Search for an Immobile Hider on a Stochastic Network

Harry hides on an edge of a graph and does not move from there. Sally, starting from a known origin, tries to find him as soon as she can. Harry's goal is to be found as late as possible. At any given time, each edge of the graph is either active or inactive, independently of the other edges, with a known probability of being active. This situation can be modeled as a zero-sum two-person stochastic game. We show that the game has a value and we provide upper and lower bounds for this value. Finally, by generalizing optimal strategies of the deterministic case, we provide more refined results for trees and Eulerian graphs.Comment: 28 pages, 9 figure

The convexity-cone approach to comparative risk and downside risk.

Based on Jewitt (1986) we try to find a characterization of comparative downside risk aversion and love. The desired characterizations involve the decomposition of the dual of the intersection of two convexity cones. The decomposition holds in the case of downside risk love, but not in the case of downside risk aversion. A counterexample is provided.Convexity cones; risk; downside risk; risk aversion; dual cones

Archimedean Copulae and Positive Dependence.

In the first part of the paper we consider positive dependence properties of Archimedean copulae. Especially we characterize the Archimedean copulae that are multivariate totally positive of order 2 (MTP2) and conditionally increasing in sequence. In the second part we investigate conditions for binary sequences to admit an Archimedean copula.Conditionally increasing, MTP2, positive lower orthant dependent, exchangeability, binary sequences.

Zonoids, Linear Dependence, and Size-Biased Distributions on the Simplex.

The zonoid of a d-dimensional random vector is used as a tool for measuring linear dependence among its components. A preorder of linear dependence is defined through inclusion of the zonoids. The zonoid of a random vector does not characterize its distribution, but it characterizes the size biased distribution of its compositional variables. This fact will allow a characterization of our linear dependence order in terms of a linear-convex order for the size-biased compositional variables. In dimension 2 the linear dependence preorder will be shown to be weaker than the concordance order. Some examples related to the Marshall-Olkin distribution and to a copula model will be presented, and a class of measures of linear dependence will be proposed.zonoid, zonotope, linear dependence, compositional variables, multivariate size biased distribution, concordance order, Marshall-Olkin distribution.

Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators

We compare estimators of the (essential) supremum and the integral of a function $f$ defined on a measurable space when $f$ may be observed at a sample of points in its domain, possibly with error. The estimators compared vary in their levels of stratification of the domain, with the result that more refined stratification is better with respect to different criteria. The emphasis is on criteria related to stochastic orders. For example, rather than compare estimators of the integral of $f$ by their variances (for unbiased estimators), or mean square error, we attempt the stronger comparison of convex order when possible. For the supremum, the criterion is based on the stochastic order of estimators.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ295 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Optimal risk sharing with background risk.

This paper examines qualitative properties of efficient insurance contracts in the presence of background risk. In order to get results for all strictly risk-averse expected utility maximizers, the concept of “stochastic increasingness” is used. Different assumptions on the stochastic dependence between the insurable and uninsurable risk lead to different qualitative properties of the efficient contracts. The new results obtained under hypotheses of dependent risks are compared to classical results in the absence of background risk or to the case of independent risks. The theory is further generalized to nonexpected utility maximizers.Efficient contracts; Stochastically increasing; Incomplete markets; Insurance;

Variance Allocation and Shapley Value

Motivated by the problem of utility allocation in a portfolio under a Markowitz mean-variance choice paradigm, we propose an allocation criterion for the variance of the sum of $n$ possibly dependent random variables. This criterion, the Shapley value, requires to translate the problem into a cooperative game. The Shapley value has nice properties, but, in general, is computationally demanding. The main result of this paper shows that in our particular case the Shapley value has a very simple form that can be easily computed. The same criterion is used also to allocate the standard deviation of the sum of $n$ random variables and a conjecture about the relation of the values in the two games is formulated.Comment: 20page

Discounted and Finitely Repeated Minority Games with Public Signals.

We consider a repeated game where at each stage players simultaneously choose one of two rooms. The players who choose the less crowded room are rewarded with one euro. The players in the same room do not recognize each other, and between the stages only the current majority room is publicly announced, hence the game has imperfect public monitoring. An undiscounted version of this game was considered by Renault et al. (2005), who proved a folk theorem. Here we consider a discounted version and a finitely repeated version of the game, and we strengthen our previous result by showing that the set of equilibrium payoffs Hausdorff-converges to the feasible set as either the discount factor goes to one or the number of repetition goes to infinity. We show that the set of public equilibria for this game is strictly smaller than the set of private equilibria.Repeated Games; Imperfect Monitoring; Public Equilibria; Private Equilibria; Discount Factor; Pareto-efficiency;