Decision making in a case of mixed uncertainty: A normative model

We consider a case of uncertainty which is frequently met in various fields, e.g., in parametric statistics: Events {θ}, θ ∈ ∵, are members of family $\text{E}$ on which the decision maker possesses no information at all; however, conditionally on the realization of {θ}, he is able to affix probabilities to all members of another family of events, $\text{F}$. We assume that the decision maker: (1) has a rational behavior under complete ignorance, for decisions whose results only depend on events of $\text{E}$; (2) with {θ} known, maximizes his conditional expected utility for decisions whose results only depend on events of $\text{F}$; (3) has (unconditional) preferences which are consistent with his conditional ones. These assumptions are shown to be sufficient to ensure an approximate representation of the decision maker's preference by a real-valued function W which has the form $\text{W(f) = v[}\text{Inf}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f),}\text{Sup}{}_{\mathrm{\theta \in \because }}\text{E}{}_{\mathrm{\theta }}\text{(u°f)]}$, where u and v, respectively, characterize the decision maker's attitudes toward risk and toward complete ignorance.     

Response mode and choice consistency: A test of unfolding theory

Coombs, Donnell, and Kirk (1978. Journal of Experimental Psychology, 4, 497–512), in a study of risk preferences, collected data using both pick $\text{1}\text{3}$ and reject $\text{1}\text{3}$ response modes. Although the preference orders derived from the two response modes were identical, the pick data contained a greater number of inconsistencies than the reject data. In the present study, predictions were derived from unfolding theory (Coombs, 1964. A theory of data. New York: Wiley) regarding the relative consistency of pick and reject response modes. An experiment performed as a test of these predictions supported the unfolding model suggesting that differences in inconsistency between response modes could be attributed to the fineness of the grid of working midpoints imposed upon the choice process by the response mode.     

On the functional form of partial masking functions in psychoacoustics

Partial masking of pure tones is often investigated in terms of matching functions which record the intensity φ(x, n) of an unmasked tone which matches, in perceived loudness, a tone of intensity x embedded in a flat, broadband noise of intensity n. Empirically, the following property of “shift invariance” is observed to hold for these matching functions: φ(λx, λ^θn) = λφ(x, n), for any λ > 0, and some θ < 1. In the context of the model $\text{φ(x, n) = F}{}_0\text{[}\text{g}{}_0\text{(x)}\text{(h}{}_0\text{(x) + k}{}_0\text{(n))}\text{]}$ which expresses the idea that the effect of a noise mask is to control the gain of a pure tone signal in a multiplicative fashion, shift invariance proves to be an extremely powerful theoretical constraint. Specifically, we show that only two parametric families of functions (F₀, g₀, h₀, k₀) are possible candidates for interpolating empirical data. These two parametric families are given by the following expressions: $\text{φ}{}_1\text{(x, n) = A(}\text{x}{}^{\mathrm{\alpha }}\text{(x}{}^{\mathrm{\alpha \prime }}\text{+ Kn}{}^{\mathrm{<mtext>\alpha \prime </mtext><mtext>\theta </mtext>}}\text{)}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>(\alpha \; ?\; \alpha \prime )</mtext>}}$, $\text{φ}{}_2\text{(x, n) = A[x}{}^{\mathrm{\alpha }}\text{(x}{}^{\mathrm{\alpha \prime }}\text{? Kn}{}^{\mathrm{<mtext>\alpha \prime </mtext><mtext>\theta </mtext>}}\text{)]}{}^{\mathrm{<mtext>1</mtext><mtext>(\alpha \; +\; \alpha \prime )</mtext>}}$. Both of these expressions are in good agreement with a large array of partial masking data.     

