Lloyd Shapley
Encyclopedia
Lloyd Stowell Shapley is a distinguished American
mathematician
and economist
. He is a Professor Emeritus at University of California, Los Angeles
, affiliated with departments of Mathematics and Economics. He has contributed to the fields of mathematical economics
and especially game theory
.
Since the work of von Neumann and Morgenstern in 1940s, Lloyd Shapley has been regarded by many experts as the very personification of game theory.
, one of the sons of the distinguished astronomer Harlow Shapley
. He was a student at Harvard when he was drafted in 1943, and in the same year, as a sergeant in the Army Air Corps in Chengdu, China, he received the Bronze Star decoration for breaking the Soviet weather code. After the war, he returned to Harvard and graduated with an A.B.
in mathematics in 1948. After working for one year at the Rand Corporation
, he went to Princeton University
where he received a Ph.D. in 1953. His thesis and post-doctoral work continued the ideas of Francis Ysidro Edgeworth
introducing the Shapley value
and the core solution concept in game theory
. After graduating, he remained at Princeton for short time before going back to the RAND corporation from 1954 to 1981. He married Marian Ludolph in 1955 (with whom he has two sons, Peter and Christopher). Since 1981 he has been a professor at UCLA.
, stochastic games, the Bondareva-Shapley theorem
(which implies that convex
games have non-empty cores), the Shapley–Shubik power index (for weighted- or block voting power), the Gale–Shapley algorithm (for the stable marriage problem
), the concept of a potential game
(with Dov Monderer), the Aumann–Shapley pricing, the Harsanyi
–Shapley solution, and the Shapley–Folkman lemma& theorem
bear his name.
Besides, his early work with R.N.Snow and Samuel Karlin
on matrix
games was so complete that little has been added since. He has been instrumental in the development of utility theory, and it was he who laid much of the groundwork for the solution of the problem of the existence of Von Neumann-Morgenstern stable sets. His work with M.Maschler
and B.Peleg on the kernel and the nucleolus, and his work with Robert Aumann
on non-atomic games and on long-term competition have all had a tremendous impact in economic theory.
In his 80s, Shapley continues publishing his old results, such as multi-person utility and authority distribution (a generalization to the Shapley–Shubik power index and useful in ranking, planning and group decision-making).
, along with Mel Hausner, John Forbes Nash
, and Martin Shubik
.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and economist
Economist
An economist is a professional in the social science discipline of economics. The individual may also study, develop, and apply theories and concepts from economics and write about economic policy...
. He is a Professor Emeritus at University of California, Los Angeles
University of California, Los Angeles
The University of California, Los Angeles is a public research university located in the Westwood neighborhood of Los Angeles, California, USA. It was founded in 1919 as the "Southern Branch" of the University of California and is the second oldest of the ten campuses...
, affiliated with departments of Mathematics and Economics. He has contributed to the fields of mathematical economics
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
and especially game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
.
Since the work of von Neumann and Morgenstern in 1940s, Lloyd Shapley has been regarded by many experts as the very personification of game theory.
Career
Lloyd Shapley was born on June 2, 1923, in Cambridge, MassachusettsCambridge, Massachusetts
Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...
, one of the sons of the distinguished astronomer Harlow Shapley
Harlow Shapley
Harlow Shapley was an American astronomer.-Career:He was born on a farm in Nashville, Missouri, and dropped out of school with only the equivalent of a fifth-grade education...
. He was a student at Harvard when he was drafted in 1943, and in the same year, as a sergeant in the Army Air Corps in Chengdu, China, he received the Bronze Star decoration for breaking the Soviet weather code. After the war, he returned to Harvard and graduated with an A.B.
Bachelor of Arts
A Bachelor of Arts , from the Latin artium baccalaureus, is a bachelor's degree awarded for an undergraduate course or program in either the liberal arts, the sciences, or both...
in mathematics in 1948. After working for one year at the Rand Corporation
RAND
RAND Corporation is a nonprofit global policy think tank first formed to offer research and analysis to the United States armed forces by Douglas Aircraft Company. It is currently financed by the U.S. government and private endowment, corporations including the healthcare industry, universities...
, he went to Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
where he received a Ph.D. in 1953. His thesis and post-doctoral work continued the ideas of Francis Ysidro Edgeworth
Francis Ysidro Edgeworth
Francis Ysidro Edgeworth FBA was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s...
introducing the Shapley value
Shapley value
In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players...
and the core solution concept in game theory
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
. After graduating, he remained at Princeton for short time before going back to the RAND corporation from 1954 to 1981. He married Marian Ludolph in 1955 (with whom he has two sons, Peter and Christopher). Since 1981 he has been a professor at UCLA.
