Abraham Neyman

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Abraham Neyman
Abraham Neyman.

Abraham Neyman (born June 14, 1949, Israel) is an Israeli mathematician and game theorist, Professor of Mathematics at the Center for the Study of Rationality[1] and the Einstein Institute of Mathematics[2] at the Hebrew University of Jerusalem in Israel. He is currently the president of the Israeli Chapter of the Game Theory Society (2014–2016).[3]

Biography[edit]

Neyman received his BSc in mathematics in 1970 and his MSc in mathematics in 1972 from the Hebrew University. His MSc thesis was on the subject of “The Range of a Vector Measure” and was supervised by Prof. Joram Lindenstrauss. His PhD thesis,[4]"Values of Games with a Continuum of Players," was completed under Prof. Robert Aumann in 1977.

Neyman has been professor of mathematics at the Hebrew University since 1982, including serving as the chairman of the institute of mathematics 1992–1994, as well as holding a professorship in economics, 1982–1990. He has been a member of the Center for the Study of Rationality at the Hebrew University since its inception in 1991. He held various positions at Stony Brook University of New York, 1985–2001. He has also held positions and has been visiting scholar at Cornell University, University of California at Berkley, Stanford University, the Graduate School of Business Administration at Harvard University, and Ohio State University.[5][6][7]

Neyman has had 12 graduate students complete Ph.D. theses under his supervision, 5 at Stony Brook University and 7 at the Hebrew University.[8] Neyman has also served as the Game Theory Area Editor for the journal Mathematics of Operations Research (1987–1993) and on the editorial board for Games and Economic Behavior (1993–2001) and the International Journal of Game Theory (2001–2007).

Awards and honors[edit]

Neyman has been a fellow of the Econometric Society since 1989.[9]

The Game Theory Society has announced a special issue of the International Journal of Game Theory in honour of Neyman, "in recognition of his important contributions to game theory", set for 2015.[10] A Festschrift conference in Neyman's honour will be held at Hebrew University in June 2015, on the occasion of Neyman's 66th birthday.[11] He gave the inaugural von-Neumann lecture[12] at the 2008 Congress of the Game Theory Society [13] as well as delivering it at the 2012 World Congress on behalf of the recently deceased Jean-Francois Mertens.[14]

His Ph.D. thesis won two prizes from the Hebrew University: the 1977 Abraham Urbach prize for distinguished thesis in mathematics and the 1979 Aharon Katzir prize (for the best Ph. D. thesis in the Faculties of Exact Science, Mathematics, Agriculture and Medicine). In addition, Neyman won the Israeli under 20 chess championship in 1966.Israeli_Chess_Championship

Research contributions[edit]

Neyman has made numerous contributions to game theory, including to stochastic games, the Shapley value, and repeated games.

Stochastic games[edit]

Together with Jean-Francois Mertens, he proved the existence of the uniform value of zero-sum undiscounted stochastic games.[15] This work is considered one of the most important works in the theory of stochastic games, solving a problem that had been open for over 20 years.[16] Together with Elon Kohlberg, he applied operator techniques to study convergence properties of the discounted and finite stage values.[17] Recently, he has pioneered a model of stochastic games in continuous time and derived uniform equilibrium existence results.[18] He also co-edited, together with Sylvain Sorin, a comprehensive collection of works in the field of stochastic games.[19]

Repeated games[edit]

Neyman has made many contributions to the theory of repeated games. One idea that appears, in different contexts, in some of his papers, is that the model of an infinitely repeated game serves also as a powerful paradigm for a long finitely repeated game. A related insight appears in a 1999 paper, where he showed that in a long finitely repeated game, an exponentially small deviation from common knowledge of the number of repetitions is enough to dramatically alter the equilibrium analysis, producing a folk-theorem-like result.[20]

Neyman is one of the pioneers and a most notable leader of the study of repeated games under complexity constraints. In his seminal paper[21] he showed that bounded memory can justify cooperation in a finitely repeated prisoner's dilemma game. His paper was followed by many others who started working on bounded memory games. Most notable was Neyman's M.Sc. student Elchanan Ben-Porath who was the first to shed light on the strategic value of bounded complexity.[22]

The two main models of bounded complexity, automaton size and recall capacity, continued to pose intriguing open problems in the following decades. A major breakthrough was achieved when Neyman and his Ph.D. student Daijiro Okada proposed a new approach to these problems, based on information theoretic techniques, introducing the notion of strategic entropy.[23][24] His students continued to employ Neyman's entropy technique to achieve a better understanding of repeated games under complexity constraints. Neyman's information theoretic approach opened new research areas beyond bounded complexity. A classic example is the communication game he introduced jointly with Olivier Gossner and Penelope Hernandez.[25]

The Shapley value[edit]

Neyman has made numerous fundamental contributions to the theory of the value. In a "remarkable tour-de-force of combinatorial reasoning",[26] he proved the existence of an asymptotic value for weighted majority games.[27] The proof was facilitated by his fundamental contribution to renewal theory.[28] In subsequent work Neyman proved that many of the assumptions made in these works can be relaxed, while showing that others are essential.

Neyman proved the diagonality of continuous values,[29] which had many implications on further developments of the theory. Together with Pradeep Dubey and Robert James Weber he studied the theory of semivalues, and separately demonstrated its importance in political economy.[30][31] Together with Pradeep Dubey [32][33] he characterized the well-known phenomenon of value correspondence, a fundamental notion in economics, originating already in Edgeworth's work and Adam Smith before him. In loose terms, it essentially states that in a large economy consisting of many economically insignificant agents, the core of the economy coincides with the perfectly competitive outcomes, which in the case of differentiable preferences is a unique element that is the Aumann–Shapley value. Another major contribution of Neyman was the introduction of the Neyman value,[34] a far-reaching generalization of the Aumann–Shapley value to the case of non-differentiable vector measure games.

