Matching, undermatching, and overmatching in studies of choice

Almost all of 103 sets of data from 23 different studies of choice conformed closely to the equation: log (B(1)/B(2)) = a log (r(1)/r(2)) + log b, where B(1) and B(2) are either numbers of responses or times spent at Alternatives 1 and 2, r(1) and r(2) are the rates of reinforcement obtained from Alternatives 1 and 2, and a and b are empirical constants. Although the matching relation requires the slope a to equal 1.0, the best-fitting values of a frequently deviated from this. For B(1) and B(2) measured as numbers of responses, a tended to fall short of 1.0 (undermatching). For B(1) and B(2) measured as times, a fell to both sides of 1.0, with the largest mode at about 1.0. Those experiments that produced values of a for both responses and time revealed only a rough correspondence between the two values; a was often noticeably larger for time. Statistical techniques for assessing significance of a deviation of a from 1.0 suggested that values of a between .90 and 1.11 can be considered good approximations to matching. Of the two experimenters who contributed the most data, one generally found undermatching, while the other generally found matching. The difference in results probably arises from differences in procedure. The procedural variations that lead to undermatching appear to be those that produce (a) asymmetrical pausing that favors the poorer alternative; (b) systematic temporal variation in preference that favors the poorer alternative; and (c) patterns of responding that involve changing over between alternatives or brief bouts at the alternatives.     

Concurrent random-interval schedules and the matching law

In Experiment I, a group of eight pigeons performed on concurrent random-interval schedules constructed by holding probability equal and varying cycle time to produce ratios of reinforcer densities of 1:1, 3:1, and 5:1 for key pecking. Schedules for a second group of seven were constructed with equal cycle times and unequal probabilities. Both groups deviated from simple matching, but the two forms of the schedules appeared to produce no consistent patterns of deviation. The data were found to be consistent with those obtained in concurrent variable-interval situations. The parameters of the matching equation in the form of Y=k X^a were estimated; the value of k was unity and a was 0.84. In Experiment II, six pigeons were exposed to two conc RI RI schedules in which one component increasingly approximated an FI schedule. The value of k was not 1.0. Concurrent RI RI schedules were shown to represent a continuum from conc FI VI to conc VI VI schedules. The use of the exponential equation in testing “matching laws” suggests that a<1 will continue to be observed, and this will set limits on the form of new laws and the assumed or rational values of the component variables in these laws.     

Conservatism as a function of bag composition

The first purpose of this bookbag and pokerchip experiment was to examine the influence of the sample size and the relative frequency of the red chips in the sample on the amount of conservatism. The second purpose was to replicate the results of DE SWART (1972), and to test his hypothesis that the slope B of the function: log Φ(T)=A+B(r−b), is linearly related to log [p_r/(1−p_r)], where p_r represents the bag composition.     

An analytic comparison of Herrnstein's equations and a multivariate rate equation

Herrnstein's equations are approximations of the multivariate rate equation at ordinary rates of reinforcement and responding. The rate equation is the result of a linear system analysis of variable-interval performance. Rate equation matching is more comprehensive than ordinary matching because it predicts and specifies the nature of concurrent bias, and predicts a tendency toward undermatching, which is sometimes observed in concurrent situations. The rate equation contradicts one feature of Herrnstein's hyperbola, viz., the theoretically required constancy of k. According to the rate equation, Herrnstein's k should vary directly with parameters of reinforcement such as amount or immediacy. Because of this prediction, the rate equation asserts that the conceptual framework of matching does not apply to single alternative responding. The issue of the constancy of k provides empirical grounds for distinguishing between Herrnstein's account and a linear system analysis of single alternative variable-interval responding.     

A representation theorem for finite random scale systems

This paper investigates necessary and sufficient conditions on choice probabilities P_a,B (of picking an element a in an offered set B), for the existence of random variables U_a, satisfying the equation P_a,B = $\text{P}${U_a = max {U_b | b ∈ B}} for all nonempty finite subsets B in a fixed set A, and all a ∈ B. A complete solution to this representation problem is obtained in the case where A is finite. The proof of the representation theorem provides an algorithm to construct the random variables U_a, up to some uniqueness properties. Investigation of these uniqueness properties show that an important part of the ordinal structure of the underlying random variables can be recovered.     

