Advanced Fixed Point Theory for Economics. Andrew McLennan

Transcription

1 Advanced Fixed Point Theory for Economics Andrew McLennan April 8, 2014

2 Preface Over two decades ago now I wrote a rather long survey of the mathematical theory of fixed points entitled Selected Topics in the Theory of Fixed Points. It had no content that could not be found elsewhere in the mathematical literature, but nonetheless some economists found it useful. Almost as long ago, I began work on the project of turning it into a proper book, and finally that project is coming to fruition. Various events over the years have reinforced my belief that the mathematics presented here will continue to influence the development of theoretical economics, and have intensified my regret about not having completed it sooner. There is a vast literature on this topic, which has influenced me in many ways, and which cannot be described in any useful way here. Even so, I should say something about how the present work stands in relation to three other books on fixed points. Fixed Point Theorems with Applications to Economics and Game Theory by Kim Border (1985) is a complement, not a substitute, explaining various forms of the fixed point principle such as the KKMS theorem and some of the many theorems of Ky Fan, along with the concrete details of how they are actually applied in economic theory. Fixed Point Theory by Dugundji and Granas (2003) is, even more than this book, a comprehensive treatment of the topic. Its fundamental point of view (applications to nonlinear functional analysis) audience (professional mathematicians) and technical base (there is extensive use of algebraic topology) arequitedifferent, butitisstillaworkwithmuchtooffertoeconomics. Particularly notable is the extensive and meticulous information concerning the literature and history of the subject, which is full of affection for the theory and its creators. The book that was, by far, the most useful to me, is The Lefschetz Fixed Point Theorem by Robert Brown (1971). Again, his approach and mine have differences rooted in the nature of our audiences, and the overall objectives, but at their cores the two books are quite similar, in large part because I borrowed a great deal. I would like to thank the many people who, over the years, have commented favorably on Selected Topics. It is a particular pleasure to acknowledge some very detailed and generous written comments by Klaus Ritzberger. This work would not have been possible without the support and affection of my families, both present and past, for which I am forever grateful. i

3 Contents 1 Introduction and Summary The First Fixed Point Theorems Fixing Kakutani s Theorem Essential Sets of Fixed Points Index and Degree Manifolds The Degree The Fixed Point Index Topological Consequences Dynamical Systems I Topological Methods 22 2 Planes, Polyhedra, and Polytopes Affine Subspaces Convex Sets and Cones Polyhedra Polytopes Polyhedral Complexes Graphs Computing Fixed Points The Lemke-Howson Algorithm Implementation and Degeneracy Resolution Using Games to Find Fixed Points Sperner s Lemma The Scarf Algorithm Homotopy Remarks on Computation Topologies on Spaces of Sets Topological Terminology Spaces of Closed and Compact Sets Vietoris Theorem Hausdorff Distance Basic Operations on Subsets ii

4 CONTENTS iii Continuity of Union Continuity of Intersection Singletons Continuity of the Cartesian Product The Action of a Function The Union of the Elements Topologies on Functions and Correspondences Upper and Lower Semicontinuity The Strong Upper Topology The Weak Upper Topology The Homotopy Principle Continuous Functions Metric Space Theory Paracompactness Partitions of Unity Topological Vector Spaces Banach and Hilbert Spaces EmbeddingTheorems Dugundji s Theorem Retracts Kinoshita s Example Retracts Euclidean Neighborhood Retracts Absolute Neighborhood Retracts Absolute Retracts Domination Essential Sets of Fixed Points The Fan-Glicksberg Theorem Convex Valued Correspondences Kinoshita s Theorem Approximation of Correspondences The Approximation Result Extending from the Boundary of a Simplex Extending to All of a Simplicial Complex Completing the Argument II Smooth Methods Differentiable Manifolds Review of Multivariate Calculus Smooth Partitions of Unity

5 CONTENTS Manifolds Smooth Maps Tangent Vectors and Derivatives Submanifolds Tubular Neighborhoods Manifolds with Boundary Classification of Compact 1-Manifolds Sard s Theorem Sets of Measure Zero A Weak Fubini Theorem Sard s Theorem Measure Zero Subsets of Manifolds Genericity of Transversality Degree Theory Orientation Induced Orientation The Degree Composition and Cartesian Product The Fixed Point Index Axioms for an Index on a Single Space Multiple Spaces The Index for Euclidean Spaces Extension by Commutativity Extension by Continuity III Applications and Extensions Topological Consequences Euler, Lefschetz, and Eilenberg-Montgomery The Hopf Theorem More on Maps Between Spheres Invariance of Domain Essential Sets Revisited Vector Fields and their Equilibria Euclidean Dynamical Systems Dynamics on a Manifold The Vector Field Index Dynamic Stability The Converse Lyapunov Problem A Necessary Condition for Stability