Conjoint Weber laws and additivity

This paper investigates the mathematical consequences of a number of related empirical laws, exemplified by

$\text{P}{}_{\mathrm{ax;by}}\text{= P(ξa)(ξx);(ξb)(ξy)}$

where a, x, b, y, and ξ are real numbers, and P_ax;by is the probability of choosing the two-dimensional object (a, x) in the set {(a, x), (b, y)}. A variety of results is derived showing that, in the presence of such laws, the class of feasible models for choice data is considerably reduced. In particular, it is shown that the above law, together with the “additive conjoint” form

$\text{P}{}_{\mathrm{ax;by}}\text{= F[l(a) + r(x), l(b) + r(y)]}$

(where F, l, and r are unspecified except for continuity and monotonicity properties), requires the choice probabilities to possess one of the following three analytic forms:

$\text{P}{}_{\mathrm{ax;by}}\text{= G}\text{a}{}^{\mathrm{\beta }}\text{+ δx}{}^{\mathrm{\beta }}\text{b}{}^{\mathrm{\beta }}\text{+ δy}{}^{\mathrm{\beta }}\text{, β ≠ 0}$

;

$\text{P}{}_{\mathrm{ax;by}}\text{= G(a}{}^{\mathrm{\beta }}\text{x}{}^{\mathrm{\gamma }}\text{/b}{}^{\mathrm{\beta }}\text{y}{}^{\mathrm{\gamma }}\text{), β + γ ≠ 0}$

;

$\text{P}{}_{\mathrm{ax;by}}\text{= Q}{}_0\text{(a/x, b/y)}$

.     

The case of Piaget's group INRC

What is it that distinguishes Piaget's transformations N, R, and C from the rest of the 16! transformations of the 16 binary propositional operations? Here Piaget's INRC is considered as a subgroup of the group $\text{M}$₂ of all automorphisms and dual automorphisms of the free Boolean algebra with two generators. This group is isomorphic to S₄ × C₂. Its elements are given explicitly. Many other psychologically relevant subgroups of $\text{M}$₂ play an important role. They are discussed and their connections shown. Particular attention is given to involutions, even if the view that they constitute the sole representation of reversibility is abandoned. Piaget's transformation R turns out not to be the inverse operation of relations. The group of automorphisms, dual automorphisms, anti-automorphisms of the algebra of binary relations on a finite set is found. A crystallographic presentation of these groups is given and related work by Bart (Journal of Mathematical Psychology, 1971, 5, 539–553), Leresche (Revue Européenne des Sciences Sociales, 1976, 14, 219–241), and Pólya (The Journal of Symbolic Logic, 1940, 5, 98–103) is discussed.     

Classification of concatenation measurement structures according to scale type

A relational structure is said to be of scale type (M,N) iff M is the largest degree of homogeneity and N the least degree of uniqueness (Narens, 1981a, Narens, 1981b) of its automorphism group.Roberts (in Proceedings of the first Hoboken Symposium on graph theory, New York: Wiley, 1984; in Proceedings of the fifth international conference on graph theory and its applications, New York: Wiley, 1984) has shown that such a structure on the reals is either ordinal or M is less than the order of at least one defining relation (Theorem 1.2). A scheme for characterizing N is outlined in Theorem 1.3. The remainder of the paper studies the scale type of concatenation structures 〈X, ?, ° 〉, where ? is a total ordering and ° is a monotonic operation. Section 2 establishes that for concatenation structures with M>0 and N<∞ the only scale types are (1,1), (1,2), and (2,2), and the structures for the last two are always idempotent. Section 3 is concerned with such structures on the real numbers (i.e., candidates for representations), and it uses general results of Narens for real relational structures of scale type (M, M) (Theorem 3.1) and of Alper (Journal of Mathematical Psychology, 1985, 29, 73–81) for scale type (1, 2) (Theorem 3.2). For M>0, concatenation structures are all isomorphic to numerical ones for which the operation can be written $\text{x°y = yf(}\text{x}\text{y}\text{)}$, where f is strictly increasing and $\text{f(x)}\text{x}$ is strictly decreasing (unit structures). The equation f(x^?)=f(x)^? is satisfied for all x as follows: for and only for ? = 1 in the (1,1) case; for and only for ?=kⁿ, k > 0 fixed, and n ranging over the integers, in the (1, 2) case; and for all ?>0 in the (2, 2) case (Theorems 3.9, 3.12, and 3.13). Section 4 examines relations between concatenation catenation and conjoint structures, including the operation induced on one component by the ordering of a conjoint structure and the concept of an operation on one component being distributive in a conjoint structure. The results, which are mainly of interest in proving other results, are mostly formulated in terms of the set of right translations of the induced operation. In Section 5 we consider the existence of representations of concatenation structures. The case of positive ones was dealt with earlier (Narens & Luce (Journal of Pure & Applied Algebra27, 1983, 197–233). For idempotent ones, closure, density, solvability, and Archimedean are shown to be sufficient (Theorem 5.1). The rest of the section is concerned with incomplete results having to do with the representation of cases with M>0. A variety of special conditions, many suggested by the conjoint equivalent of a concatenation structure, are studied in Section 6. The major result (Theorem 6.4) is that most of these concepts are equivalent to bisymmetry for idempotent structures that are closed, dense, solvable, and Dedekind complete. This result is important in Section 7, which is devoted to a general theory of scale type (2, 2) for the utility of gambles. The representation is a generalization of the usual SEU model which embodies a distinctly bounded form of rationality; by the results of Section 6 it reduces to the fully rational SEU model when rationality is extended beyond the simplest equivalences. Theorem 7.3 establishes that under plausible smoothness conditions, the ratio scale case does not introduce anything different from the (2, 2) case. It is shown that this theory is closely related to, but somewhat more general, than Kahneman and Tversky's (Econometrica47, 1979, 263–291) prospect theory.     