Toward a quantitative theory of punishment

In two experiments, pigeons' key pecking for food on concurrent variable-interval schedules was punished with electric shock according to concurrent variable-interval punishment schedules. With unequal frequencies of food but equal rates of punishment associated with the two keys and at several intensities of shock, the response and time allocation of all six pigeons overmatched the obtained relative frequency of food. The overmatching was predicted by a subtractive model of the interaction between punishment and positive reinforcement but not by two alternative models. Increases in the k and r_e parameters of the generalized matching law could not account for the observed shifts in preference.     

Alternative measures of fit for the Schönemann-carroll matrix fitting algorithm

In connection with a least-squares solution for fitting one matrix,A, to another,B, under optimal choice of a rigid motion and a dilation, Schönemann and Carroll suggested two measures of fit: a raw measure,e, and a refined similarity measure,e _s , which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix,B,e.g.,e(A,kB) ≠e(A,B) fork ≠ 1. Therefore, both measures are useless for answering questions of the type: “DoesA fitB better thanA fitsC?”. In this note two new measures of fit are suggested which do not depend upon the norms ofA andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.     

On the law of effect

Experiments on single, multiple, and concurrent schedules of reinforcement find various correlations between the rate of responding and the rate or magnitude of reinforcement. For concurrent schedules (i.e., simultaneous choice procedures), there is matching between the relative frequencies of responding and reinforcement; for multiple schedules (i.e., successive discrimination procedures), there are contrast effects between responding in each component and reinforcement in the others; and for single schedules, there are a host of increasing monotonic relations between the rate of responding and the rate of reinforcement. All these results, plus several others, can be accounted for by a coherent system of equations, the most general of which states that the absolute rate of any response is proportional to its associated relative reinforcement.     

Discrimination learning, the peak shift, and behavioral contrast

A discrimination between two successively alternating stimuli was trained under conditions that maintained equal frequencies of reinforcement in the presence of each of the discriminative stimuli (S1 and S2) but that also reduced the rate of responding to S2. These conditions included a multiple variable-interval differential-reinforcement-of-low-rate schedule and a multiple variable-interval variable-interval schedule in which responses to S2 were punished. Whenever the rate of responding to S2 was reduced, rate of responding to S1 (behavioral contrast) increased, and the peak of a subsequently obtained generalization gradient did not occur at the expected location (between S1 and S2) but was displaced away from S2, below S1. Discrimination training in which the frequencies of reinforcement earned in S1 and S2 were not equal (variable-interval 1-min variable-interval 5-min training) produced contrast and the peak shift only if the rate of responding to S2 had been reduced, as after non-differential reinforcement in which variable-interval 1-min schedules were correlated with S1 and with S2. It was concluded that a sufficient condition for the occurrence of behavioral contrast and the peak shift was reduction of the rate of responding to one of two alternating discriminative stimuli and that a peak shift will occur only if contrast had occurred during discrimination training.     

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.     

An efficient adaptive procedure for psychophysical discrimination experiments

We present an adaptive procedure to conduct psychophysical discrimination experiments. In a discrimination experiment, an observer senses (sees, hears, feels, etc.) two stimuli (separated in space or time) and is asked to order these stimuli with respect to a particular parameter (say,s). Under the usual assumption of a locally linear internal representation ofs, perturbed by additive Gaussian noise, the probabilityP(s ₂)—of judging test stimuluss ₂ “larger” than a reference stimuluss ₁—is an error function (a cumulative normal distribution). Such an error function,Erf [(s ₂μ)/σ], is parametrized by two parameters: μand σ. The parameter μ is the value for whichP(s ₂) = 50% and is related to possible bias effects in the internal representation of the observer. The parameter σ is √2 times the standard deviation of the noise distribution and is generally called thediscrimination threshold. In this paper, we present (1) an algorithm to estimate μ and σ given the data generated by such an observer, (2) an analysis of the efficiency of a stimulus presentation with respect to the estimation of μ and σ , and (3) a method that controls the specific choice of stimulus values during an experiment so that an optimal estimation of μ and σ is obtained.     

