Ranking, Grading, and Genealogy

Joseph Malkevitch
York College (CUNY)
Email Joseph Malkevitch

Introduction

People use numbers all of the time, but what exactly are numbers and what can they be used for? Furthermore, when one asks people what is mathematics about, numbers are often mentioned. Here, in an informal way I would like to explore how numbers play a role in trying to get good results when people vote on choices or elect candidates in voting situations. As I discuss voting I want you to contrast your perceptions of the way numbers are used in subjects such as physics, chemistry, and engineering and the attempts to export the success of using numbers in these fields, to using numbers in economics and social science settings. In particular, we will look at the issue of voting/elections as a way to be able to carry out tasks related to making fair decisions. Perhaps we want to find a way to take an action on the part of a group based on the differing views of the actions available by the individuals who make up the group. Another setting, which will not be treated here (but see a previous column by this author and one by David Austin), involves matching schools and students in an urban environment where the students have "preferences" for the schools and schools have "priorities" among the students. Such school priorities might be based on issues such as whether other children in that family already have individuals who attend that school or are required by a court to balance the racial composition or gender composition of the school. We will see the "tip of the iceberg" for the complexities with which we are faced when we use numbers and mathematics in the world but we will also see some wonderful mathematical ideas that were born from carrying out these applications. One important issue will be the difference between using numbers to count and using numbers to measure. I will also look at how to understand the way mathematics evolves and who creates new mathematics over time by considering the information available through the Mathematics Genealogy Project (MGP).

Counting

When you ask someone what they use numbers for they typically mention counting. Surprisingly, when I ask students in my classes, which are usually mathematics or mathematics education courses, what they use numbers for it is surprisingly hard to get them to tell me that they also use numbers to measure things. So let us begin with the ways that mathematics has explored the numbers used for counting and the numbers used for measuring.

When we count, what do we count? If you are in a supermarket and have to choose a line to check out your purchases you may see a sign saying that a line is only for people who have 10 items or less. While you may be unsure if seven identical cans of soup and a box of cereal count as two items or eight you will still need to be able to count. You also want to count how much money you have with you if you want to pay in cash for your purchases. Counting money involves some additional aspects of counting because money comes in different denominations. If you count you have six bills in your wallet, that could mean you have six dollars or, if you never have a bill greater than a fifty-dollar bill you could have as much as three hundred dollars. Or you might have a variety of amounts of money in between depending on the denominations of the six bills. And then there is the complication that you may have lots of coins in your pocket to count, which also come in several denominations.

When we count we need a collection of number names (and the names differ from English to French to German, etc.) and the notion that there is an ordering of the names, starting with what today we call 0. There is a number that comes next after zero and we call that number one, and there is a number two that comes next after one, and a number three that comes after two, and so on. With all you know you may say, wait a second, aren't there numbers between two and three, and the answer is we can create such numbers but for counting per se we don't need them. Part of what is tricky in understanding numbers is that there is also the issue of how one uses symbols to denote numbers.

The prevailing system in common use today is the Hindu-Arabic system of notation for numbers. This is a place notation system using 10 symbols: 0 ,1, 2, 3, 4, 5, 6, 7, 8, 9. Place notation means that the numbers represented by 234 and 432 are not the same even though they use the same digits. What are the pros and cons of using different numbers of symbols and changing the system that is used to represent numbers? So one can have a place notation system with two symbols (called the binary system) or, for example, sixteen symbols (hexadecimal). In binary the symbols for the two digits are usually taken as 0 and 1 and in hexadecimal the sixteen symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Table 1 shows how place notation can be used to represent the numbers from one to 20 using decimal, binary, and hexadecimal.
 

Note that the string 10 occurs in several places but in one case 10 is ten, in another it is two and in another 16. To distinguish what set of symbols is being used for representing the numbers one can write (10)₁₀, (10)₂, and (10)₁₆. The numbers 10, 2, 16 are the "base," that is the number of different symbols that are used in a place notation system to represent the different positive integers, the numbers that come one after another when one counts.

It is fascinating to look at the range of different symbols (systems) that are used to represent the counting numbers from 0 to 9 in different times and places.

 

 

Counting numbers also supports the idea that if a number comes "after" another number in the counting process we think of it as a "bigger" number. Ten comes after eight when you count so when you have ten eggs you have more eggs than if you have eight eggs.

