A simple derivation of Prelec's probability weighting function

Since Kahneman and Tversky [(1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263-291], it has been generally recognized that decision makers overweight low probabilities. Of the several weighting functions that have been proposed, that of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528] has the attractions that it is parsimonious, consistent with much of the available empirical evidence and has an axiomatic foundation. Luce [(2001). Reduction invariance and Prelec's weighting functions. Journal of Mathematical Psychology, 45, 167-179] provided a simpler derivation based on reduction invariance, rather than compound invariance of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528]. This note introduces a behavioral assumption that we call power invariance and provides a simple derivation of Prelec's function. Thus, we have three a priori different behavioral assumptions all leading to Prelec's function.     

The general psychophysical differential equation: A comparison of three specifications

The general psychophysical differential equation, dy/dx = W₂(y)/W₁(x), with the solution y = f(x), where x and y are subjective variables and W₁ and W₂ their subjective Weber functions, is (a) compared with a corresponding functional equation, and (b) studied from a stochastic point of view by error calculus, Methods for evaluating and handling divergences are proposed and illustrated for a number of combinations of Weber functions. It is shown that either the differential: and the functional equations have the same solution or the difference between the solutions is negligible compared to empirical scatter. The error calculus gives the same result: either no error at all or a negligible one.     

The magnitude effect: Temporal discount rates and restaurant tips

Decision makers show a larger subjective temporal discount rate for small magnitudes than for large ones. That is, they demand a larger percent increase in value to compensate for a delay when they are waiting for a small amount of money than for a large amount. Prelec and Loewenstein (1991; see also Loewenstein & Prelec, 1992) proposed an increasing proportional sensitivity account of this magnitude effect. This account surmises that the magnitude effect stems from the utility function for money and is consequently not unique to intertemporal choice. One study tested this prediction by demonstrating the magnitude effect in two domains: intertemporal choice and tipping for restaurant meals, haircuts, and taxi rides. In intertemporal choice, subjects showed a larger discount rate for smaller monetary amounts. They also tipped a larger percentage on small bills than on large bills. Thus, both domains showed the magnitude effect; however, the size of the effect was not well correlated between domains.     

Resolution time and the coding of arithmetic relations on supraliminally different visual extents

Comparison time for pairs of vertical-line stimuli, sufficiently different that they can be errorlessly discriminated with respect to visual extent, was examined as a function of arithmetic relations (physical ratio and difference) on members of the pair. Arithmetic relations are coded very precisely by judgment time: Responses slow as stimulus ratios approach one with difference fixed, and as stimulus differences approach zero with ratio fixed. Most models which assume a simple (Difference or Ratio) resolution rule operating on independent sensations require judgment time to depend on either ratios or on differences but not on both. Further tests showed both an index based on median judgment times and a confusion index based on pairs of observed judgment times, satisfied the requirements for a Positive Difference Structure. One representation of these data, which remains acceptable through all analyses, is a Difference resolution rule operating on sensations determined by a power psychophysical function with β < 1. Specifically, L(x, y) = F{ψ(x) ? ψ(y)} + R, where L(x, y) is the judgment time with the stimulus pair x and y, ψ(x) = Ax^β + C, R is a positive constant, and F is a continuous monotone decreasing function.     

Finding one's meaning: a test of the relation between quantifiers and integers in language development

We explored children’s early interpretation of numerals and linguistic number marking, in order to test the hypothesis (e.g., Carey (2004). Bootstrapping and the origin of concepts. Daedalus, 59-68) that children’s initial distinction between one and other numerals (i.e., two, three, etc.) is bootstrapped from a prior distinction between singular and plural nouns. Previous studies have presented evidence that in languages without singular-plural morphology, like Japanese and Chinese, children acquire the meaning of the word one later than in singular-plural languages like English and Russian. In two experiments, we sought to corroborate this relation between grammatical number and integer acquisition within English. We found a significant correlation between children’s comprehension of numerals and a large set of natural language quantifiers and determiners, even when controlling for effects due to age. However, we also found that 2-year-old children, who are just acquiring singular-plural morphology and the word one, fail to assign an exact interpretation to singular noun phrases (e.g., a banana), despite interpreting one as exact. For example, in a Truth-Value Judgment task, most children judged that a banana was consistent with a set of two objects, despite rejecting sets of two for the numeral one. Also, children who gave exactly one object for singular nouns did not have a better comprehension of numerals relative to children who did not give exactly one. Thus, we conclude that the correlation between quantifier comprehension and numeral comprehension in children of this age is not attributable to the singular-plural distinction facilitating the acquisition of the word one. We argue that quantifiers play a more general role in highlighting the semantic function of numerals, and that children distinguish between numerals and other quantifiers from the beginning, assigning exact interpretations only to numerals.     

