Died: 7 March 2008 in Berkeley, California, USA

**David Gale** was born in Manhattan. During the years in which he was growing up and attending school, his home was sometimes in New York City, sometimes in the surrounding district. He studied at Swarthmore College, Swarthmore, Pennsylvania, and was awarded a B.A. in mathematics in 1943. He then went to the University of Michigan to study for a Master's Degree which he was awarded in 1947. For his doctoral research he went to Princeton where he undertook research in game theory with Albert W Tucker as his thesis advisor. He was a Procter Fellow during 1948 and received a Ph.D. in 1949 for his thesis *Solutions of Finite Two-Person Games*.

In 1950 Gale published a number of papers with various coauthors. Related to his thesis were the papers (with S Sherman) *Solutions of finite two-person games*, and two papers written jointly with H W Kuhn and A W Tucker, *On symmetric games*, and *Reductions of game matrices*. The paper on symmetric games shows that if a zero-sum two-person game has an *m* × *n* pay-off matrix, then an optimal solution is immediately derivable from an optimal solution of a symmetric zero-sum two-person game with a square pay-off matrix of order *m*+*n*+1. The paper on reductions examines several types of transformations which can be applied to the pay-off matrix *A* of a zero-sum two-person game to yield a matrix *A**. In each case the matrix *A** has fewer rows and/or columns than *A*, so that the game with pay-off matrix *A** is in general easier to study than the matrix *A*. Also in 1950 he published *Compact sets of functions and function rings* in which he generalised S B Myers' characterization of compact sets of mappings of a space *X* (either locally compact or satisfying the first countability axiom) into a metric space *Y*.

We mention one further paper which in many ways is typical of the delightfully simple yet deep questions that Gale often investigated. In 1956 he published *Neighboring vertices on a convex polyhedron* which was summarised by Donald Coxeter as follows:-

Remarking that Kuhn has discovered(oral communication)a polytope in eleven dimensions such that every two vertices form an edge(so that there are no "diagonals"), the author proves that such a phenomenon already arises in four dimensions(for any given number of vertices). In fact, he gives an ingenious but indirect proof that for any two integers m >0, n >2m, there exists a2m-dimensional polytope having n vertices of which every m form a simplex which is an(m-1)-dimensional element(e.g. an edge when m =2).

Following the award of his doctorate from Princeton, Gale spent the year 1949-50 as an Instructor in Mathematics at Princeton. He was then appointed to the mathematics department of Brown University in Providence, Rhode Island. He spent fifteen years at Brown, but spent time away in other institutions, for example in Denmark in 1953-54 after being awarded a Fulbright Research Fellowship, and taking the year 1957-8 as a sabbatical at the RAND Corporation in Santa Monica, California. He was a Guggenheim Fellow during 1962-63, then left Brown in 1965, spending the year 1965-66 as Miller Professor at the University of California at Berkeley. In 1966 he was appointed as a full professor in both mathematics and operations research at Berkeley, being also appointed to the economics faculty in the following year. He remained at Berkeley for the rest of his career.

He met Julie B Skeby while he was in Denmark in 1953. They later married and had three daughters, Karen, Katharine, and Kirsten. The marriage ended in divorce in 1974. He died at Alta Bates Medical Center in Berkeley after a heart attack.

Let us look now at some of Gale's books since this gives a good overview of both his research interests and of courses he taught. In 1957 he published *The theory of matrix games and linear economic models* writing in the Introduction:-

These notes are based on lectures given as part of a graduate course in the Mathematics Department of Brown University ... Their purpose is to give a unified mathematical treatment of ... linear economic models..

R Solow writes in a review:-

The result is an extraordinarily neat and compact survey of zero-sum two-person games, linear programming, pure exchange models(and, implicitly, the elementary properties of finite Markov chains), and pure production models of the Leontiev and von Neumann types. Economic interpretations are usually given, but on the whole the treatment is abstract.

In 1960 Gale published *The theory of linear economic models*. R M Thrall writes about this book:-

... the author has brought together a surprising variety of linear mathematics with appropriate applications. ... An important feature of the book is the substantial collection of problems at the end of each chapter. These are not mere routine exercises but contain substantial theorems extending the development in the body of the text. Since the text itself is very lean and meaty these problems should aid the reader to add some desirable intuitive "fat".

Gale wrote the Mathematical Entertainments column of *The Mathematical Intelligencer* from 1991 to 1997. In 1998 he published these articles in the book *Tracking the automatic ant. And other mathematical explorations*.

Gale received many honours and prizes for his outstanding contributions. In particular he was awarded the Lester Ford Prize 1979-80, and the John von Neumann Theory Prize, 1980. The prestigious John von Neumann Theory Prize was awarded jointly to David Gale, Harold W Kuhn, and Albert W Tucker:-

Their research played a seminal role in laying the foundations of game theory, linear and non-linear programming - work that continues to be of fundamental importance to modern operations research and management science. The early research of the founders of the 'Princeton School,' concerned the mathematical theory of linear inequalities as applied to linear programs, matrix games, the now-classical formulations of symmetric games, games in extensive form, n-person games, games of perfect recall, and the value of information. The list of their accomplishments, not to mention distinguished students and collaborators, is too long to present in detail. Pure mathematics played a strong role in their research. ... Gale's research is known for its mathematical elegance. His contributions range from optimal assignment problems in a general setting to major contributions to mathematical economics such as his solution of the n-dimensional 'Ramsey Problem' and his important theory of optimal economic growth.

Many paid tribute to Gale after his death. Kenneth Arrow, professor emeritus of economics at Stanford University who was awarded the Nobel Prize in economics in 1972, wrote that Gale's [6]:-

... intellectual depth, originality of insight, and thorough liveliness were apparent from the beginning and have remained a source of joy and inspiration to all of us. His life's influence is a permanent heritage.

In fact Alvin Roth, professor of economics at Harvard University, had nominated Gale for the Nobel Prize in economics. He wrote [6]:-

I'm sorry that this now won't happen. But David's work will be remembered for generations to come, in books and journal articles and seminars and workshops, as well as in the very concrete allocation procedures that have been built upon his insights.

Gale's daughter Katharine [6]:-

... recalled that her father would discuss his mathematical work at the dinner table and share with his children his fascination with chess puzzles, card games, puzzle blocks and interlocking puzzles, as well as with all types of math games. Throughout his life, Gale would insist that visitors look at the newest puzzle he was working on. According to his daughter, just before his death, Gale e-mailed a colleague to discuss the mathematics of Sudoku...

In addition to his fascination with mathematical puzzles, Gale had several hobbies. He was an enthusiastic skier, tennis player, traveller and was very knowledgeable about jazz.

**Article by:** *J J O'Connor* and *E F Robertson*

**December 2008**

[http://www-history.mcs.st-andrews.ac.uk/Biographies/Gale.html]