A connection between weighted voting systems and apportionment: A topic for liberal arts mathematics classes

Brandt, Keith

ABSTRACT: This note presents an example showing the relevance of apportionment to weighted voting systems. The example is appropriate for class presentations or student projects. Also, some of the basic notions of apportionment are presented in a way different from that of standard textbooks. This approach may be useful as a supplement to the more typical treatment.

KEYWORDS: Weighted voting systems, apportionment, liberal arts mathematics courses.


Liberal arts mathematics classes have become increasingly popular in college curricula. Texts such as Malkevitch et al. [3] and Tannenbaum and Arnold [5] are used for courses that often satisfy the general education requirements at many colleges and universities in the United States. My colleagues and I have taught such a course at Missouri Western since 1996, and we have used both texts mentioned above. The emphasis of these courses is on problem solving and applications of mathematics to everyday situations. Two topics often covered, weighted voting systems and apportionment, are ideal for these courses because they are accessible and easily appreciated by students. The aim of this note is to provide a connection between these topics. Finding such connections is an important tool in all levels of mathematics instruction because it serves as a review and also helps students see mathematics as one unifying subject rather than a collection of unrelated topics. Indeed, “Recognizing and exploring interrelationships among the various parts of mathematics enhances the learning of mathematics.” [2, p. 3]

I present an example showing that the choice of apportionment method can affect the relative power of players in a weighted voting system. In this example, the concept of scaling is used to describe some of the methods of apportionment. Many of my students find this scaling approach more intuitive than the procedures described in [3] and [5].


Players are denoted by P^sub 1^, P^sub 2^, …; their weights are denoted by W^sub 1^, w^sub 2^, … respectively. A weighted voting system is denoted by [q : w^sub 1^, w^sub 2^,…, w^sub n^], where q is the quota and n is the number of players. The quota q must satisfy s/2

A classic example of a weighted voting system is the United States Electoral College. Each state controls a certain number of electoral votes based on its number of seats in the House of Representatives. There are 538 electoral votes, and the quota needed to elect a president is 270 (just over half the total). The corresponding weighted voting system is [270 54,33,32,…], where 54 is the number of electoral votes controlled by California, 33 by New York, 32 by Texas, and so on.


According to the United States Constitution (Article I, Section 2), each state’s number of seats in the House of Representatives should correspond to its “respective numbers.” The Constitution does not, however, explicitly describe how to apportion the seats. The only constraints given are “The number of Representatives shall not exceed one for every thirty thousand, but each state shall have at least one representative.” The example in this section shows the inherent difficulties in apportioning the seats to each state, and how apportionment can affect a weighted voting system.

The development below uses alternative formulations of the so called “divisor” methods of apportionment. This approach, which relies on the concept of scaling, may help students understand these methods and see why some methods tend to favor larger states while others tend to favor smaller states.

Suppose the population of a certain county is 220,000. Twenty seats in the county’s government will be apportioned to four districts. A quota of twelve votes is required to pass a motion. Once the seats are apportioned, we can study the resulting weighted voting system.

This example was chosen so that each district’s quota is at least one, thus avoiding the special case where a district might need to be assigned one seat (this case is treated in [1]). If we round the quota in the usual way, we end up missing one seat. Several well-known American scholars have suggested ways to resolve this problem. Texts [3] and [51 give detailed descriptions of these methods and their flaws. For the purposes of the example, I will briefly describe four of the standard methods.

Alexander Hamilton’s Method: For each district, round the quota down (this is called the lower quota). Then give the remaining seat(s) to the district(s) with the largest decimal part in the quota.

Using Hamilton’s method in this example, district B would receive the extra seat. Thus district B gets five seats; the other districts get their lower quotas.

The remaining three methods rely on scaling the quota. By this I mean multiplying each state’s quota by a particular constant. If, after rounding the quotas in a certain way, the desired number of seats is not obtained, we can scale the quotas (up or down) and try rounding again.

Thomas Jefferson’s Method: Scale the quota so that rounding down gives the correct number of seats.

Two other methods use scaling in a way similar to Jefferson’s method.

John Quincy Adams’ Method: Scale the quota so that rounding up gives the correct number of seats.

Daniel Webster’s Method: Scale the quota so that rounding in the usual way gives the correct number of seats.

Examples exist where all four methods produce different results 15, p. 133].