Reduction Invariance and Prelec's Weighting Functions

Within the framework of separable utility theory, a condition, called reduction invariance, is shown to be equivalent to the 2-parameter family of weighting functions that Prelec (1998) derived from the condition called compound invariance. Reduction invariance, which is a variant on the reduction of compound gambles, is appreciably simpler and more easily testable than compound invariance, and a simpler proof is provided. Both conditions are generalized leading to more general weighting functions that include, as special cases, the families of functions that Prelec called exponential-power and hyperbolic logarithm and that he derived from two other invariance principles. However, of these various families, only Prelec's compound-invariance family includes, as a special case, the power function, which arises from the simplest probabilistic assumption of reduction of compound gambles. Copyright 2001 Academic Press.     

Efficient estimation of Weber’s <Emphasis Type="Italic">W</Emphasis>

Many studies rely on estimation of Weber ratios (W) in order to quantify the acuity an individual’s approximate number system. This paper discusses several problems encountered in estimating W using the standard methods, most notably low power and inefficiency. Through simulation, this work shows that W can best be estimated in a Bayesian framework that uses an inverse (1/W) prior. This beneficially balances a bias/variance trade-off and, when used with MAP estimation is extremely simple to implement. Use of this scheme substantially improves statistical power in examining correlates of W.     

Properties of reverse hazard functions

For continuous distributions reverse hazard is defined as the probability density divided by the cumulative probability, F(x); whereas the usual hazard function is the density divided by the survivor function, 1−F(x). Reverse hazard corresponds to the conditional density of an immediate failure or state change, conditioned by the fact that the state change occurred. For example, of all the items that failed, the proportion of those items that immediately failed is reverse hazard. Reverse hazard exhibits many symmetrical properties with hazard. In this paper a set of theorems are developed that explicate the properties of reverse hazard for both continuous and discrete probability distributions. Taken together, hazard and reverse hazard are a powerful set of theoretical constructs that are valuable for understanding stochastic systems.     

Inference and exact numerical representation in early language development

How do children as young as 2 years of age know that numerals, like one, have exact interpretations, while quantifiers and words like a do not? Previous studies have argued that only numerals have exact lexical meanings. Children could not use scalar implicature to strengthen numeral meanings, it is argued, since they fail to do so for quantifiers [Papafragou, A., & Musolino, J. (2003). Scalar implicatures: Experiments at the semantics–pragmatics interface. Cognition, 86, 253–282]. Against this view, we present evidence that children’s early interpretation of numerals does rely on scalar implicature, and argue that differences between numerals and quantifiers are due to differences in the availability of the respective scales of which they are members. Evidence from previous studies establishes that (1) children can make scalar inferences when interpreting numerals, (2) children initially assign weak, non-exact interpretations to numerals when first acquiring their meanings, and (3) children can strengthen quantifier interpretations when scalar alternatives are made explicitly available.     

Affine representations in psychophysics and the near-miss to Weber’s law

A theoretical account for the near-miss to Weber’s law in the form of a power function, with a special emphasis on the interpretation of the exponent, was proposed by Falmagne [Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press] within the framework of a subtractive representation, P(x,y)=F(u(x)−g(y)). In this paper, we examine a more general affine representation, P(x,y)=F(u(x)h(y)+g(y)). We first obtain a uniqueness theorem for the affine representation. We then study the conditions that force an affine representation to degenerate to a subtractive one. Part of that study involves the case for which two different affine representations co-exist for the same data. We also show that the balance condition P(x,y)+P(y,x)=1 constrains an affine representation to be a special kind of subtractive representation, a Fechnerian one. We further show that Falmagne’s power law takes on a special form for a so-called weakly balanced system of probabilities, in which case the affine representation is Fechnerian. Finally, following Iverson [Iverson, G.J. (2006a). Analytical methods in the theory of psychophysical discrimination I: Inequalities, convexity and integration of just noticeable differences. Journal of Mathematical Psychology, 50, 271-282], we generalize the Fechner method to construct the sensory scales in a weakly balanced affine representation by integrating (derivatives of) just noticeable differences.     

