 2: The Mathematics of Power 2.1 An Introduction to Weighted Voting