Hyperbolae.

Hyperbolic relations between independent and dependent variables are ubiquitous in the experimental analysis of behavior, mentioned in over 150 articles in the Journal of the Experimental Analysis of Behavior. There are two principal forms of hyperbolae: The first describes the relation between response rate and reinforcement rate on variable-interval schedules of reinforcement; it rises asymptotically toward a maximum. The second describes the relation between the current equivalent value of an incentive and its delay or (im)probability; it falls from a maximum toward an asymptote of 0. Where do these come from? What do their parameters mean? How are they related? This article answers the first two questions and addresses the last.

From data through discount rates to the area under the curve.

The rate of discounting future goods is a crucial factor in intertemporal trade-offs, upon which depends not only individual well-being but also that of our planet: How much privation now for a temperate future for our grandchildren? What is the best way to measure how the value of future goods decreases with its delay? The most accurate discount functions involve several covarying parameters, making interpretation equivocal. A universal and robust measure is the area under the discount curve, the AuC. The AuC of a hyperbolic discount function is a logarithmic function of the discount rate, k. The same integral also approximates the area under a hyperboloid function. A simple technique converts each datum into estimates of the discount rate, eliminating rogue data points in the process. These trimmed estimates are converted into areas and tested against data, where they succeed at predicting the AuC and its relation to log(k).

Theory of reinforcement schedules.

The three principles of reinforcement are (1) events such as incentives and reinforcers increase the activity of an organism; (2) that activity is bounded by competition from other responses; and (3) animals approach incentives and their signs, guided by their temporal and physical conditions, together called the "contingencies of reinforcement." Mathematical models of each of these principles comprised mathematical principles of reinforcement (MPR; Killeen, 1994). Over the ensuing decades, MPR was extended to new experimental contexts. This article reviews the basic theory and its extensions to satiation, warm-up, extinction, sign tracking, pausing, and sequential control in progressive-ratio and multiple schedules. In the latter cases, a single equation balancing target and competing responses governs behavioral contrast and behavioral momentum. Momentum is intrinsic in the fundamental equations, as behavior unspools more slowly from highly aroused responses conditioned by higher rates of incitement than it does from responses from leaner contexts. Habits are responses that have accrued substantial behavioral momentum. Operant responses, being predictors of reinforcement, are approached by making them: The sight and feel of a paw on a lever is approached by placing paw on lever, as attempted for any sign of reinforcement. Behavior in concurrent schedules is governed by approach to momentarily richer patches (melioration). Applications of MPR in behavioral pharmacology and delay discounting are noted.

Discounting and the portfolio of desires.

The additive utility theory of discounting is extended to probability and commodity discounting. Because the utility of a good and the disutility of its delay combine additively, increases in the utility of a good offset the disutility of its delay: Increasing the former slows the apparent discount even with the latter, time-disutility, remaining invariant, giving the magnitude effect. Conjoint measurement showed the subjective value of money to be a logarithmic function of its amount, and subjective probability-the probability weighting function-to be Prelec's (1998). This general theory of discounting (GTD) explains why large amounts are probability discounted more quickly, giving the negative magnitude effect. Whatever enhances the value of a delayed asset, such as its ability to satisfy diverse desires, offsets its delay and reduces discounting. Money's liquidity permits optimization of the portfolio of desired goods, providing added value that accounts for its shallow temporal discount gradient. GTD predicts diversification effects for delay but none for probability discounting. Operations such as episodic future thinking that increase the larder of potential expenditures-the portfolio of desirable goods-increase the value of the asset, flattening the discount gradient. States that decrease the larder, such as stress, depression, and overweening focus on a single substance like a drug, constrict the portfolio, decreasing its utility and thereby steepening the gradient. GTD provides a unified account of delay, probability, and cross-commodity discounting. It explains the effects of motivational states, dispositions, and cognitive manipulations on discount gradients. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Variations on a theme by Rachlin: Probability discounting.