While in all academic areas counting is important and valuable, it is using numbers for measurements that has for many people moved science into a special place. Based on science, technologies have been developed which use numbers to make things--your cell phone, your car, the homes you live in. While theorists may argue about numbers and their natures,[ engineers, carpenters and. plumbers use numbers to make things that may not work forever but work in the short run. Bridges last many years and don't typically fall down. Without the ability of people to measure, the industrial revolution would not have been possible.

Building on the counting properties of the counting numbers one can also talk about arithmetic operations on these numbers, like addition and multiplication. Subtraction raises a problem. One can handle 12 - 4 but what about 4 - 12? Eventually, the notion of a bigger system of numbers, which now we would call the integers, results though there was opposition in some quarters to this line of thinking. When looking at the history of various aspects of numbers what is complicated is that different developments occurred over vast time scales and in different cultures--Egypt, Greece, China, and India to name but a few. All made important contributions to our current insights into numbers. However, what was discovered did not always become widely rooted in these cultures themselves nor were they "shared" or transferred to other cultures. Very often dramatic discoveries are made in mathematics but making these developments "rigorous" from a modern perspective usually comes much later and sometimes displaces the informal and intuitive ideas that might make some of the ideas easier for "beginners." Thus, the Peano Axioms or postulates are often taken today as the modern rigorous approach to the counting numbers.

One way to compare the "size" of two collections of things is to count each and use our knowledge of numbers to decide if the two collections have the same or different size. However, even if we had no way of counting one could pair the items in the two collections and if the two collections are "finite" then one can tell which collection has more elements or the same number of elements even if we don't know how many elements there are in the two collections. This notion of 1-to-1 correspondence was exploited dramatically by Georg Cantor (1845-1918).

The social and behavioral sciences are widely viewed as "less exact" than engineering and the sciences in part because they rely on numbers and mathematics less than physics, say. Yet the behavioral sciences have also moved beyond qualitative insights into quantitative ones. In economics, political science and many other behavioral science environments one is often interested in having individuals report their "preferences." Do you like Gala apples better than Granny Smith apples? Do you like apples better than oranges? For this question the answer might depend on which kind of apples and oranges you are asking about. Someone might like Gala apples better than Valencia oranges but Cara Cara oranges better than Fuji apples. You may not have heard of Cara Cara oranges and even if you have heard of them you may not have tasted them and so are unable to compare them with Fuji apples, which you have tasted. Common parlance about oranges is such that many people use different names for the same kinds of oranges.

So in providing preference information one first might be able, given a list of choices, to provide a "ranking" of these choices listed from most preferred to least preferred. I will adopt the convention that in such lists more preferred things are towards the top (Figure 1):
 

This diagram might be drawn as the mathematical model that a person used to respond to his/her 6 favorite fruits. If one asked the same question tomorrow it is natural to wonder if exactly the same set of fruits would appear on the list, or perhaps blueberry would now be present instead of blackberry. Often in situations of this kind what comes to mind when producing a long "list" depends on whether one has thought about this issue recently or there is a somewhat "noisy" quality for the items that get listed. If someone has to rank A and B without ties they might do it one way now but a different way a few hours from now, or tomorrow, having had one of these fruits for dinner, and this changes their perceptions about the merits of the fruits. Now imagine that instead of asking people to rank their 6 favorite fruits one asks the person to assign a score from one to 9 to each fruit. Two fruits might get the same score, so what is being done here "implicitly" is that some "measurement" is being made where the numbers 0 to 9 are being used to represent the score. Here for example is how a person who did the ranking in Figure 1 might score the same fruit. The individual involved has listed the six fruits above alphabetically and indicated the "score" or "grade" on the 1 to 9 scale as the number in parentheses.

Apple (7)
Banana (6)
Blackberry (7)
Figs (2)
Orange (9)
Pear (3)

or

Apple (7), banana (6), blackberry (7), fig (2), orange (9), pear(3)

which would code exactly the same information in a different "notation." Is the ranking in Figure 1 consistent with the scores above? Would you expect that it would be?

An individual presented with the scale and challenge of scoring the fruits will produce scores but they may not think very hard about the meaning of such a scale and may not "feel" that the difference of one between the scores for apple and banana represents the same difference as the difference of one between fig and pear.

The scores shown above can be used to construct a diagram (Figure 2) similar to the one in Figure 1 but where there would be a tie between apple and blackberry which could be handled as shown by listing two items at the same level. (Note that the scores provided have been "suppressed" in this ranking but could be added after the name of each fruit.) Note that this ranking is quite different from the one in Figure 1 but often when different approaches are used to elicit "preferences," these preferences can be quite different. When voters are polled about political or controversial issues slight differences in the way questions are framed often result in very different results.

