For $\ell\geq 2$ and $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ of degree $k\geq 2$, we show that every set $A\subseteq \{1,2,\dots,N\}$ lacking nonzero differences in $h(\mathbb{Z}^{\ell})$ satisfies $|A|\ll_h Ne^{-c(\log N)^{\mu}}$, where $c=c(h)>0$, $\mu=[(k-1)^2+1]^{-1}$ if $\ell=2$, and $\mu=1/2$ if $\ell\geq 3$, provided $h(\mathbb{Z}^{\ell})$ contains a multiple of every natural number and $h$ satisfies certain nonsingularity conditions. We also explore these conditions in detail, drawing on a variety of tools from algebraic geometry.

Many disciplines of scholarship are interested in the Relative Age Effect (RAE), whereby age-banding confers advantages on older members of the cohort over younger ones. Most research does not test this relationship in a manner consistent with theory (which requires a decline in frequency across the cohort year), instead resorting to non-parametric, non-directional approaches. In this article, the authors address this disconnect, provide an overview of the benefits associated with Poisson regression modelling, and two managerially useful measures for quantifying RAE bias, namely the Indices of Discrimination and Wastage. In a tutorial-like exposition, applications and extensions of this approach are illustrated using data on professional soccer players competing in the top two tiers of the "Big Five" European football leagues in the search to identify paragon clubs, leagues, and countries from which others may learn to mitigate this form of age-discrimination in the talent identification process. As with OLS regression, Poisson regression may include more than one independent variable. In this way we test competing explanations of RAE; control for unwanted sources of covariation; model interaction effects (that different clubs and countries may not all be subject to RAE to the same degree); and test for non-monotonic versions of RAE suggested in the literature.

The paper analyses two datasets of elite soccer players (top 1000 professionals and UEFA Under-19 Youth League). In both, we find a Relative Age Effect (RAE) for frequency, but not for value. That is, while there are more players born at the start of the competition year, their transfer values are no higher, nor are they given more game time. We use Poisson regression to derive a transparent index of the discrimination present in RAE. Also, because Poisson is valid for small frequency counts, it supports analysis at the disaggregated levels of country and club. From this, we conclude there are no paragon clubs or countries immune to RAE; that is clubs and countries do not differ systematically in the RAE they experience; also, that Poisson regression is a powerful and flexible method of analysing RAE data.

The Relative Age Effect (RAE) documents the inherent disadvantages of being younger rather than older in an age-banded cohort, typically a school- or competition-year, to the detriment of career-progression, earnings and wellbeing into adulthood. We develop the Tails of the Travelling Gaussian (TTG) to model the mechanisms behind RAE. TTG has notable advantages over existing approaches, which have been largely descriptive, potentially confounded, and non-comparable across contexts. In Study 1, using data from the UK's Millennium Cohort Study, we investigate the different levels of RAE bias across school-level academic subjects and "personality" traits. Study 2 concerns biased admissions to elite English Premier League soccer academies, and shows the model can still be used with minimal data. We also develop two practical metrics: the discrimination index (ID), to quantify the disadvantages facing cohort-younger children; and the wastage metric (W), to quantify the loss through untapped potential. TTG is sufficiently well-specified to simulate the consequences of ID and W for policy change.

We describe an algorithm for listing all elements of bounded height in a given number field.

Sometimes criticism of judicial decisions leads to proposals to change the method of selecting and appointing judges and magistrates. Sometimes proposals for change are prompted by other factors.

