by Linda Kirby
As a product of the Great Depression, David Harold Blackwell had planned to become an elementary school teacher—a friend of his father’s had promised him a job. That goal gradually shifted when his high school geometry teacher encouraged him to solve math problems; when a reverse discrimination policy won him a fellowship because he was black; when an adviser arranged for him go to the Institute for Advanced Study—all of these were responsible for producing an eminent statistician who specialized in probability and game theory. Although he continually cautioned against using the phrase “first of anything”, it is unavoidable here, since David Blackwell is the first African American to be hired in a tenured position at UC Berkeley, first to be inducted into the National Academy of Sciences, first to become President of the Institute of Mathematical Statistics, and first to become Vice President of the American Mathematics Society.
Unless otherwise noted, all quotations are from The Oral History Project interview [e3].
Born in 1919, Blackwell grew up in Centralia, Illinois, a railroad town of approximately 12,000, named so because it was a junction point for the railroad. His father, Grover Blackwell, had a fourth-grade education and was a hostler for the railroad (someone who delivered the engines to and from the engineer into a roundhouse), a job he started at the age of 18, which he loved and did for his whole life. Blackwell often visited his father at work and rode on the steam locomotive.
I still have a great affection for steam locomotives because of that. [My father] liked his job so much that when the working day was reduced from seven days a week to six days a week, he used to go down to the shop on his day off just to make sure that his replacement was taking care of those steam locomotives.
His mother, Mabel Johnson Blackwell, dropped out of high school in her sophomore year and was a housewife. She was active in the Baptist Church—church in his family was a women’s thing, none of the men went. Together they raised four children, David being the oldest of three brothers and a sister. His mother also managed rental properties that she inherited from her family. Her father had moved to Centralia where he had a small but successful grocery store and acquired the connecting lots: “The whole area was known as the Johnson Territory”.
Their neighborhood was mixed but primarily white, maybe 30% black, although most of his friends were black. He attended the local elementary school, which was integrated, and finished in six years rather than the normal eight, and was always with students who were older. He learned to read in his uncle’s grocery store, reading seed packages. “It was in high school that I found something that I really liked. I really liked geometry. I had always been pretty good at math and somewhat interested in it, but geometry really excited me.” He credits his first two geometry teachers for influencing him. “Mr. Huck encouraged some of us to try to solve problems in a mathematics magazine. I solved some of them and he mailed my solutions in and a few times my name appeared in the magazine.”
The Depression definitely influenced Blackwell’s career choice. Getting a job was paramount, and when one of his father’s friends promised him a teaching job after he finished college, his career path was decided: he would become an elementary school teacher. His immediate family was lucky for his father kept his job, but some of his relatives were forced to leave Centralia for Chicago or other big cities to find work. Even though neither of his parents or their friends had gone to college, there was no doubt that he and his friends would.
He felt his education was terrific, and was the reason he was a semester ahead of most of his fellow students when he started at the University of Illinois (Champaign–Urbana) about 100 miles from Centralia. As soon as he got off the train on his first day, he met a member of the black fraternity, Alpha Phi Alpha, who invited him to live there—an encounter that changed his life—and where he lived for his entire six years as a student. He was the only math major in his fraternity and eventually became president of the Math Club. Every Wednesday night he took the train home so he could deliver his dirty laundry and pick up some clean laundry, get on the train and come back. “So, I spent virtually every Wednesday night on the train!” (Since his father worked for the railroad he had a pass and didn’t cost him anything).
The university had a language requirement; the obvious choice for Blackwell was to continue with his Spanish. One of his fraternity brothers suggested that, if he had any idea of going into mathematics, he should think about German, since for a math Ph.D. it would be required to pass an exam in German. He liked math and it came easy to him, so he took every undergraduate math course that was offered and ended up switching his language option. And although he definitely planned to be an elementary school teacher, those types of courses had a “bad reputation as being a joke and really easy,” so he just kept putting them off.
Although Blackwell had won a scholarship to pay his tuition (\$35 a semester!), he had to pay his living expenses, books, etc. Blackwell discovered that his father had been borrowing to send him to college, which made him focus intensely on his academics since he wanted to be able to support himself. During the Depression, Roosevelt had created the NYA, National Youth Authority, for students (similar to the WPA, Works Progress Administration, which paid for him to work in the University etymology lab. In addition, he washed dishes and waited on tables at the fraternity house.