6 Chapter 1 Introduction and Summary The Brouwer fixed point theorem states that if C is a nonempty compact convex subset of a Euclidean space and f : C C is continuous, then f has a fixed point, which is to say that there is an x C such that f(x ) = x. The proof of this by Brouwer (1912) was one of the major events in the history of topology. Since then the study of such results, and the methods used to prove them, has flourished, undergoing radical transformations, becoming increasingly general and sophisticated, and extending its influence to diverse areas of mathematics. Around 1950, most notably through the work of Nash (1950, 1951) on noncooperative games, and the work of Arrow and Debreu (1954) on general equilibrium theory, it emerged that in economists most basic and general models, equilibria are fixed points. The most obvious consequence of this is that fixed point theorems provide proofs that these models are not vacuous. But fixed point theory also informs our understanding of many other issues such as comparative statics, robustness under perturbations, stability of equilibria with respect to dynamic adjustment processes, and the algorithmics and complexity of equilibrium computation. In particular, since the mid 1970 s the theory of games has been strongly influenced by refinement concepts defined largely in terms of robustness with respect to certain types of perturbations. As the range and sophistication of economic modelling has increased, more advanced mathematical tools have become relevant. Unfortunately, the mathematical literature on fixed points is largely inaccessible to economists, because it relies heavily on homology. This subject is part of the standard graduate school curriculum for mathematicians, but for outsiders it is difficult to penetrate, due to its abstract nature and the amount of material that must be absorbed at the beginning before the structure, nature, and goals of the theory begin to come into view. Many researchers in economics learn advanced topics in mathematics as a side product of their research, but unlike infinite dimensional analysis or continuous time stochastic processes, algebraic topology will not gradually achieve popularity among economic theorists through slow diffusion. Consequently economists have been, in effect, shielded from some of the mathematics that is most relevant to their discipline. This monograph presents an exposition of advanced material from the theory of fixed points that is, in several ways, suitable for graduate students and researchers in mathematical economics and related fields. In part the fit with the intended 2

7 1.1. THE FIRST FIXED POINT THEOREMS 3 audience is a matter of coverage. Economic models always involve domains that are convex, or at least contractible, so there is little coverage here of topics that only become interesting when the underlying space is more complicated. For the settings of interest, the treatment is comprehensive and maximally general, with issues related to correspondences always in the foreground. The project was originally motivated by a desire to understand the existence proofs in the literature on refinements of Nash equilibrium as applications of preexisting mathematics, and the continuing influence of this will be evident. The mathematical prerequisites are within the common background of advanced students and researchers in theoretical economics. Specifically, in addition to multivariate calculus and linear algebra, we assume that the reader is familiar with basic aspects of point-set topology. What we need from topics that may be less familiar to some (e.g., simplicial complexes, infinite dimensional linear spaces, the theory of retracts) will be explained in a self-contained manner. There will be no use of homological methods. The avoidance of homology is a practical necessity, but it can also be seen as a feature rather than a bug. In general, mathematical understanding is enhanced when brute calculations are replaced by logical reasoning based on conceptually meaningful definitions. To say that homology is a calculational machine is a bit simplistic, but it does have that potential in certain contexts. Avoiding it commits us to work with notions that have more direct and intuitive geometric content. (Admittedly there is a slight loss of generality, because there are acyclic that is, homologically trivial spaces that are not contractible, but this is unimportant because such spaces are not found in nature. ) Thus our treatment of fixed point theory can be seen as a mature exposition that presents the theory in a natural and logical manner. In the remainder of this chapter we give a broad overview of the contents of the book. Unlike many subjects in mathematics, it is possible to understand the statements of many of the main results with much less preparation than is required to understand the proofs. Needless to say, as usual, not bothering to study the proofs has many dangers. In addition, the material in this book is, of course, closely related to various topics in theoretical economics, and in many ways quite useful preparation for further study and research. 1.1 The First Fixed Point Theorems A fixed point of a function f : X X is an element x X such that f(x ) = x. If X is a topological space, it is said to have the fixed point property if every continuous function from X to itself has a fixed point. The first and most famous result in our subject is Brouwer s fixed point theorem: Theorem (Brouwer (1912)). If C R m is nonempty, compact, and convex, then it has the fixed point property. Chapter 3 presents various proofs of this result. Although some are fairly brief, none of them can be described as truly elementary. In general, proofs of Brouwer s

