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A History of Von Neumann and Morgenstern's Theory of Games, 12Pt Escolha Sob Incerteza Ax Marcos Thiago Graciani Axiomatic choice under uncertainty: a history of von Neumann and Morgenstern’s Theory of Games Escolha sob incerteza axiomática: uma história do Theory of Games de von Neumann e Morgenstern São Paulo 2019 Prof. Dr. Vahan Agopyan Reitor da Universidade de São Paulo Prof. Dr. Fábio Frezatti Diretor da Faculdade de Economia, Administração e Contabilidade Prof. Dr. José Carlos de Souza Santos Chefe do Departamento de Economia Prof. Dr. Ariaster Baumgratz Chimeli Coordenador do Programa de Pós-Graduação em Economia Marcos Thiago Graciani Axiomatic choice under uncertainty: a history of von Neumann and Morgenstern’s Theory of Games Escolha sob incerteza axiomática: uma história do Theory of Games de von Neumann e Morgenstern Dissertação de Mestrado apresentada ao Departamento de Economia da Faculdade de Economia, Administração e Contabilidade da Universidade de São Paulo (FEA-USP) como requisito parcial à obtenção do título de Mestre em Ciências. Área de Concentração: Teoria Econômica. Universidade de São Paulo — USP Faculdade de Economia, Administração e Contabilidade Programa de Pós-Graduação Orientador: Pedro Garcia Duarte Versão Corrigida (A versão original está disponível na biblioteca da Faculdade de Economia, Administração e Contabilidade.) São Paulo 2019 FICHA CATALOGRÁFICA Elaborada pela Seção de Processamento Técnico do SBD/FEA com dados inseridos pelo autor. Graciani, Marcos Thiago. Axiomatic choice under uncertainty: a history of von Neumann and Morgenstern’s Theory of Games / Marcos Thiago Graciani. – São Paulo, 2019. 133p. Dissertação (Mestrado) – Universidade de São Paulo, 2019. Orientador: Pedro Garcia Duarte. 1. História da teoria dos jogos 2. Axiomática 3. Incerteza 4. Von Neumann e Morgenstern I. Universidade de São Paulo. Faculdade de Economia, Administração e Contabilidade. II. Título. ACKNOWLEDGEMENTS This dissertation evolved directly from the support my parents gave me: I fully appreciate how they assisted me even if they disapproved many of my decisions so they could see me fulfill my dreams. Further, I fully acknowledge that my dissertation is not an individual accomplishment—it is undoubtedly a natural unfolding of opportunities generously placed at my hands by my parents. I wish they feel proud (even if they are not sure about what I do). Many people helped me during my graduate studies. I am most indebted to my advisor, Pedro. He offered me unbounded assistance by sharing his experiences and encouraging me whenever difficulties arose. Besides, it gratifies me that I have him as a dear friend who I long to keep throughout the years. I should also manifest my gratitude towards professors Marcos and Roberta who so carefully read preliminary versions of my dissertation. I hope my final work can measure up for how helpful they were for me. I fell grateful to them and Catherine for participating in my defense committee and providing me great insights for future research. Naturally, many other professors participated in my still short academic journey—I should thank them in person, in any hallway we eventually meet. The University awarded me many friends whom I should mention here: I thank Felipe for being such an excellent companion even if misfortune set us apart; Isabella for always being available to watch underestimated (some would unjustly say “low-quality”) movies and TV shows; Lucas for being on my side as we overcame the daily difficulties of life as graduate students; Mariana for giving me reasons to laugh constantly (even if she didn’t intend to); Tainá for encouraging me now when my future path is everything but certain. Most importantly, I thank Rafael for his love. To say I have been happy beside him would be minimizing my feelings; I hope I can convey to him how I value our relationship through different means (as I am not skilled in choosing and arranging words). I further thank our beloved Mufasa and Zazu: cats and cockatiels are great listeners (I would confidently bet they understand my research more than I do). Finally, I should thank CAPES for financially supporting me. In full compliance to their rules, I add gratefully: “This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior–Brasil (CAPES)–Finance Code 001”. RESUMO GRACIANI, M. T. Escolha sob incerteza axiomática: uma história do Theory of Games de von Neumann e Morgenstern. Dissertação (Mestrado) – Faculdade de Economia, Administração e Contabilidade, Universidade de São Paulo, São Paulo, 2019. Esta dissertação estuda a recepção imediata do Theory of Games and Economic Behavior, de von Neumann e Morgenstern. Seu foco reside em como economistas (e outros cientistas, tais como matemáticos) reagiram à axiomatização da teoria de utilidade esperada