Lloyd Shapley - Nobelist

Lloyd Shapley received the Nobel prize from King Carl XVI Gustaf of Sweden at the annual grand ceremony in December 2012. Among the Nobelists my uncle Lloyd was the tallest. At age 89, he was the second oldest. Though wobbly he was dignified and humorous. He also differed from the others by his convention-defying hair - which resembled the mathematical algorithm for which he won science’s highest honor.

At the time I posted about our family’s glorious days in Stockholm. Now that my writing focuses on high achievement in science, including that of the Shapleys, I dug a bit into the story of my talkative, funny, floppy-haired uncle.

How come Lloyd, who at age 25 was “kicking around my parents’ home” and “not knowing what to do” (his words) become “the greatest game theorist of all time” (per Nobelist Robert Aumann)? Why did Lloyd continue adding to the field for decades? After all, he was eccentric - such as granting just one interview to Sylvia Nasar for A Beautiful Mind, her biography of John Nash, who had been Lloyd’s colleague for a short time. Her book became a bestseller and movie.

**Lloyd’s grandsons Ricky and Andrew Shapley and his sons Christopher and Peter Shapley after the ceremony. Lloyd’s wife Marian Shapley died in 1997.** *Photo: Feng-hsi Liu*

Lloyd, who died in 2016, was the fourth of five children of Martha Betz Shapley and Harlow Shapley, discoverer of our place of the galaxy and Director of Harvard College Observatory from 1921-52. Lloyd was a younger brother of my father Willis. 1/

My uncle’s early life sounds like a math genius story. Actually our family doesn’t use the g-word and Lloyd would have snapped at me or anyone who called him thus. His skills at mathematics and on the piano were fostered by his mother who was proficient at both. In a 1994 oral history 2/ Lloyd said:

“I had two brilliant, straight-A students [Willis and Alan] for brothers,… Nevertheless, we would play mathematical games sometimes around the house, play with cards and multiply them, do things like that. So I had this kind of boost from trying to out-excel my brothers, who were four and six years older than I was, and I did fairly well. So I had a family reputation of being the math whiz, though the other guys were pretty good, too.”

After Phillips Exeter Academy, Lloyd entered Harvard. He was drafted in the fall of 1943 and served with the Army Air Corps in Chengdu, China. Discharged in 1945, he returned to math study at Harvard and graduated in 1948.

The Doorbell Rang

Lloyd was off at class one day when a military officer rang at his address: 60 Garden Street, Cambridge, the Residence of the Director of Harvard College Observatory.

Lloyd’s mother received the caller. She accepted the small object he had brought for Lloyd: a Bronze Star for his service in China. What Lloyd did exactly stayed secret for decades, but is described in Wikipedia as “breaking the Soviet weather code.”

When he finished Harvard in 1947, Lloyd recalled, “I didn’t know what to do. …I was sort of not well-organized. My brothers will quote my father as saying, “You know, this guy may be on our payroll for life.” … So I was sort of hanging around my parents’ home in New Hampshire, and … visiting old buddies.”

Willis worked at the Bureau of the Budget In Washington. “Willis … mentioned this Rand, not a corporation, project RAND….I got off some sort of an application.” He got back a telegram from John Williams, offering $300 a month. He was told to report Jan 1 to the RAND office in Santa Monica, California. 3/

“Inheritor of the von Neumann mantle”

"I knew the whole damn book. I knew it more or less by heart and know all the mistakes they made and so on and other turns they might have taken and didn't take." - Lloyd Shapley on von Neumann & Morgenstern Theory of Games and Economic Behaviour

Project RAND had started as a think tank for the new Air Force whose bombers armed with nuclear weapons seemed to assure US postwar dominance. But the Soviet Union was hostile. (It detonated its first A-bomb in 1949.) Project RAND urgently recruited scientists to tackle any problem to deal with this new world. John Williams head of mathematics recruited young stars.

Also, game theory had just been born. In 1947 Von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior was republished as a thick tome, spelling out the central axioms. Lloyd recalled that at RAND he and some others formed a seminar on the Bible-like tome.

“We’d get together and someone would work through another chapter.”