The paper surveys over twenty models of delay discounting (also known as temporal discounting, time preference, time discounting ), that psychologists and economists have put forward to explain the way people actually trade off time and money. Using little more than the basic algebra of powers and logarithms, I show how the models are derived, what assumptions they are based upon, and how different models relate to each other. Rather than concentrate only on discount functions themselves, I show how discount functions may be manipulated to isolate rate parameters for each model. This approach, consistently applied, helps focus attention on the three main components in any discounting model: subjectively perceived money; subjectively perceived time; and how these elements are combined. We group models by the number of parameters that have to be estimated, which means our exposition follows a trajectory of increasing complexity to the models. However, as the story unfolds it becomes clear that most models fall into a smaller number of families. We also show how new models may be constructed by combining elements of different models. The surveyed models are: Exponential; Hyperbolic; Arithmetic; Hyperboloid (Green & Myerson, Rachlin); Loewenstein and Prelec Generalized Hyperboloid; quasi-Hyperbolic (also known as β-δ discounting); Benhabib et al’s fixed cost; Benhabib et al’s Exponential / Hyperbolic / quasi-Hyperbolic; Read’s discounting fractions; Roelofsma’s exponential time; Scholten and Read’s discounting-by-intervals (DBI); Ebert and Prelec’s constant sensitivity (CS); Bleichrodt et al.’s constant absolute decreasing impatience (CADI); Bleichrodt et al.’s constant relative decreasing impatience (CRDI); Green, Myerson, and Macaux’s hyperboloid over intervals models; Killeen’s additive utility; size-sensitive additive utility; Yi, Landes, and Bickel’s memory trace models; McClure et al.’s two exponentials; and Scholten and Read’s trade-off model. For a convenient overview, a single “cheat sheet” table captures the notation and essential mathematics behind all but one of the models.

Two popular methods for assigning numerical values to a set of to-be-judged objects in order to capture their relative standing are Direct Rating (DR) and Point Allocation (PA). People using PA distribute a fixed sum of 100 points among the objects, while people using DR rate each object on a fixed scale, typically 0–10, later rescaled to sum to 100. Prior research shows that these methods exhibit distinct profiles when values are ranked from largest to smallest, with DR being more test–retest reliable. But which method best translates people’s inner judgments into outer numerical values (is more valid)? Instead of examining subjective or abstract stimuli, we use objectively verifiable perceptual tasks, namely judgments of line length presented using bar charts. We show that (i) DR is more inter-rater reliable than PA; (ii) DR is more accurate than PA at the individual level; (iii) but there is no difference in accuracy when individual judgments are combined to form group-level estimates; and (iv) DR judgments were improved by using prior knowledge of method bias, whereas PA judgments were not.

A measure of location is examined that places itself where the signed rank deviations are as close to zero as possible. A solution algorithm is sketched. The measure is robust to outliers. In three illustrative real data examples we find the measure is usually intermediate between mean and median, hence provisionally called meandian. It tracks the mean and median closely when there is little skew. It is closer to the mean when skewing is light, but closer to the median when skewing is heavy. The measure has good stability when assessed using bootstrap resampling. Its connections with Wilcoxon’s signed rank test and the Hodges-Lehmann estimator are pointed out. The meandian may be generalized in a variety of ways, and breakdown points for the simple and generalized meandians are tabulated.

The paper presents an unusually comprehensive empirical comparison of delay discounting/intertemporal choice models. A three-component model is developed, with power laws modeling subjective time, subjective money, and magnitude sensitivity. It nests several other models in the literature, among others: exponential, hyperbolic, arithmetic, hyperboloid, and Killeen’s additive utility model. The model not only leads to mathematical parsimony, but also allows all derivative models to be succinctly tested against each other using four datasets collected from three online studies. Two of the most used and discussed models in the literature, exponential and hyperbolic (also the quasi-hyperbolic model), are among the worst-fitting of those considered here, and are manifestly inferior to a new model with optimal parameter settings. Results across all studies and datasets are highly concordant, and robust to alternative re-analyses, for instance: using individual-level versus aggregate data; using nonparametric versus parametric tests; and across variants of the basic model.

We employ a behavioral-economic equation put forward by Hursh and Silberberg (2008) to explain human consumption behavior among substitutable food brands, applying a consumer-choice model the behavioral perspective model (BPM; Foxall, 1990/2004, 2005). In this study, we apply the behavioral-economic equation to human economic consumption data. We attempt to find the variation pattern of essential value across brand groups differing in utilitarian reinforcement and informational reinforcement. The BPM denotes that consumers show less price responsiveness in closed settings and more price responsiveness in open settings. We,also examine whether consumers are more sensitive to price changes in an open setting, where many alternatives are available, and vice versa. We find that (a) essential value varies across different brand groups within the same products; (b) brands with higher levels of utilitarian reinforcement showed larger essential value; (c) brands with higher levels of informational reinforcement showed larger essential value; and (d) the essential value of brands varies inversely with the degree of openness of consumer settings.