Contribution
Along with the Shapley valueShapley value
In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players...
, stochastic games, the Bondareva-Shapley theorem
Bondareva-Shapley theorem
In game theory, the Bondareva–Shapley theorem describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game. Specifically, the game's core is non-empty if and only if the game is balanced. The Bondareva–Shapley theorem implies that market games and convex...
(which implies that convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
games have non-empty cores), the Shapley–Shubik power index (for weighted- or block voting power), the Gale–Shapley algorithm (for the stable marriage problem
Stable marriage problem
In mathematics and computer science, the stable marriage problem is the problem of finding a stable matching between two sets of elements given a set of preferences for each element. A matching is a mapping from the elements of one set to the elements of the other set...
), the concept of a potential game
Potential game
A game in game theory is considered a potential game if the incentive of all players to change their strategy can be expressed in one global function, the potential function. The concept was proposed by Dov Monderer and Lloyd Shapley. Games can be either ordinal or cardinal potential games...
(with Dov Monderer), the Aumann–Shapley pricing, the Harsanyi
John Harsanyi
John Charles Harsanyi was a Hungarian-Australian-American economist and Nobel Memorial Prize in Economic Sciences winner....
–Shapley solution, and the Shapley–Folkman lemma& theorem
Shapley–Folkman lemma
In geometry and economics, the Shapley–Folkman lemma describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of...
bear his name.
Besides, his early work with R.N.Snow and Samuel Karlin
Samuel Karlin
Samuel Karlin was an American mathematician at Stanford University in the late 20th century.Karlin was born in Yanova, Poland and immigrated to Chicago as a child...
on matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
games was so complete that little has been added since. He has been instrumental in the development of utility theory, and it was he who laid much of the groundwork for the solution of the problem of the existence of Von Neumann-Morgenstern stable sets. His work with M.Maschler
Michael Maschler
Michael Bahir Maschler was an Israeli mathematician well known for his contributions to the field of game theory...
and B.Peleg on the kernel and the nucleolus, and his work with Robert Aumann
Robert Aumann
Robert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
on non-atomic games and on long-term competition have all had a tremendous impact in economic theory.
In his 80s, Shapley continues publishing his old results, such as multi-person utility and authority distribution (a generalization to the Shapley–Shubik power index and useful in ranking, planning and group decision-making).
Awards and honors
- Bronze StarBronze Star MedalThe Bronze Star Medal is a United States Armed Forces individual military decoration that may be awarded for bravery, acts of merit, or meritorious service. As a medal it is awarded for merit, and with the "V" for valor device it is awarded for heroism. It is the fourth-highest combat award of the...
, U.S. Army Air Corps, 1944 - Procter Fellow, Princeton UniversityPrinceton UniversityPrinceton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, 1951–52 - Fellow, Econometric SocietyEconometric SocietyThe Econometric Society is an international society for the advancement of economic theory in its relation with statistics and mathematics. It was founded on December 29, 1930 at the Stalton Hotel in Cleveland, Ohio....
, 1967 - Fellow, American Academy of Arts and SciencesAmerican Academy of Arts and SciencesThe American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
, 1974 - Member, National Academy of SciencesUnited States National Academy of SciencesThe National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
, 1979 - John von Neumann Theory PrizeJohn von Neumann Theory PrizeThe John von Neumann Theory Prize of the Institute for Operations Research and the Management Sciencesis awarded annually to an individual who has made fundamental and sustained contributions to theory in operations research and the management sciences.The Prize named after mathematician John von...
, 1981 - Honorary Ph.D., Hebrew University of JerusalemHebrew University of JerusalemThe Hebrew University of Jerusalem ; ; abbreviated HUJI) is Israel's second-oldest university, after the Technion – Israel Institute of Technology. The Hebrew University has three campuses in Jerusalem and one in Rehovot. The world's largest Jewish studies library is located on its Edmond J...
, 1986 - Fellow, INFORMS (Institute for Operations Research and the Management SciencesInstitute for Operations Research and the Management SciencesThe Institute for Operations Research and the Management Sciences is an international society for practitioners in the fields of operations research and management science...
), 2002 - Distinguished Fellow, American Economic AssociationAmerican Economic AssociationThe American Economic Association, or AEA, is a learned society in the field of economics, headquartered in Nashville, Tennessee. It publishes one of the most prestigious academic journals in economics: the American Economic Review...
