Editing John von Neumann (section)

== Economics ==
=== Game theory ===
Von Neumann founded the field of [[game theory]] as a mathematical discipline.<ref name="KuhnTucker">{{cite journal|last1=Kuhn|first1= H. W.|author-link=Harold W. Kuhn|last2=Tucker|first2= A. W.|author-link2=Albert W. Tucker|title=John von Neumann's work in the theory of games and mathematical economics|journal=Bull. Amer. Math. Soc.|year=1958|volume=64 (Part 2)|issue=3|pages=100–122|mr=0096572|doi=10.1090/s0002-9904-1958-10209-8|citeseerx= 10.1.1.320.2987}}</ref> He proved his [[Minimax#Minimax theorem|minimax theorem]] in 1928. It establishes that in [[zero-sum game]]s with [[perfect information]] (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of [[Strategy (game theory)|strategies]] for both players that allows each to minimize their maximum losses.<ref name="Game Theory">{{cite journal |last=von Neumann |first=J |title=Zur Theorie der Gesellschaftsspiele |language=de |journal=[[Mathematische Annalen]] |volume=100 |pages=295–320 |doi=10.1007/bf01448847|year=1928 |s2cid=122961988 }}</ref> Such strategies are called ''optimal''. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He improved and extended the [[minimax theorem]] to include games involving imperfect information and games with more than two players, publishing this result in his 1944 ''[[Theory of Games and Economic Behavior]]'', written with [[Oskar Morgenstern]]. The public interest in this work was such that ''[[The New York Times]]'' ran a front-page story.<ref>{{Cite news |last=Lissner |first=Will |date=1946-03-10 |title=Mathematical Theory of Poker Is Applied to Business Problems; GAMING STRATEGY USED IN ECONOMICS Big Potentialities Seen Strategies Analyzed Practical Use in Games |language=en-US |newspaper=The New York Times |url=https://www.nytimes.com/1946/03/10/archives/mathematical-theory-of-poker-is-applied-to-business-problems-gaming.html |access-date=2020-07-25 |issn=0362-4331 }}</ref> In this book, von Neumann declared that economic theory needed to use [[functional analysis]], especially [[convex set]]s and the [[topology|topological]] [[fixed-point theorem]], rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.<ref name="KuhnTucker"/>

Von Neumann's functional-analytic techniques—the use of [[Dual space|duality pairings]] of real [[vector space]]s to represent prices and quantities, the use of [[Supporting hyperplane|supporting]] and [[Hyperplane separation theorem|separating hyperplanes]] and convex sets, and fixed-point theory—have been primary tools of mathematical economics ever since.<ref>{{cite book |last=Blume |first=Lawrence E. |author-link=Lawrence E. Blume |contribution=Convexity |year=2008 |title=The New Palgrave Dictionary of Economics |pages=225–226 |editor1-last=Durlauf |editor1-first=Steven N. |editor1-link=Steven N. Durlauf  |editor2-last=Blume |editor2-first=Lawrence E. |publisher=Palgrave Macmillan |location=New York |edition=2nd |url=http://www.dictionaryofeconomics.com/article?id=pde2008_C000508|doi=10.1057/9780230226203.0315 |isbn=978-0-333-78676-5}}</ref>

=== Mathematical economics ===
Von Neumann raised the [[Mathematical economics|mathematical level of economics]] in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the [[Brouwer fixed-point theorem]].<ref name=KuhnTucker/> Von Neumann's model of an expanding economy considered the [[Eigendecomposition of a matrix#Generalized eigenvalue problem|matrix pencil]]&nbsp;'' '''A'''&nbsp;−&nbsp;λ'''B''''' with nonnegative matrices&nbsp;'''A''' and '''B'''; von Neumann sought [[probability vector|probability]] [[generalized eigenvector|vectors]]&nbsp;''p'' and&nbsp;''q'' and a positive number&nbsp;''λ'' that would solve the [[complementarity theory|complementarity]] equation <math>p^T (A - \lambda B) q = 0</math> along with two inequality systems expressing economic efficiency. In this model, the ([[transpose]]d) probability vector ''p'' represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution ''λ'' represents the growth factor which is 1 plus the [[economic growth|rate of growth]] of the economy; the rate of growth equals the [[interest rate]].<ref>For this problem to have a unique solution, it suffices that the nonnegative matrices&nbsp;'''A''' and&nbsp;'''B''' satisfy an [[Perron–Frobenius theorem|irreducibility condition]], generalizing that of the [[Perron–Frobenius theorem]] of nonnegative matrices, which considers the (simplified) [[Eigenvalues and eigenvectors|eigenvalue problem]]
: '''A''' − λ '''I''' ''q'' = 0,
where the nonnegative matrix''&nbsp;'''A''''' must be square and where the [[diagonal matrix]]''&nbsp;'''I''' ''is the [[identity matrix]]. Von Neumann's irreducibility condition was called the "whales and [[Wrangler (University of Cambridge)|wranglers]]" hypothesis by [[D. G. Champernowne]], who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good. Weaker "irreducibility" conditions were given by [[David Gale]] and by [[John George Kemeny|John Kemeny]], Morgenstern, and [[Gerald L. Thompson]] in the 1950s and then by Stephen M. Robinson in the 1970s.</ref><ref>{{cite book|last1=Morgenstern|first1=Oskar|author-link1=Oskar Morgenstern|last2=Thompson|first2=Gerald L.|author-link2=Gerald L. Thompson|title=Mathematical Theory of Expanding and Contracting Economies|series=Lexington Books|publisher=D. C. Heath and Company|year=1976|location=Lexington, Massachusetts|isbn=978-0-669-00089-4|url-access=registration|url=https://archive.org/details/mathematicaltheo0000morg |pages=xviii, 277}}</ref>