Blackwell explains the turning point when he realized he would get his Ph.D. in mathematics:
I had a four-year scholarship. I completed my undergraduate work in three years. So I decided to use that fourth year of my scholarship to go on and get a Master’s degree. Then I was encouraged to apply for a fellowship or a teaching assistantship to go on for a Ph.D. So my focus was gradually shifting and I did apply. And I got a fellowship. So then, that settled it. Then, I was going to go on for a Ph.D.
He went on to explain how his being awarded this fellowship ended up as a reverse-discrimination policy:
See, there were two kinds of awards, fellowships and teaching assistantships. They paid the same amount of money, but for a fellowship you didn’t have to do any teaching. So they were the preferred award. And there were maybe three fellowships and twenty teaching assistantships every year. But you submitted a single application. We all submitted applications. One of the other graduate students told me that I was going to get one of the fellowships. I said, “How do you know that?” He said, “Well, you’re good enough to be supported and they’re not going to put you in a classroom!” Because I was black, of course. He was right, sure enough I did get one of the three fellowships. And I’m sure that a partial consideration was, “Well, we need to support this fellow, and we can’t put him in a classroom, so let’s give him a fellowship.” So it was an advantage to be black.
At the suggestion of a fellow student, Blackwell asked Joe Doob, one of the founders of modern probability theory, if he would be his adviser, and Doob agreed. Paul Halmos and Warren Ambrose were also Doob’s students. “I was very lucky to have him as an adviser. The things that he told me to read and the things that he wrote were just fundamental in the future of the subject of probability.” Doob was very influential in helping Blackwell get the Rosenwald Postdoctoral Fellowship that enabled him go to the Institute for Advanced Study in Princeton, along with Halmos and Ambrose. While at the Institute, he published several papers related to Doob’s research. One of these was his thesis, which explored Markov chains.
Blackwell graduated from the University of Illinois, having earned his B.A in Mathematics in 1938, M.A. in Mathematics in 1939, and his Ph.D. in 1941, at the age of 22.
Institute for Advanced Study, Princeton,
That year at the Institute (1941–42) was a big year for Blackwell. He was learning about probabilities, real variables, point-set topology and Hilbert spaces. They would all talk about problems, concepts, ideas. “At the Institute there were two kinds of people: really great mathematicians and all the fresh new Ph.D.’s. Mostly I learned a tremendous amount from all the other young people.”
Von Neumann was there and had developed game theory, which piqued Blackwell’s interest in the subject. It was also at the Institute that Blackwell became “mildly interested in statistics”. Sam Wilks, the editor of a leading statistics journal, gave a course and “one of the main ideas that I got from that course was that I wished that I could understand statistics and the way statisticians thought, but it was too hard. But I knew there was something interesting there and I wished I could understand it.” (pg. 30)
Blackwell had many memories of other mathematicians who were at the Institute and who influenced him:
Shizuo Katatuni, I learned a lot from him. He worked in Markov chains. He went to Yale right after the war and stayed there.
Gerhard Kalisch. We used to talk and walk together. I remember admiring a beautiful brown tie that he was wearing and he took it off right then and there and gave it to me! He went to UCLA.
George Mackey was there and I learned a fair amount from him. Several years later, he and I discovered the same theorem at about the same time. And it was interesting to me because the way he came at the theorem had absolutely nothing to do with the way I came at the theorem. Somehow we both found the same theorem. It says that two countably generated sigma fields of Borel sets that have the same atoms are identical.
When he was asked about a conversation he had with Gödel (who argued that God’s existence was based on mathematics), Blackwell remembered making the comment that it seemed to him that there were some propositions which, if they were undecidable, they must be true; Gödel laughed and said, “Yes, yes, we know that”. “I was telling him something that was interesting to me, but later on found out that was page one in what logicians knew things about.”
Blackwell also remembered Dorothy Maharam and her husband Arthur Stone, who were quite knowledgeable about measure theory and that they would see each other whenever they come to Berkeley.
Jimmie Savage, with whom he later worked at RAND, was also at the Institute that year. He was one of the strongest influences on the direction of his research (see below).
