Choice between fixed-interval schedules: Graded versus step-like choice functions

Pigeons chose between two fixed-interval schedules of food reinforcement. A single peck on one of two lighted keys started the fixed-interval schedule correlated with that key. The schedule had to be completed before the next choice opportunity. The durations of the fixed intervals were varied over conditions from 15 s to 40 s. To maximize the rate of reinforcement, the pigeons had to choose exclusively the shorter of the two schedules. Nevertheless, choice was not all-or-none. Instead, relative choice, and the rates of producing the fixed intervals, varied in a graded fashion with the disparity between the two schedules. Choice ratios under this procedure (single response to choose) were highly sensitive to the ratios of the fixed-interval schedules.     

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.     

Rationalizing two-tiered choice functions through conditional choice

Set-valued choice functions provide a framework that is general enough to encompass a wide variety of theories that are significant to the study of rationality but, at the same time, offer enough structure to articulate consistency conditions that can be used to characterize some of the theories within this encompassed variety. Nonetheless, two-tiered choice functions, such as those advocated by Isaac Levi, are not easily characterized within the framework of set-valued choice functions. The present work proposes conditional choice functions as the proper carriers of synchronic rationality. The resulting framework generalizes the familiar one mentioned above without emptying it and, moreover, provides a natural setting for two-tiered choice rules.     

Prediction outcome probabilities as determinants of choice reaction time

In a two-stimulus two-response choice reaction time (RT) task in which Ss made stimulus predictions, the probability of a correct prediction was manipulated between Ss. The magnitude of the difference in RT to correctly and incorrectly predicted stimuli (i.e., the prediction outcome effect) was an increasing function of the probability of a correct prediction This finding was primarily due to a reliable decrease in RT to correctly predicted stimuli as the probability of a correct prediction increased, since RT to incorrectly predicted stimuli was not affected by prediction outcome probability. These results were interpreted as partially supporting a continuous expectancy notion which involves facilitory and inhibitory mechanisms winch are differentially influenced by the probability of a correct prediction.     

Coherent choice functions under uncertainty

We discuss several features of coherent choice functions—where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility.     

On estimating inter-subject variability of choice probabilities under observability constraints

Methods are presented for estimating inter-subject variability of the probability of a given event defined in terms of subject's behavior (e.g., probability of a given choice in a discrimination experiment). The constraints consist of using no more than two independent observations for each subject. Estimators are provided for assessing the inter-subject “variance” of the analyzed probabilities; also, a method is given for testing the hypothesis that the average probability is the same for two groups of subjects.     

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.     

Global and local control of choice behavior by cyclically varying outcome probabilities

Subjects asked to simulate decision makers in a competitive bidding situation chose repeatedly between two alternatives; reward probability varied according to a sine wave function of time for one alternative but was held constant over time for the other. Learning functions for choice probability exhibited the wavelike pattern predicted by a statistical learning model. However, on later transfer trials, when success probability was independent of subjects' choices, their choice behavior continued to follow a wavelike function rather than approaching the constant .5 level predicted by the learning model. A possible basis in memory for the transfer performance was revealed in subjects' sketches of the remembered pattern of variation in reward probabilities. It is concluded that choice performance is controlled by a mixture of local feedback, in a manner described by the learning model, and more global information encoded in an abstract memory representation, with the balance of the influence shifting toward the latter when local feedback becomes uninformative.     

Choice functions in multistage optimization

This paper is devoted to the multistage optimization problem with a vector-valued gain function. The notion of optimality is determined here by the choice function. The conditions for the choice function that guarantee the applicability and efficiency of dynamic programming are found.     

Preference-based choice functions: a generalized approach

Although choice and preference are distinct categories, it may in some contexts be a useful idealization to treat choices as fully determined by preferences. In order to construct a general model of such preference-based choice, a method to derive choices from preferences is needed that yields reasonable outcomes for all preference relations, even those that are incomplete and contain cycles. A generalized choice function is introduced for this purpose. It is axiomatically characterized and is shown to compare favourably with alternative constructions.     

Stability and coherence of health experts' upper and lower subjective probabilities about dose-response functions

As part of a method for assessing health risks associated with primary National Ambient Air Quality Standards. T. B. Feagans and W. F. Biller (Research Triangle Park, North Carolina. EPA Office of Air Quality Planning and Standards, May 1981) developed a technique for encoding experts' subjective probabilities regarding dose--response functions. The encoding technique is based on B. O. Koopman's (Bulletin of the American Mathematical Society, 1940, 46, 763-764; Annals of Mathematics, 1940, 41, 269-292) probability theory, which does not require probabilities to be sharp, but rather allows lower and upper probabilities to be associated with an event. Uncertainty about a dose--response function can be expressed either in terms of the response rate expected at a given concentration or, conversely, in terms of the concentration expected to support a given response rate. Feagans and Biller (1981, cited above) derive the relation between the two conditional probabilities, which is easily extended to upper and lower conditional probabilities. These relations were treated as coherence requirements in an experiment utilizing four ozone and four lead experts as subjects, each providing judgments on two separate occasions. Four subjects strongly satisfied the coherence requirements in both conditions. and three more did no in the second session only. The eighth subject also improved in Session 2. Encoded probabilities were highly correlated between the two sessions, but changed from the first to the second in a manner that improved coherence and reflected greater attention to certain parameters of the dose--response function.     

Conditions under which thurstone case III representations for binary choice probabilities are also Fechnerian

By a Thurstone Case III representation for binary symmetric choice probabilities P_x,y we mean that there exist functions F, μ, σ > 0 such that $\text{P}{}_{\mathrm{x,y}}\text{= F[}\text{(μ(x) ? μ(y))}\text{(σ}{}^2\text{(x) + σ}{}^2\text{(y))}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{]}$. We show that the constraint σ = constant, or μ = ασ + β, α ≠ 0, is both necessary and sufficient for a Thurstone Case III representation to be Fechnerian, i.e., to be reexpressable as as P_x,y = G(u(x) ? u(y)) for some suitably chosen functions G, u.