Condorcet domains of tiling type

A Condorcet domain (CD) is a collection of linear orders on a set of candidates satisfying the following property: for any choice of preferences of voters from this collection, a simple majority rule does not yield cycles. We propose a method of constructing "large" CDs by use of rhombus tiling diagrams and explain that this method unifies several constructions of CDs known earlier. Finally, we show that three conjectures on the maximal sizes of those CDs are, in fact, equivalent and provide a counterexample to them.Comment: 16 pages. To appear in Discrete Applied Mathematic

The a-MEU Model: A Comment

In [7] Ghirardato, Macheroni and Marinacci (GMM) propose a method for distinguishing between perceived ambiguity and the decision-maker's reaction to it. They study a general class of preferences which they refer to as invariant biseparable. This class includes CEU and MEU. They axiomatize a subclass of a-MEU preferences. If attention is restricted to finite state spaces, we show that any a-MEU preference relation, satisfies GMM's axioms if and only if a = 0 or 1, that is, the preferences must be either maxmin or maxmax. We show by example that these axioms may be satisfied when the state space is [0,1].Ambiguity, multiple priors, invariant biseparable, Clarke derivative, ambiguity-preference.

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

Equilibria with indivisible goods and package-utilities

We revisit the issue of existence of equilibrium in economies with indivisible goods and money, in which agents may trade many units of items. In [5] it was shown that the existence issue is related to discrete convexity. Classes of discrete convexity are characterized by the unimodularity of the allowable directions of one-dimensional demand sets. The class of graphical unimodular system can be put in relation with a nicely interpretable economic property of utility functions, the Gross Substitutability property. The question is still open as to what could be the possible, challenging economic interpretations and relevant examples of demand structures that correspond to other classes of discrete convexity. We consider here an economy populated with agents having a taste for complementarity; their utilities are generated by compounds of specific items grouped in 'packages'. Simple package-utilities translate in a straightforward fashion the fact that the items forming a package are complements. General package-utilities are obtained as the convolution (or aggregation) of simple packageutilities. We prove that if the collection of packages of items, that generates the utilities of agents in the economy, is unimodular then there exists a competitive equilibrium. Since any unimodular set of vectors can be implemented as a collection of 0-1 vectors ([3]), we get examples of demands for each class of discrete convexity

Lift expectations of random sets

It is known that the distribution of an integrable random vector $\xi$ in $\mathbb{R}^d$ is uniquely determined by a $(d+1)$-dimensional convex body called the lift zonoid of $\xi$. This concept is generalised to define the lift expectation of random convex bodies. However, the unique identification property of distributions is lost; it is shown that the lift expectation uniquely identifies only one-dimensional distributions of the support function, and so different random convex bodies may share the same lift expectation. The extent of this nonuniqueness is analysed and it is related to the identification of random convex functions using only their one-dimensional marginals. Applications to construction of depth-trimmed regions and partial ordering of random convex bodies are also mentioned.Comment: 13 pages, 1 figur

Planar flows and quadratic relations over semirings

Adapting Lindstr\"om's well-known construction, we consider a wide class of functions which are generated by flows in a planar acyclic directed graph whose vertices (or edges) take weights in an arbitrary commutative semiring. We give a combinatorial description for the set of "universal" quadratic relations valid for such functions. Their specializations to particular semirings involve plenty of known quadratic relations for minors of matrices (e.g., Pl\"ucker relations) and the tropical counterparts of such relations. Also some applications and related topics are discussed.Comment: 35 pages. This is the revised version accepted for publication in J. Algebraic Comb. (The final publication is available at springerlink.com.) Also in the Appendix we add the assertion that the function of minors of any matrix over a field is generated (using Lindstr\"om method) by flows in a weighted planar acyclic directed grap