Lloyd Shapley | |
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Shapley in 1980 | |

Born | Lloyd Stowell Shapley June 2, 1923 Cambridge, Massachusetts, U.S. |

Died | March 12, 2016 92) Tucson, Arizona, U.S. | (aged

Nationality | American |

Alma mater | Princeton University Harvard University |

Known for | Shapley value Shapley–Shubik power index stochastic games Bondareva–Shapley theorem Shapley–Folkman lemma & theorem Gale–Shapley algorithm potential game core, kernel and nucleolus market games authority distribution multi-person utility non-atomic games |

Awards | Nobel Memorial Prize in Economic Sciences (2012) Golden Goose Award (2013) John von Neumann Theory Prize (1981) |

Scientific career | |

Fields | Mathematics, economics |

Institutions | University of California, Los Angeles RAND Corporation Princeton University |

Doctoral advisor | Albert W. Tucker |

Influences | John von Neumann Martin Shubik Jon Folkman |

Influenced | Martin Shubik Jon Folkman |

**Lloyd Stowell Shapley** ( /ˈʃæpli/ ; June 2, 1923 – March 12, 2016) was an American mathematician and Nobel Prize-winning economist. He contributed to the fields of mathematical economics and especially game theory. Shapley is generally considered one of the most important contributors to the development of game theory since the work of von Neumann and Morgenstern.^{ [1] } With Alvin E. Roth, Shapley won the 2012 Nobel Memorial Prize in Economic Sciences "for the theory of stable allocations and the practice of market design."^{ [2] }^{ [3] }

Lloyd Shapley was born on June 2, 1923, in Cambridge, Massachusetts, one of the sons of astronomers Harlow Shapley and Martha Betz Shapley, both from Missouri.^{ [4] } He attended Phillips Exeter Academy and was a student at Harvard when he was drafted in 1943. He served in the United States Army Air Corps in Chengdu, China and received the Bronze Star decoration for breaking the Soviet weather code.^{ [5] }

After the war, Shapley 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.^{ [6] } His thesis and post-doctoral work introduced the Shapley value and the core solution in game theory. Shapley defined game theory as "a mathematical study of conflict and cooperation." After graduating, he remained at Princeton for a short time before going back to the RAND corporation from 1954 to 1981. In 1950, while a graduate student, Shapley invented the board game * So Long Sucker *, along with Mel Hausner, John Forbes Nash, and Martin Shubik.^{ [7] } Israeli economist Robert Aumann said Shapley was "the greatest game theorist of all time."^{ [8] }

From 1981 until his death, Shapley was a professor at University of California, Los Angeles (UCLA), serving at the time of his death as a professor emeritus there, affiliated with departments of Mathematics and Economics. He died on March 12, 2016, in Tucson, Arizona, after suffering from a broken hip, at the age of 92.^{ [2] }

Shapley was an expert Kriegspiel player, and an avid baseball fan.^{ [8] }

Along with the Shapley value, 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, the Snow–Shapley theorem for matrix games, and the Shapley–Folkman lemma & theorem bear his name.^{ [9] } According to * The Economist *, Shapley "may have thought of himself as a mathematician, but he cannot avoid being remembered for his huge contributions to economics".^{ [10] } The American Economic Association noted that Shapley was "one of the giants of game theory and economic theory".^{ [9] }

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 appeared in economic theory.^{ [11] }

Shapley argued with his sons about whether he should accept the Nobel Prize at all. He opined that his father, the astronomer Harlow Shapley, deserved it more. His sons persuaded him to accept it and accompanied him to Stockholm.^{ [12] }

- Bronze Star, U.S. Army Air Corps, 1944
^{ [5] } - Procter Fellow, Princeton University, 1951–52
^{ [13] } - Fellow, Econometric Society, 1967
^{ [13] } - Fellow, American Academy of Arts and Sciences, 1974
^{ [13] } - Member, National Academy of Sciences, 1978
^{ [13] } - John von Neumann Theory Prize, 1981
^{ [13] } - Honorary Ph.D., Hebrew University of Jerusalem, 1986
^{ [13] } - Fellow, INFORMS (Institute for Operations Research and the Management Sciences), 2002
^{ [14] } - Distinguished Fellow, American Economic Association, 2007
^{ [15] } - Fellow, American Mathematical Society, 2012
^{ [16] } - Sveriges Riksbank Nobel Memorial Prize in Economic Sciences, 2012
^{ [6] } - Golden Goose Award, 2013
^{ [17] }

- 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. doi : 10.1073/pnas.39.10.1095 - A Method for Evaluating the Distribution of Power in a Committee System [1954] (with Martin Shubik),
*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 Shubik),
*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 Shubik),
*International Journal of Game Theory*Vol. 1, pp. 111–130. *Values of Non-Atomic Games*[1974] (with Robert Aumann), Princeton University Press.- Mathematical Properties of the Banzhaf Power Index [1979] (with Pradeep Dubey),
*Mathematics of Operations Research*Vol. 4, pp. 99–132. - Long-Term Competition – A Game-Theoretic Analysis [1994] (with Robert Aumann), in
*Essays in Game Theory: In Honor of Michael Maschler*, Nimrod Megiddo (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 Xingwei Hu),
*Games and Economic Behavior*Vol. 45, pp. 132–152, 153–170. - Multiperson Utility [2008] (with Manel Baucells).
*Games and Economic Behavior*Vol. 62, pp. 329–347.