8 4 CHAPTER 1. INTRODUCTION AND SUMMARY theorem are closely related to algorithmic procedures for finding approximate fixed points. Chapter 3 discusses the best known general algorithm due to Scarf, a new algorithm due to the author and Rabee Tourky, and homotopy methods, which are the most popular in practice, but require differentiability. The last decade has seen major breakthroughs in computer science concerning the computational complexity of computing fixed points, with particular reference to (seemingly) simple games and general equilibrium models. These developments are sketched briefly in Section 3.7. In economics and game theory fixed point theorems are most commonly used to prove that a model has at least one equilibrium, where an equilibrium is a vector of endogenous variable for the model with the property that each individual agent s predicted behavior is rational, or utility maximizing, if that agent regards all the other endogenous variables as fixed. In economics it is natural, and in game theory unavoidable, to consider models in which an agent might have more than one rational choice. Our first generalization of Brouwer s theorem addresses this concern. If X and Y are sets, a correspondence F : X Y is a function from X to the nonempty subsets of Y. (Ontherareoccasions when theyarise, we use theterm set valued mapping forafunction fromx toall thesubsets ofy, including theempty set.) We will tend to regard a function as a special type of correspondence, both intuitively and in the technical sense that we will frequently blur the distinction between a function f : X Y and the associated correspondence x {f(x)}. If Y is a topological space, F is compact valued if, for all x X, F(x) is compact. Similarly, if Y is a subset of a vector space, then F is convex valued if each F(x) is convex. The extension of Brouwer s theorem to correspondences requires a notion of continuity for correspondences. If X and Y are topological spaces, a correspondence F : X Y is upper semicontinuous if it is compact valued and, for each x 0 X and each neighborhood V Y of F(x 0 ), there is a neighborhood U X of x 0 such that F(x) V for all x U. It turns out that if X and Y are metric spaces and Y is compact, then F is upper semicontinuous if and only if its graph Gr(F) := {(x,y) X Y : y F(x)} is closed. (Proving this is a suitable exercise, if you are so inclined.) Thinking of upper semicontinuity as a matter of the graph being closed is quite natural, and in economics this condition is commonly taken as definition, as in Debreu (1959). In Chapter 5 we will develop a topology on the space of nonempty compact subsets of Y such that F is upper semicontinuous if and only if it is a continuous function relative to this topology. A fixed point of a correspondence F : X X is a point x X such that x F(x ). Kakutani (1941) was motivated to prove the following theorem by the desire to provide a simple approach to the von Neumann (1928) minimax theorem, which is a fundamental result of game theory. This is the fixed point theorem that is most commonly applied in economic analysis.

9 1.2. FIXING KAKUTANI S THEOREM 5 Theorem (Kakutani s Fixed Point Theorem). If C R m is nonempty, compact, and convex, and F : C C is an upper semicontinuous convex valued correspondence, then F has a fixed point. 1.2 Fixing Kakutani s Theorem Mathematicians strive to craft theorems that maximize the strength of the conclusions while minimizing the strength of the assumptions. One reason for this is obvious: a stronger theorem is a more useful theorem. More important, however, is the desire to attain a proper understanding of the principle the theorem expresses, and to achieve an expression of this principle that is unencumbered by useless clutter. When a theorem that is too weak is proved using methods that happen to work there is a strong suspicion that attempts to improve the theorem will uncover important new concepts. In the case of Brouwer s theorem the conclusion, that the space has the fixed point property, is a purely topological assertion. The assumption that the space is convex, and in Kakutani s theorem the assumption that the correspondence s values are convex, are geometric conditions that seems out of character and altogether too strong. Suitable generalizations were developed after World War II. A homotopy is a continuous function h : X [0,1] Y where X and Y are topological spaces. It is psychologically natural to think of the second variable in the domain as representing time, and we let h t := h(,t) : X Y denote the function at time t, so that h is a process that continuously deforms a function h 0 into h 1. Another intuitive picture is that h is a continuous path in the space C(X,Y) of continuous function from X to Y. As we will see in Chapter 5, this intuition can be made completely precise: when X and Y are metric spaces and X is compact, there is a topology on C(X,Y) such that a continuous path h : [0,1] C(X,Y) is the same thing as a homotopy. We say that two functions f,g : X Y are homotopic if there is a homotopy h with h 0 = f and h 1 = g. This is easily seen to be an equivalence relation: symmetry and reflexivity are obvious, and to establish transitivity we observe that if e is homotopic to f and f is homotopic to g, then there is a homotopy between e and g that follows a homotopy between e and f at twice its original speed, then follows a homotopy between f and g at double the pace. The equivalence classes are called homotopy classes. AspaceX iscontractibleiftheidentityfunctionid X ishomotopictoaconstant function. That is, there is a homotopy c : X [0,1] X such that c 0 = Id X and c 1 (X) is a singleton; such a homotopy is called a contraction. Convex sets are contractible. More generally, a subset X of a vector space is star-shaped if there is x X (the star) such that X contains the line segment {(1 t)x+tx : 0 t 1} between each x X and x. If X is star-shaped, there is a contraction (x,t) (1 t)x+tx.