Rachlin and colleagues laid the groundwork for treating the discounting of probabilistic goods as a variant of the discounting of delayed goods. This approach was seminal for a large body of subsequent research. The present paper finds the original development problematic: In converting probability to delay, the authors incorrectly dropped trial duration. The subsumption of probability by delay is also empirically questionable, as those are different functions of variables such as magnitude of outcome and commodity versus money. A variant of Rachlin's theme treats human discounting studies as psychophysical matching experiments, in which one compound stimulus is adjusted to equal another. It is assumed that a function of amount (its utility) is multiplied by a function of probability (its weight). Conjoint measurement establishes the nature of these functions, yielding a logarithmic transform on amount, and a Prelec function on probability. This model provides a good and parsimonious account of probability discounting in diverse data sets. Variant representations of the data are explored. By inserting the probabilistically discounted utility into the additive utility theory of delay discounting, a general theory of probabilistic intertemporal choice is achieved.

A trace theory of time perception.

Many comparisons involve sequentially presented stimuli, as perforce the case in comparisons of temporal intervals. Interactions of such stimuli are as inevitable as the spatial interactions that yield color and brightness contrast. A memory-trace theory of perception (TToP) is developed and applied to time perception. Duration is estimated based on the memorial strength of the stimuli that signal the initiation of an interval at the time of its termination. Memorial persistence depends on modality and character of the signals, which condition the response to them. When the constant difference limen on the memorial continuum is back-translated to the temporal one it yields a generalized Weber function. Memory traces interact as a function of generalization gradients: Memories of stimuli that are similar enough are aggregated-feature-bound-some veridically, others as illusory conjunctions. The resulting representations may then be judged in a discrimination paradigm, or translated back to the physical domain as reproductions of the intervals. The presentation of a standard stimulus affects the perception of the comparison stimulus, warping the ruler by which it is measured. Complementary effects are predicted for discrimination and adjustment paradigms. Thus configured, the TToP accounts for multiple special effects, variously referred to as distortions, anomalies, and illusions, that are observed with classical psychophysical methods: Scalar and nonscalar timing, modality effects, time-order errors, masking, time warping, lengthening, and Vierordt's law. Similar processes affect the perception of nontemporal stimuli whenever they are presented in sequential proximity to one another. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Applied Quantitative Analysis of Behavior: What It Is, and Why We Care-Introduction to the Special Section.

Science evolves from prior approximations of its current form. Interest in changes in species over time was not a new concept when Darwin made his famous voyage to the Galapagos Islands; concern with speciation stretches back throughout the history of modern thought. Behavioral science also does and must evolve. Such change can be difficult, but it can also yield great dividends. The focus of the current special section is on a common mutation that appears to have emerged across these areas and the critical features that define an emerging research area-applied quantitative analysis of behavior (AQAB). In this introduction to the "Special Issue on Applications of Quantitative Methods," we will outline some of the common characteristics of research in this area, an exercise that will surely be outdated as the research area continues to progress. In doing so, we also describe how AQAB is relevant to theory, behavioral pharmacology, applied behavior analysis, and health behaviors. Finally, we provide a summary for the articles that appear in this special issue. The authors of these papers are all thinking outside the Skinner box, creating new tools and approaches, and testing them against relevant data. If we can keep up this evolution of methods and ideas, behavior analysis will regain its place at the head of the table!

Howard Rachlin (1935-2021).