**Game theory** is the study of mathematical models of strategic interaction among rational decision-makers. It has applications in all fields of social science, as well as in logic, systems science and computer science. Originally, it addressed zero-sum games, in which each participant's gains or losses are exactly balanced by those of the other participants. Today, game theory applies to a wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers.

* Theory of Games and Economic Behavior*, published in 1944 by Princeton University Press, is a book by mathematician John von Neumann and economist Oskar Morgenstern which is considered the groundbreaking text that created the interdisciplinary research field of game theory. In the introduction of its 60th anniversary commemorative edition from the Princeton University Press, the book is described as "the classic work upon which modern-day game theory is based."

**Oskar Morgenstern** was a economist. In collaboration with mathematician John von Neumann, he founded the mathematical field of game theory and its application to economics.

**David Gale** was an American mathematician and economist. He was a professor emeritus at the University of California, Berkeley, affiliated with the departments of mathematics, economics, and industrial engineering and operations research. He has contributed to the fields of mathematical economics, game theory, and convex analysis.

In game theory, the **core** is the set of feasible allocations that cannot be improved upon by a subset of the economy's agents. A coalition is said to *improve upon* or *block* a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first except that every member of the coalition has a different consumption bundle that is part of an aggregate consumption bundle that can be constructed from publicly available technology and the initial endowments of each consumer in the coalition.

**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. He also holds a visiting position at Stony Brook University, and is one of the founding members of the Stony Brook Center for Game Theory.

**Hobart Peyton Young** is an American game theorist and economist known for his contributions to evolutionary game theory and its application to the study of institutional and technological change, as well as the theory of learning in games. He is currently centennial professor at the London School of Economics, James Meade Professor of Economics Emeritus at the University of Oxford, professorial fellow at Nuffield College Oxford, and research principal at the Office of Financial Research at the U.S. Department of the Treasury.

**Martin Shubik** was an American economist, who was Professor Emeritus of Mathematical Institutional Economics at Yale University.

**Pradeep Dubey** is an Indian game theorist. He is Professor of Economics at State University of New York, Stony Brook and a member of the Stony Brook Center for Game Theory. He also holds a visiting position at Cowles Foundation, Yale University. He did his schooling from the St. Columba's School, Delhi. He received his Ph.D. in Applied Mathematics from Cornell University and B.Sc. from the University of Delhi. His areas of research interests are game theory and mathematical economics. He has published, among others, in *Econometrica*, *Games and Economic Behavior*, *Journal of Economic Theory* and *Quarterly Journal of Economics*. He is a Fellow of *The Econometric Society* and a member of the council of *Game Theory Society*.

**Yair Tauman** is a Professor of Economics at State University of New York, Stony Brook and the Director of the Stony Brook Center for Game Theory. He studied at the Hebrew University of Jerusalem where he obtained his B.Sc. in Mathematics and Statistics and M.Sc. and Ph.D. in Mathematics, the latter two under the supervision of Robert Aumann. His areas of research interests are game theory and industrial organization. He has published, among others, in *Econometrica*, *Games and Economic Behavior*, *Journal of Economic Theory*, *Quarterly Journal of Economics* and *RAND Journal of Economics*.

**Shapley** is a **surname** that might refer to one of the following:

**Alvin Elliot Roth** is an American academic. He is the Craig and Susan McCaw professor of economics at Stanford University and the Gund professor of economics and business administration emeritus at Harvard University. He was President of the American Economics Association in 2017.

**Mathematical economics** is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.

**Ehud Kalai** is a prominent Israeli American game theorist and mathematical economist known for his contributions to the field of game theory and its interface with economics, social choice, computer science and operations research. He was the James J. O’Connor Distinguished Professor of Decision and Game Sciences at Northwestern University, 1975-2017, and currently is a Professor Emeritus of Managerial Economics and Decision Sciences.

The **Shapley–Folkman lemma** is a result in convex geometry with applications in mathematical economics that 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 zero, one, and two:

In **economics**, **non-convexity** refers to violations of the convexity assumptions of elementary economics. Basic economics textbooks concentrate on consumers with convex preferences and convex budget sets and on producers with convex production sets; for convex models, the predicted economic behavior is well understood. When convexity assumptions are violated, then many of the good properties of competitive markets need not hold: Thus, non-convexity is associated with market failures, where supply and demand differ or where market equilibria can be inefficient. Non-convex economies are studied with nonsmooth analysis, which is a generalization of convex analysis.