Life at The Institute was casual—small offices, libraries, rooms with blackboards where one could go and talk, maybe bump into someone that would clarify something. The director was Frank Aydelotte who staunchly supported Blackwell’s becoming an honorary faculty member at Princeton. Blackwell tells the following story,
I didn’t find out about it until years later. There was a custom that all members of the Institute would be made honorary faculty members at Princeton. And so when I was invited to become a member of the Institute, that meant that I would be appointed an honorary faculty member at Princeton. Well, the president of Princeton did not want any black honorary faculty members at Princeton. And, as I understand it, he notified the director of the Institute and there was a big fuss over this. And several of the professors in the Institute complained about it and threatened to disconnect the Institute from Princeton unless I was accepted (Oswald Veblen being one). And I guess it wasn’t a big thing, so the President of Princeton backed down. I never knew anything about that. Of course, it was all settled before I got there. And I was just welcomed cordially along with everybody else. It was only much later that I found out that there had been all of this to-do.
Blackwell said that never experienced any racial prejudice while he was at the Institute. In fact, Blackwell says that he may have been denied access to things because he was black, but never among his colleagues or people he associated with. He does describe a time in Washington, DC, when he and his wife wanted to buy tickets to a play, but the clerk was unable to sell him tickets because she would lose her job. He defines this as “institutional racism, something that is stronger than the particular people that are involved in it”.
Although Blackwell downplays any racism in his life, professionally and personally, he makes a very telling statement in an interview with Morris DeGroot [e2]: “Being black shaped my expectations from the very beginning. It never occurred to me to think about teaching in a major university since it wasn’t in my horizon at all.”
After his year at the Institute, Blackwell was quite concerned about being drafted. He had heard that the Office of Price Administration in Washington, DC, had a position that was classified as an “essential occupation”, i.e., protected from the draft. When he discovered that it did not offer protection, he resigned and went on to Southern State University, where he had already been hired. Years later, in 1943–44, he was called. Blackwell was worried, but one of his students mentioned his own rejection for psychoneurosis. “I am one, I know one when I see one, and your are one!” Blackwell was indeed examined by a psychiatrist, and several weeks later he was rejected because of “anxiety neurosis”. He was quite pleased! ”But a cousin and one of his brothers, Joe, did volunteer for the army after the war was over, and the GI Bill supported them both through college.”
In the interim, Blackwell had begun to apply for jobs after his year at the Institute. Although he said he never experienced discrimination, he did acknowledge that no white college would hire him, so he sent applications to the 105 historically black colleges. In addition, he did something unusual, he went on an automobile tour of about thirty colleges.
I just drove up to an institution and looked for the mathematics department and went in and introduced myself to the head of the math department and told him I was looking for a job. Crazy way to do things! But I didn’t know any better. Mostly I got cordial receptions but it was made clear to me that well, often, I was referred to the president of the institution because the head of the math department didn’t make appointments to the math department; the president of the institution made appointments. So, sometimes I got to see the president. And out of all this activity came three job offers.
I was having fun, just getting to see what the country was like and what the black colleges were like.
His efforts resulted in three job offers, the first being from Southern State University in Baton Rouge, Louisiana, which he accepted. He was one of two mathematicians, the other being the chair of the department. It was the first time he had been in an institution that was all black. The other two offers were from Clark College in Atlanta (where he taught the following year), and West Virginia State.
In another first, Baton Rouge was the first time Blackwell lived in the south. He relays this funny story:
The first time I got on a streetcar in New Orleans there’s a little board that you plugged into the top of the seat, and on the front of it said “White” and on the back of it, it said “Colored”. The idea of it was that if the board was here and all the Colored seats were taken, and the next row above was vacant, you moved the separation board up one row and then sat there, and vice-versa. I thought that board was rather funny. And when I got off the streetcar or the bus, I took the board with me. [laughs] I took it back to my room, posted it for a while. So I, of course, accepted segregation but I didn’t take it very seriously. Just another one of those silly customs.
After one year, he left Southern to go to Clark in Atlanta, as he saw it as a potentially richer academic community, because Morehouse, Morris Brown and Atlanta University were all in the same area. He started a joint seminar where students from all four institutions would participate.
One of the colleges that Blackwell visited during his automobile tour was Howard University, where the head of the math department was clearly uninterested. However, a man sitting at a nearby desk became chair two years later and, remembering Blackwell, offered him a job, which he accepted. That man was Dudley Woodward, with whom Blackwell was quite impressed. At the age of 45, Woodward gave up a deanship and chair of the Math Department at Howard to get his Ph.D. in mathematics. He returned to Howard at the professorship level, set up a special room for a math library and started a math seminar.