Characteristics of forgetting functions in delayed matching to sample

Performance of pigeons in delayed matching-to-sample procedures was measured in terms of an index of discriminability derived from the difference between logarithms of ratios of choice responses to comparison stimuli following the different sample stimuli. Forgetting functions that plotted discriminability as a function of delay-interval duration were well described by a simple negative exponential function with two parameters, one describing initial discriminability of sample stimuli at zero delay (log d₀) and the other describing rate of decrement in discriminability with increasing delay-interval duration (b). With the difference between wavelength values of the comparison stimuli held constant, a large difference between wavelengths of the sample stimuli resulted in a higher log d₀ value than that for a small difference between sample stimuli, without changing the rate of decrement in discriminability, b. An increase in the fixed-ratio requirement for sample-key responding produced an increase in log d₀ without affecting b, and interpolation of ambient illumination in the delay interval increased b without influencing log d₀. Both parameters changed when intertrial-interval duration was varied. The result of variation in the point of interpolation of ambient illumination in the delay interval indicated that levels of discriminability at longer delays were independent of discriminability levels at earlier delays, consistent with the properties of the exponential function. Functions relating performance to delay-interval duration were suggested to have two characteristics: discriminability of the sample stimuli in the absence of a delay between the stimuli and the behavior they occasion, and rate of attenuation in discriminability with increasing delay-interval duration.     

Resolution of Algebraic Systems of Equations in the Variety of Cyclic Post Algebras

There is a constructive method to define a structure of simple k-cyclic Post algebra of order p, L _p,k , on a given finite field F(p ^k ), and conversely. There exists an interpretation ??₁ of the variety ${\mathcal{V}(L_{p,k})}$ generated by L _p,k into the variety ${\mathcal{V}(F(p^k))}$ generated by F(p ^k ) and an interpretation ??₂ of ${\mathcal{V}(F(p^k))}$ into ${\mathcal{V}(L_{p,k})}$ such that ??₂??₁(B) =  B for every ${B \in \mathcal{V}(L_{p,k})}$ and ??₁??₂(R) =  R for every ${R \in \mathcal{V}(F(p^k))}$ . In this paper we show how we can solve an algebraic system of equations over an arbitrary cyclic Post algebra of order p, p prime, using the above interpretation, Gröbner bases and algorithms programmed in Maple.     

A relational differential outcomes effect: pigeons can classify outcomes as “good” and “better”

In a conditional discrimination each of two sample stimuli indicates which of two comparison stimuli is correct. When correct choice following each conditional stimulus is followed by a different outcome (one kind of food following one, a different kind of food following the other) it often facilitates acquisition and improves memory. In transfer designs, in which two different conditional discriminations are followed by the same two differential outcomes, outcome expectation can be shown to be sufficient for comparison choice. That is, the samples from one conditional discrimination are matched to comparisons from the other conditional discrimination based on the common outcomes alone. In the present study we asked if for pigeons the relative value of the differential outcomes (higher versus lower value) can serve as the basis for comparison choice, independent of other characteristics of the outcomes and of differential sample responding. That is, would different outcomes that could be described as “good” and “better” form two stimulus classes. For one conditional discrimination, the differential outcomes involved differential probability of reinforcement for choice of the correct comparison stimulus (0.80 vs. 0.20 for correct choice of the two comparisons, respectively). For the other conditional discrimination, the differential outcomes involved differential responding to the two comparison stimuli (5 pecks vs. 20 pecks to the correct comparisons, respectively). On test trials, when conditional stimuli from the two conditional discriminations were interchanged and the relative value of the differential outcomes could serve as the only basis for comparison choice, we found positive transfer. The results indicate that relational attributes of outcomes can serve as effective cues for comparison choice.     

A nonsimple conjoint measurement model

The nonsimple conjoint measurement model examined in this paper maps each (a₁, a₂, a₃) in A₁ × A₂ × A₃ into ω₁(a₁)φ₂(a₂) + ω₂(a₁)φ₃(a₃), where each of ω₁, ω₂, φ₂, and φ₃ is a real-valued function, so as to preserve a binary relation ≥ on A₁ × A₂ × A₃ by ≧ in the numerical system.The principle structure of the present model is similar to the structure of the simple models. But in the nonsimple model, we define two different identity elements of A₁ for its different multiplicative effects on the other two components A₂ and A₃, whereas in the simple model, one identity element for each component is defined.     