Lloyd wrote up a mathematic paper based on a bit of it. A mimeographed copy was shown to von Neumann, who was in Princeton but often out at Rand in Santa Monica.

“Von Neumann was all excited about this and said he’ll publish it, and he’ll sponsor it in any journal you name.” He was “enthusiastic”, Lloyd recalled, “because the mathematicians had not really risen to his great work he had put out.” 4/

So Lloyd arrived at Princeton in 1949 to work toward a Ph.D. in mathematics. “He was immediately recognized as brilliant and quite sophisticated in his thinking,” Nazar writes in A Beautiful Mind. 4/

“He was the son of one of the most famous and revered scientists in America, the Harvard astronomer Harlow Shapley. The senior Shapley was a public figure known to every household.” Lloyd was “a war hero” with a Bronze Star for his service in China while a Harvard undergraduate. He was well-read. He played the piano beautifully. At 26 he was older than most students.

His roommate was Martin Shubik. John Nash was in the same dormitory suite. Lloyd also met David Gale and Herbert Scarf. Another friend was John McCarthy. (McCarthy founded artificial intelligence. Meanwhile von Neumann, having birthed game theory, was devising the computer.)

It was not foregone that Lloyd would return to Rand - which he did in 1954. Yet in his Princeton years at years he “spent every summer.. .at Rand…So I had a continuous thing.”

Nasar writes, “The consensus at Princeton at the time was that Shapley was the real star of the next generation inheritor of the von Neumann mantle. “

"The justification of all mathematical models is that, however simplified, unrealistic or even false as they may be in some respects, they force analysts to confront possibilities they would not have occurred to them otherwise." - Sylvia Nasar

A Career of Contributions
The lives of people who are prolific with their minds often seem uneventful on the outside.

Lloyd’s production of ideas continued, supported by National Science Foundation grants and others. It may be fitting that his long, stable career at RAND and marriage to Marian Ludolph in 1955 eventually earned him the Nobel for his 1962 paper on stable marriages.

His Nobel co-winner Alvin E. Roth said in a fine brief summary of Lloyd’s work 5/ that, while he won for his paper with Gale on stable matching, “he could have won a Nobel prize for any of a number of his papers that spawned whole literatures.”

Roth’s summary of Lloyd’s contributions was published as a teaching module for Lindau-Nobel Institute in 2016. Also see below Game Theory Math.

**Lloyd Shapley at the time he won the John Von Neumann Theory Prize.**
*Photo: Konrad Jacobs, Erlangen, Mathematisches Institut Oberwolfach (MFO).*

Though Lloyd stuck with mathematics “he contributed deep insights about…the structure of markets,” Roth said. Lloyd’s 1953 paper proposing the “Shapley value” became such a “crown jewel” in the field that, in 1988 Roth arranged the publication of The Shapley Value: Essays in Honor of Lloyd S. Shapley.” This was for Lloyd’s 1000001st binary birthday - what most people would call his 65th. In 2020, for the 65th anniversary of the 1953 paper a group of editors published Handbook of the Shapley Value. 6/

In 1981 Lloyd was awarded the highest prize in his field the Von Neumann Theory Prize. It is “for a body of work, rather than a single piece….to reflect contributions that have stood the test of time.” 7/
Story Continues ⥥

BACK TO STORY ⥣

Game theory math spreads to many different fields

“The mathematical discovery is the really exciting part,” Lloyd said in his 1994 oral history. “You create conjectures, but you discover results. I think that’s the way. [In mathematics] you don’t create facts. Of course the facts are always there but you discover them.”

Mathematical models of different types of games drove applications - to markets, military and diplomatic negotiations, policies, proposed laws. Game theory applications exploded at RAND and elsewhere; theorists were in high demand. “[G]ame theory served very well in that way because it spreads out so quickly into so many different fields,” Lloyd explained. He loved “exciting” mathematics, but he said “if there’s simply an interesting application, well, maybe someone else can do it.”