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We consider Apollonian circle packings of a half Euclidean plane. We give necessary and sufficient conditions for two such packings to be related by a Euclidean similarity (that is, by translations, reflections, rotations and dilations) and describe explicitly the group of self-similarities of a given packing. We observe that packings with a non-trivial self-similarity correspond to positive real numbers that are the roots of quadratic polynomials with rational coefficients. This is reflected in a close connection between Apollonian circle packings and continued fractions which allows us to completely classify such packings up to similarity.

This paper makes the first attempt to present explicit empirical evidence that market inefficiency can be multi-dimensional. Testing the Efficient Market Hypothesis (EMH) over 76 stock indices using 17 best established indicators (e.g. runs test), we show that most indices exhibit some type(s) of anomaly and that indicators differ from each other in terms of statistical power and/or the type of anomaly detected. A principal components analysis (PCA) demonstrates that indicators group along orthogonal dimensions, and hence a market can exhibit short-term memory, long-term memory and/or calendar effects, which are all distinct sources of possible inefficiency. This research presents statistical evidence on the extent and nature of market inefficiency, offers possible explanations for conflicting previous findings, and provides new insights into studying market efficiency. (C) 2012 Elsevier B.V. All rights reserved.

This paper shows that tests of Random Number Generators (RNGs) may be used to test the Efficient Market Hypothesis (EMH). It uses the Overlapping Serial Test (OST), a standard test in RNG research, to detect anomalous patterns in the distribution of sequences of stock market movements up and down. Our results show that most stock markets exhibit idiosyncratic recurrent patterns, contrary to the efficient market hypothesis; also that OST detects a different kind of non-randomness to standard econometric long- and short-memory tests. Exposure of these anomalies should contribute to making markets more efficient. (C) 2012 Elsevier B.V. All rights reserved.

Essential value is defined by Hursh and Silberberg (2008) as the value of reinforcers, presented in an exponential model (Equation 1). This study extends previous research concerned with animal behavior or human responding in therapeutic situations. We applied 9 available demand curves to consumer data that included 10,000+ data points collected from 1,600+ consumers during 52 weeks. The exponential approach of Hursh and Silberberg is highly predictive and capable of comparing the strength of reinforcers among different products. Moreover, this study proposes a second exponential model (Equation 2) that incorporates utilitarian and informational reinforcement, posited by the behavioral perspective model (BPM; Foxall 1990/2004), into the Hursh–Silberberg model, to ascertain the influence of these variables on demand. Results for Equation 2 indicate greater predictive capacity than results for Equation 1 across products. Hence, utilitarian and informational reinforcement influence the consumption of economic goods by human consumers. Both models, calculated for different products, fit the data well, and their parameters show good reliability across time periods.

Delay discounting is often considered a universal feature of human choice behavior, but there is controversy over whether it is an individual difference that reflects an underlying psychological trait or a domain-specific behavior. Trait influence on discounting would manifest in (a) highly correlated discount rates for all decisions, regardless of context, and (b) the reflection of discounting behavior in psychometric measures of individual difference. We examined these propositions for consumers making hypothetical decisions with respect to financial returns, health outcomes, and vacation alternatives. Questionnaires were employed to assess discounting rates, and respondents' (N = 74) cognitive styles were measured by the Kirton Adaption-Innovation Inventory (KAI). Results suggested that discounting is a domain-specific behavior rather than a single trait. Individual discounting rates differed markedly among product contexts; moreover, individual differences in cognitive style were not related to discounting behavior.

Recent work suggests that sound symbolism in brand names conveys marketing–relevant messages. However, if the customer sees a brand name rather than hears it, visual characteristics of the letters may convey messages of their own. These may conflict with or reinforce the message conveyed by sound symbolism of the name. Study 1 replicates the essence of the sound symbolism effect claimed in recent work. Study 2 shows that the visual characteristics of letters provide a plausible alternative explanation of these findings. Study 3 manipulates the visual characteristics in the brand name letters and reverses the previously found direction of sound symbolism effects. The findings suggest that powerful visual messages are present in brand names, and that because of confounding, the contribution sound symbolism makes to the brand name may not always act as thought.