On a functional equation related to Thurstone models

If f and g are nonvanishing characteristic functions the functional equation g(s)g(t)g(?s ? t) = f(as)f(at)f(?as ? at) implies g(s) = e^ibsf(as), i.e., f and g corresponding to probability distributions of the same type. It is shown here that when f and g are allowed to vanish this equation also has solutions in which f and g correspond to distributions of different types. The practical implication is that there are nonequivalent. Thurstone models which cannot be discriminated by any choice experiment with three objects.     

Lexicographic additive differences

Several authors have identified sets of axioms for a preference relation ? on a two-factor set A × X which imply that ? can be represented by specific types of numerical structures. Perhaps the two best-known of these are the additive representation, for which there are real valued functions f_A on A and f_X on X such that (a, x) ? (b, y) if and only if f_A(a) + f_X(x) > f_A(b) + f_X(y), and the lexicographic representation which, with A as the dominant factor, has (a, x) ? (b, y) if and only if f_A(a) > f_A(b) or {f_A(a) = f_A(b) and f_X(x) > f_X(y)}. Recently, Duncan Luce has combined the additive and lexicographic notions in a model for which A is the dominant factor if the difference between a and b is sufficiently large but which adheres to the additive representation when the difference between a and b lies within what might be referred to as a lexicographic threshold. The present paper specifies axioms for ? which lead to a numerical model which also has a lexicographic component but whose local tradeoff structure is governed by the additive-difference model instead of the additive model. Although the additive-difference model includes the additive model as a special case, the new lexicographic additive-difference model is not more general than Luce's model since the former has a “constant” lexicographic threshold whereas Luce's model has a “variable” lexicographic threshold. Realizations of the new model range from the completely lexicographic representation to the regular additive-difference model with no genuine lexicographic component. Axioms for the latter model are obtained from the general axioms with one slight modification.     

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.     

Interval representations for interval orders and semiorders

This paper discusses two types of real interval representations for interval orders and semiorders ? on a set X of arbitrary cardinality. In each type, each x in X is mapped into a real interval F(x). The first model is: x ? y iff a < b for all a in F(x) and all b in F(y). The second is: x ? y iff sup F(x) < infF(y). Necessary and sufficient countability conditions are presented for the second model for interval orders and for semiorders; simpler sets of these conditions are shown to be sufficient for the first model. Some special properties for the representations are noted, including two monotonicity properties for the semiorder representation.     

Additional-delay schedules: A continuum of temporal contingencies by varying food delay

Pigeons performed on discrete-trial, temporally defined schedules in which the food delay (D) was adjusted according to the latency of the key peck (X) and two schedule parameters (t and A). The schedule function was D = A(t − X), where D is the experienced delay between a response and a reinforcer. The schedule parameter t is the maximum value below which the present contingencies occur. A is the additional delay to reinforcement for each second the response latency is shorter than the t value. When A = 0 s, the schedule is a continuous reinforcement schedule with immediate reinforcement. When A = 1 s, the schedule is a conjunctive fixed-ratio 1 fixed-time t-s schedule. When A approaches infinity, the schedule becomes a differential reinforcement of long latency schedule. The latencies for subjects with t = 10 s and t = 30 s were observed with the present schedules having seven values for A between 0 s and 11 s. In addition, the latencies for subjects for which t = 30 s were observed at an A value of 31 s to 41 s. As the A value increased, the latencies approached the t value for subjects for which t = 10 s. The latencies for 30-s-t subjects did not approach t, even when the A value was 41 s. The latencies for 10-s-t subjects at 11-s A value were longer than those under yoked conditions having exactly the same delays/interreinforcement intervals. These results demonstrated a continuum of latency related to the schedule continuum (value of A) at a small t value.     

Noticing and responding in a discrimination based on a distinguishing element

In learning a successive go/no-go discrimination between a positive display consisting of the elements A and B, and negative display consisting of A-alone, pigeons first trained to peck A shift to pecking the distinguishing B element. In order to learn whether or not the shift to B is facilitated by B's function as a signal of the reinforcer, apart from the direct reinforcement of B responses, six arrangements of the elements with respect to the food reinforcer were used. Discrete trials were terminated by a single peck or after 4 sec. The A and B elements were dots of different color. The most critical comparison was between two groups, both of which received no reinforcement for a response directed to B. In one case, B signaled the reinforcer: An A response was reinforced when B was present but not when A appeared alone. In the other, B signaled the absence of a reinforcer: An A response was nonreinforced when B was present, while the response to A-alone was reinforced. During training and in extinction many more reponses were made to B when it signaled the reinforcer than when it signaled its absence. It is concluded that in a discrimination between AB positive and A negative the shift from pecking A to pecking B is facilitated by B's role as a signal for the reinforcer even on trials in which the peck is not made to B. Results from certain other groups showed that, unless pecks shifted to B within the positive display, pecks to A continued on the negative, A-alone trials as well as on the positive AB trials.     

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.     