On the basis of early transitivity judgments

This research had two aims. The first was to test three explanations of performance on N-term series tasks by young children: the labeling model of B.DeBoysson-Bardies and K. O'Regan (1973), Nature (London), 246, 531–534, the sequential-contiguity model of L. Breslow (1981, Psychological Bulletin, 89, 325–351), and the ordered array or image model of C. A. Riley and T. Trabasso (1974, Journal of Experimental Child Psychology, 17, 187–202). In the first experiment, 5-year-old children were taught additional premises which would interfere with labeling and sequential-contiguity processes, but not with forming an ordered array. Reasoning performance was essentially comparable to previous results with the paradigm, thus supporting the ordered array model. The second aim was to reexamine children's ability to learn sets of premises which can be assembled into an ordered array, since there was reason to believe that previous studies had created false positives. In the second experiment, 3- to 7-year-old children were taught either overlapping (a > b, b > c, …) or nonoverlapping (a > b, c > d, …) premises. Overlapping premises can be integrated into an ordered array (a, b, c, d, e), but nonoverlapping premises cannot. However, the overlapping condition proved more difficult, and the success rate for preschoolers ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-olds}$) was of zero order. This raises doubts about their ability to learn a set of premises of the kind required for transitive inference. These doubts were strengthened by the third experiment which showed that when premises were not presented in serial order, preschool ($\text{3}\text{1}\text{2}\text{-}\text{to 4}\text{1}\text{2}\text{-year-old}$) children could not learn the premises of an N-term series task.     

Elkind on egocentrism

The purpose of this paper is to respond to D. Elkind's (1985, Developmental Review, 5, 218–226) reply to D. Lapsley and M. Murphy's (1985, Developmental Review, 5, 201–217) critique of the adolescent egocentrism theory. After a reprise of the issues in contention, we address the problem of theoretical consistency and self-other differentiation in Dr. Elkind's (1967, Child Development, 38, 1025–1034) theory. The role of formal operations in the Lapsley and Murphy (1985) account of the imaginary audience and personal fable is revised, and the empirical support for Elkind's theory is reviewed. We conclude that there are good conceptual and empirical grounds for doubting the major assumptions of the adolescent egocentrism theory, and that the Lapsley and Murphy (1985) framework has promise in suggesting theoretical integration with other approaches to the self.     

The case of the disappearing disagreements: A reply to Yussen

S. R. Yussen (1984, Developmental Review, 4, 139–144) and I have few disagreements. The major one seems to be in what the purpose of my article (R. J. Sternberg, 1984, Developmental Review, 4, 113–138) should have been. I believe the article fulfills my own purpose, if not Yussen's.     

