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Spektabiles,
### Michel Balinski: Vita

### Augsburg: Anwendungsorientierte Mathematik

Thus having innoccuously found our way to Augsburg, let us stay here for a
while. Founded by the Romans, the city of Augsburg looks back on more than
2000 years of history. In the middle ages, Augsburg was the premier
banking place of the Old World, a place you would necessarily turn to, if
you wanted to be elected emperor and were in need of a few thousands, or
hundred thousands, of *Gulden* to bribe your electors. Strangely,
though, the bourgeois elite of the Freie Reichsstadt never contemplated
investing their money in a University.
### Contributions to linear and to combinatorial optimization

Balinski's 1959 Princeton PhD thesis, directed by Albert W. Tucker, was
entitled *An Algorithm for Finding All Vertices of Convex Polyhedral
Sets*. Mathematicians have always been interested in finding good
algorithms, that is, descriptions which calculations have to be executed in
order to solve a problem. After World War II, the advent of computers gave
rise to a renewed interest in algorithms, and in particular those
algorithms that are suitable for machine calculations.
### Contributions to discrete model building

Problems of matching and allocation, as just outlined, are categorized as
*discrete mathematics*. The attribute »discrete« is used for the
very reason why we term a person discrete, as somebody who honors the
individual, who acknowledges that there are features peculiar to an
individual rather than being shared by many, as somebody who refrains from
generalizing inappropriately. In social life the opposite is indiscretion.
In mathematics, however, the opposite of discrete mathematics is continuous
mathematics.
### Contributions to the analysis of proportional representation systems

Balinski's research into the mathematics of apportionment and proportional
representation originates from the early 70's, much of it together with
his junior co-author Peyton Young. The collaboration of the two culminated
in the 1982 monograph on *Fair Representation: Meeting the Ideal of One
Man, One Vote*. The first edition from Yale University Press was followed
by a 1987 Japanese translation and, in 2001, by a second edition whose
pagination is identical with that of the first edition. The book centers
around the apportionment problem as it manifests itself for the US-American
House of Representatives. There the issue is to apportion (zuteilen) the
435 house seats among the 50 States of the Union, proportionally to the
population counts of the decennial census data.
Of that there is no matter of doubt--

No probable, possible, shadow of doubt--

No possible doubt whatever.

sehr geehrter Herr Prorektor,

hohe Festversammlung,

liebe Damen Balinska Junior,

und -- last but eigentlich first -- verehrter Kollege Michel Balinski,

soon to become *Doctor rerum naturalium honoris causa Universitatis
Augustanae*. Soon, *in spe*, it is my pleasure to emphasize, because
Spektabiles Jungnickel will bestow the honorary degree on you only when, or
shall I say: if, I have come to an end with my laudatio, which has barely
even begun.

In fact, right away we are facing a certain problem. In case that of the many achievements of Professor Balinski there were too many or, mathematically speaking, infinitely many, my laudatio would never ever come to an end, and Professor Balinski would have to wait for the honorary degree infinitely long. Luckily, Michel Balinski distinguished himself in a subfield of mathematics that goes unter the name of discrete mathematics, and that is characterized by dealing with finitely many items and finite sets. Hence, my dear Michel, I am confident that you will tolerate my saying that of your lasting achievements there are finitely many.

Since all good things come in threes, I shall concentrate, firstly, on your contributions to linear and to combinatorial optimization, secondly, on your contributions to discrete model building and, thirdly, on your contributions to the analysis of proportional representation systems. To begin with, however, I would like to introduce you to the audience as a human being, by telling a little bit from your vita, and to introduce us to you, by reviewing parts of the short history of our Institute.

Michel Balinski was born some seventy years ago, in Geneva, into a polyglott family. His father being occupied by his position as a Polish diplomat with the League of Nations, much of Michel's education laid in the hands of his grandparents. In fact, his grandfather was Ludwik Rajchman (1881-1965) who, then, was the Director of the League of Nations' Health Organization and, later, in December 1946, became the founder of Unicef, the United Nations International Children's Emergency Fund.

The Geneva start was followed by a few years in France which, fleeing from the invading German troops, ended in the exodus, via Lisbon, to the US. In the New World his mother was to continue to speak to him in French, thus helping the child to maintain the French roots.