 

I like the notation for preferences in Figures 1 and 2 because it tends to be self-explanatory, but it is not pleasing to people who have to do page design!

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Ballot

One way to conceptualize about preferences is that one has some set of choices to evaluate and does this with a ballot. From a mathematical modeling perspective (modeling is the branch of mathematics which is concerned with how to effectively use mathematics in areas outside of mathematics) one can imagine that one is given a "ballot" and must use it to evaluate the choices. What complicates the process is the variety of issues related to the knowledge base of the person making the decision and the skills this person has in evaluating the desirability of the different alternatives.

Here are some examples of ballots that have have been looked at from a mathematical perspective: The ballot involved might be the input to an election or voting situation where one or more choices are involved. In the discussion below think that one is choosing one candidate from a collection of several candidates, but it is intriguing to see how one might want to change the ballot type for an election where several winners will be selected. To be more specific one might use a ballots to solicit voter opinion to elect the chairperson of an academic department versus a ballot to select the 3 members of a curriculum committee for a mathematics department in a college. Here is a sample of commonly used ballots for the case of picking a single choice from a list of choices:

a. Standard ballot

Choices are listed and each "voter" selects exactly one.

b. Yes/No ballot

Choices are listed and one must choose yes or no for each of the choices.

c. Approval ballot

In the elections context one votes for any person or persons one is willing to have serve.

d. Ordinal or preferential ballot (no ties, no truncations)

Truncation refers to not requiring that all candidates be listed. Typical reasons for truncation are that a voter not know anything about one or more of the candidates or because the voter feels that not "mentioning" a candidate will not give assistance to that candidate in winning.

Choices are ranked from top choice to bottom choice, with no ties and one can't omit any choices even if perhaps one knows nothing about these choices.

e. Ordinal or preferential ballot with ties

f. Ordinal or preferential ballot with ties allowed and one can truncate the ballot.

Choices are ranked from top to bottom but one can leave off certain choices (either because one does not know who or what they are or one merely wants not to list them) and one can rank several choices at the same level.

g. Cardinal ballot

One has a scale, perhaps 0 to 10, and each choice is assigned a number from the measurement scale, where bigger numbers mean that one is more enthusiastic about that choice.

It is important to be aware that it may make a difference in who might win an election as to what scale is used in a cardinal ballot. Instead of using 0 to 10 one might use 0 to 9 or A, B, C, D, E (A most preferred), or the scale is from 0 to 99. Sometimes there are verbal choices: most preferred,..., neutral, least preferred.

Ballots based on using a scale of some sort can be used to construct rankings but such a ranking would be artificial without the capability of allowing ties within the ranking framework. We can ask an individual to "grade" the 6 fruits above using the numbers 1 to 10. Try ranking the 6 fruits, ties allowed, and then try grading the same fruits using the measuring scale of 1 to 10. Which was harder? Were your results consistent?

Social choice theory is the branch of mathematics that concerns itself with taking individual preferences and combining them into a collective decision for a group or society. While there are many intrinsically mathematical questions associated with social choice theory there is also the issue of using the knowledge obtained for mankind's benefit. Within the domain of particular countries (institutions) that function in a democratic way, what is the best or optimal way to carry out various tasks for society's benefit? A particularly important issue is how to conduct voting and elections. Voting involves using ballots to solicit voter opinion and using that information to select a group of winners or provide a ranking. Kenneth Arrow (1921-2017)
 

was a pioneer in relatively modern times in studying such questions. Through his writings and research he transformed our insights into this interface between individual choices and how society might build on them.

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Ranking versus grading

The brief discussion of ballots above raises an important conceptual or philosophical distinction about ways that we can acquire information from a group of people in a situation where a decision has to be made on the basis of the input from that group. This distinction is between the ideas of ranking and grading. When teaching a discrete mathematics course in high school or college to 25 students, typically you as the teacher must provide a grade for every student in the class. In high school, the grading system is usually based on numbers, each student getting a grade of 0 to 100. In college, students would also get a grade but perhaps on a scale of A, B, C, D, F or these same letters with only plus to modify them, thus B+ but not C-, or only minuses to modify the grades, A- but not C+ or allowing plus and minus grades (C+ or B-).