On the relations between random walk models for two-choice response times

We state a condition on the moment-generating functions of the random walk model derived by Link and Heath (1975), which is necessary and sufficient to yield the sequential probability ratio test of Wald (1947) and the reaction-time model proposed by Stone (1960). A restricted form of this condition can also be obtained by imposing the types of symmetries on the moment-generating functions considered by Link and Health (1975) and Thomas (1975). The various conditions are contrasted and their implications for empirical investigation are discussed.     

Why Reading? Because it's there

Three papers (V. A. Mann, Reading skill and language skill. Developmental Review, 1984, 4, 1–15; G. Wolford & C. A. Fowler, Differential use of partial information by good and poor readers. Developmental Review, 1984, 4, 16–35; F. J. Morrison, Reading disability: A problem in rule learning and word decoding. Developmental Review, 1984, 4, 36–47) are critiqued from the standpoint of their adequacy in advancing our understanding of a problem as complex as reading disability. Experimenters should be explicit about their guiding theoretical assumptions, and should think through the relations, if any, between their laboratory tasks and the actual processes of reading. The reading protocol of a dyslexic child is provided, and is interpreted within the frameworks of the Mann, Wolford and Fowler, and Morrison viewpoints.     

Positive-difference structures and bilinear utility functions

Let (M₁, f), (M₂, g) be mixture sets and let ? be a binary preference relation on M₁ × M₂. By using the concept of positive-difference structures, necessary and sufficient conditions are given for the existence of a real-valued utility function u on M₁ × M₂ which represents ? and possesses the bilinearity property

$\text{u(?(α, x}{}_1\text{,x}{}_2\text{),g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{αu(x}{}_1\text{, g(βy}{}_1\text{, y}{}_2\text{))}\text{+}\text{(1 ? α) u(x}{}_2\text{, g(β, y}{}_1\text{, y}{}_2\text{))}\text{=}\text{βu(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_1\text{)}\text{+}\text{(1 ? β) u(?(α,x}{}_1\text{, x}{}_2\text{),y}{}_2\text{)}$

, for all α, β ∈ [0, 1], all x₁, x₂ ∈ M₁ and all y₁, y₂ ∈ M₂. Moreover, uniqueness up to positive linear transformations can be proved for those utility functions. Finally an outline is given of applications of these results in expected utility theory.     

Subjective utility,anticipated utility,and the Allais paradox

One of the most notable counterexamples to expected utility theory is the “Allais paradox” (M. Allais, 1953, Econometrica, 31, 503–546). A number of alternative theories have been proposed in an attempt to resolve this paradox, notably including Karmarkar, 1978, Karmarkar, 1979, 24, 67–72). It is shown that SWU theory necessarily involves violations of dominance, but that the theory can be modified to avoid these violations. The result is a special case of J. Quiggin's anticipated utility theory (1982, Journal of Economic Behaviour and Organisation, 3, 323–343).     

The abstraction of visual prototypes by children

The present studies demonstrated that children as young as second graders could form prototypical representations from a set of highly similar stimuli. The prototype-plus-transformation model of this performance (Franks & Bransford, 1971) was contrasted with an attribute-frequency model (Neumann, 1974). Results supported the latter model when trigram stimuli were used to test differential predictions from the models.     

On mediation

The mediation-nonmediation dichotomy proposed by the Kendlers (Psychological Review, 1962, 69, 1–16) and subscribed to by Cole and Medin (Journal of Experimental Child Psychology, 1973, 15, 352–355) is reviewed and critiqued. It is proposed that the appropriate question for research is how children mediate, not when they mediate.     