The first of Lloyd’s key papers in 1953 proposed “the Shapley value” 6/ - a way to derive the payoff to participants (players) in cooperative games. The value is the amount of money or utility that a coalition of players in a larger group can distribute among the coalition’s members. With Martin Shubik Lloyd extended this to “political games” (1954). With John Milnor he initiated “oceanic games” (1961). Lloyd fostered “dynamic multistage games” (1953). With Don Monderer he worked out “potential games” which consider that a prospective player may choose not to play the game. Lloyd and Robert Aumann proposed a “grand unification of ideas from game theory with ideas from economics” (1974).

Early on, Lloyd “introduced the idea of the core” into game theory. The core is an outcome in which “no coalition of players that can go off on their own and do better than they could do in the marketplace.” The core became key to many marketplace designs.

Mathematical modeling assumed terrifying importance in Rand’s nuclear war planning. Rand’s Herman Kahn and Thomas Schelling could show outcomes of “noncooperative games” - such as between the US and recently nuclear-armed Soviet bomber forces - that top brass might not have thought of.

Game theory spread into economics. RAND’s young economists included Kenneth Arrow, Paul Samuelson and Herbert Simon. Nasar writes, “It was at Rand, more than anywhere else, that game theory in particular and mathematical modeling in general entered the mainstream of postwar economics.” BACK TO STORY ⥣

Sources: Alvin E Roth, Lindau-Nobel Institute, 2016. Lloyd Shapley Oral History, 1994.

**Lloyd Shapley enjoys a joke with his wife Marian and sons Christopher 17 and Peter 18 in 1977.** *Photo: Peter Shapley.*

Marriage Matching: Yearning, Deferment, Rejection
Lloyd’s best-known work in the outside world was co-inventing the Gale-Shapley Algorithm for Stable Matching. It is also known as the Deferred Acceptance algorithm. Gale had been at Princeton in 1949-50, where he met Lloyd. Gale’s was at Brown in the early 1960s when he and Lloyd collaborated. They took up the Stable Matching Problem (SMP); how two groups of people -such as men and women in a “marriage market” - a be matched in a way that is stable.

College admissions had the same problem: each applicant must apply, but each college can reject an applicant in favor of another they prefer. By deferring all parties’ acceptances until the end, this market structure (game) produces matches where no one can do better with another choice. 8/

**Gale-Shapley algorithm**. Also see Wiki article animation how a stable match is reached. *Graphic: Deborah Shapley*

Roth and Next-Generation Uses of Theory

The other 2012 Nobel in Economic Sciences went to Alvin E. Roth. He was born in 1951 and earned his Ph.D. from Stanford in 1974. 9/ Appearing with Lloyd at ceremonies and lectures Roth told audiences that he stood on the shoulders of the first giants of game theory. Roth said this standing beside his tall, somewhat fearsome-looking co-winner.

Roth enlarged the kit of mathematical game scenarios in real world. In 1984 he noted that the US clearinghouse for assigning medical school graduates to residency positions was equivalent to the Gale-Shapley deferred acceptance algorithm of 1962. This led him to compare residency matching systems in the US and Britain: why some worked and others didn’t. Among the fixes: In the 1990s he and Eliot Peranson adapted the NRMP for couples who sought medical residency jobs in the same city, They tweaked the system for other needs.

Roth and the Director of New York City Schools redesigned that system’s method for assigning the 90,000 students who applied to high schools each year. They created a new, stable clearinghouse (marketplace in games-speak). This assigned nearly all students to a high school they had preferred on the rounds. Before, administrators had to assign 30,000 applicants each year, via office work. The new algorithm left just 10% or 3,000 needing administrative placement. Boston, Denver and New Orleans public schools also adopted the system.

Gale had proposed “top trading cycles” by which the players express their preferences in sequence, or “cycle.” Lloyd and Herbert Scarf (1974) showed that this algorithm could be applied to create chains of matching pairs. This cyclical process led to a stable ending when all players’ preferences were met.

This abstract math led to kidney exchanges in which compatible donors and recipients are matched, in pairs and even in series’ of preferences or chains. This novel approach has rescued many patients from costly and life-threatening renal failure.

A kidney can’t be donated in advance or sold legally and ethically. Roth and colleagues pioneered how donors and recipients can be matched.