Extensions of Narens' theory of ratio magnitude estimation

Stevens postulated that the responses of a participant in a ratio scaling experiment can be used directly to construct a psychophysical function. Today, it is generally accepted that the axioms of commutativity and multiplicativity are crucial for the interpretation of the subjects' ratio scaling behaviour. Empirical findings provide evidence that commutativity holds, whereas multiplicativity fails to hold across different sensory modalities. This shows that, in principle, Stevens' direct scaling methods yield measurements on a ratio scale level, but that the numerals occurring in a ratio scaling experiment cannot be taken at face value. Thus, Narens and others introduced a transformation function f, which converts the numerals used in an experiment into the latent mathematical numbers. The aim of the present paper is to specify the (unknown) shape of the transformation function f, by analysing different extensions of the multiplicative property. The results provide evidence that f is either a power function or a logarithmic function.     

Judgments of differences and ratios of numerals

Each subject performed two tasks, dividing a line segment so that either (a) theratio of subjective lengths corresponded to the ratio of the magnitudes of two numerals or (b) thedifference in length was proportional to the numerical difference. Had subjects actually performed two operations on the same scale, the responses would have been nonmonotonically related. Instead, data for the two tasks were nearly identical and ordinally compatible with either a ratio or a subtractive model. The ratio model implied scale values for numerals that were a positively accelerated function of numerical value, inconsistent with previous results. With a nonlinear response function for graphic length, the subtractive model fit well, yielding scale values that were a negatively accelerated function of numerical value and a linear function of previously obtained scales. These results, together with other recent findings, suggest that subjects may perform the same operation in spite of instructions to judge “ratios” or “differences” and that this operation can be best represented by a subtractive model.     

Similarity of rectangles: An analysis of subjective dimensions

Two defining properties of psychological dimensions (intradimensional subtractivity and interdimensional additivity) are introduced and their consequences, formulated in terms of an ordinal dissimilarity scale, are derived. These consequences are investigated using dissimilarity judgments between rectangles to determine which of two alternative dimensional structures area (A) and shape (S), or width (W) and height (H), satisfies additivity and/or subtractivity. The results show that neither dimensional structure is acceptable, although A × S provides a better account for the data of most Ss than does W × H. Tests of relative straightness show that A is the least “curved” of the four attributes. Methodological and substantive implications of the study are discussed.     

On minima of discrimination functions

A discrimination function ψ(x,y) assigns a measure of discriminability to stimulus pairs x,y (e.g., the probability with which they are judged to be different in a same-different judgment scheme). If for every x there is a single y least discriminable from x, then this y is called the point of subjective equality (PSE) for x, and the dependence h(x) of the PSE for x on x is called a PSE function. The PSE function g(y) is defined in a symmetrically opposite way. If the graphs of the two PSE functions coincide (i.e., g≡h⁻¹), the function is said to satisfy the Regular Minimality law. The minimum level functions are restrictions of ψ to the graphs of the PSE functions. The conjunction of two characteristics of ψ, (1) whether it complies with Regular Minimality, and (2) whether the minimum level functions are constant, has consequences for possible models of perceptual discrimination. By a series of simple theorems and counterexamples, we establish set-theoretic, topological, and analytic properties of ψ which allow one to relate to each other these two characteristics of ψ.     

Digit identity influences numerical estimation in children and adults

Learning the meanings of Arabic numerals involves mapping the number symbols to mental representations of their corresponding, approximate numerical quantities. It is often assumed that performance on numerical tasks, such as number line estimation (NLE), is primarily driven by translating from a presented numeral to a mental representation of its overall magnitude. Part of this assumption is that the overall numerical magnitude of the presented numeral, not the specific digits that comprise it, is what matters for task performance. Here we ask whether the magnitudes of the presented target numerals drive symbolic number line performance, or whether specific digits influence estimates. If the former is true, estimates of numerals with very similar magnitudes but different hundreds digits (such as 399 and 402) should be placed in similar locations. However, if the latter is true, these placements will differ significantly. In two studies (N = 262), children aged 7–11 and adults completed 0–1000 NLE tasks with target values drawn from a set of paired numerals that fell on either side of “Hundreds” boundaries (e.g., 698 and 701) and “Fifties” boundaries (e.g., 749 and 752). Study 1 used an atypical speeded NLE task, while Study 2 used a standard non‐speeded NLE task. Under both speeded and non‐speeded conditions, specific hundreds digits in the target numerals exerted a strong influence on estimates, with large effect sizes at all ages, showing that the magnitudes of target numerals are not the primary influence shaping children's or adults’ placements. We discuss patterns of developmental change and individual difference revealed by planned and exploratory analyses.     