For Blackwell, being at an all black institution was not very important, but Howard was a black scholar’s dream, ranked the highest among black colleges. This was an extremely important time for him; his interests were basically formed while he was at Howard. He taught undergrads who were interested in becoming high-school math teachers or getting a civil-service job in Washington, not to become a math professor—similar to Blackwell when he started college and before his focus shifted.
Blackwell’s two colleagues at Howard were Elbert Cox, an instructor and the first black man to get a Ph.D. in math, and Woodward. Blackwell got tenure surprisingly quickly, after three years. It was a question of how much research one did. When Woodward was forced to retire because of a mandatory retirement age, Blackwell became chair at the young age of 28. As chair, it was his responsibility to make sure all the classes ran. It was right after World War II, 1947–49, the GI influx was big, and there were more students than regular faculty could teach. He borrowed instructors from other local institutions and also hired two or three faculty, one of whom was Claytor, a student of Woodward, and “one of the best lecturers I ever heard in my life, very clear and well organized.”
The Howard library carried the leading math-statistical journals but, for mathematical contacts, he went outside the university. Abe Girshick was a statistician in the Department of Agriculture. Blackwell was a consultant at the Operations Research Office in Washington, where he was doing mostly game theory and optimization theory, and where one of his better mathematical ideas came about by thinking about wars and how to fight them.
[It was] thinking about that kind of conflict that led to what I call “approachability.” Other people have called it “approachability theory,” I just introduced the concept of a set being approachable by one player or excludable by another player. And it’s been used by a number of people working in game theory.
Blackwell didn’t think this theory would be militarily useful, but for him it was mathematically interesting:
If you can look at a real situation and translate that into something that’s interesting mathematically, that’s sort of what I like to do. Not going the other way, taking mathematics and interpreting it to the real world. But taking something from the real world and see if it suggested some interesting mathematics.
This philosophy has been consistent throughout Blackwell’s professional life; his main focus has always been to find a mathematically interesting situation.
Although he didn’t believe that his paper “On multi-component attraction games” [Naval Research Logistics Quarterly, 1954] has ever been cited, it did lead to some of his other theorems, which have been cited and used in game theory, such as the concepts of approachable and excludable sets [“An Analog of the Minimax Theorem for Vector Payoff”].
It was Abe Girshick’s lecture that transitioned him into statistics. The lecture was sponsored by the Washington Chapter of the American Statistical Association, and Girshick spoke about Wald’s work in sequential analysis and mentioned Wald’s Equation. Blackwell found a counterexample and sent it to Girshick. Although Blackwell’s example was incorrect, he and Girshick talked about it and it was during that conversation that Blackwell became interested in sequential analysis. That was the beginning of their collaboration. Girshick was also instrumental in Blackwell going to RAND where they continued to work together.
The Rao–Blackwell Theorem from 1946 (published in 1947) shows how to turn crude guesses into good estimates. The theorem stemmed from the Girshick–Mosteller–Savage group’s theorem that found a way to get unbiased estimates from a sequential sample. In studying this concept, Blackwell realized that it is calculating a “conditional expectation”, which results in another unbiased estimate that is better than the first, sequential, sampling. However, two years before, C. R. Rao, who was a student of R. A. Fischer in Cambridge, England, had published his thesis and this result was one of many results buried in his thesis, to which not many people paid attention until Blackwell rediscovered it.
Rao, of course, was not particularly happy that Blackwell’s name is now attached to Rao’s theorem. As Blackwell says, “He shouldn’t be [particularly happy], because he has the priority by two years. It’s just that somehow, when I did it, it got publicity.” Blackwell says he probably never cited that paper afterward.
By the time Blackwell left Howard, he had published 20 papers. The one he likes the best was a paper he wrote in 1953, under (government) contract, “On Optimal Systems”.
Blackwell’s relationship with Girshick continued during the academic year 1950–1951, when Blackwell went as a visiting professor to Stanford where Girshick was a professor. He, Girshick and Kenneth Arrow wrote an important paper that caused some major personal problems. Wald and Wolfowitz had started research on the same subject, and Blackwell’s group developed it further but did not give them proper credit. Wolfowitz was so angry that he didn’t speak to Blackwell for about 25 years. Blackwell acknowledges that they were wrong to not give adequate acknowledgment to Wolfowitz and Wald.