The diagram shows one pair of donations which requires 4 surgeries. The photos below show a 4-surgery exchange in real-time: Donor A and Recipient A are in one operating room (Cincinnati, OH); Donor B and Recipient B are in another (Toledo, OH). 10/ Marketplace math has created chains of 60 people (30 donors and 30 recipients); the average chain involves 10 patients and 5 surgeries.

Exchange of kidneys among pairs of people requires simultaneous surgeries to avoid unethical trading or sale. Above is a 4-way exchange underway. see 10/. *Images: Roth Nobel Lecture.*

A Fraught Relationship

To the public, Lloyd will always be associated with John Forbes Nash, Jr. Nash co-won the 1994 Nobel in economics for inventing “Nash equilibrium “when he was a young Ph.D. student at Princeton in 1948-50. Nash equilibrium is to this day the central concept in “noncooperative games” - the nuclear arms race and myriad other situations where players get to a standoff: they do not agree but can’t get any better outcome for themselves. Thus the game or conflict is “solved.” 10/

Nasar’s 1998 biography of Nash A Beautiful Mind offered a harrowing account of Nash’s growing paranoid schizophrenia after his brilliant early work. When at Princeton in 1948 at age 20, his behavior was strange even by the standards of eccentric math students.

Upon meeting Lloyd - older, a war hero, son of the famous astronomer - “Nash acted like a thirteen-year-old having his first crush,” Nazar writes. He disrupted the Kriegspiel games Lloyd loved - sweeping the pieces to the ground. “He rifled through his mail.” Nash played aggressive pranks on Lloyd and out of jealousy on Lloyd’s other friends.

Lloyd reacted “with amused tolerance.” He helped Nash when he was required to produce a concrete example of an equilibrium point for his thesis, Lloyd “spent weeks” working out an example “involving three-handed poker, another Shapley specialty.”

When Nash’s brilliant Ph.D. was done, Princeton did not offer him a post - some faculty were disturbed by him. He went to Rand, but for a year. At 23 he got a job as “a lowly math instructor” at MIT. He published no more on game theory.

**John Nash on the cover of the first (1998) edition of *A Beautiful Mind* by Sylvia Nazar**. Lloyd granted her one interview notwithstanding his large influence on Nash and the field of game theory.

“A Chance to Do Something for Nash”

By the 1970s Nash had retreated from active life. Nasar describes “a phantom” hanging around the Princeton math department, avoided by others and concerning the secretaries.

“In 1978, largely thanks to the kindness of his old classmate from graduate school and RAND, Lloyd Shapley, Nash was finally awarded a mathematical prize: the John von Neumann Theory Prize, awarded by the Institute for Operations Research and Management for a body of work.

Lloyd was on the prize committee. It was his idea.

“Here’s a chance to do something for Nash,” he told Nasar. “I felt sentiment and nostalgia.” He hoped the award would help Nash’s wife and son to picture “he’s great. his work is being recognized.

“He was an immense genius.”

Nash’s “beautiful, logical mind”

Lloyd could be intolerant himself. He refused to discuss his work with amateurs who would not understand it and try his patience - including me. Nasar’s book refers to just one interview she had with Lloyd Shapley.

He did recall to Nasar his impressions of Nash in 1949-50. “‘He was immature, he was obnoxious, he was a brat. What redeemed him was a keen, beautiful, logical mind.’”

She goes on: “So Now you know to whom I owe the title of this biography.” 11/

Notes and Banner photo credit are below pictures.

**Martha Betz Shapley at the piano with two sons in the Director’s Residence at 60 Garden Street, Cambridge, MA.** *Photo: Shapley-Matthews Colletion*

Notes:

1/ The subject of this article was named after his uncle, Lloyd Stowell Shapley (1875-1939). The elder Lloyd was son of Col. Calvin Harlow Shapley and Kate Stowell. He was a Naval captain and later governor of Guam. Wikipedia lists this man by his full name. Wiki lists his distinguished nephew as Lloyd Shapley. See box “More than you want to know about Shapleys & Stowells” in earlier post “Harlow Shapley - Unlikely Liberal.”

