# Andy McLennan's Research Papers

### [A.] "A Geometric Vietoris-Begle Theorem, with an Application to Convex Subsets of Riesz Spaces"

Abstract: We show that a surjective map between compact connected ANR's is a homotopy equivalence if the fibers are contractible and either the domain is simply connected or the fibers are AR's. This is a geometric analogue of the Vietoris-Begle theorem. We use it to show that if $L$ is a Hausdorff locally convex Riesz space, $x \in L$, the function $u_x : y \mapsto x \vee y$ is continuous, and $C \subset L$ is compact, convex, and metrizable, then $u_x|_C$ is a homotopy equivalence, so $u_x(C)$ is a compact AR.

### [B.] "The Morse-Sard Theorem Demystified"

Abstract: We provide a result that is stronger than the Morse-Sard theorem in its classic form. The proof is centered on an argument of Dubovitskii (1967) that clearly illuminates the driving force of the result. It does not require advanced background, and its demands on the reader are comparable to those of Milnor's (1965) proof of the $C^\infty$ case of the classic version. Applications of the Morse-Sard theorem in economic theory are explained, and Harsanyi's (1973) application to noncooperative game theory is proved. In anticipation of submission to the American Mathematical Monthly, their style files have been used to prepare the document, but it has not yet been submitted there.

### [C.] "The Generalized Morse-Sard Theorem"

Abstract: The Morse-Sard theorem gives conditions under which the set of critical values of a function between Euclidean spaces has Lebesgue measure zero. Over the years the result has been extended and strengthened in various ways. We present a result, along with a simple proof, that subsumes many of these generalizations. We also review methods of constructing examples showing that differentiability hypotheses cannot be weakened, and we construct a complete set of examples for our result.

### [D.] "Efficient Disposal Equilibrium"

Abstract: For an economy with compact consumption and production sets and some goods that can be freely disposed, an efficient disposal equilibrium specifies prices, consumptions, and production plans such that: a) each agent maximizes utility among bundles costing no more than her income and minimizes expenditure among bundles providing the same utility; b) an unsated agent consumes a bundle that is at least as valuable as her income; c) each producer maximizes profits; d) the aggregate endowment plus aggregate production minus aggregate consumption is a nonnegative bundle of disposable goods; e) disposable goods that are not completely consumed have the minimal price of disposable goods. We prove an existence of equilibrium result that nests those of Hylland and Zeckhauser (1979), Mas-Colell (1992), and Budish, Che, Kojima, and Milgrom (2013). It significantly improves the latter by increasing flexibility and relaxing assumptions that are not satisfied by applications such as course allocation. Open problems concerning generic finiteness of the set of equilibria and efficient algorithms for computing equilibria are described.

### [E.] "The Index +1 Principle"

Abstract: In the simplest (generic) case the fixed point index assigns an index of +1 or -1 to each fixed point of a function or correspondence, and these indices sum to +1. If an isolated equilibrium is stable with respect to any process of adjustment to equilibrium that is natural,'' in the sense that the agents adjust their strategies in directions that increase utility, or prices adjust in the general direction of excess demand, then the index of the equilibrium is +1. The index +1 principle asserts that consequently only index +1 equilibria are empirically relevant; we argue that this should be regarded as a fundamental principle of economic analysis. Since the fixed point index has a general axiomatic characterization, the index +1 principle is universally applicable to economic models in which equilibria are topological fixed points. It does not depend on insignificant details of model specification. The index +1 principle itself, and the hypothesis that processes of adjustment to equilibrium are natural, are strongly supported by experimental evidence. The index +1 principle can be seen as the multidimensional extension of Samuelson's correspondence principle.

### [F.] "Some People Never Learn, Rationally" (Joint with Simon Loertscher)

Abstract: A Bayesian decision maker does not know which of several parameters is true. In each period she chooses an action from an open subset of $\Re^n$, observes an outcome, and updates her beliefs. There is an action $a^*$ that is uninformative in the sense that when it is chosen all parameters give the same distribution over outcomes, and consequently beliefs do not change. We give conditions under which a policy specifying an action as a function of the current belief can result in a positive probability that the sequence of beliefs converge to a belief at which $a^*$ is chosen, so that learning is asymptotically incomplete. Such a policy can be optimal even when the decision maker is not myopic and values experimentation.

### [G.] "The Computational Complexity of Games and Markets: An Introduction for Economists"

Abstract: This is an expository survey of recent results in computer science related to the computation of fixed points, with the central one being that the problem of finding an approximate Nash equilibrium of a bimatrix game is PPAD-complete. This means that this problem is, in a certain sense, as hard as any fixed point problem. Subsequently many other problems have been shown to be PPAD-complete, including finding Walrasian equilibria in certain simple exchange economies. We also comment on the scientific consequences of complexity as a barrier to equilibration, and other sorts of complexity, for our understanding of how markets operate. It is argued that trading in complex systems of markets should be analogized to games such as chess, go, bridge, and poker, in which the very best players are much better than all but a small number of competitors. These traders make positive rents, and their presence is a marker of complexity. Consequences for the efficient markets hypothesis are sketched.

### [H.] "On the Mean Number of Facets of a Gaussian Polytope"

Abstract: A Gaussian polytope is the convex hull of normally distributed random points in a Euclidean space. We give an improved error bound for the expected number of facets of a Gaussian polytope when the dimension of the space is fixed and the number of points tends to infinity. The proof applies the theory of the asymptotic distribution of the top order statistic of a collection of independently distributed $N(0,1)$ random variables.

### [I.] "Fixed Point Theorems"

Abstract: This draft of an entry in The New Palgrave Dictionary of Economics (Second Edition) gives statements of the Tarski fixed point theorem and the main versions of the topological fixed point principle that have been applied in economic theory. Pointers are given to literature concerned with proofs of Brouwer's theorem, and with algorithms for computing approximate fixed points. The topological results are all consequences of a slightly weakened version of the Eilenberg-Montgomery (1946) fixed point theorem. The axiomatic characterization of the Leray-Schauder fixed point index (which is even more powerful) is also stated, and its application to issues concerning robustness of sets of equilibria is explained.

This brief piece should be accessible (and, I hope, enjoyable) if you've taken an undergraduate course in real analysis, but it gives a much more complete view of the subject than you'll find in any economics text, at essentially maximal generality.