Von Neumann's results have been viewed as a special case of [[linear programming]], where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists.<ref>{{cite book |last=Rockafellar |first=R. T. |author-link=R. Tyrrell Rockafellar |title=Convex analysis |publisher=Princeton University Press |year=1970 |isbn=978-0-691-08069-7 |oclc=64619 |pages=i, 74}} {{pb}} {{cite conference |last=Rockafellar |first=R. T. |author-link=R. Tyrrell Rockafellar |chapter=Convex Algebra and Duality in Dynamic Models of production |title=Mathematical Models in Economics |conference=Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972 |editor1-first=Josef |editor1-last=Loz |editor2-first=Maria |editor2-last=Loz |publisher=Elsevier North-Holland Publishing and Polish Academy of Sciences |location=Amsterdam |year=1974 |oclc=839117596 |pages=351–378}}</ref><ref>{{cite book |last=Ye |first=Yinyu |author-link=Yinyu Ye |year=1997 |url=https://books.google.com/books?id=RQZd7ru8cmMC&pg=PA277 |contribution=The von Neumann growth model |title=Interior point algorithms: Theory and analysis |publisher=Wiley |location=New York |isbn=978-0-471-17420-2 |oclc=36746523 |pages= 277–299}}</ref> This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, [[Linear inequality|linear inequalities]], [[Linear programming#Complementary slackness|complementary slackness]], and [[Duality (optimization)|saddlepoint duality]].{{sfn|Dore|Chakravarty|Goodwin|1989|p=xi}} In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.<ref>{{cite book|editor1-last=Bruckmann|editor1-first=Gerhart|editor2-last=Weber|editor2-first=Wilhelm|date=September 21, 1971|doi=10.1007/978-3-662-24667-2|title=Contributions to von Neumann's Growth Model|series=Proceedings of a Conference Organized by the Institute for Advanced Studies Vienna, Austria, July 6 and 7, 1970|publisher=Springer–Verlag|isbn=978-3-662-22738-1}}</ref> The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of [[Nobel prize]]s in 1972 to [[Kenneth Arrow]], in 1983 to [[Gérard Debreu]], and in 1994 to [[John Forbes Nash Jr.|John Nash]] who used fixed point theorems to establish equilibria for [[non-cooperative game]]s and for [[bargaining problem]]s in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureates [[Tjalling Koopmans]], [[Leonid Kantorovich]], [[Wassily Leontief]], [[Paul Samuelson]], [[Robert Dorfman]], [[Robert Solow]], and [[Leonid Hurwicz]].{{sfn|Dore|Chakravarty|Goodwin|1989|p=234}}

Von Neumann's interest in the topic began while he was lecturing at Berlin in 1928 and 1929. He spent his summers in Budapest, as did the economist [[Nicholas Kaldor]]; Kaldor recommended that von Neumann read a book by the mathematical economist [[Léon Walras]]. Von Neumann noticed that Walras's [[General Equilibrium Theory]] and [[Walras's law]], which led to systems of simultaneous linear equations, could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced his paper.{{sfn|Macrae|1992|pp=250–253}}

=== Linear programming ===
Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when [[George Dantzig]] described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.<ref>{{cite book | last=Dantzig | first=G. B. | author-link=George Dantzig | year=1983 | contribution=Reminiscences about the origins of linear programming. | title=Mathematical Programming The State of the Art: Bonn 1982 | editor1-last=Bachem | editor1-first=A. | editor2-last=Grötschel |editor2-first=M. | editor3-last=Korte | editor3-first=B. | location=Berlin, New York | publisher=Springer-Verlag | pages=78–86 | isbn=0387120823 | oclc=9556834}}</ref>

Later, von Neumann suggested a new method of [[linear programming]], using the homogeneous linear system of [[Paul Gordan]] (1873), which was later popularized by [[Karmarkar's algorithm]]. Von Neumann's method used a pivoting algorithm between [[simplex|simplices]], with the pivoting decision determined by a nonnegative [[least squares]] subproblem with a convexity constraint ([[Projection (linear algebra)#Orthogonal projections|projecting]] the zero-vector onto the [[convex hull]] of the active simplex). Von Neumann's algorithm was the first [[interior point method]] of linear programming.<ref name="George B 2003">{{cite book | last1 = Dantzig | first1 = George |author1-link=George Dantzig |last2=Thapa |first2=Mukund N. | title = Linear Programming : 2: Theory and Extensions | publisher = [[Springer Science+Business Media|Springer-Verlag]]| location = New York, NY | year = 2003| isbn = 978-1-4419-3140-5}}</ref>