2/ Lloyd S Shapley, Oral History with Martin Collins, UCLA, 1994.

3/ Nasar, Sylvia A Beautiful Mind, Has one of many accounts of Project RAND.. It was set up in 1945 under the Douglas Aircraft Corp. In November 1948 it became a separate entity, RAND Corporation. Lloyd’s quotes in this passage are from L. S. Shapley Oral History. TK

4/ L. S. Shapley Oral History pp. TK Nasar pp.TK

5/ Roth summarized Lloyd’s contributions as a teaching module for Lindau-Nobel Institute in 2016. https://www.lindau-nobel.org/lloyd-shapley-a-founding-giant-of-game-theory/ See story insert “Game theory math spreads out in many directions.”

6/ Roth, A. E. The Shapley Value: Essays in Honor of Lloyd S. Shapley, Editor A.E. Roth, Cambridge University Press, 1988. In 2020 editors Encarnacion Algaba, Vito Fragnelli, and Joaquin Sanchez-Soriano edited Handbook of the Shapley Value,, CRC Press, Taylor & Francis Group LLC, 2020. See Foreward to that volume: Roth, Alvin E. “The Shapley Value, A Giant Legacy, and Ongoing Research Agenda” xvii - xx.
7/ John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciences (INFORMS)

8/ Wiki article “David Gale” lays out the Gale–Shapley algorithm. See also MathMate Animation of Gale-Shapley game https://en.wikipedia.org/wiki/Gale%E2%80%93Shapley_algorithm#/media/File:Gale-Shapley.gif UCLA Newsroom “UCLA Professor Wins Nobel Memorial Prize in Economics” Oct 15, 2012, as updated Oct 17

http://newsroom.ucla.edu/portal/ucla/ucla-professor-wins-nobel-memorial-239684.aspx

The 2013 Golden Goose Award was made to Lloyd Shapley, David Gale, and Alvin E. Roth. Gale could be recognized because the award can be made researchers even if deceased, unlike the Nobels. Golden Goose Award, 2013[19]

9/ Roth’s paper on the National Resident Matching Program said Gale and Shapley’s 1962 “simple insight” had “enormous power.” Roth, A.E., "New Physicians: A Natural Experiment in Market Organization," Science, 250, 1990, 1524-1528.

NSF press release. “Kidney Exchange A Life-Saving Application of Matching Theory” Oct 5, 2005 https://www.nsf.gov/discoveries/disc_summ.jsp?cntn_id=104404&org=NSF

National Kidney Registry, a system for optimizing the search for live kidney-donor transfers. https://www.kidneyregistry.org/for-patients/paired-exchange/. See also https://en.wikipedia.org/wiki/National_Kidney_Registry

10/ Two pair exchange diagram from Roth, Sonmez, and Unver; and Ashlagi, Frank Delmonico and Susan Saidman and Mike Rees.

Operating room photos. Shown in Cincinnati OH are surgeons are Steve Woodle and Rino Munda. Alvin Roth is in yellow gown. Not shown is the other half of the exchange which is underway at the same time in Toledo OH land ed by surgeon Mike Rees. Image: Roth Nobel lecture.

11/ Nasar, Beautiful Mind, pp. 338-339 for Lloyd arranging Von Neumann Prize to do something for Nash. “He had immense genius” is from LSS 1994 Oral History. “To whom I owe the title” from Nasar 1%. Nasar also relates how the Nobel economics committee chose Nash as a co-winner of the 1994 prize, having decided not to make the award for Lloyd’s branch of game theory (co-operative games) on that round.

Banner photo: Lloyd S. Shapley accepting Sveriges Riksbank Nobel Memorial Prize in Economic Sciences, Dec 10, 2012. Stockholm, Sweden.
Photo: A Mahmoud, Nobel Foundation

Lloyd Shapley - Nobelist

Lloyd Shapley - Nobelist

Notes:

About HARLOW SHAPLEY

HARLOW SHAPLEY PROJECT GOALS

2-Step Search

Of blog Posts:

TO exit search

Harlow Shapley On Social

MORE Posts

HOME CONTACT BACK TO TOP