Applications of Topoi Theory and Non-standard Analysis in Mathematical Economics (shortened version)
[This text represents a shortened version of a Project proposed to POSDRU European Commission - Romania Funds in 2010 by: Dr. Angela Pasarescu, University "Politehnica" of Bucharest, and Dr. Ovidiu Pasarescu, "Simion Stoilow" Institute of Mathematics of the Romanian Academy(IMAR)]
The present Project is centered on applications in Mathematical Economics of methods coming, mainly, from Mathematical Logic. Shortly, the approaches are described in 3 parts. The first one is based on the so-called Non-standard Analysis (built by Abraham Robinson), the second one is based on Modal Logic (Types Theory), and Intuitionistic Logic (having several truth values), and the third one is at a higher level of abstraction, using Topoi Theory (notion introduced by Alexander Grothendieck in Algebraic Geometry as a formal extention in the Theory of Cathegories of the classical notion of Topological Space). The combined approach using these 3 manners is not used, at our knowledge in the literature related to Mathematical Economics, although there ar some (few) separated approaches.
A) THE NON-STANDARD APPROACH.
The Non-standard Analysis has been rigorously introduced, in the years '60, by Abraham Robinson. One of its purposes was to formalize the notions already introduced by Leibniz and Newton when building the differential calculus, called by them infinite real numbers (small and large). In particular, a new notion appear in arithmetic, namely the notion of hyperfinite natural number, representing the cardinal of a hyper-finite set, that is, an infinite set having similar proprieties with the usual finite sets at the level of internal functions (in the context, one have two kinds of sets, namely internal and external). So, the hyper-natural numers are infinitely large real numbers, looking, in some sense, like a natural number.
i) Hyper-finite Economies. In the literature, there are discussions on how to take in consideration all the participants to economical activities from a large economy (like that of the European Union, or USA, or China,...). For example, as the Nobel laureate Robert Aumann pointed out, the small pruducers from USA contributes with one third of the total economy. The employee from Whashington DC feel their activities like a continuous one, not in a discrete way (like for big companies), like the water from an ocean, not like the molecules (finite in number) generating the water. So, R. Aumann considered that it is better to introduce the notion of so called "continuum economy", whose number of players (producers, consumers, financiers,...) is of the power of continuum, that is, in bijection with the set of real numbers. Other economists, like the American K. Prasad, considered, on the opposite sense, that one should to never consider infinite sets when studying economical problems, because everything is finite in this context. This means that one should put ourselves in the context of the so-called, intuitionistic logic, having many truth values (which, in particular, denies the existence of the infinite sets). The problem is that, some statements, known to be true in the context of continuum economies (for instance the existence of Nash or Walras equilibrium , Pareto optimum,... when we consider the setting of non-cooperative games as a model) become false or indecidable when one denies the existence of infinite entities. So, when taking an economical decision, should we use these kinds of statements, or not ? If we consider the hyper-finite economies (that is, with a hyper-finite number of participants) we are exactly in between: we consider an infinite number of players which looks like a finite one. Studying some standard topics in this new setting could help to better undurstand which statemets should be used and which ones don't, when taking economical decisions in big economies (and to avoid a possible crisis, generated by wrong decisions).
ii) Transfer. The second important advantage of the use of non-standard analysis consist in the fact that it has several proprieties of transfer and permanence (relating the standard and non-standard Universes). These allow us to deduce that some results, proved to be true for sufficienly small hyper-finite economies to remain true for finite large enough economies. On the other hand, it is possible, in some situations, to see the continuum economies of Aumann as standard shadows (a notion from non-standard analysis) of some hyper-finite economies, thus, by the Idealisation principle (another notion from non-standard analysis) to deduce that they are true. So, these statements are true both for continuum economies, and for sufficiently large finite economies. Such Theorems can be used when taking decisions in the context of large economies. On the other hand, one can find statements from continuum economies, whose hyper-finite analogues are false. The previous argument does not work for such statements. So, it is better to not use these statements when taking decisions (although they are known to be true for continuum economies). In such a way we built a test allowing us to decide which statements for Aumann economies can be used. This test is possible due to the existence of the so-called Loeb measure, a probabilistic measure on hyper-finite sets, whose shadow is the usual Lebesgue measure on the closed interval [0,1].
B) GAME THEORY VERSUS MODEL THEORY.
