Classics in Game Theory

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Classics in Game Theory Edited by Harold W. Kuhn PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY CONTENTS Permissions vii H. W. KUHN Foreword ix DAVID KREPS AND ARIEL RUBINSTEIN An Appreciation xi 1. JOHN F. NASH, JR. Equilibrium Points in n-Person Games. PNAS 36 (1950) 48-49. 3 2. JOHN F. NASH, JR. The Bargaining Problem. Econometrica 18 (1950) 155-162. 5 3. JOHN NASH Non-Cooperative Games. Annais of Mathematics 54 (1951) 286-295. 14 4. JULIA ROBINSON An Iterative Method of Solving a Game. Annah of Mathematics 54 (1951) 296-301. 27 5. F. B. THOMPSON Equivalence of Games in Extensive Form. RAND Memo RM-759 (1952). 36 6. H. W. KUHN Extensive Games and the Problem of Information. Contributions to the Theory of Games II (1953) 193-216. 46 7. L. S. SHAPLEY A Value for n-Person Games. Contributions to the Theory of Games II (1953) 307-317. 69 8. L. S. SHAPLEY Stochastic Games. PNAS 39 (1953) 1095-1100. 80 VI CONTENTS 9. H. EVERETT Recursive Games. Contributions to the Theory of Games III (1957) 47-78. 87 10. R. J. AUMANN AND B. PELEG Von Neumann-Morgenstern Solutions to Cooperative Games without Side Payments. Bulletin AMS 66 (1960) 173-179. 119 11. GERARD DEBREU AND HERBERT SCARF A Limit Theorem on the Core of an Economy. International Economic Review 4 (1963) 235-246. 127 12. ROBERT J. AUMANN AND MICHAEL MASCHLER The Bargaining Set for Cooperative Games. Advances in Game Theory (1964) 443-477. 140 13. ROBERT J. AUMANN Existence of Competitive Equilibria in Markets with a Continuum of Traders. Econometrica 34 (1966) 1-17. 170 14. HERBERT E. SCARF The Core of an n-Person Game. Econometrica 35 (1967) 50-69. 192 15. JOHN C. HARSANYI Games with Incomplete Information Played by "Bayesian" Players. Part I: The Basic Model. Man. Sei. 14 (1967) 159-182 216 Part II: Bayesian Equilibrium Points. Man. Sei. 14 (1968) 320-334 247 Part III: The Basic Probability Distribution of the Game. Man. Sei. 14 (1968) 486-502. 268 16. DAVID BLACKWELL AND T. S. FERGUSON The Big Match. Ann. Math. Stat. 39 (1968) 159-163. 289 17. LLOYD S. SHAPLEY AND MARTIN SHUBIK On Market Games. J.E. T. 1 (1969) 9-25. 296 18. R. SELTEN Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games. Int. J. Game Th. 4 (1975) 25-55. 317 List of Contributors 355 Index 357 .

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An Analog of the Minimax Theorem for Vector Payoffs
Pacific Journal of Mathematics AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL Vol. 6, No. 1 November 1956 AN ANALOG OF THE MINIMAX THEOREM FOR VECTOR PAYOFFS DAVID BLACKWELL 1. Introduction* The von Neumann minimax theorem [2] for finite games asserts that for every rxs matrix M=\\m(i, j)\\ with real elements there exist a number v and vectors P=(Pi, •••, Pr)f Q={QU •••> Qs)f Pi, Qj>β, such that i> 3) for all i, j. Thus in the (two-person, zero-sum) game with matrix Λf, player I has a strategy insuring an expected gain of at least v, and player II has a strategy insuring an expected loss of at most v. An alternative statement, which follows from the von Neumann theorem and an appropriate law of large numbers is that, for any ε>0, I can, in a long series of plays of the game with matrix M, guarantee, with probability approaching 1 as the number of plays becomes infinite, that his average actual gain per play exceeds v — ε and that II can similarly restrict his average actual loss to v-he. These facts are assertions about the extent to which each player can control the center of gravity of the actual payoffs in a long series of plays. In this paper we investigate the extent to which this center of gravity can be controlled by the players for the case of matrices M whose elements m(i9 j) are points of ΛΓ-space. Roughly, we seek to answer the following question.