How accurate is size and distance perception for very far terrestrial objects? Function and causality

This study investigated absolute estimation of size and distance for natural and artificial objects at viewing distances of 1.1–15.3 km (Experiments 1 and 2) and 0.4–5.0 m (Experiment 3). The main results were that, regardless of distance range, size and distance estimates (S′ andD′) were related to objective size and distance (S andD), respectively, by a power function with an exponent of unity, but great individual differences in exponent were obtained for the far objects. The ratioS′/D′ was reasonably represented byS′/D′ =Kθ ⁿ andS′/D′ = tan(aθ +b), rather thanS′/D′ = tan θ, where θ is the visual angle. Partial correlations were obtained to examine whether (1) apparent size is determined by taking apparent distance into account or (2) both apparent size and apparent distance are determined directly by external stimuli. The combined data for the far objects and the data for the close objects showed that there were high correlations betweenS andS′ and betweenD andD′ and a low correlation betweenD′ andS′. The data of Experiment 2 showed that bothD′ andS′ were highly correlated withS D, and θ, and there was a high positive correlation betweenD′ andS′. It was suggested that the direct-perception model is valid under some situations, but the taking-into-account model is not supported in any set of data.     

Utility of extension of functional equations—when possible

Jean-Claude Falmagne observed in 1981 [On a recurrent misuse of classical functional equation result. Journal of Mathematical Psychology, 23, 190-193] that, even under regularity assumptions, not all solutions of the functional equation k(s+t)=k(s)+k(t), important in many fields, also in the theory of choice, are of the form k(s)=Cs. This is certainly so when the domain of the equation (the set of (s,t) for which the equation is satisfied) is finite. We mention an example showing that this can happen even on some infinite, open, connected sets (open regions). The more general equations k(s+t)=?(s)+n(t) and k(s+t)=m(s)n(t), called Pexider equations, have been completely solved on R². In case they are assumed valid only on an open region, they have been extended to R² and solved that way (the latter if k is not constant). In this paper their common generalization

k(s+t)=?(s)+m(s)n(t)     

Pseudocontingencies derived from categorically organized memory representations

Pseudocontingencies (PCs) allow for inferences about the contingency between two variables X and Y when the conditions for genuine contingency assessment are not met. Even when joint observations X _i and Y _i about the same reference objects i are not available or are detached in time or space, the correlation r(X _i ,Y _i ) is readily inferred from base rates. Inferred correlations are positive (negative) if X and Y base rates are skewed in the same (different) directions. Such PC inferences afford useful proxies for actually existing contingencies. While previous studies have focused on PCs due to environmental base rates, the present research highlights memory organization as a natural source of PC effects. When information about two attributes X and Y is represented in a hierarchically organized categorical memory code, as category-wise base rates p(X) and p(Y), the reconstruction of item-level information from category base rates will naturally produce PC effects. Three experiments support this contention. When the yes base rates of two respondents in four questionnaire subscales (categories) were correlated, recalled and predicted item-level responses were correlated in the same direction, even when the original responses to specific items within categories were correlated in the opposite direction.     

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.     

Binary choice probabilities: on the varieties of stochastic transitivity

Let {P_λ} denote the family of decisiveness relations {$\text{P}{}_{\mathrm{\lambda }}\text{:}\text{1}\text{2}\text{≤ λ < 1}$} with aP_λb if and only if P(a,b) > λ, where P is a binary choice probability function. Families in which all decisiveness relations are of the same type, such as all strict partial orders or all semiorders, are characterized by stochastic transitivity conditions. The conditions used for this purpose differ in various ways from the traditional forms of strong, moderate, and weak stochastic transitivity. The family {P_λ} is then examined from the viewpoint of interval representation models, the most general of which is aP_λb if and only if I(a, λ) > I(b, λ), where the I's are real intervals with I(a, λ) > I(b, λ) if and only if the first interval is completely to the right of the second. With I(a, λ) = [f(a, λ), f(a, λ) + σ(a, λ)], the specializations of the interval model that are discussed include those where the location function f (for left end-points) depends only on the set A of alternatives or stimuli and where the length function σ depends only on A or on λ or neither.     

Experimentally induced phonological paraphasias.

A motor aphasic patient had to repeat word lists that contained, among other things, words with the sequences r - r - l and l - l - r. These sequences were often repeated as l - r - l (anticipatory distant assimilation) and r - l - r (same type, but less frequently), respectively. These assimilations violate the cross-over constraint (or relevancy condition or adjacency principle) and thus represent another instance of a flagrant contradiction between pathological and normal (natural) phonological processes.     

Real Interval Representations

This paper presents several necessary and sufficient conditions for real interval representability of biorders, interval orders, and semiorders. Let A and X be nonempty sets. We consider two types of interval representations for P⊂A×X. The first concerns the existence of two mappings, F: A→J_↓ and F: X→J_↑, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)∩F (x)= ∅, where J_↓ and J_↑ respectively denote the set of all real intervals that are unbounded below and the set of all real intervals that are unbounded above. The second yields two mappings, F: A→J_↓ and G: X→J_↓, such that, for all (a, x)∈A×X, (a, x)∈P⇔F(a)⊂G(x). Specializations of those representations include the cases of A=X for interval orders and semiorders.     