Michel, having grown up outside New York City, spent most of his formative years and his academic career on the East coast. In 1954 he graduated from Williams College with a Bachelor's Degree in Mathematics, followed by a Master's Degree in Economics from the Massachusetts Institute of Technology and, in 1959, a PhD degree in Mathematics from Princeton University.

The academic affiliations of Michel Balinski include Princeton University, the Wharton School at the University of Pennsylvania, the City University of New York, Yale University, and the State University of New York at Stony Brook. The latter ran in parallel with a position of a Directeur de Recherche with the CNRS and the Laboratoire d'Econométrie of the Ecole Polytechnique. Over time the double appointment across the Atlantic proved too much of a burden. Hence in 1989 Michel Balinski moved permanently to the Ecole Polytechnique, and this is the position from which he retired to the status of emeritus, in 1999.

In the French system -- other than in the German system, I enviously add -- there is also a life after retirement, of which Michel Balinski has made and is still making good use, in his position as a Directeur de Recherche de classe exceptionnelle émérité. Judging from the rate of output of papers, attendance of conferences, and other academic activities I am happy to report that Professor Balinski must be alive and well.

In the forty years of professional career Michel Balinski distinguished
himself with an exceptional number of successful activities beyond what a
Professor normally is paid for, research and teaching. He founded the
journal *Mathematical Programming*, which became a leading journal of
the field. He served as president of professional societies, visited an
impressive number of international universities as a guest lecturer, and
served as the System and Decision Sciences Chairman of IIASA, the
International Institute for Applied Systems Analysis in Laxemburg near
Wien. In 1965 Balinski was awarded the prestiguous Lanchester Prize of the
Operations Research Society of America, which is a particular pleasure to
mention since the Augsburg Mathematics Faculty includes with Karl-Heinz
Borgwardt another Lanchester Prize winner.

We therefore have to admit that the University is less of a product of Augsburg's glorious past. Rather, founded in 1970, it is more of a political reaction to the post-68 syndrome. Which, on the other hand, makes the University a young lady, crispy and attractive.

When, in 1981, mathematics was added to the growing University, the
challenge was to develop an image which would convey to the general public
the idea of how modern, useful, and profitable mathematics is, both in the
real world that we live in, as well as the complex world that we think in.
The challenge was met by letting the Augsburg Mathematics Institute sail
under the heading of *anwendungsorientierte Mathematik*, and this
strategic orientation has proved to be extremely successful since.

It is difficult to properly translate *anwendungsorientierte
Mathematik* into English. Applied mathematics would be too narrow, besides
being already confined to the connotation of Angewandte Mathematik and,
inappropriately, suggesting a contraposition with Pure Mathematics. I am
afraid I cannot do better than translating *anwendungsorientierte
Mathematik* rather literally into *application oriented mathematics*
which, indeed, embraces abstract research such as pure mathematics, while
at the same time prominently emphasizes the practical use to which
mathematics is put.

Under the label *anwendungsorientierte Mathematik* a novel degree in
*Wirtschaftsmathematik* was devised and was then, and still is today,
the curriculum that most of our students choose to enroll in. In 1981
there was only a handful of mathematics departments at German universities
offering such a degree. With the success saga spreading this has changed,
and we have lost our unique position. The anwendungsorientierte concept
carried not only beyond Augsburg to other campusses. On this campus, it
also carried beyond mathematics to other fields. It provided an extremely
fruitful starting point for the Augsburg Physics Department and, as of
recently, of the Computer Science Department. The current strategic plan
of the University provides some renewed visibility for the concept, by
presenting it under the timely label of *innovative technologies*.

Ladies and gentlemen, I have reviewed part of the University history on this occasion, of conferring an honorary PhD degree on Michel Balinski, because he and his scientific œuvre testify in a prime way that the anwendungsorientierte interplay, of real world problems and complex academic solutions, is fascinating, fruitful, and never ending. As mentioned in the beginning, I will exemplify this claim by marking three of the fields where Balinski's scientific achievements stand out.