Other[edit]

Neyman has made contributions to other fields of mathematics, usually motivated by problems in game theory. Among these contributions are a renewal theorem for sampling without replacement (mentioned above as applied to the theory of the value), contributions to embeddings of Lp spaces,[35] contributions to the theory of vector measures,[36] and to the theory of non-expansive mappings.[37]

Business involvements[edit]

Neyman previously served (2005–8) as director at Tradus (previously named QXL).[38][39] He also held a directorship (2004–5) at Gilat Satellite Networks.[40] In 1999, Neyman co-founded Bidorbuy, the first online auction company to operate in India and in South Africa, and serves the chairman of the board.[41] Since 2013, he has held a directorship at the Israeli bank Mizrahi-Tefahot.[42]

References[edit]

  1. ^ Center for the Study of Rationality Members
  2. ^ Einstein Institute of Mathematics Faculty
  3. ^ Game Theory Society, announced April 9, 2014
  4. ^ Mathematics Genealogy Project
  5. ^ The Division for Development and Public Relations, Hebrew University of Jerusalem [1]
  6. ^ Bloomberg Business Week Executive Profile
  7. ^ Personal CV
  8. ^ Mathematics Genealogy Project
  9. ^ Econometric Society Fellows
  10. ^ Game Theory Society, announced May 29, 2014
  11. ^ Festschrift conferences in honor of Abraham Neyman and Sergiu Hart on the occasion of their 66th birthday [2]
  12. ^ The John von Neumann Lecture, given at each World Congress of the Game Theory Society, presents important developments in game theory that are of significant mathematical interest. [3]
  13. ^ 2008 World Games Conference Program
  14. ^ 2012 World Games Conference Program
  15. ^ Mertens, J.F., and Neyman, A. (1981). "Stochastic Games," International Journal of Game Theory, 10: 53–66.
  16. ^ Review by Tijs, H.S., MathSciNet [4]
  17. ^ Kohlberg, E. and Neyman, A (1981)., "Asymptotic Behavior of Nonexpansive Mappings in Normed Linear Spaces," Israel Journal of Mathematics, 38 , pp. 269–275.
  18. ^ Neyman, A. (2013), "Continuous-Time Stochastic Games," Center for the Study of Rationality Discussion Paper, #616 [5]
  19. ^ Nato Science Series: Mathematical and Physical Sciences, Volume 570, Proceedings of the NATO Advanced Study Institute on Stochastic Games and Applications (Neyman, A. and Sorin, S. (eds)), held in Stony Brook, NY during July 7–17, 1999.
  20. ^ Neyman, A. (1999), "Cooperation in Repeated Games when the Number of Stages is not Commonly Known," Econometrica, 67: 45–64.
  21. ^ Neyman, A. (1985) "Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma." Economics Letters, 19(3), 227–229.
  22. ^ Ben-Porath, E. (1993) "Repeated games with finite automata." Journal of Economic Theory, 59(1), 17–32.
  23. ^ Neyman, A. and Okada, D. (1999). "Strategic entropy and complexity in repeated games." Games and Economic Behavior, 29(1), 191–223.
  24. ^ Neyman, A., & Okada, D. (2000). "Repeated games with bounded entropy." Games and Economic Behavior, 30(2), 228–247.
  25. ^ Gossner, O., Hernandez, P., and Neyman, A. (2006). "Optimal use of communication resources." Econometrica, 74(6), 1603–1636.
  26. ^ Aumann, R.J. (1980), "Recent Developments in the Theory of the Shapley Value", Proceedings of the International Congress of Mathematicians, Helsinki, 1978, pp. 995–1003, Academia Scientiarum Fennica
  27. ^ Neyman, A., 1981, "Singular games have asymptotic values," Mathematics of Operations Research, 6, pp 205–212.
  28. ^ Neyman, A., 1982, "Renewal theory for sampling without replacement," Annals of Probability, 10, pp 464–481.
  29. ^ Neyman, A., 1977, "Continuous values are diagonal," Mathematics of Operations Research, 2, pp 338–342
  30. ^ Dubey, P., Neyman, A., and Weber, R.J. , 1981, "Value theory without efficiency," Mathematics of Operations Research, 6, pp 122–128
  31. ^ Neyman, A., 1985, "Semi-values of political economic games," Mathematics of Operations Research, 10, pp 390–402
  32. ^ Dubey. P. and Neyman, A., 1984, "Payoffs in nonatomic economies: An axiomatic approach," Econometrica, 52, pp 1129–1150
  33. ^ Dubey, P. and Neyman, A., 1997, "An equivalence principle for perfectly competitive economies," Journal of Economic Theory, 75, pp 314–344
  34. ^ Neyman, A., 2001, "Values of non-atomic vector measure games," Israel Journal of Mathematics, 124, pp 1–27
  35. ^ Neyman, A. (1984), “Representation of Lp-Norms and Isometric Embedding in Lp–Spaces,” Israel Journal of Mathematics, 48, pp. 129–138.
  36. ^ Neyman, A. (1981) “Decomposition of Ranges of Vector Measures,” Israel Journal of Mathematics, 40, pp. 54–64
  37. ^ Kohlberg, E. and Neyman, A. (1999), “A Strong Law of Large Numbers for Nonexpansive Vector-Valued Stochastic Processes,” Israel Journal of Mathematics, 111, pp. 93–108
  38. ^ Daily Mail (London), September 24th, 2005, "Professors Hammer out Path to Riches with QXL", via Highbeam Research
  39. ^ Profile at Opencorporates
  40. ^ Wikinvest
  41. ^ FE Investigate
  42. ^ Mizrahi Tefahot Bank Ltd, Officers and Directors

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