10 6 CHAPTER 1. INTRODUCTION AND SUMMARY It seems natural to guess that a nonempty compact contractible space has the fixedpointproperty. Whether thisisthecasewasanopenproblemforseveral years, but it turns out to be false. In Chapter 7 we will see an example due to Kinoshita (1953) of a nonempty compact contractible subset of R 3 that does not have the fixed point property. Fixed point theory requires some additional ingredient. If X is a topological space, a subset A X is a retract if there is a continuous function r : X A with r(a) = a for all a A. Here we tend to think of X as a simple space, and the hope is that although A might seem to be more complex, or perhaps crumpled up, it nonetheless inherits enough of the simplicity of X. A particularly important manifestation of this is that if r : X A is a retraction and X has the fixed point property, then so does A, because if f : A A is continuous, then so is f r : X A X, so f r has a fixed point, and this fixed point necessarily lies in A and is consequently a fixed point of f. Also, a retract of a contractible space is contractible because if c : X [0,1] X is a contraction of X and r : X A X is a retraction, then (a,t) r(c(a,t)) is a contraction of A. A set A R m is a Euclidean neighborhood retract (ENR) if there is an open superset U R m of A and a retraction r : U A. If X and Y are metric spaces, anembedding of X in Y is a functione : X Y that is a homeomorphism between X and e(x). That is, e is a continuous injection 1 whose inverse is also continuous when e(x) has the subspace topology inherited from Y. An absolute neighborhood retract (ANR) is a separable 2 metric space X such that whenever Y is a separable metric space and e : X Y is an embedding, there is an open superset U Y of e(x) and a retraction r : U e(x). This definition probably seems completely unexpected, and it s difficult to get any feeling for it right away. In Chapter 7 we ll see that ANR s have a simple characterization, and that many of the types of spaces that come up most naturally are ANR s, so this condition is quite a bit less demanding than one might guess at first sight. In particular, it will turn out that every ENR is an ANR, so that being an ENR is an intrinsic property insofar as it depends on the topology of the space and not on how the space is embedded in a Euclidean space. An absolute retract (AR) is a separable metric space X such that whenever Y is a separable metric space and e : X Y is an embedding, there is a retraction r : Y e(x). In Chapter 7 we will prove that an ANR is an AR if and only if it is contractible. Theorem If C is a nonempty compact AR and F : C C is an upper semicontinuous contractible valued correspondence, then F has a fixed point. An important point is that the values of F are not required to be ANR s. 1 We will usually use the terms injective rather than one-to-one, surjective rather than onto, and bijective to indicate that a function is both injective and surjective. An injection is an injective function, a surjection is a surjective function, and a bijection is a bijective function. 2 A metric space is separable if it has a countable dense subset.

11 1.3. ESSENTIAL SETS OF FIXED POINTS 7 For practical purposes this is the maximally general topological fixed point theorem, but for mathematicians there is an additional refinement. There is a concept called acyclicity that is defined in terms of the concepts of algebraic topology. A contractible set is necessarily acyclic, but there are acyclic spaces (including compact ones) that are not contractible. The famous Eilenberg-Montgomery fixed point theorem is: Theorem (Eilenberg and Montgomery (1946)). If C is a nonempty compact AR and F : C C is an upper semicontinuous acyclic valued correspondence, then F has a fixed point. 1.3 Essential Sets of Fixed Points It might seem like we have already reached a satisfactory and fitting resolution of The Fixed Point Problem, but actually (both in pure mathematics and in economics) this is just the beginning. You see, fixed points come in different flavors s t 1 Figure 1.1 The figure above shows a function f : [0,1] [0,1] with two fixed points, s and t. If we perturb the function slightly by adding a small positive constant, s disappears in the sense that the perturbed function does not have a fixed point anywhere near s, but a function close to f has a fixed point near t. More precisely, if X is a topological space and f : X X is continuous, a fixed point x of f is essential if, for any neighborhood U of x, there is a neighborhood V of the graph of f such that any continuous f : X X whose graph is contained in V has a fixed point in U. If a fixed point is not essential, then we say that it is inessential. These concepts were introduced by Fort (1950). There need not be an essential fixed point. The function shown in Figure 1.2

12 8 CHAPTER 1. INTRODUCTION AND SUMMARY has an interval of fixed points. If we shift the function down, there will be a fixed point near the lower endpoint of this interval, and if we shift the function up there will be a fixed point near the upper endpoint. This example suggests that we might do better to work with sets of fixed points. A set S of fixed points of a function f : X X is essential if it is closed, it has a neighborhood that contains no other fixed points, and for any neighborhood U of S, there is a neighborhood V of the graph of f such that any continuous f : X X whose graph is contained in V has a fixed point in U. The problem with this concept is that large connected sets are not of much use. For example, if X is compact and has the fixed point property, then the set of all fixed points of f is essential. It seems that we should really be interested in sets of fixed points that are either essential and connected 3 or essential and minimal in the sense of not having a proper subset that is also essential Figure 1.2 In Chapter 8 we will show that any essential set of fixed points contains a minimal essential set, and that minimal essential sets are connected. The theory of refinements of Nash equilibrium (e.g., Selten (1975); Myerson (1978); Kreps and Wilson (1982); Kohlberg and Mertens (1986); Mertens (1989, 1991); Govindan and Wilson (2008)) has many concepts that amount to a weakening of the notion of essential set, insofar as the set is required to be robust with respect to only certain types of perturbations of the function or correspondence. In particular, Jiang (1963) pioneered the application of the concept to game theory, defining an essential!nash equilibrium and an essential set of Nash equilibria in terms of robustness with respect to perturbations of the best response correspondence induced by perturbations of the payoffs. The mathematical foundations of such 3 We recall that a subset S of a topological space X is connected if there do not exist two disjoint open sets U 1 and U 2 with S U 1 S U 2 and S U 1 U 2.

