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Fixing the point: the contribution of early game theory to the tool-box of modern economics

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The paper aims at reconstructing the historical sequence of mathematical works through which the fixed-point technique entered the tool-box of modern economics and at establishing a link between this sequence and the neoclassical approach to economic
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  This article was downloaded by: [Universita Di Pisa]On: 27 September 2012, At: 06:04Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK Journal of EconomicMethodology Publication details, including instructions for authorsand subscription information:http://www.tandfonline.com/loi/rjec20 Fixing the point: thecontribution of early gametheory to the tool-box ofmodern economics Nicola Giocoli aa University of Pisa, E-mail:Version of record first published: 12 Dec 2010. To cite this article: Nicola Giocoli (2003): Fixing the point: the contribution of earlygame theory to the tool-box of modern economics, Journal of Economic Methodology,10:1, 1-39 To link to this article: http://dx.doi.org/10.1080/1350178032000042040 PLEASE SCROLL DOWN FOR ARTICLEFull terms and conditions of use:http://www.tandfonline.com/page/terms-and-conditionsThis article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expresslyforbidden.The publisher does not give any warranty express or implied or make anyrepresentation that the contents will be complete or accurate or up todate. The accuracy of any instructions, formulae, and drug doses should beindependently verified with primary sources. The publisher shall not be liablefor any loss, actions, claims, proceedings, demand, or costs or damages  whatsoever or howsoever caused arising directly or indirectly in connectionwith or arising out of the use of this material.    D  o  w  n   l  o  a   d  e   d   b  y   [   U  n   i  v  e  r  s   i   t  a   D   i   P   i  s  a   ]  a   t   0   6  :   0   4   2   7   S  e  p   t  e  m   b  e  r   2   0   1   2  The contribution of early game theory to the tool-box of modern economics 1 Fixing the point: the contribution of earlygame theory to the tool-box of moderneconomics Nicola Giocoli  Abstract The paper aims at reconstructing the sequence of works throughwhich the fixed-point technique entered the tool-box of modern economics and atestablishing a link between this sequence and the neoclassical approach to eco-nomic modeling. The focus is on the change in the demonstration techniquescaused by the spread of the so-called formalist approach to mathematicaleconomics; this change was embodied by the fixed-point technique. The mainconclusions of the paper are that the formalist revolution marked a dramaticdiscontinuity in the history of economic theory and that early game theory–despitehaving been the gateway through which the fixed-point entered economics–wasonly partly responsible for such a discontinuity. Keywords: fixed-point theorems, demonstration techniques, John Nash, Johnvon Neumann, equilibrium point, history of game theory 1INTRODUCTION The technique most frequently used in economics for establishing theexistence of solutions to an equilibrium system of equations is that of searching for a fixed point of a suitably constructed function or corre-spondence  f  : A → A from a set A ⊂ℜ N  into itself. A vector x ∈ A is said tobe a fixed point of   f  (  ) if  x =  f  ( x )–in case of a function–or x ∈  f  ( x )–incase of a correspondence. This means that the vector is mapped by  f  ( • )into itself, and so remains ‘fixed’ despite the application of the mapping.The main reason for proceeding in this often roundabout way is thatimportant mathematical theorems proving the existence of fixed pointsare readily available. Thus, if we manage to transform our equilibriumproblem into a fixed point problem we can easily derive existence resultsfor economic equilibrium. 1 The application of fixed-point techniques in economics is one of thelegacies of the game-theoretic revolution brought forth by mathematicians Journal of Economic Methodology ISSN 1350-178X print/ISSN 1469-9427 online © 2003 Taylor & Francis Ltdhttp://www.tandf.co.uk/journalsDOI:10.1080/1350178032000042040   Journal of Economic Methodology 10:1, 1–39 2003    D  o  w  n   l  o  a   d  e   d   b  y   [   U  n   i  v  e  r  s   i   t  a   D   i   P   i  s  a   ]  a   t   0   6  :   0   4   2   7   S  e  p   t  e  m   b  e  r   2   0   1   2  2 Articles such as John von Neumann and John F. Nash. A well-defined sequence of contributions led to that outcome. The sequence started with von Neumann’sfirst paper on game theory (von Neumann 1959 [1928]), passed throughhis 1937 general equilibrium paper (von Neumann 1968 [1937, 1945–46])and the 1944 Theory of Games and Economic Behavior 2 , written in col-laboration with Oskar Morgenstern (von Neumann and Morgenstern1953 [1944, 1947]), then featured Nash’s new equilibrium concept (Nash1950, 1951) and culminated with Arrow and Debreu’s existence proof,which relied on game theory and Nash equilibrium (Arrow and Debreu1954). This sequence is arguably one of the most important in the wholehistory of economics. Beside the fixed-point techniques, the above-mentionedworks offered the economists’ community a score of other tools and methodsthat became the backbone of modern mathematical economics: convexanalysis, linear algebra, duality theory, etc.The history of when, how and why all this happened has been recon-structed by Weintraub, and Ingrao and Israel, while historians of gametheory, such as Leonard, Mirowski and the Dimands, have investigatedboth the context in which the new discipline was born and the linksbetween its founders and the economists’ community. 