The type of problems that are exceptionally well suited to be handled by a
machine came to be known under the name of *linear programs*.
Balinski's dissertation dealt with particular geometric structures that
submit themselves to this approach, convex polyhedral sets (Vielecke),
whose shape is determined by flat sides meeting in straight line edges, and
edges meeting in vertices, *Eckpunkten*. Any such set can be described
from an internal, primal point of view, or alternatively, from an external,
dual view point. Just as we can describe this lecture hall by looking to
the walls from the inside where we are now -- which is the primal approach
to the problem, or else by walking around on the outside and tell what we
see then -- the dual approach. At times the dual approach is quite
persuasive, just think of the drinks that may be waiting outside. However,
Balinski approached the problem more from an academic point of view and,
together with his Doktorvater Al Tucker, published a long article on the
*Duality Theory of Linear Programs* in the 1969 *SIAM Review*.

Another important subclass of problems are those, where the vertices of the convex bodies (konvexe Körper) have integer coordinates (ganzzahlige Koordinaten). Balinski was one of the first to extend the theory to such models. Nowadays there are many textbooks devoted to this type of problem, but in 1965 there was none. The subject was new, and Balinski's seventy-page 1965 overview article served as a welcome reference and first textbook for the new field. The article enjoyed the fate, rather rare in mathematics, of being reprinted twice, in 1968 and in 1970.

Another subset of combinatorial results comes under the inviting name of
*stable marriage theorems* (stabile Heiratssätze), proving by
terminology better than by anything else that mathematics is so utterly
anwendungsorientiert. The problem is standard. There is a set of ladies
and a set of men. Each lady likes certain of the men and has her
preferences among them, but detest the others. Similarly, each men has his
preferences among the women he likes, but cares not a whit about the
others. The mathematical question is this: Can men and women be happily
married given their respective preferences?

Imagine what could go wrong if Monika and Friedrich, say, were married to others, and yet, Monika preferred him to her current mate, while at the same time Friedrich preferred Monika over his current mate. They would be unhappy and, provided the two couples meet too often, they would abandon their current mates for each other. Though I grant that marriage is a very serious business, it is a peculiar trait of mathematicians to go to the simplest possible situation that reveals the essence of a problem, however frivilous it may sound. There are also polyandrous versions of the question, where every lady may have several husbands, and polygamous versions, starring men with many wives.

Are there marriages which would avoid the sad instability of our example?
The mathematical theorem answers: Yes, there always are! Of course, the
mathematical results are independent of the narrative frame in which they
are presented. There are important practical problems submitting
themselves to just the same analysis: admitting students to universities,
appointing candidates to jobs, or allocating hours of work on different
tasks to workers of different qualifications. All these instances come
under what now is called »two-sided markets«, for which the mathematical
approach provides a vital aid how to match, allocate, or apportion
resources. As with other problems, mathematics focusses on two issues: Can
it be done at all and, if so, how to do it efficiently. Balinski and
co-authors have contributed considerably to the realm of marriage theorems
and, in a recent 2003 *American Mathematical Monthly* note exemplify
its usefulness for *Admissions and Recruitment*.

Discrete mathematics counts items; it rules out continuous transitions from one to the other. Each candidate stands for herself or himself, each institution is recognized as a unit on its own, each seat in parliament is honored as a valuable entity by itself. Such constructs as fractional candidates, or fractional institutions, or fractional seats in parliament are meaningless. The friction between mathematical terms and practical needs becomes particularly apparent in consulting, when the -- mathematical -- consultant wants to persuade the -- non-mathematical -- consultee of the usefulness of the approach. While model building itself may be more of an art than a science, the mathematical problems thereby generated are abundant, and challenging.

Michel Balinski's experience to apply abstract mathematical concepts to concrete problems of economics and decision making is based on an impressive experience as a mathematical consultant, for such companies as de Borden Mills Inc., the RAND Corporation, Mathematica Inc., Mobile Oil Research Laboratories, ORTF, Econ Inc., and others. His pointed opinions on where mathematics can contribute to the solution of practical problems is all too noticeable throughout his technical papers, be it matching problems as covered by the type of marriage theorems mentioned above, or problems of assignment, allocation, or apportionment. As a third group of examples, I would like to finally turn to his work on apportionment methods for proportional representation.