13 1.4. INDEX AND DEGREE 9 concepts are treated in Section Index and Degree There are different types of essential fixed points. Figure 1.3 shows a function with three fixed points. At two of them the function starts above the diagonal and goes below it as one goes from left to right, and at the third it is the other way around. For any k it is easy to imagine a function with k fixed points of the first type and k 1 fixed points of the second type. This phenomenon generalizes to higher dimensions. Let D m = {x R m : x 1} and S m 1 = {x R m : x = 1} be the m-dimensional unit disk and the (m 1)-dimensional unit sphere, and suppose that f : D m D m is a C function. In the best behaved case each fixed point x is in the interior D m \ S m 1 of the disk and regular, which means that Id R m Df(x ) is nonsingular, where Df(x ) : R m R m is the derivative of f at x. We define the index of x to be 1 if the determinant of Id R m Df(x ) is positive and 1 if this determinant is negative. We will see that there is always one more fixed point of index 1 than there are fixed points of index 1, which is to say that the sum of the indices is 1. What about fixed points on the boundary of the disk, or fixed points that aren t regular, or nontrivial connected sets of fixed points? What about correspondences? What happens if the domain is a possibly infinite dimensional ANR? The most challenging and significant aspect of our work will be the development of an axiomatic theory of the index that is general enough to encompass all these possibilities. The work proceeds through several stages, and we describe them in some detail now Figure 1.3

14 10 CHAPTER 1. INTRODUCTION AND SUMMARY Manifolds First of all, it makes sense to expand our perspective a bit. An m-dimensional manifold is a topological space that resembles R m in a neighborhood of each of its points. More precisely, for each p M there is an open U R m and an embedding ϕ : U M whose image is open and contains p. Such a ϕ is a parameterization and its inverse is a coordinate chart. The most obvious examples are R m itself and S m. If, in addition, N is an n-dimensional manifold, then M N is an (m+n)- dimensional manifold. Thus the torus S 1 S 1 is a manifold, and this is just the most easily visualized member of a large class of examples. An open subset of an m-dimensional manifold is an m-dimensional manifold. A 0-dimensional manifold is just a set with the discrete topology. The empty set is a manifold of any dimension, including negative dimensions. Of course these special cases are trivial, but they come up in important contexts. A collection {ϕ i : U i M} i I of parameterizations is an atlas if its images cover M. The composition ϕ 1 j ϕ i (with the obvious domain of definition) is called a transition function. If, for some 1 r, all the transition functions are C r functions, then the atlas is a C r atlas. An m-dimensional C r manifold is an m-dimensional manifold together with a C r atlas. The basic concepts of differential and integral calculus extend to this setting, leading to a vast range of mathematics. In our formalities we will always assume that M is a subset of a Euclidean space R k called the ambient space, and that the parameterizations ϕ i and the coordinate charts ϕ 1 i are C r functions. This is a bit unprincipled for example, physicists see only the universe, and their discourse is more disciplined if it does not refer to some hypothetical ambient space but this maneuver is justified by embedding theorems due to Whitney that show that it does not entail any serious loss of generality. The advantages for us are that this approach bypasses certain technical pathologies while allowing for simplified definitions, and in many settings the ambient space will prove quite handy. For example, a function f : M N (where N is now contained in some R l ) is C r for our purposes if it is C r in the standard sense: for any S R k a function h : S R l is C r, by definition, if there is an open W R k containing S and a C r function H : W R l such that h = H S. Having an ambient space around makes it relatively easy to establish the basic objects and facts of differential calculus. Suppose that ϕ i : U i M is a C r parameterization. If x U i and ϕ i (x) = p, the tangent space of M at p, which we denote by T p M, is the image of Dϕ i (x). This is an m-dimensional linear subspace of R k. If f : M N is C r, the derivative Df(p) : T p M T f(p) N of f at p is the restriction to T p M of the derivative DF(p) of any C r function F : W R l defined on an open W R k containing M whose restriction to M is f. (In Chapter 10 we will show that the choice of F doesn t matter.) The chain rule holds: if, in addition, P is a p-dimensional C r manifold and g : N P is a C r function, then g f is C r and D(g f)(p) = Dg(f(p)) Df(p) : T p M T g(f(p)) P.