The (Mathematical) Game theory is very important when studying economical phenomena. This hapens because the traders, buyers, stock workers,... plays, in fact, some games which they want to win (or, at least, to put themselves in a better situation). These games could be cooperative (for example between the branches of the same company) or non-cooperative. The last ones could be either with complete information (when one knows all the actions of the competitors, like chess, for instance), or with incomplete information (when not all the actions and possibilities of the competitors are known (like most of the cards games, for instance; bridge is an example). The last ones could be static (when the decision is taken from the beginning for a period), or dynamic (when one takes decitions at any moment), and so long.The number of players is finite, and a game can me interpreted as a graph and then as a landmark in Modal Logic. One knows that various proprieties with economical applications can be equivalently reformulated in Modal Logic. For instance, the best answer, the Nash equilibrium,...So, one can use the theorems appearing in Modal Logic. The Finite Model Theory is a powerful technical support also (as examples, finite fields, graphs, games, seen as theories). However, in Finite Model Theory some important results of Model Theory does not hold. An important example is the Compactness Theorem. Passing from finite to hiper-finite games helps in this setting also. The model completness of the Theory of real closed fields is related to the Nash equilibrium.
C) TOPOI THEORY
The notion of topoi has been introduced by A. Grothendieck for the use of Algebraic Geometry. As examples are the flat topos, the etal topos,.... A more general notion (elementary topos) have been worked out later, and it became a poweful tool in Model Theory (Categorial Logic). In Topoi it is natural to built theories with intuitionistic logic (this means that, if a sentence P is false, it does not automatically follow that its negation is true; so, one have sevaral truth valus; in particular, the existence of infinite sets is not allowed). In this context (of Prasad kind, see A)i)), we'll try to understand what happens with those Theorems for Aumann economies which does not pass the test from A)ii)(i.e. they does not come from a hyper-finite economy with a Loeb measure). We also study what can be done when using the context of non-standard analysis type in topoi.
I) j) Reproving Theorems from the Theory of continuum economics using hyper-finite economies (these represents good models for large economies).
jj) Find Theorems for Aumann economies which does not pass the test from A) i). Study them in the context of elementary topoi.
II) Built a model of non-standard analysis in topoi. Extend at this level of abstraction the results from I).
NOTE: The subject of this Project is related to Decision Theory in Mathematical Economics.
2. Scientific Description
2.1 The General Objective
The general Objective of this Project is: CONTRIBUTIONS TO DECISION THEORY IN LARGE ECONOMIES; REFORMULATIONS OF STRATEGIES IN SOME SITUATIONS; EXTENSIONS TO A HIGHER LEVEL OF ABSTRACTION. One have to divide the possible approches in three sub-objectives. The first sub-objective is to detect statements from the theory of Aumann Economies passing the hyper-finite test described before, and to produce a list of such statements; they can be used when taking economical decisions in LARGE economies. The same sub-objective require to also detect statements from the Aumann Theory of continuum economics which do not pass the the hyper-finite test, and to produce a list of such statements; they cannot be used when taking economical decisions for large economies (although they cannot be excluded ab initio for small enough economies). So, the first sub-objective is to produce these two kinds of lists. They will not be, certainely exhaustive ones. They only represent a beginning. The use of countinuum economies allow us to use the topological, differential, integral, variational, etc. calculus, avoiding in such a way the complexity of the discrete calculus when dealing with a huge number of players. So, for the new Theorems related to Aumann economies which will be produced in the future, we have to add at the end the hyper-finite test. Aumann himself, when introduced his notion, gave an example of statement true for continuum economies, but false foar any finite economy, regardless on its magnitude. We will never have 1 Euro for each real number ! Not so-many Euros ! The second sub-objective consists in reconsidering the topics from the second list in the contexts of modal, and intuitionistic logic, and to try to reformulate the statements in order to become valid in this new setting. These new statements can, now, be used when taking economical decisions. The third, and last, sub-objective consist in reconsidering some of the topics from the 3rd list in a more abstarct and general setting of a kind of non-standard analysis in topoi. This part represents mainly a theoretical study.
2.2 The Scientific Importance and Relevance.
The method of ¨hyper-finite test¨ previously described is, at our knowledge, new in Mathematical Economics (and not only). It applies at a various results from the literature, like extensive games, repetitive games, with complete or incomplete information, cooperative or non-cooperative, models of cooperative and non-cooperative bargainings, the core of an economy, both in the perfect and imperfect case, differential games, problems of equilibrium, the theories of producer and of the consumer, the oligopol theory, and so long. Each case has its own personality, becoming different subject of the research project. The hyper-finite test equally applies in continuum mechanics, biology, sociology, problems of climate change and evolutionary biology (evolutionary repetitive games), the content of the proposed research becaming easily an interdisciplinary one. Moreover, one could discover unexpected analogies between apparently completely different arias of research. And, when passing to the higher level of abstraction, in topoi and multivalued (intuitionistic) logic, the things could become even more interesting.