**John Geanakoplos** is an American economist, and the current James Tobin Professor of Economics at Yale University.

**Abraham Neyman** is an Israeli mathematician and game theorist, Professor of Mathematics at the Federmann Center for the Study of Rationality and the Einstein Institute of Mathematics at the Hebrew University of Jerusalem in Israel. He served as president of the Israeli Chapter of the Game Theory Society (2014–2018).

The **Game Theory Society (GTS)** is a society for the promotion of research, teaching and application of game theory. It was founded in 1999 by Ehud Kalai and Robert Aumann and is registered in the Netherlands.

- ↑ Roth, A.E., Introduction to the Shapley Value, in "The Shapley Value: Essays in Honor of Lloys S. Shapley", Cambridge University Press, 1988.
- 1 2 "Lloyd Shapley, a Nobel laureate in economics, has died".
*The Economist*. ISSN 0013-0613 . Retrieved March 13, 2016. - ↑ Roth, Al (March 12, 2016). "Lloyd S. Shapley 1923– 2016".
*Market Design*. Retrieved March 13, 2016. - ↑ "MARTHA BETZ SHAPLEY".
*The New York Times*. January 27, 1981. - 1 2 "Lloyd S. Shapley – Interview". Nobel Media AB. Retrieved March 13, 2016.
- 1 2 "Princeton alumnus Shapley wins Nobel Prize". Princeton University. October 15, 2012. Retrieved March 13, 2016.
- ↑ Hausner, M., Nash, J. F., Shapley, L. S. & Shubik, M., (1964), "So Long Sucker, A Four-Person Game",
*Game Theory and Related Approaches to Social Behavior*, John Wiley & Sons, Inc., New York. - 1 2 Hagerty, James, Lloyd Shapley: 1923-2016, Wall Street Journal, March 19–20, 2016, p. A7.
- 1 2 "Lloyd Shapley" (PDF). American Economic Association. Archived from the original (PDF) on March 16, 2016. Retrieved March 13, 2016.
- ↑ "Matchmaker in heaven – Lloyd Shapley, a Nobel laureate in economics, has died".
*The Economist*. March 13, 2016. Retrieved March 13, 2016. - ↑ Diertele, David A. (August 8, 2013).
*Economic Thinkers: A Biographical Encyclopedia*. p. 385. ISBN 9780313397479 . Retrieved March 13, 2016. - ↑ Hagerty, James, Lloyd Shapley: 1923-2016, Wall Street Journal, March 19–20, 2016, p.A7
- 1 2 3 4 5 6 "Lloyd Stowell Shapley – Vita". UCLA . Retrieved March 13, 2016.
- ↑ "INFORMS – Fellows Class of 2002". Institute for Operations Research and the Management Sciences. Archived from the original on March 14, 2016. Retrieved March 13, 2016.
- ↑ "Distinguished Fellows". American Economic Association . Retrieved March 13, 2016.
- ↑ List of Fellows of the American Mathematical Society, retrieved July 18, 2013.
- ↑ "Market Design". The Golden Goose Award. Retrieved May 27, 2015.

*Stable Marriage and Its Relation to Other Combinatorial Problems: An Introduction to the Mathematical Analysis of Algorithms*, Donald E. Knuth, American Mathematical Society, 1997 (English Translation.)

- Home Page
- Lloyd Shapley at the Mathematics Genealogy Project
- The Shapley Value
- Citation of von Neumann Theory Prize on L.S.Shapley's work: "Lloyd Shapley has dominated game theory for the thirty-seven years since von Neumann and Morgenstern published their path-breaking book,
*The Theory of Games and Economic Behavior*." - Albert Tucker's comment on L.S.Shapley's work. In 1995, Albert W. Tucker mentioned in his passing that Shapley was second only to Von Neumann as the most important researcher in theory of games so far. Philip Wolfe Interview by Irv Lustig, May 4, 2001. Video by Irv Lustig, Short Hills, NJ.
- Robert Aumann's Nobel lecture, also see here .
- UCLA - In Memoriam
- Biography of Lloyd S. Shapley from the Institute for Operations Research and the Management Sciences
- Lloyd Shapley publications indexed by Google Scholar
- "Lloyd S. Shapley". EconPapers.
- "Lloyd Shapley". JSTOR.

Awards | ||
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Preceded by Thomas J. Sargent Christopher A. Sims | Laureate of the Nobel Memorial Prize in Economics 2012 Served alongside: Alvin E. Roth | Succeeded by Eugene F. Fama Lars Peter Hansen Robert J. Shiller |

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