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Blackwell, David; Diaconis, Persi a Non-Measurable Tail Set
Blackwell, David; Diaconis, Persi A non-measurable tail set. Statistics, probability and game theory, 1--5, IMS Lecture Notes Monogr. Ser., 30, Inst. Math. Statist., Hayward, CA, 1996. Blackwell, David Operator solution of infinite $G\sb \delta$ games of imperfect information. Probability, statistics, and mathematics, 83--87, Academic Press, Boston, MA, 1989. 60040 Blackwell, David; Mauldin, R. Daniel Ulam's redistribution of energy problem: collision transformations. Lett. Math. Phys. 10 (1985), no. 2-3, 149--153. Blackwell, David; Dubins, Lester E. An extension of Skorohod's almost sure representation theorem. Proc. Amer. Math. Soc. 89 (1983), no. 4, 691--692. 60B10 Blackwell, David; Maitra, Ashok Factorization of probability measures and absolutely measurable sets. Proc. Amer. Math. Soc. 92 (1984), no. 2, 251--254. Blackwell, David; Girshick, M. A. Theory of games and statistical decisions. Reprint of the 1954 edition. Dover Publications, Inc., New York, 1979. xi+355 pp. ISBN: 0-486-63831-6 90D35 (62Cxx) Blackwell, David On stationary policies. With discussion. J. Roy. Statist. Soc. Ser. 133 (1970), no. 1, 33--37. 90C40 Blackwell, David The stochastic processes of Borel gambling and dynamic programming. Ann. Statist. 4 (1976), no. 2, 370--374. Blackwell, David; Dubins, Lester E. On existence and non-existence of proper, regular, conditional distributions. Ann. Probability 3 (1975), no. 5, 741--752. Blackwell, David; MacQueen, James B. Ferguson distributions via Pólya urn schemes. Ann. Statist. 1 (1973), 353--355. Blackwell, David; Freedman, David On the amount of variance needed to escape from a strip. Ann. Probability 1 (1973), 772--787. Blackwell, David Discreteness of Ferguson selections.
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Rational Inattention with Sequential Information Sampling∗
Rational Inattention with Sequential Information Sampling∗ Benjamin Hébert y Michael Woodford z Stanford University Columbia University July 1, 2016 Preliminary and Incomplete. WARNING: Some results in this draft are not corrected as stated. Please contact the authors before citing. Comments welcome. Abstract We generalize the rationalize inattention framework proposed by Sims [2010] to allow for cost functions other than Shannon’s mutual information. We study a prob- lem in which a large number of successive samples of information about the decision situation, each only minimally informative by itself, can be taken before a decision must be made. We assume that the cost required for each individual increment of in- formation satisﬁes standard assumptions about continuity, differentiability, convexity, and monotonicity with respect to the Blackwell ordering of experiments, but need not correspond to mutual information. We show that any information cost function sat- isfying these axioms exhibits an invariance property which allows us to approximate it by a particular form of Taylor expansion, which is accurate whenever the signals acquired by an agent are not very informative. We give particular attention to the case in which the cost function for an individual increment to information satisﬁes the ad- ditional condition of “prior invariance,” so that it depends only on the way in which the probabilities of different observations depend on the state of the world, and not on the prior probabilities of different states. This case leads to a cost function for the overall quantity of information gathered that is not prior-invariant, but “sequentially prior-invariant.” We offer a complete characterization of the family of sequentially prior-invariant cost functions in the continuous-time limit of our dynamic model of in- formation sampling, and show how the resulting theory of rationally inattentive choice differs from both static and dynamic versions of a theory based on mutual information.
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An Annotated Bibliography on the Foundations of Statistical Inference
• AN ANNOTATED BIBLIOGRAPHY ON THE FOUNDATIONS OF STATISTICAL INFERENCE • by Stephen L. George A cooperative effort between the Department of Experimental • Statistics and the Southeastern Forest Experiment Station Forest Service U.S.D.A. Institute of Statistics Mimeograph Series No. 572 March 1968 The Foundations of Statistical Inference--A Bibliography During the past two hundred years there have been many differences of opinion on the validity of certain statistical methods and no evidence that ,. there will be any general agreement in the near future. Also, despite attempts at classification of particular approaches, there appears to be a spectrum of ideas rather than the existence of any clear-cut "schools of thought. " The following bibliography is concerned with the continuing discussion in the statistical literature on what may be loosely termed ''the foundations of statistical inference." A major emphasis is placed on the more recent works in this area and in particular on recent developments in Bayesian analysis. Invariably, a discussion on the foundations of statistical inference leads one to the more general area of scientific inference and eventually to the much more general question of inductive inference. Since this bibliography is intended mainly for those statisticians interested in the philosophical foundations of their chosen field, and not for practicing philosophers, the more general discussion of inductive inference was deliberately de-emphasized with the exception of several distinctive works of particular relevance to the statistical problem. Throughout, the temptation to gather papers in the sense of a collector was resisted and most of the papers listed are of immediate relevance to the problem at hand.
