The classification of p-compact groups and homotopical group theory

We survey some recent advances in the homotopy theory of classifying spaces, and homotopical group theory. We focus on the classification of p-compact groups in terms of root data over the p-adic integers, and discuss some of its consequences e.g. for finite loop spaces and polynomial cohomology rings.Comment: To appear in Proceedings of the ICM 2010

Group cohomology and control of p-fusion

We show that if an inclusion of finite groups H < G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.Comment: 15 pages; v3: title and exposition shortened, and a short section on Lannes' T-functor removed, following comments by referees. Final version, to appear in Invent. Mat

Preference for Flexibility and the Opportunities of Choice

A decision-maker exhibits preference for flexibility if he always prefers any set of alternatives to its subsets, even when two of them contain the same best element. Desire for flexibility can be explained as the consequence of the agent's uncertainty along a two-stage process, where he must first preselect a subset of alternatives from which to make a final choice later on. We investigate conditions on the rankings of subsets that are compatible with the following assumptions: (1) the agent is endowed with a VN-M utility function of alternatives, (2) the agent attaches a subjective probability to the survival of each subset of alternatives, and (3) the agent will make a best choice out of any set which becomes available, and ranks sets ex-ante in terms of the expected utility of the best choices within them. We first prove that any total ordering respecting set inclusion is rationalizable in these terms. This result is essentially the same obtained by Kreps (1979) under an alternative interpretation. We also show that we cannot learn anything about the underlying utilities of agents unless we impose further restrictions on the admissible distributions of survival probabilities. Then we investigate the additional consequences of assuming that the survival probabilities of individual alternatives are independently distributed. We prove that this reduces significantly the class of set rankings which can be rationalized and that then one can infer some of the characteristics of the agentís preferences. We offer a full characterization for the case of three alternatives. We also provide necessary conditions for rationalizability in the general case.NULL

Preference for Flexibility and the Opportunities of Choice

A decision-maker exhibits preference for flexibility if he always prefers any set of alternatives to its subsets, even when two of them contain the same best element. Desire for flexibility can be explained as the consequence of the agent’s uncertainty along a two-stage process, where he must first preselect a subset of alternatives from which to make a final choice later on. We investigate conditions on the rankings of subsets that are compatible with the following assumptions: (1) the agent is endowed with a VN-M utility function on alternatives, (2) the agent attaches a subjective probability to the survival of each subset of alternatives, and (3) the agent will make a best choice out of any set which becomes available, and ranks sets ex-ante in terms of the expected utility of the best choices within them. We first prove that any total ordering respecting set inclusion is rationalizable in these terms. This result is essentially the same obtained by Kreps (1979) under an alternative interpretation. We also show that we cannot learn anything about the underlying utilities of agents unless we impose further restrictions on the admissible distributions of survival probabilities. Then we investigate the additional consequences of assuming that the survival probabilities of individual alternatives are independently distributed. We prove that this reduces significantly the class of set rankings which can be rationalized and that then one can infer some of the characteristics of the agent’s preferences. We offer a full characterization for the case of three alternatives. We also provide necessary conditions for rationalizability in the general case.

The classification of 2-compact groups

We prove that any connected 2-compact group is classified by its 2-adic root datum, and in particular the exotic 2-compact group DI(4), constructed by Dwyer-Wilkerson, is the only simple 2-compact group not arising as the 2-completion of a compact connected Lie group. Combined with our earlier work with Moeller and Viruel for p odd, this establishes the full classification of p-compact groups, stating that, up to isomorphism, there is a one-to-one correspondence between connected p-compact groups and root data over the p-adic integers. As a consequence we prove the maximal torus conjecture, giving a one-to-one correspondence between compact Lie groups and finite loop spaces admitting a maximal torus. Our proof is a general induction on the dimension of the group, which works for all primes. It refines the Andersen-Grodal-Moeller-Viruel methods to incorporate the theory of root data over the p-adic integers, as developed by Dwyer-Wilkerson and the authors, and we show that certain occurring obstructions vanish, by relating them to obstruction groups calculated by Jackowski-McClure-Oliver in the early 1990s.Comment: 47 page

Objectives of an Imperfectly Competitive Firm: A Surplus Approach.

We consider a firm acting strategically on behalf of its shareholders. The price normalization problem arising in general equilibrium models of imperfect competition can be overcome by using the concept of real wealth maximization. This concept is based on shareholders' aggregate demand and does not involve any utility comparisons. We explore the efficiency properties of real wealth maxima for the group of shareholders. A strategy is called S-efficient (S stands for shareholders) if there is no other strategy such that shareholders' new total demand can be redistributed in a way that all shareholders will be better off. Our main result states that the set of real wealth maximizing strategies coincides with the set of S-efficient strategies provided that shareholders' social surplus is concave. Thus, even if a firm does not know the preferences of its shareholders it can achieve S-efficiency by selecting a real wealth maximizing strategy.

Small Income Effects Destroy the Constrained Efficiency of All Equilibria in Finance Economies with Production.

We consider economies with incomplete markets, one good per state, private ownership of initial endowments, a single firm, and no assets other than shares in this firm. In this simple framework, arbitrarily small income effects can render every market equilibrium resulting from some production decision constrained inefficient. Thus, even if all utility functions are approximately quasilinear, the stock market can be unable to achieve a constrained efficient allocation given the agents' characteristics. Moreover, the phenomenon persists when the efficiency requirements are substantially weakened.Incomplete markets with production; constrained efficiency; Drèze equilibria

Cournot-Nash Competition in a General Equilibrium Model of International Trade.

We use the two-factor, two-sector, two-country model of Melvin and Warne (1973) and Markusen (1981), in which the production of one good is monopolized in each country, in order to investigate the role of the price normalization. We illustrate several puzzling effects that occur if the price normalization is changed. However, we show that Markusen’s result on the direction of the trade flow between two proportional countries with constant returns to scale is robust with respect to the choice of the normalization rule. To overcome the price normalization problem in international trade we suggest to use the concept of real wealth maximization.international trade with imperfect competition; price normalization; real wealth maximization

Are Incomplete Markets Able to Achieve Minimal Efficiency?

We consider economies with incomplete markets, one good per state, two periods, t = 0; 1, private ownership of initial endowments, a single firm, and no assets other than shares in this firm. In Dierker, Dierker, Grodal (2002), we give an example of such an economy in which all market equilibria are constrained ineffcient. In this paper, we weaken the concept of constrained effciency by taking away the planner’s right to determine consumers’ investments. An allocation is called minimally constrained efficient if a planner, who can only determine the production plan and the distribution of consumption at t = 0, cannot find a Pareto improvement. We present an example with arbitrarily small income effects in which no market equilibrium is minimally constrained effcient.incomplete markets with production; constrained efficiency; Drèze equilibria

Are Incomlete Markets Able to Achieve Minimal Efficiency?

We consider economies with incomplete markets, one good per state, two periods, t=0,1, private ownership of initial endowments, a single firm, and no assets other than shares in this firm. In Dierker, Dierker, Grodal (2002), we give an example of such an economy in which all market equilibria are constrained inefficient. In this paper, we weaken the concept of constrained efficiency by taking away the planner's right to determine consumers' investments. An allocation is called minimally constrained efficient if a planner, who can only determine the production plan and the distribution of consumption at t=0, cannot find a Pareto improvement. We present an example with arbitrarily small income effects in which no market equilibrium is minimally constrained efficient.