José-Manuel Rey

On the role of Shannon's entropy as a measure of heterogeneity

Ž . Ibanez et al. 1998 proposed the use of Shannon’s entropy to analyze the ̃ diversity of the world pedosphere on the basis of data compiled by the F.A.O. at the scale 1:5,000,000. Here we will try to provide some mathematically founded arguments to justify the use and interpretation of Shannon’s information entropy as a measure of diversity and homogeneity. Information entropy h is computed from a discrete probability distribution 4 p : is1, 2, . . . , N via Shannon’s formula hsyÝ p log p . This quantity i i i i was originally proposed by Shannon as a measure of the average information content that is gained from observing the realization of an experiment with N possible outcomes with probabilities of occurence given by p , p , . . . , p . 1 2 N Ž . Well-known mathematical facts are that a h attains its maximum value log N Ž . only in the equiprobable case, that is p s1rN for all i’s, and b h vanishes in i Ž . the case that some p s1 and thus p s0 if i/ j . These two extreme j i Ž . Ž . situations, respectively, correspond with a the most informative case and b Ž . the least informative case, since observing the actual outcome provides a much Ž . rich information being all outcomes equally probable and b very poor information in the case that outcome j has all the chances to occur. Also, the number h depends continuously on the probabilities p so that similar distributions render i close values of h.

Singularity of self-similar measures with respect to Hausdorff measures

Besicovitch (1934) and Eggleston (1949) analyzed subsets of points of the unit interval with given frequencies in the figures of their base-p expansions. We extend this analysis to self-similar sets, by replacing the frequencies of figures with the frequencies of the generating similitudes. We focus on the interplay among such sets) self-similar measures? and Hausdorff measures3. We give a fine-tuned classification of the Hausdorff measures according to the singularity of the self-sitnilar meas;ures with respect to those measures. We show that the self-similar measures are concentrated on sets whose frequencies of similitudes obey the Law of the Iterated Logarithm.

A Mathematical Model of Sentimental Dynamics Accounting for Marital Dissolution

BACKGROUND Marital dissolution is ubiquitous in western societies. It poses major scientific and sociological problems both in theoretical and therapeutic terms. Scholars and therapists agree on the existence of a sort of second law of thermodynamics for sentimental relationships. Effort is required to sustain them. Love is not enough. METHODOLOGY/PRINCIPAL FINDINGS Building on a simple version of the second law we use optimal control theory as a novel approach to model sentimental dynamics. Our analysis is consistent with sociological data. We show that, when both partners have similar emotional attributes, there is an optimal effort policy yielding a durable happy union. This policy is prey to structural destabilization resulting from a combination of two factors: there is an effort gap because the optimal policy always entails discomfort and there is a tendency to lower effort to non-sustaining levels due to the instability of the dynamics. CONCLUSIONS/SIGNIFICANCE These mathematical facts implied by the model unveil an underlying mechanism that may explain couple disruption in real scenarios. Within this framework the apparent paradox that a union consistently planned to last forever will probably break up is explained as a mechanistic consequence of the second law.

Numerical analysis of a time allocation model accounting for choice overload

The cost of time has been suggested as a factor associated with the choice overload problem. This term refers to the discomfort or paralysis experienced by individuals when facing a choice within a large set of alternatives, as it has been evidenced in experiments by behavioural and social psychologists. We introduce a rational model of time allocation to analyse how increasing the number of options of a given product may change consumers allocation of time and in turn affect her welfare. Under some standard assumptions, the numerical analysis of the model reproduces two key experimental findings, namely choice paralysis – i.e. the choice problem is abandoned if the number of options is too large – and choice dissatisfaction – that is, the apparent paradox that increasing the number of considered options beyond certain limit, in turn choosing better, eventually diminishes welfare. The model analysis provides specific threshold values for the occurrence of both phenomena.

Choice Overload, Satisficing Behavior, and Price Distribution in a Time Allocation Model

Recent psychological research indicates that consumers that search exhaustively for the best option of a market product—known as maximizers—eventually feel worse than consumers who just look for something good enough—called satisficers. We formulate a time allocation model to explore the relationship between different distributions of prices of the product and the satisficing behavior and the related welfare of the consumer. We show numerically that, as the number of options becomes large, the maximizing behavior produces less and less welfare and eventually leads to choice paralysis—these are effects of choice overload—whereas satisficing conducts entail higher levels of satisfaction and do not end up in paralysis. For different price distributions, we provide consistent evidence that maximizers are better off for a low number of options, whereas satisficers are better off for a sufficiently large number of options. We also show how the optimal satisficing behavior is affected when the underlying price distribution varies. We provide evidence that the mean and the dispersion of a symmetric distribution of prices—but not the shape of the distribution—condition the satisficing behavior of consumers. We also show that this need not be the case for asymmetric distributions.

Sentimental equilibria with optimal control

Abstract The ubiquitous phenomenon of marital dissolution in the West poses a major theoretical problem that is not well understood. The fact that most couples declare that they are committed to an enduring and happy relationship allows us to model their sentimental relationship as a simple optimal control problem. A state variable monitors the wellness of the relationship whose natural decay in time must be counteracted with effort–the control variable–according to a widely accepted principle in marital psychology. Equilibria are basic desirable solutions of this control problem. In this note key properties of equilibria–in particular, existence, dynamical instability and sensitivity analysis–are obtained for a class of decaying sentimental dynamics. An underlying mechanism combining the existence of an effort stress with the instability of the model may be identified as a main cause of distress for the couple relationship.

Pricing Strategy versus Heterogeneous Shopping Behavior under Market Price Dispersion

We consider the ubiquitous problem of a seller competing in a market of a product with dispersed prices and having limited information about both his competitors’ prices and the shopping behavior of his potential customers. Given the distribution of market prices, the distribution of consumers’ shopping behavior, and the seller’s cost as inputs, we find the computational solution for the pricing strategy that maximizes his expected profits. We analyze the seller’s solution with respect to different exogenous perturbations of parametric and functional inputs. For that purpose, we produce synthetic price data using the family of Generalized Error Distributions that includes normal and quasiuniform distributions as particular cases, and we also generate consumers’ shopping data from different behavioral assumptions. Our analysis shows that, beyond price mean and dispersion, the shape of the price distribution plays a significant role in the seller’s pricing solution. We focus on the seller’s response to an increasing diversity in consumers’ shopping behavior. We show that increasing heterogeneity in the shopping distribution typically lowers seller’s prices and expected profits.