15 1.4. INDEX AND DEGREE 11 The inverse and implicit function theorems have important generalizations. The point p is a regular point of f if the image of Df(p) is all of T f(p) N. We say that f : M N is a C r diffeomorphism if m = n, f is a bijection, and both f and f 1 are C r. The generalized inverse function theorem asserts that if m = n, f : M N is C r, and p is a regular point of f, then there is an open U M containing p such that f(u) is an open subset of N and f U : U f(u) is a C r diffeomorphism. If 0 s m, a set S R k is an s-dimensional C r submanifold of M if it is an s-dimensional C r submanifold that happens to be contained in M. We say that q N is a regular value of f if every p f 1 (q) is a regular point. The generalized implicit function theorem, which is known as the regular value theorem, asserts that if q is a regular value of f, then f 1 (q) is an (m n)-dimensional C r submanifold of M The Degree The degree is closely related to the fixed point index, but it has its own theory, which has independent interest and significance. The approach we take here is to work with the degree up to the point where its theory is more or less complete, then translate what we have learned into the language of the fixed point index. We now need to introduce the concept of orientation. Two ordered bases v 1,...,v m and w 1,...,w m of an m-dimensional vector space have the same orientation if the determinant of the linear transformation taking each v i to w i is positive. It is easy to see that this is an equivalence relation with two equivalence classes. An oriented vector space is a finite dimensional vector space with a designated orientation whose elements are said to be positively oriented. If V and W are m-dimensional oriented vector spaces, a nonsingular linear transformation L : V W is orientation preserving if it maps positively oriented ordered bases of V to positively oriented ordered bases of W, and otherwise it is orientation reversing. For an intuitive appreciation of this concept just look in a mirror: the linear map taking each point in the actual world to its position as seen in the mirror is orientation reversing, with right shoes turning into left shoes and such. In our discussion of degree theory nothing is lost by working with C objects rather than C r objects for general r, and smooth will be a synonym for C. An orientation for a smooth manifold M is a continuous specification of an orientation of each of the tangent spaces T p M. We say that M is orientable if it has an orientation; the most famous examples of unorientable manifolds are the Möbius strip and the Klein bottle. (From a mathematical point of view 2-dimensional projective space is perhaps more fundamental, but it is difficult to visualize.) An oriented manifold is a manifold together with a designated orientation. If M and N are oriented smooth manifolds of the same dimension, f : M N is a smooth map, and p is a regular point of f, we say that f is orientation preserving at p if Df(p) : T p M T f(p) N is orientation preserving, and otherwise f is orientation reversing at p. If q is a regular value of f and f 1 (q) is finite, then the degree of f over q, denoted by deg q (f), is the number of points in f 1 (q) at which f is orientation preserving minus the number of points in f 1 (q) at which f is orientation reversing.

16 12 CHAPTER 1. INTRODUCTION AND SUMMARY We need to extend the degree to situations in which the target point q is not a regular value of f, and to functions that are merely continuous. Instead of being abletodefinethedegreedirectly, aswedidabove, we willneedtoproceedindirectly, showing that the generalized degree is determined by certain of its properties, which we treat as axioms. The first step is to extend the concept, giving it a local character. For a compact C M let C = C (M \C) be the topological boundary of C, and let intc = C\ C be its interior. A smooth function f : C N with compact domain C M is said to be smoothly degree admissible over q N if f 1 (q) C = and q is a regular value of f. As above, for such a pair (f,q) we define deg q (f) to be the number of p f 1 (q) at which f is orientation preserving minus the number of p f 1 (q) at which f is orientation reversing. Note that deg q (f) = deg q (f C ) whenever C is a compact subset of C and f 1 (q) has an empty intersection with the closure of C \ C. Also, if C = C 1 C 2 where C 1 and C 2 are compact and disjoint, then deg q (f) = deg q (f C 1 )+deg q (f C 2 ). From the point of view of topology, what makes the degree important is its invariance under homotopy. If C M is compact, a smooth homotopy h : C [0,1] N is smoothly degree admissible over q if h 1 (q) ( C [0,1]) = and q is a regular value of h 0 and h 1. In this circumstance deg q (h 0) = deg q (h 1). ( ) Figure 1.4 illustrates the intuitive character of the proof t = 0 t = 1 Figure 1.4