3 However, the storyreconstructed so far has focused on the events internal to the small groupsof mathematicians and economists who were directly involved in thesequence, namely, the researchers belonging to Karl Menger’s Math-ematical Colloquium, to the Princeton Mathematics Department or to theCowles Commission. These people were the leading characters in producingwhat is now commonly called the  formalist revolution in economics (seeBlaug 1999). Less has been done, instead, to try to link the new ideasemerging from these groups with the views of the majority of economistsworking in the same time span, i.e. from the late 1930s to the early1950s. This is curious, since in the same period economics underwent aconsiderable development thanks to the implementation of the wealth of new ideas which emerged during the so-called ‘years of high theory’(Shackle 1967). How could it happen that a discipline which had just gonethrough one of its most fruitful periods in terms of theoretical inventivenessand which was just starting to enjoy the benefits of such innovationsfound itself on the verge of being revolutionized in both its methods andgoals by a restricted number of economists who imported into it a newapproach developed in a different field? It is when we come to the issue of appraising the extent of the formalist revolution in economics that therole played by game theory in general, and the fixed-point techniques inparticular, turns out to have been absolutely crucial.This paper aims at investigating this role, that is, at analyzing theextent of the change caused by the fixed point in the way neoclassicaleconomists thought of economic theory and equilibrium. I will argue thatthe formal techniques the mathematical economists borrowed from    D  o  w  n   l  o  a   d  e   d   b  y   [   U  n   i  v  e  r  s   i   t  a   D   i   P   i  s  a   ]  a   t   0   6  :   0   4   2   7   S  e  p   t  e  m   b  e  r   2   0   1   2  The contribution of early game theory to the tool-box of modern economics 3 von Neumann and Nash implied a new approach to the interpretationand justification of equilibrium and the abandonment of what constitutedthe central question for most previous economists (especially during the1930s), namely, the question of the out-of-equilibrium revision of theagents’ plans and expectations. My main thesis is that, while the majorityof neoclassical economists formulated economic models as a way to describe the economic reality and viewed equilibrium as a state of rest in a dynamicmulti-period system whose laws of movement were the main object of research, the adoption of the formalist approach led to a new generationof non-descriptive axiomatic models, where the equilibrium was consid-ered a static concept validating the internal consistency of the modelitself. A corollary of this thesis is that the change was so radical that thevehicle through which formalism entered economics, i.e. game theory,could not meet an immediate success in the economists’ community. Formany years game theory, in both von Neumann’s and Nash’s versions,was considered just a useful tool-box, and this only by those very feweconomists who shared the formalist creed and were determined to applyit to transform economic theory.The events surrounding the introduction of fixed point techniques ineconomics provide a useful lesson concerning the discipline’s evolutionand its methodology. My narrative shows that, to borrow the terminologyof Velupillai (1996: 255–8), when economics meets the decisive forks in itshistorical development it tends to select one path or the other accordingto the availability of an analytical technology  –that is, a set of tools andformal methods almost always imported from the outside–capable of allowing the economists to achieve a satisfactory intellectual representationand solution of at least some of the discipline’s open issues. This attitudeentails however that the problems that do not fit in the analytical technologytend to be abandoned and, moreover, that if the technology proves effectiveit tends also to drive the formulation of new theoretical questions. At thebottom lies of course the wider methodological issue of the non-neutralityof the tools that economics borrows from other disciplines. It often happensthat behind the tools lies a whole system of value judgements, if not afull-fledged methodology of scientific analysis, that can conflict with theone prevailing in economics. This turned out to be true even in the caseof game theory, despite it being the first and only example of a math-ematical theory explicitly developed for the social sciences. As a result,game theory, after having contaminated economics with the ‘virus’ of the fixed-point logic, disappeared for almost 30 years, only to make asurprising comeback in the 1980s, when the ‘virus’ had eventually beeneffective in transforming the way economics was done. This was so muchso that the majority of economic theorists were ready to welcome gametheory and Nash equilibrium as the main language and concept toexpress no less than the keystone of the whole neoclassical edifice,    D  o  w  n   l  o  a   d  e   d   b  y   [   U  n   i  v  e  r  s   i   t  a   D   i   P   i  s  a   ]  a   t   0   6  :   0   4   2   7   S  e  p   t  e  m   b  e  r   2   0   1   2
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