Multilinear representations for ordinal utility functions

An ordinal utility function u over two attributes X₁, X₂ is additive if there exists a strictly monotonic function ϕ such that ϕ(u) = v₁(x₂) + v₂(x₂) for some functions v₁, v₂. Here we consider the class of ordinal utility functions over n attributes for which each pair of attributes is additive, but not necessarily separable, for any fixed levels of the remaining attributes. We show that while this class is more general than those that are ordinally additive, the assessment task is of the same order of difficulty, and involves a hierarchy of multilinear rather than additive decompositions.     

Christmas and Subjective Well-Being: a Research Note

According to Holmes and Rahe, Journal of Psychosomatic Research, 11(2), 213–218, (1967), Christmas is a critical life event that may cause feelings of stress that, in turn, can lead to reduced subjective well-being (SWB) and health problems. This study uses a quantitative approach and large-scale survey data to assess whether or not respondents in European countries indicate lower SWB before and around Christmas. Precisely, respondents interviewed in the week before Christmas or at Christmas holidays are compared to respondents who are questioned at other times throughout the year. Moreover, the assumption is tested if religious denomination and religiousness moderate the association between Christmas and SWB. Main findings suggest that the Christmas period is related to a decrease in life satisfaction and emotional well-being. However, Christians, particularly those with a higher degree of religiousness, are an exception to this pattern.     

The parameters in the near-miss-to-Weber's law

Many empirical data support the hypothesis that the sensitivity function grows as a power function of the stimulus intensity. This is usually referred to as the near-miss-to-Weber's law. The aim of the paper is to examine the near-miss-to-Weber's law in the context of psychometric models of discrimination. We study two types of psychometric functions, characterized by the representations P_a(x)=F(ρ(a)x^γ(a)) (type A), and P_a(x)=F(γ(a)+ρ(a)x) (type B). A central result shows that both types of psychometric functions are compatible with the near-miss-to-Weber's law. If a representation of type B exists, then the exponent in the near-miss is necessarily a constant function, that is, does not depend on the criterion value used to define “just noticeably different”. If, on the other hand, a representation of type A exists, then the exponent in the near-miss-to-Weber's law can vary with the criterion value. In that case, the parameters in the near-miss co-vary systematically.     

Further evidence for addition and numerical competence by a Grey parrot (Psittacus erithacus)

A Grey parrot (Psittacus erithacus), able to quantify sets of eight or fewer items (including heterogeneous subsets), to sum two sequentially presented sets of 0–6 items (up to 6), and to identify and serially order Arabic numerals (1–8), all by using English labels (Pepperberg in J Comp Psychol 108:36–44, 1994; J Comp Psychol 120:1–11, 2006a; J Comp Psychol 120:205–216, 2006b; Pepperberg and Carey submitted), was tested on addition of two Arabic numerals or three sequentially presented collections (e.g., of variously sized jelly beans or nuts). He was, without explicit training and in the absence of the previously viewed addends, asked, “How many total?” and required to answer with a vocal English number label. In a few trials on the Arabic numeral addition, he was also shown variously colored Arabic numerals while the addends were hidden and asked “What color number (is the) total?” Although his death precluded testing on all possible arrays, his accuracy was statistically significant and suggested addition abilities comparable with those of nonhuman primates.     

Comma N' cents in pricing: The effects of auditory representation encoding on price magnitude perceptions

Numbers and prices can be processed and encoded in three different forms: 1) visual [based on their written form in Arabic numerals (e.g., 72)], 2) verbal [based on spoken word-sounds (e.g., “seventy” and “two”), and 3) analog (based on judgments of relative “size” or amount (e.g., more than 70 but less than 80)]. In this paper, we demonstrate that including commas (e.g., $1599 vs. $1599) and cents (e.g., $1599.85 vs. $1599) in a price's Arabic written form (i.e., how it is perceived visually) can change how the price is encoded and represented verbally in a consumer's memory. In turn, the verbal encoding of a written price can influence assessments of the numerical magnitude of the price. These effects occur because consumers non-consciously perceive that there is a positive relationship between syllabic length and numerical magnitude. Three experiments are presented demonstrating this important effect.