On linear classifications under varying choice probabilities

The paper deals with certain problems connected with the assumption that choice probabilities p_s(x, y) depend on the subject s. A set of postulates is given, which implies the existence of sequences of “classification standards”, i.e., sequences {z_j} such that whenever we have 0 < p_s0(x, z_i) < 1 for some s₀ and i, then p_s(z_i+k, x) = p_s(x, z_i?k) = 1 for all s, and k ≥ 1. Elements of any such sequence {z_j} can serve as boundaries between successive categories of classification based on the following rule: Assign x to jth category if you feel it is “to the right” of z_j and “to the left” of z_j+1. Under the condition stated above this rule is unambiguous, and the resulting classification has the property that every element is assigned to one of the two neighboring categories, regardless who performs the classification.Next, the postulates are enriched so as to imply the existence of “tightest” among such sequences {z_j}, hence leading to a classification with largest number of categories.     

Some counterexamples in multidimensional scaling

Given a set X with elements x, y,… which has a partial order < on the pairs of the Cartesian product X², one may seek a distance function ? on such pairs (x, y) which satisfies ?(x₁, y₁) < ?(x₂, y₂) precisely when (x₁, y₁) < (x₂, y₂), and even demand a metric space (X, ?) with some such compatible ? which has an isometric imbedding into a finite-dimensional Euclidean space or a separable Hilbert space. We exhibit here systems (X, <) which cannot meet the latter demand. The space of real m-tuples (ξ₁,…,ξ_m) with either the “city-block” norm Σ_i ∥ξ_i∥ or the “dominance” norm max_i, ∥ξ_i∥ cannot possibly become a subset of any finite-dimensional Euclidean space. The set of real sequences (ξ₁, ξ₂,…) with finitely many nonzero elements and the supremum norm sup_i, ∥ξ_i∥ cannot even become a subset of any separable Hilbert space.     

Specifically objective stochastic latency mechanisms

A two parameter additive model for psychological response latencies f_vi(t) = (θ_v + ?_i) exp(?(θ_v + ?_i)t) is suggested. Conditional inference procedures are derived for the psychometric latency model. If the model is considered as a theory for speed test construction the unrestricted comparability of measurements obtained under “speed” and “power” conditions is an attractive feature of the model. An extension of the parametric structure gives the “linear exponential model” f_vi(t) = (θ_v + Σ_ι^ma_iιη_ι) exp(?(θ_v + Σ_ι^ma_iιη_ι)t), where the time constants of the latencies are conceived as linear combinations of “basic parameters” corresponding to the experimental conditions of the latency situations. The linear exponential model includes as special cases some well known series latency mechanisms. Some generalizations of the distributional assumptions are discussed, especially the Weibull distributions. The estimation of parameters by conditional inference procedures and the evaluation of the fit of the model by conditional likelihood ratio tests is illustrated by means of the numerical data from a free recall experiment.     

The child's memory for actional, locational, and serial scenes

Three-, four-, and six-year-old children (N = 180) were tested for recall of a toy missing from a previously seen display of four toys. Toys were displayed initially as a series (StoS), as a unit (UtoU), or moved from a series to a unit as S watched (StoU). There was a significant increase in memory for the missing toy with age. In addition, children viewing StoU displays remembered significantly more toys than did children viewing the other displays, independent of age. Twenty-five percent of the 6-year-olds rehearsed spontaneously while viewing the displays. Results indicated that the nature of a memory unit does not change with age, that memory improves with age, and that separate visual and verbal memories are operative in Ss in this age range.     

Triadic judgment models and Weber's law

Let a, b, c, with a?b?c, be positive real numbers indicating the intensities of physical stimuli in a psychophysical experiment; let P_abc be the probability that b is judged to be more similar to a (“closer to”) a than to c. This paper investigates the following representation and its subcases for triadic judgments

P_abc=F[u(a)-u(b),u(b)-u(c)],     

Strategies used by children when solving simple subtractions

Ten children, 9–11 years old, solved all subtraction problems in the form of M?N=…, where 0 ? M ? 13, 0 ? N ? 13 and M ? N. The solution times were analysed and used for the formulation of a process model for subtraction. The model involves memory processes on two different levels, called reproductive and reconstructive respectively. When M=N, N=1, and M=2N the answers were quickly retrieved in reproductive memory processes. The reconstructive processes were found to be analogous to one of two counting procedures, viz. counting up and counting down. In general, the counting process starts either on N (when M < 2N) and counts up to reach the answer, or on M (when M > 2N) and counts down to reach the answer. This may reflect an effort to minimize the number of steps to be counted. However, when M > 10 and N < 10 a problem is always solved in a decrementing counting process. When M=10 many subtractions are solved in a reproductive memory process and the number 10 is also important as a point of reference for solving subtractions when M > 10 and N < 10.