17 1.4. INDEX AND DEGREE 13 The notion of an m-dimensional manifold with boundary is a generalization of the manifold concept in which each point in the space has a neighborhood that is homeomorphic to an open subset of the closed half space {x R m : x 1 0}. Aside from the half space itself, the closed disk D m = {x R m : x 1} is perhaps the most obvious example, but for us the most important example is M [0,1] where M is an (m 1)-dimensional manifold without boundary. Note that any m- dimensional manifold without boundary is (automatically and trivially) a manifold with boundary. All elements of our discussion of manifolds generalize to this setting. In particular, the generalization of the regular value theorem states that if M is an m-dimensional smooth manifold with boundary, N is an n-dimensional (boundaryless) manifold, f : M N is smooth, and q N is a regular value of both f and the restriction of f to the boundary of M, then f 1 (q) is an (m n)-dimensional manifold with boundary, its boundary is its intersection with the boundary of M, and at each point in this intersection the tangent space of f 1 (q) is not contained in the tangent space of the boundary of M. In particular, if the dimension of M is the dimension of N plus one, then f 1 (q) is a 1-dimensional manifold with boundary. If, in addition, f 1 (q) is compact, then it has finitely many connected components. Suppose now that h : C [0,1] N is smoothly degree admissible over q, and that q is a regular value of h. The consequences of applying the regular value theorem to the restriction of h to intc [0,1] are as shown in Figure 1.4: h 1 (q) is a 1-dimensional manifold with boundary, its boundary is its intersection with C {0,1}, and h 1 (q) is not tangent to C {0,1} at any point in this intersection. In addition h 1 (q) is compact, so it has finitely many connected components, each of which is compact. A connected compact 1-dimensional manifold with boundary is either a circle or a line segment. (It will turn out that this obvious fact is surprisingly difficult to prove!) Thus each component of h 1 (q) is either a circle or a line segment connecting two points in its boundary. If a line segment connects two points in C {0}, say (p,0) and (p,0), then it turns out that h 0 is orientation preserving at p if and only if it is orientation reversing at p. Similarly, if a line segment connects two points (p,1) and (p,1) in C {1}, then h 1 is orientation preserving at p if and only if it is orientation reversing at p. On the other hand, if a line segment connects a point (p 0,0) in C {0} to a point (p 1,1) in C {1}, then h 0 is orientation preserving at p 0 if and only if h 1 is orientation preserving at p 1. Equation ( ) is obtained by summing these facts over the various components of h 1 (q). This completes our discussion of the proof of ( ) except for one detail: if h : C [0,1] N is a smooth homotopy that is smoothly degree admissible over q, q is not necessarily a regular value of h. Nevertheless, Sard s theorem (which is the subject of Chapter 11, and a crucial ingredient of our entire approach) implies that h has regular values in any neighborhood of q, and it is also the case that deg q (h 0) = deg q (h 0) and deg q (h 1) = deg q (h 1) when q is sufficiently close to q. It turns out that the smooth degree is completely characterized by the properties we have seen. That is, if D (M,N) is the set of pairs (f,q) in which f : C N is smoothly degree admissible over q, then (f,q) deg q (f) is the unique function from D (M,N) to Z satisfying: ( 1) deg q (f) = 1 for all (f,q) D (M,N) such that f 1 (q) is a singleton {p}

18 14 CHAPTER 1. INTRODUCTION AND SUMMARY and f is orientation preserving at p. ( 2) deg q (f) = r i=1 deg q (f Ci ) whenever (f,q) D (M,N), the domain of f is C, and C 1,...,C r are pairwise disjoint compact subsets of C such that f 1 (q) intc 1... intc r. ( 3) deg q (h 0) = deg q (h 1) whenever C M is compact and the homotopy h : C [0,1] N is smoothly degree admissible over q. We note two additional properties of the smooth degree. The first is that if, in addition to M and N, M and N are m -dimensional smooth functions, (f,q) D (M,N), and (f,q ) D (M,N ), then (f f,(q,q )) D (M M,N N ) and deg (q,q ) (f f ) = deg q (f) deg q (f ). Since (f f ) 1 (q,q ) = f 1 (q) f 1 (q ), this boils down to a consequence of elementary facts about determinants: if (p,p ) (f f ) 1 (q,q ), then f f is orientation preserving at (p,p ) if and only if f and f are either both orientation preserving or both orientation reversing at p and p respectively. The second property is a strong form of continuity. A continuous function f : C N with compact domain C M is degree admissible over q N if f 1 (q) C =. If this is the case, then there is a neighborhood U C N of the graph of f and a neighborhood V N \f( C) of q such that deg q (f ) = deg q (f ) whenever f,f : C N are smooth functions whose graphs are contained in U, q,q V, q is a regular value of f, and q is a regular value of q. We can now define deg q (f) to be the common value of deg q (f ) for such pairs (f,q ). Let D(M,N) be the set of pairs (f,q) in which f : C N is a continuous function with compact domain C M that is degree admissible over q N. The fully general form of degree theory asserts that (f,q) deg q (f) is the unique function from D(M,N) to Z such that: (D1) deg q (f) = 1 for all (f,q) D(M,N) such that f is smooth, f 1 (q) is a singleton {p}, and f is orientation preserving at p. (D2) deg q (f) = r i=1 deg q(f Ci ) whenever (f,q) D(M,N), the domain of f is C, and C 1,...,C r are pairwise disjoint compact subsets of U such that f 1 (q) C 1... C r \( C 1... C r ). (D3) If (f,q) D(M,N) and C is the domain of f, then there is a neighborhood U C N of the graph of f and a neighborhood V N \f( C) of q such that deg q (f ) = deg q (f ) whenever f,f : C N are continuous functions whose graphsarecontained in U and q,q V.