composta por von Neumann e Morgenstern. Tal estudo se vale de resenhas do Theory of Games, artigos autorados por leitores proficientes em matemática que seguiram a deixa dos autores de axiomatizar teoria de escolha sob incerteza e, por fim, artigos cujas citações incluem trabalhos destes leitores habilidosos. Há três conclusões principais. Primeiro, para entender a história de recepção do Theory of Games, é importante considerar que fontes secundárias agiram como disseminadores de premissas, resultados e o próprio método do Theory of Games. Segundo, muitos leitores capazes refletiram sobre o livro de von Neumann e Morgenstern. A maioria dos que usaram tal literatura a fizeram de acordo com o método axiomático, citanto aqueles artigos para reproduzir ou adaptar hipóteses. Dentre os que os citaram para aplicar seus resultados diretamente usaram ferramentas matemáticas menos sofisticadas e não tinham como objetivo a produção de demonstrações formais, em geral. Terceiro, enquanto o axioma de independência é uma condição necessária para a teoria de utilidade esperada, economistas tiveram dificuldades em compreender como von Neumann e Morgenstern usaram-no. Não estava claro para eles onde o Theory of Games o havia escondido. Uma vez que os economistas descobriram o axioma, não encontraram uso imediato para ele. Palavras-chave: história da teoria dos jogos; axiomática; incerteza; Von Neumann; Mor- genstern. ABSTRACT GRACIANI, M. T. Axiomatic choice under uncertainty: a history of von Neu- mann and Morgenstern’s Theory of Games. Dissertation (Master) – School of Economics, Administration, and Accounting, University of São Paulo, São Paulo, 2019. This dissertation studies the immediate reception of von Neumann and Morgenstern’s Theory of Games and Economic Behavior. It focuses on how economists (and other scientists, such as mathematicians) reacted to von Neumann and Morgenstern’s axiomatization of expected utility theory. Such study employs book reviews the Theory of Games received, articles authored by mathematically-proficient readers who followed von Neumann and Morgenstern’s lead of axiomatizing choice under uncertainty, and articles that cited the later. The main conclusions are threefold. First, to understand the history of the Theory of Games’ reception it is unavoidable to consider how secondary sources acted as disseminators of its premises, results, and method. Second, many skilled authors reflected on von Neumann and Morgenstern’s book. Most economists who used that literature in an axiomatic framework cited such contributions to borrow and adapt assumptions. Those who applied results directly generally used less-sophisticated mathematical tools and were not proof-driven. Third, while the independence axiom is a necessary condition for expected utility theory, economists struggled to understand how von Neumann and Morgenstern used it. It was not clear where the Theory of Games hid that assumption. After economists discovered the independence axiom, they did not find an immediate use for it. Keywords: history of game theory; axiomatics; uncertainty; Von Neumann; Morgenstern. LIST OF FIGURES Figure 1 – Column Maxima and Row Minima .................... 49 Figure 2 – Mixed Strategies and Saddle-Points.................... 52 Figure 3 – Visual Approach to Intransitivity..................... 56 Figure 4 – Graphical Model of One Seller and Two Buyers ............. 57 Figure 5 – Cowles’ Logo: “Science is Measurement” ................. 67 Figure 6 – Marschak’s “Domain” and “Feasible Set”................. 82 Figure 7 – Marschak’s “Postulate IV1” and “Postulate IV2”............. 83 Figure 8 – The “Theorem of Mixtures”........................ 97 LIST OF TABLES Table 1 – The Theory of Games’ Book Reviews ................... 22 Table 2 – Cowles Commission Seminars, January 1, 1949–June 30, 1949 . 71 Table 3 – Cowles Commission Seminars, July 1, 1950–February 15, 1951 . 72 LIST OF SYMBOLS ¬ The negation operation. N The set of natural numbers (zero excluded). R The set of real numbers. R+ The set {x ∈ R : x > 0}. n An element of N representing a number of commodities or outcomes. n X A commodity space or a set of outcomes (a subset of R+ ∪ {0}). n L A space of simple lotteries (a subset of {(p1, ..., pn) ∈ R+ ∪ {0} : Pn i=1 pi = 1}). g A relation on X or L (meaning “at least as good as”). p A relation on X or L (meaning “strictly preferred than”). i A relation on X or L (meaning “indifferent to”). gT The transpose of g. pT The transpose of p. Ij Given Lj ∈ L , the set {L ∈ L : LiLj}. u A utility function or index u : X → R. U An expected utility function U : L → R. ◦ Function composition. CONTENTS Introduction ............................... 17 1 A TORRENT OF EARLY REACTIONS................ 21 1.1 A Path Through Book Reviews ...................... 23 1.2 The
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