Decision policies minimizing risk in a multistage betting game

Considered in this paper is a decision task which has been employed to study multistage betting behavior. When the task commences, a decision maker (DM) is provided with some capital x (x > 0) which he is required to allocate over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_i=1^mp_i = 1). If the amount y_i is allocated to alternative i (y_i ≥ 0, Σ_i=1^my_i = x) and alternative i obtains, DM's capital for the next stage of the game becomes y_ir_i, where r_i (r_i > 0) is the return per unit allocated to alternative i. The task consists of N stages.Defining risk in terms of the mean and variance of DM's bets, and assuming that the minimization of risk is DM's objective, decision policies satisfying this objective are derived in closed form and their testable properties are briefly discussed.     

On the consistency of intramodal intensity matching in olfaction

Magnitude matchings of odor intensity were obtained for three chemical compounds (hydrogen sulfide, pyridine, and dimethyl disulfide) in four experiments. An equal-sensation function in the form of a power function described the data well. This equation can be written neglecting the multiplicative constant, Ф_i = Ф_k ^b _ik, where i and k represent two different odorants. The exponent b_ik, where the first index stands for matching continuum and the second index for reference continuum, was fairly well predicted according to the formulae, (ie, 264, 2) where i, j, and k are three different odorants. The results validate the equal-sensation function and indicate implicitly that intensity matchings between different intramodal continua are approximately transitive and symmetric with respect to odor.     

On two types of deviation from the matching law: bias and undermatching

Data on choice generally conform closely to an equation of the form: log(B₁/B₂)=a log(r₁/r₂+log k, where B₁ and B₂ are the frequencies of responding at Alternatives 1 and 2, r₁ and r₂ are the obtained reinforcement from Alternatives 1 and 2, and a and k are empirical constants. When a and k equal one, this equation is equivalent to the matching relation: B₁/B₂=r₁/r₂. Two types of deviation from matching can occur with this formulation: a and k not equal to one. In some experiments, a systematically falls short of one. This deviation is undermatching. The reasons for undermatching are obscure at present. Some evidence suggests, however, that factors favoring discrimination also favor matching. Matching (a=1) may represent the norm in choice when discrimination is maximal. When k differs from one, its magnitude indicates the degree of bias in choice. The generalized matching law predicts that bias should take this form (adding a constant proportion of responding to the favored alternative). Data from a variety of experiments indicate that it generally does.     

How one gambles if one must: Effects of differing return rates on multistage betting decisions

In the multistage betting game (MBG), a decision maker (DM) is provided with some capital x which he is required to bet over m (m > 1) mutually exclusive and collectively exhaustive alternatives, each of which occurs with probability p_i (p_i > 0, i = 1,…, m; Σ_{i = 1}^mp_i = 1). If y_i is bet on alternative i (y_i ≥ 0, Σ_{i = 1}^my_i = x) and alternative i obtains, the DM's capital for the next stage is y_ir_i, (r_i > 0). The MBG lasts until either the DM loses his capital or N stages elapse, whichever comes first. Each of six subjects participated in six sessions consisting of several hundred 3-alternative MBG stages. A within-subject design assigned negative expected value (EV) bets to the first three sessions and positive EV bets to three more sessions. Significant effects were found due to return rate, capital size, homogeneous runs of either wins or losses, and individual differences. Four maximization of expected utility and two minimization of risk models were presented and tested. A modified logarithmic utility model is proposed, which provides the best fit to the data. The implications of the results and directions for further research are briefly discussed.     

Tone-on-tone binaural masking with an antiphasic masker

One pure tone (500 Hz) was used to mask another pure tone of the same frequency and duration. The signal and masker were presented in three binaural stimulus configurations, Mo-So, Mo-Sπ, and Mπ -So. The Mo-So condition is a diotic condition; the Mo-Sβ condition is a dichotic condition in which the masker is homophasic and the signal is antiphasic; and the Mπ-So condition is a dichotic condition in which the masker is antiphasic and the signal homophasic. The signal-to-masker ratio required for detection was measured in each condition as a function of the signal-plus-masker phase angle, α. The data showed that the difference in detection between the Mo-Sπ and Mπ-So conditions varied between 0 dB when α=0 deg and 11 dB when α=90 deg. The difference in detection between the Mo-Sπ and Mπ-So conditions is due to the Os’ sensitivities to the interaural phase difference present in the Mo-Sπ and Mπ-So conditions. The results are similar to those obtained in investigations involving lateralization. The difference between detection in either the Mo-Sπ or Mπ-So condition and that in the Mo-So condition (the MLD) was variable due to differences in the Os’ sensitivities in the Mo-So condition.     