Memorializes Howard Rachlin (1935-2021). Rachlin was born to Irving and Gussie Kugler Rachlin in New York City on March 10, 1935. He died 86 years later of cancer, leaving his wife Nahid, daughter Leila, and grandson Ethan. He received numerous recognitions: the Med Associates Distinguished Contributions to Basic Behavioral Research award from Division 25 of the American Psychological Association, the Impact of Science on Application award from the Association for Behavior Analysis, a James McKeen Cattell Fellowship, continuous funding from the National Science Foundation and the National Institutes of Mental Health (from which he received the MERIT award), visiting scholar at the Russell Sage Foundation, and invited speaker at the Nobel symposium on Behavioral and Experimental Economics. Of himself Rachlin wrote: "He obtained a bachelor of mechanical engineering degree from Cooper Union in New York City [1957], where he learned to treat all scientific and practical questions as asking for answers rather than for self-expression; masters in philosophy and psychology from The New School of Social Research in New York City [1962], where he learned that the whole may be greater than the sum of its parts; and a PhD from Harvard University [1965], where B. F. Skinner and Richard Herrnstein taught him how to be a behaviorist." After teaching at Harvard, he joined Stony Brook University in New York in 1969, rising to the position of Distinguished Research Professor. Rachlin studied choice and decision-making; he was one of the founders of behavioral economics. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Addendum to Killeen's (2019) Bidding for Delayed Rewards.

Killeen (2019) portrayed an intimate relation between diverse economic indices, in particular compensation functions, discount functions, and demand functions. The article bemused some experts, however, by its counterintuitive prediction of an increase in the amount bid as the delay increased. Furthermore, the article failed to provide an explicit treatment of the small-soon versus large-late choice paradigm, to cite several papers that provided precedent for the current work, and to demonstrate the adequacy of his expenditure functions for data on purchasing decisions. These shortcomings are remedied in the current note, and some additional extensions offered.

Bidding for delayed rewards: Accumulation as delay discounting, delay discounting as regulation, demand functions as corollary.

In the accumulation paradigm animals press one manipulandum to accumulate pellets or seconds of access to food, and then press another manipulandum, or run some distance, to collect it. The accumulation may be interpreted as delay discounting, with the animals adjusting the distal amount to compensate for its distance or delay. The amount accumulated before being collected is a linear function of the distance or time that the experimental paradigm stipulates for collecting it. That linear function follows from the Unit-Price/Unit-Amount axiom. The inverse of the linear compensation functions gives a delay-discount hyperbola. The advantages of the accumulation paradigm and analytic framework for delay discounting studies are noted. Compensation functions are then derived from a behavioral regulation model, which generalizes them to contexts where the individual's budget for response cost becomes over-taxed. In turn, such compensation/regulation models lead directly to representative demand functions. In sum, regulation models provide a theoretical grounding for demand functions, compensation functions, and delay discounting hyperbolas. The parameter that links them is the unit amount k, the slope of the compensation function and of the discount function, the setpoint for consumption in a regulation model, and the ideal quantity consumed at minimal price in a demand analysis.

Timberlake's theories dissolve anomalies.

Two of Timberlake's major contributions, amongst numerous other good notes, are Behavior Regulation Theory (BRT), and Behavior Systems Theory (BST). BRT was a refinement of the Premack Principle. What both got right was that reinforcers are responses, not stimuli. For BRT, they were responses that were occurring below the rate at which they otherwise would given free access to them. BST was a larger ethological framework for our science of behavior. We have always needed it, as it opens an important window on our field. With that window closed, it is easy to stumble over a half-dozen anomalies in the dark, ones that we say humph to, scratch our heads, and then move on. When illuminated by BST, however, such anomalies become keys to a deeper understanding of our subject. This paper reviews numerous anomalies that make sense within the joint framework of BST and BRT, and Dickinson's Dual-Process theory of learned behavior. No longer anomalous in that context, all that is now left to do is test the validity and productivity of this general framework for those many strange cases.

Models of attention-deficit hyperactivity disorder.

One of the most notable aspects of the behavior of individuals with Attention Deficit Hyperactivity Disorder (ADHD) is increased variability in many aspects of their behavior, including response times and attentional focus. Among the many theories of ADHD is one that identifies its material cause as phasic malnutrition of the neurons required to maintain constancy of performance. Of the diverse predictions issuing from this theory, one concerns ubiquitous data: response times and their variance in decision tasks. This paper reviews that behavioral neuroenergetics theory and model, shows how they predict representative data, and suggests their relevance to researchers studying animal models of ADHD.

