A Hyper-Relation Characterization of Weak Pseudo-Rationalizability

I provide a characterization of weakly pseudo-rationalizable choice functions---that is, choice functions rationalizable by a set of acyclic relations---in terms of hyper-relations satisfying certain properties. For those hyper-relations Nehring calls extended preference relations, the central characterizing condition is weaker than (hyper-relation) transitivity but stronger than (hyper-relation) acyclicity. Furthermore, the relevant type of hyper-relation can be represented as the intersection of a certain class of its extensions. These results generalize known, analogous results for path independent choice functions

Solution Concepts for Games with General Coalitional Structure (Replaces CentER DP 2011-025)

We introduce a theory on marginal values and their core stability for cooperative games with arbitrary coalition structure. The theory is based on the notion of nested sets and the complex of nested sets associated to an arbitrary set system and the M-extension of a game for this set. For a set system being a building set or partition system, the corresponding complex is a polyhedral complex, and the vertices of this complex correspond to maximal strictly nested sets. To each maximal strictly nested set is associated a rooted tree. Given characteristic function, to every maximal strictly nested set a marginal value is associated to a corresponding rooted tree as in [9]. We show that the same marginal value is obtained by using the M-extension for every permutation that is associated to the rooted tree. The GC-solution is defined as the average of the marginal values over all maximal strictly nested sets. The solution can be viewed as the gravity center of the image of the vertices of the polyhedral complex. The GC-solution differs from the Myerson-kind value defined in [2] for union stable structures. The HS-solution is defined as the average of marginal values over the subclass of so-called half-space nested sets. The NT-solution is another solution and is defined as the average of marginal values over the subclass of NT-nested sets. For graphical buildings the collection of NT-nested sets corresponds to the set of spanning normal trees on the underlying graph and the NT-solution coincides with the average tree solution. We also study core stability of the solutions and show that both the HS-solution and NT-solution belong to the core under half-space supermodularity, which is a weaker condition than convexity of the game. For an arbitrary set system we show that there exists a unique minimal building set containing the set system. As solutions we take the solutions for this building covering by extending in a natural way the characteristic function to it by using its Möbius inversion.Core;polytope;building set;nested set complex;Möbius inversion;permutations;normal fan;average tree solution;Myerson value

Matroids on convex geometries (cg-matroids)

AbstractWe consider matroidal structures on convex geometries, which we call cg-matroids. The concept of a cg-matroid is closely related to but different from that of a supermatroid introduced by Dunstan, Ingleton, and Welsh in 1972. Distributive supermatroids or poset matroids are supermatroids defined on distributive lattices or sets of order ideals of posets. The class of cg-matroids includes distributive supermatroids (or poset matroids). We also introduce the concept of a strict cg-matroid, which turns out to be exactly a cg-matroid that is also a supermatroid. We show characterizations of cg-matroids and strict cg-matroids by means of the exchange property for bases and the augmentation property for independent sets. We also examine submodularity structures of strict cg-matroids

Path Independence and a Persistent Paradox of Population Ethics

In the face of an impossibility result, some assumption must be relaxed. The Mere Addition Paradox is an impossibility result in population ethics. Here, I explore substantially weakening the decision-theoretic assumptions involved. The central finding is that the Mere Addition Paradox persists even in the general framework of choice functions when we assume Path Independence as a minimal decision-theoretic constraint. Choice functions can be thought of either as generalizing the standard axiological assumption of a binary “betterness” relation, or as providing a general framework for a normative (rather than axiological) theory of population ethics. Path Independence, a weaker assumption than typically (implicitly) made in population ethics, expresses the idea that, in making a choice from a set of alternatives, the order in which options are assessed or considered is ethically arbitrary and should not affect the final choice. Since the result establishes a conflict between the relevant ethical principles and even very weak decision-theoretic principles, we have more reason to doubt the ethical principles

Minimality and comparison of sets of multi-attribute vectors

In a decision-making problem, there is often some uncertainty regarding the user preferences. We assume a parameterised utility model, where in each scenario we have a utility function over alternatives, and where each scenario represents a possible user preference model consistent with the input preference information. With a set A of alternatives available to the decision-maker, we can consider the associated utility function, expressing, for each scenario, the maximum utility among the alternatives. We consider two main problems: firstly, finding a minimal subset of A that is equivalent to it, i.e., that has the same utility function. We show that for important classes of preference models, the set of possibly strictly optimal alternatives is the unique minimal equivalent subset. Secondly, we consider how to compare A to another set of alternatives B , where A and B correspond to different initial decision choices. This is closely related to the problem of computing setwise max regret. We derive mathematical results that allow different computational techniques for these problems, using linear programming, and especially, with a novel approach using the extreme points of the epigraph of the utility function