Comparatal dispersion,a measure of accuracy of judgment

It is suggested that the ambiguity of aset of paired comparison judgments may be measured by the quantity _i ² + _j ² – 2r _{ij ii}, This quantity is termed thecomparatal dispersion. A simultaneous solution for scale values and ratios of comparatal dispersions has been presented and applied to some data on food preferences.This research was jointly supported in part by Princeton University, the Office of Naval Research under contract Nonr-1858(15), and the National Science Foundation under grant NSF G-642, and in part by Educational Testing Service. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Stochastic processes and the guttman simplex

The problem of whether a precise connection exists between stochastic processes of the type considered in mathematical learning theory and the Guttman simplex is investigated. The approach is to consider a class of models that characterize the sequential properties of discrete data, and to derive a set of conditions which a model must satisfy in order to generate inter-trial correlationsr _ij with the ‘perfect simplex’ property:r _ik =r _ij r _jk for all trialsi<j<k. It is shown that the Chapman-Kolomogorov Equations provide a necessary and sufficient condition for this property to hold. It follows that a process which is Markovian in the errors and successes will have this property. It is also shown that, if certain reliability assumptions are introduced, then the all-or-none model has the simplex property if the appropriate correlation coefficients are corrected for attenuation. This research was supported by a grant from the United States Public Health Service, NIMH 07722.     

Generalized procrustes analysis

SupposeP _i ⁽ⁱ⁾ (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ _i=1 ^m Σ _i=1 ⁿ Δ²(P _j ⁽ⁱ⁾ G _i ), whereG _i is the centroid of them pointsP _i ⁽ⁱ⁾ (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.     

Energy,compressibility and covalency in the carbon subgroup

It is shown that a scaling relation of the type K V ₀/ E ₀ constant, where K₀, V₀ and E₀ are the bulk modulus, volume and total energy respectively at normal pressure, holds for the C subgroup of the Mendeleev periodic table of elements, which consists of C itself, Si, Ge, Sn and Pb. This means that so-called covalency does not contribute significantly to the compressibility of the corresponding substances. The extreme high value of the bulk modulus of diamond is clearly the result of its highest known atomic density.     

The role of binaural and fundamental frequency difference cues in the identification of concurrently presented vowels

The relative importance of voice pitch and interaural difference cues in facilitating the recognition of both of two concurrently presented synthetic vowels was measured. The interaural difference cues used were an interaural time difference (400 μsec ITD), two magnitudes of interaural level difference (15 dB and infinite ILD), and a combination of ITD and ILD (400 μsec plus 15 dB). The results are analysed separately for those cases where both vowels are identical and those where they are different. When the two vowels are different, a voice pitch difference of one semitone is found to improve the percentage of correct reports of both vowels by 35.8% on average. However, the use of interaural difference cues results in an improvement of 11.5% on average when there is a voice pitch difference of one semitone, but only a non-significant 0.1% when there is no voice pitch difference. When the two vowels are identical, imposition of either a voice pitch difference or binaural difference reduces performance, in a subtractive manner. It is argued that the smaller size of the interaural difference effect is not due to a “ceiling effect” but is characteristic of the relative importance of the two kinds of cues in this type of experiment. The possibility that the improvement due to interaural difference cues may in fact be due to monaural processing is discussed. A control experiment is reported for the ITD condition, which suggests binaural processing does occur for this condition. However, it is not certain whether the improvement in the ILD condition is due to binaural processing or use of the improvement in signal-to-noise ratio for a single vowel at each ear.     