19 1.4. INDEX AND DEGREE The Fixed Point Index Although the degree can be applied to continuous functions, and even to convex valued correspondences, it is restricted to finite dimensional manifolds. For such spaces the fixed point index is merely a reformulation of the degree. Its application to general equilibrium theory was initiated by Dierker (1972), and it figures in the analysis of the Lemke-Howson algorithm of Shapley (1974). There is also a third variant of the underlying principle, for vector fields, that is developed in Chapter 15, and which is related to the theory of dynamical systems. Hofbauer (1990) applied the vector field index to dynamic issues in evolutionary stability, and Ritzberger (1994) applies it systematically to normal form game theory. However, it turns out that the fixed point index can be generalized much further, due to the fact that, when we are discussing fixed points, the domain and the range are the same. The general index is developed in three main stages. In order to encompass these stages in a single system of terminology and notation we take a rather abstract approach. Fix a metric space X. An index admissible correspondence for X is an upper semicontinuous correspondence F : C X, where C X is compact, that has no fixed points in C. An index base for X is a set I of index admissible correspondences such that: (a) f I whenever C X is compact and f : C X is an index admissible continuous function; (b) F D I whenever F : C X is an element of I, D C is compact, and F D is index admissible. Definition Let I be an index base for X. An index for I is a function Λ X : I Z satisfying: (I1) (Normalization) If c : C X is a constant function whose value is an element of intc, then Λ X (c) = 1. (I2) (Additivity) If F : C X is an element of I, C 1,...,C r are pairwise disjoint compact subsets of C, and FP(F) intc 1... intc r, then Λ X (F) = i Λ X (F Ci ). (I3) (Continuity) For each element F : C X of I there is a neighborhood U C X of the graph of F such that Λ X (ˆF) = Λ X (F) for every ˆF I whose graph is contained in U. For each m = 0,1,2,... an index base for R m is given by letting I m be the set of index admissible continuous functions f : C R m. Of course (I1)-(I3) parallel (D1)-(D3), and it is not hard to show that there is a unique index Λ R m for I m given by Λ R m(f) = deg 0 (Id C f). We now extend our framework to encompass multiple spaces. An index scope S consists of a class of metric spaces S S and an index base I S (X) for each X S S such that

20 16 CHAPTER 1. INTRODUCTION AND SUMMARY (a) S S contains X X whenever X,X S S ; (b) F F I S (X X ) whenever X,X S S, F I S (X), and F I S (X ). These conditions are imposed in order to express a property of the index that is inherited from the multiplicative property of the degree for cartesian products. The index also has an additional property that has no analogue in degree theory. Suppose that C R m and C R m are compact, g : C C and g : C C are continuous, and g g and g g are index admissible. Then Λ R m( g g) = Λ R m(g g). When g and g are smooth and the fixed points in question are regular, this boils down to a highly nontrivial fact of linear algebra (Proposition ) that was unknown prior to the development of this aspect of index theory. This property turns out to be the key to moving the index up to a much higher level of generality, but before we can explain this we need to extend the setup a bit, allowing for the possibility that the images of g and g are not contained in C and C, but that there are compact sets D C and D C with g(d) C and g( D) C that contain the relevant sets of fixed points. Definition A commutativity configuration is a tuple (X,C,D,g, ˆX,Ĉ, ˆD,ĝ) where X and ˆX are metric spaces and: (a) D C X, ˆD Ĉ ˆX, and C, Ĉ, D, and ˆD are compact; (b) g C(C, ˆX) and ĝ C(Ĉ,X) with g(d) intĉ and ĝ(ˆd) intc; (c) ĝ g D and g ĝ ˆD are index admissible; (d) g(fp(ĝ g D )) = FP(g ĝ ˆD ). After all these preparations we can finally describe the heart of the matter. Definition An index for an index scope S is a specification of an index Λ X for each X S S such that: (I4) (Commutativity) If (X,C,D,g, ˆX,Ĉ, ˆD,ĝ) is a commutativity configuration with X, ˆX S S, (D,ĝ g D ) I S (X), and (ˆD,g ĝ ˆD) I S ( ˆX), then The index is said to be multiplicative if: Λ X (ĝ g D ) = Λ ˆX (g ĝ ˆD ). (M) (Multiplication) If X,X S S, F I S (X), and F I S (X ), then Λ X X (F F ) = Λ X (F) Λ X(F ).