Subject: toplogy question From: Amit Kumar Gandhi Date: Tue, 25 Oct 2005 13:12:58 -0500 I am a graduate student in Economics at the University of Chicago, and my research has produced an algebraic topological problem of sorts that has had me stuck for a while The underlying Economic problem is a very foundational problem in the theory of games and Nash equilibrium and concerns the new topic of the "learnability" of equilibrium. Its resolution would be a highly publishable contribution. I have provided a description of the problem below, and I was wondering if you would be willing to post it on your topology discussion board - I am in search of a topologist who might be interested in collaborating on the problem. Thanks alot for your help, Amit Below is a description of a fixed point problem I am facing. There is a simple game theoretic intepretation of things that I can also give, but I will just describe the mathematical construction. I can show the proposition is true in 2 dimensions using a geometric argument, but I am searching for a proof of the proposition for the general case. First let me just recall the definition of a "strictly quasiconcave" real valued function. Let A be a convex subset of a Euclidean space. We say a function f is "strictly quasiconcave" if for any x,y in A and any 0 < \theta < 1, f(\theta*x + (1-\theta)*y) > min{f(x),f(y)}. Thus strict quasiconcavity captures the idea of a real valued f being "single humped", or unimodal over the domain. With this definition in mind, I can explain how my set is constructed. For i = 1,...,n, let A_i be a nonempty, closed, bounded rectangle of a Euclidean space. It would suffice for present purposes to let each A_i be a closed interval of the real line. Furthermore, for each i, let f_i be a real valued continuous function over A = A_1 x...x A_n. Let me establish the following notation. Consider a point a in A. Then a = (a_1,...,a_n) = (a_i, a_-i). Thus for x in A_i, the point (x, a_-i) = (a_1,...,a_(i-1), x, a_(i+1),...,a_n). Now let us make one further assumption about the functions f_i. For each i, and for any a_-i, f_i( . , a_-i) is a strictly quasiconcave function over A_i. For each i, we can define the relation >i over A : we say that x >i y if x_-i = y_-i and f_i(x) > f_i(y). Now for any a in A, define the set: B_i(a) = {a} UNION {x in A : x >i a}. By the strict quasiconcavity assumption, it follows that B_i(a) is a nonempty convex subset of A. This is not hard to show just using the definitions, and is an important property. To help visualize things, since we are letting each A_i be a nonempty closed interval of the reals, then B_i(a) is always a line segment going in the direction of one of ith coordinate axis with the point "a" as the closed endpoint, the other endpoint being open. Now finally I can describe how my set of interest is constructed. The proposition of interest is that this set has a nonzero Euler characteristic, or more generally that this set is contractible. I will define the set inductively Pick any arbitrary a in A. Let S1 = {a}. Now we define S^m for m > 2 inductively : S^m = { x in A : x in B_i(y) for some y in S^(m-1) }, where i = (m Mod n) + 1. So i goes from 1,2,3,...,n back to 1,2,3,...,n and so forth. Moreover each S^m is contained in S^(m+1) Let S_infinity = infinite union over S^m for m=0,1,2,..., etc, and let S_bar = closure of S_infinity. S_bar is the set whose Euler characteristic I wish to compute. It is closed and bounded and easily shown to be connected. Moreover S_bar is closed under each set-valued B_i function, i.e., for any x in S_bar, B_i(x) is a subset of S_bar for any i. Actually a stronger statement holds true : for any i we have that B_i(S_bar)=S_bar. This closure of S_bar under the B_i functions implies the following - S_bar has the property that for any two points in the set that differ in at most one coordinate, the line segment joining the two points is also in S_bar. I have a continuous map b from S_bar into S_bar that is homotopic to the identity. I wish to show that b has a fixed point. From what I understand, this amounts to showing that S_bar has nonzero Euler characteristic. Anyways - this is my total situation - and I have been working on it figuring it for a while. Any insights or questions - please send them along. Amit