Is the number system better or the letter system? Is it reasonable that during the course of a semester a teacher can tell whether a particular student is a C+ or B- student or that a student deserves a 73 rather than a 74? In high school, is more accuracy achieved because there are 101 possible grades, while in the letter system with pluses and minuses there are only 9 grades (using F but not F- and F+). When one grades it is implicit that one has a "preexisting" scale by which to judge the "contestants." Each contestant can be graded without taking into account the other contestants since one does not have to have seen the performance of all the contestants to grade a particular one. Once one grades each contestant one can now "rank" the contestants (often there will be ties) based on the grades so that if Ann has a higher grade than Betty, then Ann is ranked higher than Betty. One can grade students in a class, individuals or teams participating in an ice skating competition, or the candidates seeking the Presidency. Again, implicit in grading is that when judge A grades contestant C higher than contestant D, that C is "superior" to D. What conclusion should be drawn if judge A gives C 20 and 30 to D while judge B gives D a 20 and C a 30? Do these grades say something about contestants C and D or something about judges A and B? Even when judges give C and D identical scores does that mean they think they are equally accomplished?

Ranking by comparison requires the idea that between any pair of candidates one can determine that the two candidates are tied or that one of them is "stronger" than or more preferred to the other. Either one has to be able to look at all of the contestants at once and produce such a ranking or one must be able to make a decision about every pair but have the skill that the ranking of pairs makes possible a consistent ranking for the whole group. To be consistent one does not have paired comparisons where A is preferred to B, B is preferred to C but C is preferred to A. In general if a relation R has this property, it is said to be a transitive relationship.

Grading seems to require more skill than ranking but perhaps there are advantages to using a system of ranking rather than grading which, though seemingly "cruder," might be carried out in a way that is more reliable.

Where one grades or ranks candidates in an election one needs a decision method for arriving at a choice for society. You can find out lots about different methods here.

Having briefly examined issues related to how one can use numbers or some ordered scale to judge or rank people, perhaps a few words are in order about how questions of this kind came to be part of the mathematics landscape and how people interested in the history of mathematics can get a window onto the way a mathematical subject evolves.

Roots

While the roots of interest in how to translate individual opinions into a course of action for a group (society) go back hundreds of years (work of Borda and Condorcet) it has only been relatively recently that a full array of mathematical tools has been used to study the issues. Behavioral scientists have tried to use ideas that helped get insights into physics in political science and economics. In recent times no one has done more to promote the systematic study of elections and voting systems than Kenneth Arrow, who died recently at age 95. Arrow started out as a mathematics major at City College (now part of the City University of New York) and eventually wrote a doctoral dissertation at Columbia University in the area of economics under the direction of Harold Hoteling, known for his work in statistics. Arrow, according to the Mathematics Genealogy Project (MGP) had 27 doctoral students, some at Stanford University and others at Harvard. In addition to winning the Nobel Memorial Prize in Economics four of his doctoral students--Eric Maskin, John Harsanyi, Michael Spence and Roger Myerson--also won Nobel Memorial Prizes. All four of these scholars border mathematics, applied mathematics, economics, and philosophy in their scholarly interests and contributions. One of Arrow's academic grandchildren, Jean Tirole, also won a Nobel Memorial Prize in Economics.

It is interesting to list Arrow and his students who won Nobel Memorial Prizes in the order of the year they earned their doctoral dissertation and show the titles of their doctoral thesis:

Kenneth Joseph Arrow (1951): Social Choice and Individual Values

Arrow's academic descendants:

John Charles Harsanyi (1958): A Bargaining Model for the Cooperative n-Person Game

A. Michael Spence (1972): Market Signaling: The Informational Structure of Job Markets and Related Phenomena

Eric Stark Maskin: (1976) Social Choice on Restricted Domains

Roger Myerson (1976): A Theory of Cooperative Games

Jean Tirole (1981): Essays in Economic Theory

These individuals have taught in economics departments as well as in some cases mathematics or applied mathematics departments. They make strong use of mathematical ideas, approaches and tools.

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The Mathematics Genealogy Project

The Mathematics Genealogy Project (MGP) attempts to understand the development of mathematics from an historical perspective in part by looking at those who have achieved (in relatively recent years) high attainment in mathematics by getting a doctorate degree, though of course in different countries the doctorate degree has had its own history. For example, in what is now Germany, doctoral degrees were offered at universities much earlier than in, for example, Great Britain. So Arthur Cayley (1821-1895), certainly a very distinguished mathematician, did not earn a doctorate though he contributed in many ways to many parts of mathematics. More recently, the distinguished American mathematician Andrew Gleason also did not earn a doctorate. Cayley did not supervise the doctoral dissertations of any students though he had "mathematical descendants" in the sense there were people who studied with him or were significantly "trained" by him. By contrast, Cayley's good friend James Joseph Sylvester (1814-1897), a Jew, had difficulty getting even bachelor's and master's degrees because many English universities in the 19th century had requirements regarding religion in order to be awarded a degree. Similarly, in tracing the influence of women mathematicians many countries did not allow women to get advanced degrees until comparatively recently.

