Celebratio Mathematica

David H. Blackwell

Statistics  ·  UC Berkeley


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].

Early years

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, 1941–42

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.”

Job search

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.

Howard University, 1944–54

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”.

Stanford, 1950–51

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, 1948–50

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.

Berkeley, 1954–2010

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 was Bayes’ buri­al place where he took pic­tures of his tomb.

He also vis­ited the school where Wolfgang Döblin was 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.

Black­well was:

  • 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.

Re­gard­ing race:

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.


Black­well met his wife, Ann, while he was re­gis­ter­ing stu­dents for a phys­ics course that she thought a friend of her fam­ily’s would be teach­ing. A friend of Black­well’s told Ann that the sub­sti­tute (Black­well) would be a reas­on­able teach­er, and she should take the course. They mar­ried right after he went to Howard, and ul­ti­mately had eight chil­dren, none of whom have gone in­to math­em­at­ics. None of his kids went to Howard or his­tor­ic­ally black col­leges. Like his par­ents raised him with the as­sump­tion he would go to col­lege, so did the Black­wells raise theirs. After his ap­point­ment to Berke­ley, they bought a home in Berke­ley, as well as a 40-acre re­treat in Men­d­o­cino (north of Berke­ley) where they ex­pec­ted to re­lax, but have been foiled by all man­ner of house­hold re­pairs and needs.