In the example above, Hamilton’s method produces the weighted voting system [12 : 8,5,4,3]; Jefferson’s method produces [12 : 9,4,4,3]. The point of this example is that these two systems are not equivalent. When Jefferson’s method is used, player 1 (district A) has veto power, but when Hamilton’s method is used, player 1 does not have veto power. Although this is a simple example, it reveals the significant (and seemingly arbitrary) impact apportionment has on weighted voting systems.

The following exercises provide additional examples where different apportionment methods produce weighted voting systems that are not equivalent.

Exercise 1:


As we emphasize the important relationships within mathematics, we need to find topics and examples that reveal these relationships. Since liberal arts mathematics texts often cover the two topics in this note, it is natural to look for connections between them. Both books [3] and [51 include an example on the landmark legal case involving the Nassau County Board of Supervisors in New York [5, p. 51] that led to the Banzhaf power index. The questionable apportionment of the House of Representatives in 1872 and subsequent election of Rutherford B. Hayes (instead of Samuel Tilden) is also mentioned [5, p. 144]. The example in this note serves as a fitting follow-up to these examples. It again forces students to evaluate the notion of equity in government and everyday life.

My students have had difficulty calculating divisors. I offer them the equivalent descriptions involving scaling as an alternative way to view the various methods. The trial and error process of finding the correct scaling factor seems to be clearer to my students than the corresponding process for finding divisors. I attribute this difference to the simple idea that multiplication is intuitively easier than division. The scaling approach has also helped my students to see why some methods tend to favor certain states and can violate the quota rule. After working several exercises, students usually offer the explanation themselves: scaling affects larger states more than smaller states. Scaling can also be used to present two other apportionment methods which use “exotic” rounding rules. Dean’s Method uses the “harmonic mean.” The Huntington-Hill method, which is currently used to apportion seats in the House of Representatives, uses the “geometric mean.” The latest edition of Malkevitch et al. [41 uses an approach similar to the one presented in this note: critical multipliers are used to give a more mechanical way to apply the divisor methods.

I have presented this example in class; I have also assigned it as a writing project. In the assignment, I give the population of the four districts and ask students to apportion the county seats according to the various methods. I then ask them to determine if the resulting weighted voting systems are equivalent. I was surprised to learn that many of my students did not immediately see the seemingly obvious connection between the two topics (hence this paper). On the assignment, several students simply apportioned the seats accordingly and never even wrote down the corresponding weighted voting systems. The better students had no trouble finding the weighted voting systems and were quick to point out their differences. Samples of my students’ papers are available upon request.


While preparing this note, I had several helpful conversations with Mark J. Johnson. I thank Mark for his input and encouragement. I would also like to thank the referees for making several valuable suggestions.


1. Balinski, Michael L. and H. Peyton Young. 1982. Fair Representation. New Haven CT: Yale University Press.

2. Leitzel, James R. C., ed. 1991. A Call for Change: Recommendations for the Mathematical Preparation of Teachers of Mathematics. Washington


3. Malkevitch, Joseph et al. 1997. For All Practical Purposes: Introduction to Contemporary Mathematics, 4th ed. New York: W. H. Freeman and Company.

4. Malkevitch, Joseph et al. 2000. For All Practical Purposes: Introduction to Contemporary Mathematics, 5th ed. New York: W. H. Freeman and Company.

5. Tannenbaum, Peter and Robert Arnold. 1995. Excursions in Modern Mathematics, 3rd ed. Upper Saddle River NJ: Prentice Hall.

6. Young, H. Peyton. 1994. Equity. Princeton NJ: Princeton University Press.

Keith Brandt*

ADDRESS: Mathematics, Computer Sciences, and Physics Department, Rockhurst University, 1100 Rockhurst Road, Kansas City MO 64110 USA. keith.brandt@rockhurst.edu

* This paper was written while the author was a member of the faculty at the Department of Computer Science, Mathematics, and Physics, Missouri Western State College, St. Joseph MO 64507 USA.


Keith Brandt received his bachelor’s degree from the University of California at Irvine. He received his PhD from the University of Wisconsin, where he studied hyperplane arrangements under the shared guidance of Peter Orlik and Hiroaki Terao. He held a two-year temporary position at the University of Kansas before joining Missouri Western in 1994. His mathematical interests include algebra, combinatorics, and undergraduate mathematics education. When not at work (when is that?) he enjoys many activities including bicycling, juggling, hiking, skiing, and traveling.

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