Abstract. When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.Abrégé. Le choix de métrique de probabilité est une décision très importante lorsqu'onétudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs métriques/distances de probabilité couramment utilisées par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux résultats qui se rapportentà leurs bornes. Avoir connaissance d'autres métriques peut vous fournir avec un moyen de dériver des bornes pour une autre métrique dans un problème appliqué. Le fait de prendre en considération plusieurs métriques vous permettra d'approcher des problèmes d'une manière différente. Ainsi, nous vous démontrons que les taux de convergence peuvent dépendre de façon importante sur votre choix de métrique. Il est donc important de tout considérer lorsqu'on doit choisir une métrique.

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Abstract. When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.Abrégé. Le choix de métrique de probabilité est une décision très importante lorsqu'onétudie la convergence des mesures. Nous vous fournissons avec un sommaire de plusieurs métriques/distances de probabilité couramment utilisées par des statisticiens(nes) at par des probabilistes, ainsi que certains nouveaux résultats qui se rapportentà leurs bornes. Avoir connaissance d'autres métriques peut vous fournir avec un moyen de dériver des bornes pour une autre métrique dans un problème appliqué. Le fait de prendre en considération plusieurs métriques vous permettra d'approcher des problèmes d'une manière différente. Ainsi, nous vous démontrons que les taux de convergence peuvent dépendre de façon importante sur votre choix de métrique. Il est donc important de tout considérer lorsqu'on doit choisir une métrique.

Every weighted tree corresponds naturally to a cooperative game that we call a tree game; it assigns to each subset of leaves the sum of the weights of the minimal subtree spanned by those leaves. In the context of phylogenetic trees, the leaves are species and this assignment captures the diversity present in the coalition of species considered. We consider the Shapley value of tree games and suggest a biological interpretation. We determine the linear transformation M that shows the dependence of the Shapley value on the edge weights of the tree, and we also compute a null space basis of M. Both depend on the split counts of the tree. Finally, we characterize the Shapley value on tree games by four axioms, a counterpart to Shapley's original theorem on the larger class of cooperative games. We also include a brief discussion of the core of tree games.

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed. 

We prove the following conjecture of Atanassov (Studia Sci. Math. Hungar. 32 (1996), 71-74). Let T be a triangulation of a d-dimensional polytope P with n vertices v 1 ; v 2 ; . . . ; v n : Label the vertices of T by 1; 2; . . . ; n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if v j is on F: Then there are at least n À d full dimensional simplices of T; each labelled with d þ 1 different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a pebble set of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes, as in

Abstract. We develop a procedure for implementing an e‰cient and envy-free allocation of m objects among n individuals with the possibility of monetary side-payments, assuming that players have quasi-linear utility functions. The procedure eliminates envy by compensating envious players. It is fully descriptive and says explicitly which compensations should be made, and in what order. Moreover, it is simple enough to be carried out without computer support. We formally characterize the properties of the procedure, show how it establishes envy-freeness with minimal resources, and demonstrate its application to a wide class of fair-division problems.

Abstract. Fix α ∈ [0, 1). Consider the random walk on the circle S 1 which proceeds by repeatedly rotating points forward or backward, with probability , by an angle 2πα. This paper analyzes the rate of convergence of this walk to the uniform distribution under "discrepancy" distance. The rate depends on the continued fraction properties of the number ξ = 2α. We obtain bounds for rates when ξ is any irrational, and a sharp rate when ξ is a quadratic irrational. In that case the discrepancy falls as k − 1 2 (up to constant factors), where k is the number of steps in the walk. This is the first example of a sharp rate for a discrete walk on a continuous state space. It is obtained by establishing an interesting recurrence relation for the distribution of multiples of ξ which allows for tighter bounds on terms which appear in the Erdős-Turán inequality.