An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

A method of multidimensional mapping is described which constructs a configuration of points {P_i} in a Euclidean map of Riemannian space of constant curvature (hyperbolic, Euclidean, and elliptic) from the dissimilarity matrix (d_ij). The method was applied to the distance matrix in visual space where stimulus points Q_i were either small light points in the dark or small black points in the illuminated field surrounded by white curtains and d_ij represent scaled values of perceptual distances. Configuration of points {Q_i} were at intersections of parallel or distance alleys and horopters for the subject in the horizontal plane of the eye level. In contrast to the theoretical equations for {Q_i} by Luneburg and Blank, no a priori assumption on mapping functions between {Q_i} and {P_i} is necessary in this procedure to fit theoretical curves to {P_i} in the Euclidean map. The data were accounted for better by equations in the hyperbolic plane than by ones in the Euclidean plane. Discussions are made on robustness of Euclidean representation and on how to approach geometry of visual space as a dynamic entity under more natural conditions than the traditional frameless condition for alley and horopter experiments.     

An evaluation of the effectiveness of countermeasures for improving the safety of dilemma zones: A driving simulator study

The main objective of this driving simulator study is to analyze the behavior of the driver at the start of the yellow signal of a signalized rural intersection and identify the most effective countermeasures for tackling the dilemma zone, namely an area on the intersection approach where vehicles at the start of the yellow phase can neither safely stop before the stop line nor cross the intersection. The following countermeasures were tested in the study on a sample of 48 drivers: green signal countdown timers, GSCT (C₁); a new pattern of vertical and horizontal warning signs (C₂); and an advanced on-board driver assistance system based on augmented reality (AR) and connected vehicle technologies (C₃). These countermeasures were tested and compared to a baseline condition (B) where no countermeasures were applied. Based on the results of this study, the C₂ and C₃ countermeasures have proven to be valid tools for reducing driver indecision when approaching signalized intersections at the start of the yellow signal. In fact, using C₂ and C₃, the length of the dilemma zone was equal to 30 m and 36 m, respectively, with a reduction of about 50%, as compared to the baseline condition (B). Moreover, a reduced number of false behaviors was recorded, as well as a greater consistency in driver decision-making behaviors. Conversely, the C₁ countermeasure did not lead to a significant improvement in the dilemma zone: an unnecessary increase in early stop rates was recorded, resulting in reduced intersection efficiency and operations.     

A strong developmental theory of field dependence-independence

Assume an X-linked gene in two alleles mediates performance on field dependent-independent tests such as the rod-and-frame test. Only the recessive gene with relative frequency q facilitates field independence. Other genotypes lead to field dependence. Under a simple genetic model, field dependence-independence may be viewed as outcomes of a discrete random variable B with field independent and dependent probabilities π_iq and 1 ? π_iq for men, and π_iq² and 1 ? π_iq² for women, respectively. The parameter π_i is a maturational ageindexed parameter, 0 < π_i ≤ 1, monotonically increasing with development until maturity when π_k = 1. Observations of performance are made on a random variable W of the form W = B + N, where N is normal in distribution independent of B; N represents a composite of influences including error. The model implies testable age-related between- and within-sex predictions regarding E(W) and Var(W), predictions which appear to coincide with major empirical findings; it also generates novel predictions. For instance, W is a mixture of normals distribution. The model is briefly evaluates in two data sets.     

On the size of refutation Kripke models for some linear modal and tense logics

LetL be any modal or tense logic with the finite model property. For eachm, definer _L (m) to be the smallest numberr such that for any formulaA withm modal operators,A is provable inL if and only ifA is valid in everyL-model with at mostr worlds. Thus, the functionr _L determines the size of refutation Kripke models forL. In this paper, we will give an estimation ofr _L (m) for some linear modal and tense logicsL.     