Efficient sets,decision heuristics,and single-peaked preferences

The origins of single-peaked preferences in multiattribute choice behavior are analyzed. The starting point is (a), (b), 16, 261–266) theory which explains single-peakedness as the joint effect of task and behavior characteristics. Whereas the task conditions—so-called efficient sets—are retained, and their importance is even more stressed, it is argued that Coombs and Avrunin's behavioral assumptions—classical marginally decreasing preference functions—may not be representative of actually displayed choice behavior. Rather, the information processing approach to decision research has shown that multiattribute choices are often based on simplifying heuristic rules which, due to their possible intransitivity, are inconsistent with Coombs and Avrunin's behavioral requirements. Tversky's (Psychological Review, 1969, 76, 31–48) additive difference model is used as an algebraic framework for analyzing such so-called attribute-based choice rules. Sufficient (and in the continuous case also necessary) conditions are presented which guarantee that the additive difference model will also be single-peaked on efficient sets. These conditions show that the behavioral principles yielding single-peaked preferences are far more general than Coombs and Avrunin assumed and include most of the empirically substantiated forms of attribute-based choice rules. Further, a necessary and sufficient condition is derived which insures that elimination rules, like the conjunctive rule, for instance, and generalizations thereof are single-peaked. Finally, making use of a result by Edgeworth, (Mathematical Psychics, London, Paul, 1881), it is shown that efficient sets per se may originate in quite different ways. It is concluded that the phenomenon of single-peakedness is mainly a function of the task rather than of some specific behavior.     

Traits,processes, and the purpose of templates

The template method proposed by D. J. Bem and D. C. Funder 1978, Psychological Review, 85, 485–501) can have two separate purposes. First, template-behavior pairs can be used as a type of explanation in which situational influences are incorporated within a trait theory. Second, template matching can be used as a method for evaluating process theories. Recent discussion (D. J. Bem, 1983; Psychological Review, 90, 390–393; D. C. Funder, 1983; Psychological Review, 90, 283–289; W. Mischel & P. K. Peake, 1982, Psychological Review, 89, 730–755) confuses these two purposes. If used as a type of explanation, then templates can be criticized on the grounds of blunt empiricism; but if templates are used as a method for evaluating process theories, then process theorists should welcome this new technique.     

Occupational prestige: Exploration of a theoretical basis

Drawing on the theory and research of psychophysics, a nonlinear model is hypothesized to explain the connection between education and income and occupational prestige. To achieve this, Weber's (R. L. Gregory, 1981, Mind in Science, Cambridge, Cambridge Univ. Press, pp. 501–503) and Stevens' (S. S. Stevens, 1970, Science170, 1043–1050) laws are brought together in an intrinsically nonlinear model. Guided by the earlier work of R. L. Hamblin (1971, Sociometry, 34, 423–452) and others, the work of O. D. Duncan (1961, in A. J. Reiss, Jr., O. D. Duncan, P. K. Hatt, & C. C. North (Eds.), Occupations and Social Status, New York, Free Press) is reanalyzed testing the possibility that work on the socioeconomic index can be understood as a prestige allocation process which follows psychophysical principles. That is, prestige is assigned to occupations, given specifiable levels of educational and income attainment, in a manner parallel to the way in which individuals respond to changes in the intensity of other stimuli. Using first the data developed by Duncan (1961) to test the model and the 1963 NORC data (R. W. Hodge, P. M. Siegel, & P. H. Rossi, 1964, American Journal of Sociology, 70, 286–302) to replicate it, a measurement model consistent with the theoretical model is evaluated. Comparing the results of the nonlinear model to that of the linear, it is concluded that a model is obtained yielding theoretical confirmation with no loss in predictive accuracy. The resultant nonlinear model yields alternative substantive implications concerning the relative influence of income and education on occupational prestige to those to be inferred from linear models. Perhaps most important, however, is the candidacy given by these results to psychophysics as the explanatory mechanism in the prestige allocation process.     

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.     

Causal attributions and information seeking to explain success and failure

Previous studies of attributions for success and failure have relied upon a theoretically derived set of causal attributions, either luck versus skill (derived from Rotter, 1966) or ability, effort, luck, and task difficulty (Weiner et al., 1971; Weiner, 1972). The same tendency of relying upon a set of logically derived cues has been evident in investigations of information-processing in making attributional judgments for success and failure (e.g., Fontaine, 1972; Frieze & Weiner, 1971). Two studies were conducted which employed an open-ended format to determine the relevance of the causal categories and information which have been utilized in previous studies. Results of these studies supported the validity of previously employed causal categories and information cues as well as establishing other important but previously ignored causes and sources of information.