Positive value of information in games

We exhibit a general class of interactive decision situations in which all the agents benefit from more information. This class includes as a special case the classical comparison of statistical experiments à la Blackwell. AMS 2000 Subject Classification:Primary 91A35.The work of Bruno Bassan and Marco Scarsini was partially supported by MIUR-COFIN. The authors express their thanks to Sylvain Sorin for enlightening comments.     

Characterization of a Marshall-Olkin type class of distributions

Summary A class of bivariate distributions that generalize Marshall-Olkin's one is characterized through a functional equation which involves two associative operations. The obtained distributions concentrate positive mass on the linex=y when the two associative operations coincide; otherwise a positive mass is concentrated on a continuous monotone function. Work performed while the authors were members of CNR-GNAFA.     

Hessian orders and multinormal distributions

Several well known integral stochastic orders (like the convex order, the supermodular order, etc.) can be defined in terms of the Hessian matrix of a class of functions. Here we consider a generic Hessian order, i.e., an integral stochastic order defined through a convex cone of Hessian matrices, and we prove that if two random vectors are ordered by the Hessian order, then their means are equal and the difference of their covariance matrices belongs to the dual of . Then we show that the same conditions are also sufficient for multinormal random vectors. We study several particular cases of this general result.     

Lower bounds for the distribution function of a k-dimensionaln-extensible exchangeable process

A lower bound for the distribution function of a k-dimensional, n-extensible exchangeable process is provided when the marginals are uniform on the unit segment. The result is obtained by means of standard linear programming techniques. The lower bound for infinitely extendible exchangeable processes is the distribution of independent random variables.     

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.     

Positive dependence orderings and stopping times

We study the positive dependence of pairs of stochastic processes and examine its relation with the properties of certain stopping times. Some special cases, such as dependent random walks, Gaussian processes and exchangeable sequences of elliptically contoured random variables, are taken into account.     

Ordering distributions by scaled order statistics

Motivated by applications in reliability theory, we define a preordering (X ₁, ...,X _n) (Y ₁ ...,Y _n) of nonnegative random vectors by requiring thek-th order statistic ofa ₁ X ₁,..., a _n X _n to be stochastically smaller than thek-th order statistic ofa ₁ Y ₁, ...,a _n Y _n for all choices ofa _i >0,i=1, 2, ...,n. We identify a class of functionsM _{k, n} such that if and only ifE(X)E(Y) for allM _k,n. Some preservation results related to the ordering are obtained. Some applications of the results in reliability theory are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205.     

Discounted and finitely repeated minority games with public signals

We consider a repeated game where at each stage players simultaneously choose one of the 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. [Renault, J., Scarlatti, S., Scarsini, M., 2005. A folk theorem for minority games. Games Econom. Behav. 53 (2), 208–230], 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.     

Stochastic Comparisons of Symmetric Sampling Designs

We compare estimators of the integral of a monotone function f that can be observed only at a sample of points in its domain, possibly with error. Most of the standard literature considers sampling designs ordered by refinements and compares them in terms of mean square error or, as in Goldstein et al. (2011), the stronger convex order. In this paper we compare sampling designs in the convex order without using partition refinements. Instead we order two sampling designs based on partitions of the sample space, where a fixed number of points is allocated at random to each partition element. We show that if the two random vectors whose components correspond to the number allocated to each partition element are ordered by stochastic majorization, then the corresponding estimators are likewise convexly ordered. If the function f is not monotone, then we show that the convex order comparison does not hold in general, but a weaker variance comparison does.     

Total positivity order and the normal distribution

Unlike the usual stochastic order, total positivity order is closed under conditioning. Here we provide a general formulation of the preservation properties of the order under conditioning; we study certain properties of the order including translation properties and the implications of having equality in the inequality defining the order. Specializing to the multivariate normal distribution, the study of total positivity order leads to new cones defined in terms of covariance M-matrices related to positive dependence, whose properties we study.