Dependent Random Choice

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique has had several striking applications to Extremal Graph Theory, Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this survey we discuss some of them.Comment: 32 page

Mating order-dependent female mate choice in the polygynandrous common lizard Lacerta vivipara.

Recent studies indicate that directional female mate choice and order-dependent female mate choice importantly contribute to non-random mating patterns. In species where females prefer larger sized males, disentangling different hypotheses leading to non-random mating patterns is especially difficult, given that male size usually correlates with behaviours that may lead to non-random mating (e.g. size-dependent emergence from hibernation, male fighting ability). Here we investigate female mate choice and order-dependent female mate choice in the polygynandrous common lizard (Lacerta vivipara). By sequentially presenting males in random order to females, we exclude non-random mating patterns potentially arising due to intra-sexual selection (e.g. male-male competition), trait-dependent encounter probabilities, trait-dependent conspicuousness, or trait-dependent emergence from hibernation. To test for order-dependent female mate choice we investigate whether the previous mating history affects female choice. We show that body size and body condition of the male with which a female mated for the first time were bigger and better, respectively, than the average body size and body condition of the rejected males. There was a negative correlation between body sizes of first and second copulating males. This indicates that female mate choice is dependent on the previous mating history and it shows that the female's choice criteria are non-static, i.e. non-directional. Our study therefore suggests that context-dependent female mate choice may not only arise due to genotype-environment interactions, but also due to other female mating strategies, i.e. order-dependent mate choice. Thus context-dependent female mate choice might be more frequent than previously thought

Random Utility, Repeated Choice, and Consumption Dependence

We study consumption dependence in the context of random utility and repeated choice. We show that, in the presence of consumption dependence, the random utility model is a misspecified model of repeated rational choice. This misspecification leads to biased estimators and failures of standard random utility axioms. We characterize exactly when and by how much the random utility model is misspecified when utilities are consumption dependent. As one possible solution to this problem, we consider time disaggregated data. We offer a characterization of consumption dependent random utility when we observe time disaggregated data. Using this characterization, we develop a hypothesis test for consumption dependent random utility that offers computational improvements over the natural extension of Kitamura and Stoye (2018) to our setting

Discrete choice non-response

Missing values are endemic in the data sets available to econometricians. This paper suggests a unified likelihood-based approach to deal with several nonignorable missing data problems for discrete choice models. Our concern is when either the dependent variable is unobserved or situations when both dependent variable and covariates are missing for some sampling units. These cases are also considered when a supplementary random sample of observations on all covariates is available. A unified treatment of these various sampling structures is presented using a formulation of the nonresponse problems as a modification of choice-based sampling. Extensions appropriate for nonresponse are detailed of Imbens' (1992) effcient generalized method of moments (GMM) estimator for choice-based samples. Simulation evidence reveals very promising results for the various GMM estimators proposed in this paper.

Bounds for the price of discrete arithmetic Asian options.

In this paper the pricing of European-style discrete arithmetic Asian options with fixed and floating strike is studied by deriving analytical lower and upper bounds. In our approach we use a general technique for deriving upper (and lower) bounds for stop-loss premiums of sums of dependent random variables, as explained in Kaas, Dhaene and Goovaerts (2000), and additionally, the ideas of Rogers and Shi (1995) and of Nielsen and Sandmann (2003). We are able to create a unifying framework for discrete Asian options through these bounds, that generalizes several approaches in the literature as well as improves the existing results. We obtain analytical and easily computable bounds. The aim of the paper is to formulate an advice of the appropriate choice of the bounds given the parameters, investigate the effect of different conditioning variables and compare their efficiency numerically. Several sets of numerical results are included. We also show that the hedging using these bounds is possible. Moreover, our methods are applicable to a wide range of (pricing) problems involving a sum of dependent random variables.Asian option; Choice; Efficiency; Framework; Hedging; Methods; Options; Premium; Pricing; Problems; Random variables; Research; Stop-loss premium; Variables;

Estimating the Maximum Expected Value: An Analysis of (Nested) Cross Validation and the Maximum Sample Average

We investigate the accuracy of the two most common estimators for the maximum expected value of a general set of random variables: a generalization of the maximum sample average, and cross validation. No unbiased estimator exists and we show that it is non-trivial to select a good estimator without knowledge about the distributions of the random variables. We investigate and bound the bias and variance of the aforementioned estimators and prove consistency. The variance of cross validation can be significantly reduced, but not without risking a large bias. The bias and variance of different variants of cross validation are shown to be very problem-dependent, and a wrong choice can lead to very inaccurate estimates

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

Quantum and random walks as universal generators of probability distributions

Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.Comment: 8 pages, 2 figures; revised and enlarged versio