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David Blackwell Instance for Activities That Could Run Sometime Between Now and the End of June 2012
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David Blackwell, 1919–2010: an Explorer In
RETROSPECTIVE David Blackwell, 1919–2010: An explorer in mathematics and statistics RETROSPECTIVE Peter J. Bickela,1 David Blackwell, a pioneering explorer who made foundational contributions to several branches of mathematics and statistics, passed away on July 8, 2010. He was born in Centralia, Illinois, on April 24, 1919, and, as his mathematical talents were recog- nized early, he entered the University of Illinois at Urbana–Champaign at age 16. Although racial dis- crimination affected his life and career in painful ways, his accomplishments were eventually rewarded with the honors they deserved, including election to the National Academy of Sciences in 1965 as the first Black member and the American Academy of Arts and Sciences in 1968. Nevertheless, his love of math- ematics, science, people, and his sunny personality prevailed. Blackwell left contributions that bear his name and other major ideas in five quite different areas of mathematics, statistics, and operations research. With limited opportunities available to him, Black- well initially thought of becoming an elementary school teacher. However, his professors at the Univer- sity of Illinois soon recognized his talent for mathe- matics and encouraged him to pursue graduate studies in the Illinois Mathematics program instead. During his graduate studies, Blackwell worked with Joseph Doob, one of the founding figures of modern David Blackwell. Image credit: The Blackwell family. probability theory, a National Academy of Sciences member, and National Medal of Science winner. In sequential analysis, game theory, and decision the- 1941, at age 22, he completed a doctoral thesis in the ory. They included: what is now known as the Rao- theory of Markov chains, a set of ideas to which he Blackwell theorem, a fundamental improvement frequently returned in his later work.
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Bayesian Statistics: Thomas Bayes to David Blackwell
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Rationally Inattentive Behavior: Characterizing and Generalizing
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Math Times, Summer 2021
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Obituary: David Blackwell 1919–2010
14 . IMs Bulletin Volume 39 . Issue 8 Obituary: David Blackwell 1919–2010 The last Olympian figure of the statistics elected a member of the National Academy Statistics pantheon residing in the University of of Sciences in 1965, and a member of the of California at Berkeley has fallen. David American Academy of Arts and Sciences in Blackwell passed away on July 8, 2010, at 1968. Department the age of 91. David Blackwell was the W.W. Rouse David Blackwell was born on April 24, Ball Lecturer at Cambridge University, Berkeley 1919, in Centralia, Illinois, where he went UK; Wald Lecturer of the Institute of UC David Blackwell at his 90th birthday party at Berkeley through the public school system. He was Mathematical Statistics; and Faculty awarded the AB (1938), AM (1939), and Research Lecturer at UC-Berkeley. He is a classic on this subject matter, and his PhD(1941) degrees, in mathematics, from was awarded the TIMS/ORSA John Von book on elementary statistics, Basic Statistics the University of Illinois; Joseph Doob was Neumann Theory Prize, the R.A. Fisher (1969), is a gem and a breeze of fresh air his thesis advisor. Award by the Committee of Presidents of in the existing curriculum. Between 1955 After three-year appointments as a the Statistical Societies, and the Berkeley and 1981, David Blackwell was the advisor Fellow at the Institute of Advanced Studies Citation. of 65 students. In a single year alone, he (Princeton), and Instructor at Southern During his professional tenure, David graduated 7 students! University (Baton Rouse, LA) and Clark Blackwell served the UC System in many The hallmark of David Blackwell was College (GA), he joined Howard University ways, including the directorship of the elegance and simplicity over mathematical (Washington DC) in 1944 as an Assistant UK-Ireland UC Study Center (1973–75) abstraction.
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An Interview with Kenneth J. Arrow Author(S): J
An Interview with Kenneth J. Arrow Author(s): J. S. Kelly and Kenneth J. Arrow Source: Social Choice and Welfare, Vol. 4, No. 1 (1987), pp. 43-62 Published by: Springer Stable URL: http://www.jstor.org/stable/41105852 . Accessed: 27/09/2013 14:15 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Springer is collaborating with JSTOR to digitize, preserve and extend access to Social Choice and Welfare. http://www.jstor.org This content downloaded from 150.135.135.70 on Fri, 27 Sep 2013 14:15:11 PM All use subject to JSTOR Terms and Conditions " Soc Choice Welfare 4 : 43-62 Z (1987) Social# Choice ^Welfare © Springer-Verlag 1987 An Interviewwith Kenneth J. Arrow J. S. Kelly Departmentof Economics,Syracuse University, Maxwell Hall, Syracuse,NY 13210, USA The followingis an editedtranscript of an interviewconducted on March 4, 1986 withProfessor Arrow while he was visitingSyracuse University to deliverthe Frank W. Abrams LectureSeries to be publishedas The UncertainFuture and Present Action by Syracuse UniversityPress. This interviewwas to elaborate on his description,presented in Volume 1 of his CollectedPapers (Harvard University Press,1983), of the originsof his work in collectivechoice theory.
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