, 2007
Selected publications
- A Value for n-person Games [1953], In Contributions to the Theory of Games volume II, H.W. Kuhn and A.W. Tucker (eds.).
- Stochastic Games [1953], Proceedings of National Academy of Science Vol. 39, pp. 1095–1100.
- A Method for Evaluating the Distribution of Power in a Committee System [1954] (with Martin ShubikMartin ShubikMartin Shubik is an American economist, who is Professor Emeritus of Mathematical Institutional Economics at Yale University. He was educated at the University of Toronto and Princeton University...
), American Political Science Review Vol. 48, pp. 787–792. - College Admissions and the Stability of Marriage [1962] (with David Gale), The American Mathematical Monthly Vol. 69, pp. 9–15.
- Simple Games : An Outline of the Descriptive Theory [1962], Behavioral Science Vol. 7, pp. 59–66.
- On Balanced Sets and Cores [1967], Naval Research Logistics Quarterly Vol. 14, pp. 453–460.
- On Market Games [1969] (with Martin ShubikMartin ShubikMartin Shubik is an American economist, who is Professor Emeritus of Mathematical Institutional Economics at Yale University. He was educated at the University of Toronto and Princeton University...
), Journal of Economic Theory Vol. 1, pp. 9–25. - Utility Comparison and the Theory of Games [1969], La Decision, pp. 251–263.
- Cores of Convex Games [1971] International Journal of Game Theory Vol. 1, pp. 11–26.
- The Assignment Game I: The Core [1971] (with Martin ShubikMartin ShubikMartin Shubik is an American economist, who is Professor Emeritus of Mathematical Institutional Economics at Yale University. He was educated at the University of Toronto and Princeton University...
), International Journal of Game Theory Vol. 1, pp. 111–130. - Values of Non-Atomic Games [1974] (with Robert AumannRobert AumannRobert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
), Princeton University Press. - Mathematical Properties of the Banzhaf Power Index [1979] (with Pradeep DubeyPradeep DubeyPradeep Dubey, born January 9, 1951 in Patna, Bihar, India, is an American game theorist. He is Professor of Economics at State University of New York, Stony Brook and a member of at Stony Brook. He also holds a visiting position at Cowles Foundation, Yale University. He did his schooling from...
), Mathematics of Operations Research Vol. 4, pp. 99–132. - Long-Term Competition – A Game-Theoretic Analysis [1994] (with Robert AumannRobert AumannRobert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
), In Essays in Game Theory: In Honor of Michael Maschler Nimrod MegiddoNimrod MegiddoNimrod Megiddo is a mathematician and computer scientist. He is research scientist at the IBM Almaden Research Center.His interests include optimization, algorithm design and analysis, game theory, and machine learning....
(ed.), Springer-Verlag. - Potential Games [1996] (with Dov Monderer), Games and Economic Behavior Vol. 14, pp. 124–143.
- On Authority Distributions in Organizations [2003] (with X.Hu), Games and Economic Behavior Vol. 45, pp. 132–152, 153-170.
Trivia
In 1950, Lloyd Shapley invented the board game So Long SuckerSo Long Sucker
So Long Sucker is a board game invented in 1950 by John Forbes Nash, Mel Hausner, Lloyd S. Shapley and Martin Shubik. It is a four-person bargaining/economic strategy game. Each player begins the game with 7 chips, and in the course of play, attempts to acquire all the other players' chips. This...
, along with Mel Hausner, John Forbes Nash
John Forbes Nash
John Forbes Nash, Jr. is an American mathematician whose works in game theory, differential geometry, and partial differential equations have provided insight into the forces that govern chance and events inside complex systems in daily life...
, and Martin Shubik
Martin Shubik
Martin Shubik is an American economist, who is Professor Emeritus of Mathematical Institutional Economics at Yale University. He was educated at the University of Toronto and Princeton University...
.
External links
- Home Page
- Mathematical Genealogy Project
- The Shapley Value
- Citation of von Neumann Theory Prize on L.S.Shapley's work
- Albert Tucker's comment on L.S.Shapley's work. In 1995, Albert W. TuckerAlbert W. TuckerAlbert William Tucker was a Canadian-born American mathematician who made important contributions in topology, game theory, and non-linear programming....
mentioned in his passing that Shapley was second only to Von Neumann as the most important researcher in theory of games so far. - Robert Aumann's Nobel lecture, also see here.