Charles Stein was the intellectual leader of the statistics group. Blackwell says that, when there was a problem that he couldn’t solve, he would talk to other people about it. He just wanted the problem solved. An example he cites is that he kept mentioning the “comparison of experiments” problem to Stein, who eventually solved it. Blackwell was then happy because he didn’t have to continue thinking about it and could go on to other things.
Although he liked Stanford, he decided to return to Howard and Washington, DC.
RAND, three summers,
Blackwell’s summers at RAND proved to be quite productive. During his first summer, Jimmie Savage was the lynchpin who converted Blackwell to Bayesianism. Blackwell tells the story:
A RAND economist came in one day to talk to me while I was visiting there. And he said, “I need a number. I need to know the probability of a major war within the next five years.” And he explained to me why he needed to know that number and it made a lot of sense. But, I turned him off. I said, “The concept of probability makes sense only in a long sequence of events under identical conditions.” And the occurrence of a war in the next five years is a unique phenomenon and the probability is either zero or one and we won’t know for five years. And he looked at me, and he said, “Thank you”. He said that he had spoken with several other statisticians and they’d all told him the same thing, and he left.
Blackwell was bothered by his flip response to an economist’s serious, reasonable question. Shortly thereafter, he relayed the exchange to Jimmie Savage, who proceeded to explain the subjective theory of probability and that the question asked made perfectly good sense. Earlier in his life Blackwell had considered the probability of single events, the natural way to think about probability, but regretted it wasn’t the correct way. Now Jimmie Savage was saying that, indeed, it was the correct way. “For me, that was a very important intellectual shift. The idea that probability does not apply just to events that occur under identical conditions, but that the concept of probability applies to single events, unique events.”
The statistician who made the original inquiry turned out to be Abraham Wald (of Wald-equations fame). Since he had been told that applying the concept of probability to single events was inappropriate, he called them “weight functions”, giving them the formal name of Bayes solutions. This approach has become popular not only among statisticians, but engineers, economists and biologists—real people making decisions about single events.
Blackwell had known Jimmie Savage since 1941, when they were at the Institute and both attended Wilks’ lectures. They met again several years later at a conference at Brown, and Blackwell remembers Savage turning a previous comment that Blackwell had made into a theorem: “Blackwell’s theorem is: every suitcase can be closed.”
I had said that once. We were in a room and somebody was trying to pack a suitcase and put a lot of stuff in it. And I simply said, “Every suitcase can be closed.” And Jimmie remembered that and quoted it back to me a month later as a theorem.
It was at RAND that Blackwell began to focus on game theory, researching the best timing for two duelists to fire as they approach each other with loaded pistols.
In a general context, anything that has to do with warfare is interesting. So, you start thinking about conflicts in general. As I say, a duel is clearly a very special kind of fighting, and this comparison of experiments was put in terms of comparisons of reconnaissances in a game theory context. And, of course, games are clearly conflict situations. So, that meant that RAND would be interested in them.
In conjunction with his interest in games, Blackwell organized a daily noon-time Kriegspiel (war game) in his office. This is a variation of chess where you don’t see the opponent’s board. There was a steady stream of mathematicians/statisticians who came to play, one of whom was Norbert Wiener. According to Blackwell, Wiener was a terrible Kriegspiel player, who continuously lost but promptly and happily showed up daily. Blackwell remembers Abe Girshick was annoyed because he felt you “shouldn’t spend an hour playing Kriegspiel when you could spend it doing statistics”.
In another study, Blackwell proved that consensus was unnecessary. He, his supervisor and others wanted to combine mathematical opinions into an overall math opinion that was more reliable than any one expert. They compiled a questionnaire and then combined all the individual opinions into group opinion that was more reliable than any expert. The most reliable consensus they had was 81%. But, it turned out that there was one individual that was right 80% of the time; so therefore, the consensus was superfluous, you just had to ask this one person!
It was Richard Bellman who got Blackwell interested in dynamic programming while at RAND. Bellman saw that sequential analysis had far-reaching applications beyond simple sequential decision theory; this turned into a new field called dynamic programming. “So, again, in trying to understand Bellman’s research, I found several new theorems. As a teacher, your job is to teach people and help other people understand things. Well, if you’re going to … you have to understand it well yourself.”
Another area he was studying was three-person games and loyalty. He relays the following story about Julia Robinson and Marian Shapley that somewhat annoyed him. He was studying how people behaved in a three-person game and discovered that, once two people form a pair against a third, the third “had better back out of the game as soon as he can”.