Of course, parliamentary seats are in the first place more of a political than of a mathematical nature. However, when one of the seats was contested in 1992, the United States Supreme Court found it appropriate to include in the decision a six-page review of the Balinski/Young monograph. I would bet that theirs is the only mathematics book that ever got read and reviewed by any supreme court throughout the world. The judges, while not questioning the mathematical correctness of the book, did improve on its political correctness, by turning the untimely Balinski/Young motto of »one man, one vote« into the more equal principle of »one person, one vote«. It may have escaped their attention that, in mathematics and statistics, the data unit »man« refers to what in the old days the great Latin writers would have worded as »homo«, and not as »vir«. As down-to-earth scientists we would never set up a theory that visibly excludes half of mankind, or politically more correct, half of personkind.

The Balinski/Young *Fair representation* work is actually two books in
one. The first half carefully reviews the historical experience that
accumulated over more than 200 years of US history. I find these first
hundred pages a gem of scientific writing, always lucid, occasionally
thrilling, and at times entertaining. The authors aim, and succeed, in
extracting from the historical experience general rules which, in
mathematical language, may serve as an axiomatic foundation of a theory of
apportionment, which then is laid out in the second half of the book.

For instance, one such axiom demands that changes in representation agree with changes in population. If, relative to several competing groups, one is growing larger, then the number of its representatives should increase. But does it? Not in the system that we use for the election of the Deutschen Bundestag where, weird as it may sound, more Zweitstimmen votes for a party may result in fewer Bundestag seats.

Another axiom stipulates that a parliamentary body that grows in size will never see a party shrink in its number of representatives. But does it? Not in Germany. When, in 1989, the steering committee of the Wetteraukreis in Hessen was formed, the two major parties raised the number of seats from nine to ten. Why? Because this made one of the minor parties drop from one representative to none. That is, not only was a new seat created by enlarging the committee size from nine to ten, but, due to the pecularities of the apportionment method used, an old seat was taken away from an unwanted competitor who was thereby pushed out. Not surprisingly, the two seats thus generated benefitted the two major parties.

Laying down general principles, or axioms as we say in mathematics, enables us to classify the many methods that are available to convert population or vote counts into numbers of representatives. These mathematical classifications are carried out in the second half of the Balinski/Young monograph, where the authors scrutinize the apportionment methods that are in use all over the world against the general axioms that appear so compelling and hard to deny.

Michel Balinski pursued the subject into many different directions. One of them includes biproportional representation methods, wherein proportionality is achieved in two directions, one along the regional subdivision of the population, the other, along party lines. I had the privilege of proposing one such biproportional method to the Kantonsrat Zürich who, in fact, adopted it almost unanimously to include it in their new electoral law.

Another line of Balinski's research aims at the districting problem, that is, achieving electoral equality in electoral districts (Wahlkreise, Stimmkreise). We will hear more about this problem in a minute or two from Professor Balinski himself.

I have subsumed Michel Balinski's research work under the Augsburg standard
of Anwendungsorientiertheit, and I would like to end by drawing your
attention to another point of what Anwendungsorientiertheit includes.
Namely, proliferating the mathematical findings, not only as technical
papers in academic journals, but also as nontechnical articles in the
public science press. Who otherwise would publicize the findings, if not
those who generate them? You will not be surprised to hear that Michel
Balinski has done so, in *Le Monde, Pour la science, Spektrum der
Wissenschaft* and other press products.

Naturally it always remains a challenge to translate dry academic truths into juicy public stories, and a welcome trick is to embellish the presentation by drawing on quotes from literary and intellectual authorities. If my laudatio were in German I certainly would have taken recourse to Goethe and Schiller. As it is in English, I have to switch to the British G&S-counterpart, Gilbert and Sullivan, and end with a three-liner quoted by Professor Balinski in order to dismiss any doubt that his proposed mechanism for a stable marriage assignment is optimal. As for me, the quote is to imply that the mechanism of the Mathematisch-Naturwissenschaftliche Fakultät der Universität Augsburg to nominate their PhD laureates is -- doubtlessly -- equally optimal:

No probable, possible, shadow of doubt--

No possible doubt whatever.

(Gilbert and Sullivan, *The Gondoliers*)

Ich danke für Ihre Aufmerksamkeit.

*(Professor Dr. Friedrich Pukelsheim)*

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- 10.07.2007