The MGP classifies the mathematical topics of people who wrote doctoral dissertations with mathematical content using the Mathematical Subject Classification. This classification breaks down the parts of mathematics into various pieces so that research and expository articles listed in databases related to mathematics can work efficiently. Individuals who get degrees in departments other than mathematics such as computer science or mathematics education sometimes appear in MGP. Thus, a person might have gotten an Ed.D. rather than a doctor of philosophy degree and in this case MGP will show that the thesis they wrote is in category 97, mathematics education.

The current classification system dates to 2010, but steps are being taken to revise the classification in order for an updated system to be available in 2020. Interested people can look at proposed changes and offer ideas for what should be done. The parts of mathematics in this system are coded using two digits. Some items in the classification data back to when the system used was put in place (1940) but the classification for social choice comes under: 91 Game theory, economics, social and behavioral science, which was added to the classification system in 2000. 00 is used for general content while 99 is used for miscellaneous. Computer science is 68. Many of these two-digit codes are subdivided further into more narrowly defined topics.

Another complication is the issue of whether mathematics develops only via the scholars who get degrees in mathematics. What about people who get degrees in physics or economics and do important work in mathematics? Here an example is offered by Edward Witten. Witten won a Fields Medal, which some consider mathematics' most prestigious award, but his Ph.D. is not in mathematics. Witten started as a graduate student in economics at the University of Wisconsin in Madison but left to study applied mathematics at Princeton, though eventually he moved into physics and earned his doctorate in physics, with a dissertation entitled Some Problems in the Short Distance Analysis of Gauge Theories. Witten's thesis was directed by David Gross whose MGP ancestors lead back to the very well-known physicist Enrico Fermi, going back several "academic ancestor" steps. There is little doubt of the important role Witten has had for mathematics including his work in what has come to be called Gromov-Witten Theory. While Witten is trained as a physicist, Mikhail Gromov's academic ancestors are mathematicians. Gromov's thesis advisor (Vladimir Abramovich Rokhlin) is "descended" academically from mathematicians.

 

These complications notwithstanding there is much to be learned from a database such as the MGP. At the current time for those people who recently have gotten doctoral degrees related to mathematical topics, this project attempts to list all such people and create a "record" for them. The information it displays draws upon: the updates from schools that award doctorate degrees sent to the MGP on a regular basis, the input of particular individuals with an interest in the history of mathematics who report information they have obtained that does not appear in the MPG, the reporting to the MPG of individuals who supervise doctoral dissertations, and the self-reporting of individuals about having earned a doctorate. All of this is done using the "honor system" in that while the MGP tries to make sure of the accuracy of the information it posts, it has no reason to assume that information that is submitted is "fake" in any sense. What information does a "record" on the MPG provide?

Here, for example, the text below Nash's picture is the substance of the entry for the great game theorist and mathematician, John Forbes Nash, Jr. on the Mathematical Genealogy Project.

John Forbes Nash, Jr.

Biography MathSciNet

Ph.D. Princeton University 1950 United States

Dissertation: Non-Cooperative Games

Advisor: Albert William Tucker

Student
Name:                School                        Year
Patinkin, Seth    Princeton University     2003

According to our current on-line database, John Nash, Jr. has 1 student and 1 descendant.

As you can see, each "record" in this database has certain information: an individual's name, whether the individual wrote something which was picked up by the American Mathematical Society to be listed on MathSciNet and whether this person has a bibliographic record as part of the series of biographical essays available on the web at MacTutor; the kind of degree that the person achieved, the school where this degree was obtained, the person who served as "advisor" for this degree, and additional information about students (typically, who earned a doctoral degree that was supervised by this person); finally, a count of the number of students this individual had who had this individual advisor, and the total number of "mathematical descendants" of this individual.

A few general comments. As noted above for mathematicians of the distant past the individual may or may not have gotten a doctorate degree and the advisor was perhaps more a person of great influence than that person's thesis advisor. Thus, Isaac Newton (1643-1727) shows two advisors, Isaac Barrow (1630-1677) and Benjamin Pulleyn (died 1690).
 