Changeover delay and concurrent-schedule performance in domestic hens

Six domestic hens were exposed to a series of five pairs of two-key concurrent variable-interval schedules with a range of changeover delays from no delay to 15 s. Times spent responding on each alternative and total, within_, and post-changeover-delay response ratios were analyzed in terms of the generalized matching law. The sensitivity parameters, a, for response and time data were generally low when no changeover delay was programmed but were not 0.0. They were higher for all other changeover-delay values, with some tendency to increase as the changeover delay lengthened at very short delays. Within-delay responding was insensitive to reinforcement-rate differences at all changeover delays (a values close to 0.0). As a result of this insensitivity, post-changeover-delay responding was more sensitive to reinforcement-rate changes than was total responding. Interchangeover intervals increased systematically with changeover-delay duration. Responding, particularly after the changeover delay, was well predicted by an equation based on a reinforcer-loss model.     

Cognitive style and elaboration of logical abilities in hearing-impaired children

Ninety-seven hearing-impaired children ages 8–12 were given the Raven Coloured Progressive Matrices (CPM) and a Piagetian battery under one of six conditions of testing: (a₁) standard, (a₂) simple feedback, (a₃) “verbalization” after problem solution (a₄) “verbalization” during and after solution, (a₅) differential feedback, and (a₆) conditions a₄ and a₅ combined. Kagan's Matching Familiar Figures Test (MFFT) and the Nowicki-Strickland Internal-External Control scale were also administered. Performance under conditions a₅ and a₆ was greater than performance under conditions a₃ and a₄, which in turn exceeded performance in conditions a₁ or a₂. MFFT errors, MFFT latency, and locus of control score were differentially related to performance in the different testing conditions. A significant prediction equation was found for both the CPM and Piagetian battery under a₂, a₃, and a₆ and for the CPM under a₄. All analyses indicated consistency of findings across both criterion measures. The findings of this investigation may be used as an aid in the development of diagnostic instruments and procedures which take individual differences along dimensions of personality and cognitive style into account.     

The optimal number of alternatives at a choice point with travel time considered

The problem of optimal design is considered for tasks that can be represented as n choice points with a_i alternatives per choice point. Examples are mazes or multiple choice tests. Some fixed time t′ elapses between choice points, say traveling from one to the other or reading a question, and additional time t is spent considering each alternative at the choice point. For fixed total time, the goal is to find the optimum number of choice points and the optimum number of alternatives at each choice point. In general, the longer the time spent between choice points, the greater the optimum number of alternatives at each point.     

Two-key concurrent responding: response-reinforcement dependencies and blackouts

Two-key concurrent responding was maintained for three pigeons by a single variable-interval 1-minute schedule of reinforcement in conjunction with a random number generator that assigned feeder operations between keys with equal probability. The duration of blackouts was varied between keys when each response initiated a blackout, and grain arranged by the variable-interval schedule was automatically presented after a blackout (Exp. I). In Exp. II every key peck, except for those that produced grain, initiated a blackout, and grain was dependent upon a response following a blackout. For each pigeon in Exp. I and for one pigeon in Exp. II, the relative frequency of responding on a key approximated, i.e., matched, the relative reciprocal of the duration of the blackout interval on that key. In a third experiment, blackouts scheduled on a variable-interval were of equal duration on the two keys. For one key, grain automatically followed each blackout; for the other key, grain was dependent upon a response and never followed a blackout. The relative frequency of responding on the former key, i.e., the delay key, better approximated the negative exponential function obtained by Chung (1965) than the matching function predicted by Chung and Herrnstein (1967).     

Phason elastic constants of the icosahedral Al-Pd-Mn phase derived from diffuse scattering measurements

The diffuse scattering in the diffraction pattern of the icosahedral Al-Pd-Mn quasicrystalline phase has been measured on an absolute scale by X-ray and neutron scattering on single-grain samples. Most of the diffuse scattering can be interpreted in the framework of the elasticity theory of icosahedral quasicrystals considering only phason fluctuations. At room temperature the absolute values of the K₁/k_BT and K₂/k_B T phason elastic constants are of the order of 0.06 and 0.031atom^-1. The amount of diffuse scattering intensity is insensitive to the sample annealing treatment.