Virtual Modality

Model-theoretic 1-types overa given first-order theory T may be construed as natural metalogical miniatures of G. W. Leibniz' ``complete individual notions', ``substances' or ``substantial forms'. This analogy prompts this essay's modal semantics for an essentiallyundecidable first-order theory T, in which one quantifies over such ``substances' in a boolean universe V(C), where C is the completion of the Lindenbaum-algebra of T.More precisely, one can define recursively a set-theoretic translate of formulae _N of formulae of a normal modal theory T_m based on T, such that the counterpart `<sub>i</sub>' of a the modal variable `x_i' of L(T_m) in this translation-scheme ranges over elements of V(C) that are 1-types of T with value 1 (sometimes called `definite' C-valued 1-types of T).The article's basic completeness-result (2.13) then establishes that varphi; is a theorem of T<sub>m</sub> iff [[ <sub>N</sub> () is aconsequenceof <sub>N</sub> (T<sub>m</sub>) for each extension N of T which is a subtheory of the canonical generic theory (ultrafilter) u]] = 1 – or equivalently, that T<sub>m</sub> is consistent iff[[there is anextension N of T such that N is a subtheory of the canonical generic theory u, and <sub>N</sub>() for all in T<sub>m</sub>]] > 0.The proof of thiscompleteness-result also shows that an N which provides a countermodel for a modally unprovable – or equivalently, a closed set in the Stone space St(T) in the sense of V(C) – is intertranslatable with an `accessibility'-relation of a closely related Kripke-semantics whose `worlds' are generic extensions of an initial universe V via C.This interrelation providesa fairly precise rationale for the semantics' recourse to C-valued structures, and exhibits a sense in which the boolean-valued context is sharp.     

Transcending Lindemann's predictions for the melting temperatures of metals and for the long-wavelength limit of their liquid structure factors

Lindemann's melting criterion remains useful. However, one prediction it makes for liquid metals (our focus here) is that the long-wavelength limit of the structure factor S(q) at freezing, S _{T m} (0), where T _m is the melting temperature, is a universal constant. For 34 metals we have calculated S _{T m} (0) from input data, which is essentially the measured T _m and the surface thickness L, defined near freezing as the product of isothermal compressibility and surface tension. To complete the characterization of S _{T m} (0) we fit to one metal, chosen as Rb, for which S _{T m} (0) is well established experimentally. For a wide variety of metals considered, S _{T m} (0) is then found to vary by a factor of 10.     

Conjunctions,Disjunctions and Lewisian Semantics for Counterfactuals

Consider the reasonable axioms of subjunctive conditionals (1) if p q ₁ and p q ₂ at some world, then p (q ₁ & q ₂) at that world, and (2) if p ₁ q and p ₂ q at some world, then (p ₁ ∨ p ₂) q at that world, where p q is the subjunctive conditional. I show that a Lewis-style semantics for subjunctive conditionals satisfies these axioms if and only if one makes a certain technical assumption about the closeness relation, an assumption that is probably false. I will then show how Lewisian semantics can be modified so as to assure (1) and (2) even when the technical assumption fails, and in fact in one sense the semantics actually becomes simpler then.     

The optimum decision rules for the oddity task

This paper presents the optimum decision rule for an m-interval oddity task in whichm-1 intervals contain the same signal and one is different or odd. The optimum decision rule depends on the degree of correlation among observations. The present approach unifies the different strategies that occur with “roved” or “fixed” experiments (Macmillan & Creelman, 1991, p. 147). It is shown that the commonly used decision rule for anm-interval oddity task corresponds to the special case of highly correlated observations. However, as is also true for thesame-different paradigm, there exists a different optimum decision rule when the observations are independent. The relation between the probability of a correct response andd’ is derived for the three-interval oddity task. Tables are presented of this relation for the three-, four-, and five-interval oddity task. Finally, an experimental method is proposed that allows one to determine the decision rule used by the observer in an oddity experiment.     

Masking of tones by tones and of noise by noise

Using a two-alternative temporal forced-choice technique, two binaural detection experiments were performed. In the first, the detectability of a 250-Hz 128-msec tonal signal masked by a gated 70-dB SPL tone of the same frequency and duration was measured as a function of the level of the signal, the phase angle at which the signal was added to the masker, and the interaural phase difference of the signal. In the second experiment, the signal was a wideband (100-3,000 Hz) 128-msec Gaussian noise masked by a continuous Gaussian noise of the same bandwidth and coherent with the signal. The detectability of this noise signal was measured as a function of the same variables investigated in the first experiment. In both experiments detectability was found to follow a simple energy- or power-detection model when the interaural phase difference was 0 deg. When the interaural phase difference was 180 deg, the function relating the signal level required for a constant level of performance to the signal-masker phase angle is such that neither the Webster-Jeffress hypothesis nor Durlach’s E-C model accounts for the data. The data are reasonably well fit by a model proposed by Hafter and Carrier.