While Barrow did work in the area of geometry and also knew some ideas related to the Calculus, Pulleyn, who by his name being listed in the MGP as an ancestor of Newton might be construed as having a mathematical influence on him. This was probably not the case. Pulleyn was a Greek scholar, and while he was a tutor of Newton, he probably provided input regarding Greek rather than mathematics. Benjamin Pulleyn was also part of the infrastructure which made sure students who won degrees could pass exams on the "required" curriculum of the time which included Euclid's Elements. It appears as if Pulleyn sent Newton to Barrow to make sure that he was properly versed in Euclid. Perhaps Newton was more versed in Descartes' mathematics than Euclid's mathematics when Barrow looked into the matter. Yet, Pulleym has descendants, 14621 mathematical descendants by MGP's reckoning, and with time this number will probably grow! Newton is listed with two students, Roger Cotes (1682-1716) (many descendants) and William Whiston (1667-1752) who has no descendants but who is often discussed in the history of science (and philosophy).
 

Portrait of William Whiston, Courtesy of Wikipedia.

Contributions are made to mathematics by people who "call themselves" members of many academic disciplines, ranging from mathematicians, mathematics educators, computer scientists, physicists, chemists and economists, to give but a sample. Clearly having a doctorate degree in mathematics, applied mathematics or mathematics education conveys a different message from having a doctorate in political science but pursuing a career which entails important contributions to game theory. Doctoral degrees are awarded by departments or sometimes interdisciplinary programs. Thus, there are at some schools separate departments of mathematics and applied mathematics. The pattern of department names where one studies mathematics varies somewhat from one country to another. The MGP indicates, using a flag, the country where a listed person had his/her roots but there are certainly very different traditions of higher education from one country to another. Not only does this entail matters such as when a country started offering a doctorates in mathematics but there are also issues about whether academic degrees could be obtained by members of particular religions.

To understand the mathematics of voting issues one might look at the MGP using various strategies. One could enter names of particular individuals like Kenneth Arrow and Eric Maskin who have published research and expository articles about elections and voting and see if theses they directed were about voting/elections. One might enter a string into thesis key words such as voting or elections or social choice theory, a more general title than voting and/or elections that encompasses voting/elections. Similarly, the string "game theory" might be entered into theses key words. Certainly game theory is a much broader topic than voting/elections but many people who work in the area of game theory (such as Donald Saari) have also done work about elections. Using MGP one can often see "chains" of people who devoted their mathematical talents to social choice and more generally economics and behavior sciences topics.

Mathematics has a rich history of practitioners whose work created a web of ideas which allowed mathematics to expand its domain into areas such as social choice and game theory. These practitioners have also contributed tools, ideas, and students who continue to develop new mathematical tools and whose work has influenced others to expand the reach of mathematics.

Comments?

References

Arrow, K., Social Choice and Individual Values, Wiley, New York, 1963.

Arrow, K., and A. Sen, A. K. Suzumura, eds., Handbook of Social Choice and
Welfare (Vol. 1). Elsevier., 2002.

Arrow, K., and A. Sen, A. K. Suzumura, eds., Handbook of Social Choice and
Welfare (Vol. 2). Elsevier., 2010.

Aumann, R. and S. Hart (Eds.) Handbook of Game Theory with Economic Applications, North-Holland, New York, Volume 1, (1992), Volume 2, (1994).

Balinski, M. and R. Laraki, Majority Judgment, MIT Press, Cambridge, 2010.

Black, D., Theory of Committees and Elections, Cambridge U. Press, Cambridge, l958.

Brams, S., and P. Fishburn, Approval Voting, American Political Science Review, 72 (1978) 831-47.

Brams, S., and P. Fishburn, Approval Voting, Birkhauser, Boston, 1983.

Kreps, D., Game Theory and Economic Modeling, Clarendon Press, Oxford, 1990.

Kuhn, H., (ed.), Classics in Game Theory, Princeton U. Press, Princeton, 1997.

Luce, R. and H. Raiffa, Games and Decisions, Wiley, New York, 1957.

Saari, D., Geometry of Voting, Springer-Verlag, New York, 1994.

Saari, D., Basic Geometry of Voting, Springer-Verlag, New York, 1995.

Saari, D., Chaotic Elections! A Mathematician Looks at Voting, American Mathematical Society, Providence, 2001.

Young, H., Equity, Princeton U. Press, Princeton, 1994.

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some of these materials.

Joseph Malkevitch
York College (CUNY)
Email Joseph Malkevitch