The non-Darwinian evolution of behavers and behaviors.

Many readers of this journal have been schooled in both Darwinian evolution and Skinnerian psychology, which have in common the vision of powerful control of their subjects by their sequalae. Individuals of species that generate more successful offspring come to dominate their habitat; responses of those individuals that generate more reinforcers come to dominate the repertoire of the individual in that context. This is unarguable. What is questionable is how large a role these forces of selection play in the larger landscape of existing organisms and the repertoires of their individuals. Here it is argued that non-Darwinian and non-Skinnerian selection play much larger roles in both than the reader may appreciate. The argument is based on the history of, and recent advances in, microbiology. Lessons from that history re-illuminate the three putative domains of selection by consequences: The evolution of species, response repertoires, and cultures. It is argued that before, beneath, and after the cosmically brief but crucial epoch of Darwinian evolution that shaped creatures such as ourselves, non-Darwinian forces pervade all three domains.

Predict, Control, and Replicate to Understand: How Statistics Can Foster the Fundamental Goals of Science.

Scientists abstract hypotheses from observations of the world, which they then deploy to test their reliability. The best way to test reliability is to predict an effect before it occurs. If we can manipulate the independent variables (the efficient causes) that make it occur, then ability to predict makes it possible to control. Such control helps to isolate the relevant variables. Control also refers to a comparison condition, conducted to see what would have happened if we had not deployed the key ingredient of the hypothesis: scientific knowledge only accrues when we compare what happens in one condition against what happens in another. When the results of such comparisons are not definitive, metrics of the degree of efficacy of the manipulation are required. Many of those derive from statistical inference, and many of those poorly serve the purpose of the cumulation of knowledge. Without ability to replicate an effect, the utility of the principle used to predict or control is dubious. Traditional models of statistical inference are weak guides to replicability and utility of results. Several alternatives to null hypothesis testing are sketched: Bayesian, model comparison, and predictive inference (p rep). Predictive inference shows, for example, that the failure to replicate most results in the Open Science Project was predictable. Replicability is but one aspect of scientific understanding: it establishes the reliability of our data and the predictive ability of our formal models. It is a necessary aspect of scientific progress, even if not by itself sufficient for understanding.

Suing for Peace in the War Against Mentalism.

The antimentalists' war against mentalism has not vanquished it. To examine why, we focus on two theses-mind as causal and internal-and three standard attacks against mentalism as defined by both theses: 1) mentalism implies dualism; 2) mind is unobservable, which hinders its scientific study; and 3) mentalism is impractical. These salients fail because: 1) if the mind is causal and internal, it must be material; 2) the observable/unobservable distinction is too problematic, with antimentalists equivocal about where to draw that line, with some even embracing publicly unobservable behavior as causally relevant; and 3) mentalism has not been demonstrated to be less practical than antimentalism. For the war on mentalism to succeed, stronger attacks must be devised, both scientific and philosophical. We contemplate some possibilities, while expressing doubts as to the wisdom of continuing the war. Peace may be better than war, and the resulting intellectual commerce may be good for both sides.

The basis of behavioral momentum in the nonlinearity of strength.

The persistence of operant responding in the context of distractors and opposing forces is of central importance to the success of behavioral interventions. It has been successfully analyzed with Behavioral Momentum Theory. Key data from the research inspired by that theory are reanalyzed in terms of more molecular behavioral mechanisms: the demotivational effects of disruptors, and their differential impacts on the target response and other responses that interact with them. Behavioral momentum is regrounded as a nonlinear effect of motivation and reinforcement rate on response probability and persistence. When response probabilities are high, more energy is required to further increase or to decrease them than when they are low. Classic Behavioral Momentum Theory effects are reproduced with this account. Finally, it is shown how the new account involving motivation and competition is closely related to the metaphor of force and action that is at the core of Behavioral Momentum Theory.