 AUMANN, R., Markets with a Continuum of Traders, Econometrica, 34(1964), no. 1-2, pp. 39-50.
 AUMANN, R., Existence of Competitive Equilibria in Markets with a Continuum of Traders, Econometrica, 34(1966), pp. 1-17.
 AUMANN, R., A Survey of Cooperative Games without Side Payments, in Essays in Math. Ec. In Honour of Oskar Morgenstern, Princeton Univ. Press(1967), pp. 3-27.
 AUMANN, R., HART, S., Handbook of Game Theory (with Economical Applications), vol. 1-3, Elsevier Science Publ.
 AUMANN, R., MASCHLE, R., The Bargaining Set of Cooperative Games, Adv. in Game Theory, Ann. of Math. Studies 52(1964), pp. 443-476.
NISAN,N., ROUGHGARDEN,T., TARDOS,E., VAZIRANI,V. (edt.), Algorithmic Game Theory, Cambridge Univ. Press, 2007.
BLACKBURN,P., et all., Handbook of Modal Logic, Elsevier Sci. Publ., 2007.
 BROWN, D., LOEB, P., The values of Nonstandard Exchange Economies, Israel J.of Math., 25 (1976), pp. 71-86.
 BROWN, D., ROBINSON, A., A Limit Theorem on the Cores of Large Standard Exchange Economies, Proc. Nat. Acad. Sci. USA, 69(1972), no.5, pp. 1258 -1260.
 BROWN, D., ROBINSON, A., Nonstandard Exchange Economies, Econometrica, 43 (1975), no. 1, pp. 41-55.
 CHANG,C.C., KEISLER,H.J., Model Theory, North-Holland, 1992.
 CHICHILNISHY, G.,ZHOU, Y., Smooth Infinite Economies, J.of Math. Ec., 29(1998), pp. 27- 42.
 CONWAY,J.H., On Numbers and Games, Acad. Press, 1976.
 CUTLAND, N., KOPP, E., A Nonstandard Approach to Option Pricing, Math. Fin., 1 (1994), no.4, pp. 1-38.
 FITTING,M.C., Intuitionistic Logic, Model Theory, and Forcing, North Holl., 1992.
 GOODSTEIN,R.L., Recursive Analysis, North Holl., 1961.
 GRADEL,E., et all., Finite Model Theory, Springer, 1998.
 HILDEBRAND, W., SONNENSCHEIN, H., Handbook of Mathematical Economics, vol. 1-4, Elsevier Science Publ.
 JOHNSTONE,P.T., Topos Theory, Acad. Press, 1977.
 KHAN, A., SUN., Y., Nonatomic Games on Loeb Spaces, Proc. Nat. Acad. Sci. USA, 93 (1996), pp. 15518-15521.
 LAMBEK,J., SCOTT,P.J., Introduction to Higher Order Categorial Logic, Cambridge Univ. Press, 1986.
 LURIE,J., Higher Topos Theory, Ann. of Math., Studies, 170(2009).
 MAKKAI,M., REYES,E., First Order Categorial Logic, LNM 611, Springer, 1977.
 MINTS,G., A Short Introduction in Intuitionistic Logic, Kluwer Acad. Publ., 2002.
 MINTA,G., MISKENS,R., (edt.), Games, Logic, and Constructive Sets, Center for the study of language and info, Stanford, CA, 2003.
 OPRIŞ,D., FERRARA, M.,UDRIŞTE, C., Economic Geometric Dynamics, Monographs and Text books 6(2004), Geometry Balkan Press.
 PĂSĂRESCU, A., Nonstandard Analytic Methods in the Study of Analytic Spaces, APPS Monograph Ser. 2(2003), Geometry Balkan Press.
 PĂSĂRESCU, A., PĂSĂRESCU, O., Study on Formal Convergent Series Using Non-standard Analysis, Bull. Sci. Univ. Pitesti 5(2000), pp. 85-103.
 PRASAD, K., Constructive and Classical Models for Results in Economics and Game Theory, Metroeconomica, 55(2004), no. 2&3, pp. 141-154.
 POHLERS,W., Proof Theory, an Introduction, LNM 1407, 1989.
 RASIOVA, SIKORSKI,R., The Mathematics of Metamathematics, Panstwowe Wid. Naukowe, Warshaw, 1963.
 ROBINSON,A., Non-standard Analysis, Princeton Univ. Press, 1996.