Julia chose Marian and Marian chose Julia. So it didn’t matter who I chose, I didn’t have any partner. So I had to pay them each a dollar. And, after that, I offered one of them a bribe to be my partner, but it didn’t work. We played it again and again they chose each other and I paid each one of them a dollar. And finally, I offered one of them two dollars just to choose me. But no, no, she wouldn’t do it. They stuck to each other. So, they laughed about it and I sort of laughed about it. I finally recognized that once two people form a pair, it was very hard to break up. And I resented that with Julia Robinson and Marian Shapley for about a week, I’d say. It took me a week to get over that.
Early in his job search in 1941, Jerzy Neyman interviewed Blackwell for a position at Berkeley but, due to race-based objections, Blackwell was not appointed. Interestingly, in 1937, while Blackwell was a student at the University of Illinois, he attended a lecture given by Jerzy Neyman and was later introduced to him. Blackwell’s recollection was: “I was honored to be introduced to this very distinguished man. It never occurred to me that I would have any later contact with him, but that incident stuck in my mind.”
Ten years later, however, in 1954, Neyman again offered Blackwell a job (no interview necessary this time!). Blackwell and Neyman had remained friends and colleagues over the years and Blackwell describes some of Neyman’s political and social philosophies. Blackwell told Donald Alpers in 1983 that,
My blackness was a plus for Neyman. He had a tremendous amount of sympathy for anyone who had been oppressed or mistreated in any way. He always favors the underdog. It would have given him a special pleasure to appoint me just because I was black.
After Neyman stepped down, Blackwell became the next chair of the Statistics Department (three years after he arrived). He hated making a specific budget, so he calculated generally about yearly departmental expenses and monetary needs and let the dean figure it out. He believes his major contribution as chair was to give the faculty freedom to choose their own textbooks and design new courses. And, after stepping down, he has said he doesn’t miss administrative work at all!
When Alpers asked Blackwell what makes teaching fun for him, Blackwell’s response was, “Why do you want to share something beautiful with somebody else? It’s because of the pleasure he will get, and in transmitting it you appreciate its beauty all over again.”
Less well known facts
During his European travels, Blackwell visited places that were associated with people he admired. One wasburial place where he took pictures of his tomb.
He also visited the school wherewas a student because he learned a lot about Markhov chains from a paper that Doblin had written. Blackwell said because he admired and respected him so much, he wrote a paper for Döblin, even though he (Döblin) never got to read it.
Döblin proved many beautiful results for normal chains, and said he didn't believe there were not normal chains. Well, I constructed one and wrote a paper. The only one who would have been interested was Döblin, and he was dead.
Blackwell’s thinking toward probability was the same as Döblin’s: never leave probabilistic thinking vs using the techniques of analysis to solve a probability.
- anti-union: His black father came to Centralia in 1912 as a strike breaker. The union didn’t accept blacks, and the black strike-breakers were the ones that kept the railroads running.
- anti-war: He admired non-violence, notably Martin Luther King and Ghandi. He believed they had the right idea, and would have liked to see more non-violence today.
It is important for black people to see opportunities that are there, specifically math. There are many things other than teaching mathematicians can do, and this is important because teaching has a low priority among college black people. What is important is that each person is free to go in his own direction. In my time, that wasn't true, i.e., I was limited to black-only colleges. That has changed now.
He thought race should be taken into consideration-one needs to look at the whole student and race is an important part of him/her. He wants to see a time when every student who wants to go to college can, like grade school and high school.
Regarding how the mathematical mind operates differently: He believed that they think differently, but otherwise they are no different from other people, and told a funny story about tax evaluating:
A man named Merrill Flood was once asked by some tax commissioner in the state of West Virginia, I believe, what to do about evaluating property for tax purposes. People were constantly complaining that their taxes were too high. And Merrill Flood’s suggestion was, let each person evaluate his own property but then give the state the option of buying the property at the specified value. Now, that’s sort of a self-enforcing plan, you see. You won’t evaluate your property too low. Otherwise the state might buy it. And, of course, you won’t evaluate it too high, because you would be paying more taxes than you ought to. Now, if I hadn’t known who proposed that method, I would say that a mathematician did that. It has a kind of neat, self-enforcing simple-mindedness that would appeal to a mathematician.