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  • 1
    Electronic Resource
    Electronic Resource
    Measurement structures with archimedean ordered translation groups (1987)
    Springer
    Order 4 (1987), S. 165-189 
    Details
    ISSN: 1572-9273
    Keywords: 08A02 ; 08A35 ; Archimedean ; automorphism ; measurement ; ordered relational structure ; translation group
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper focuses on three problems of generalizing properties of concatenation structures (ordered structures with a monotonic operation) to ordered structures lacking any operation. (1) What is the natural generalization of the idea of Archimedeaness, of commensurability between large and small? (2) What is the natural generalization of the concept of a unit concatenation structure in which the translations (automorphisms with no fixed point) can be represented by multiplication by a constant? (3) What is the natural generalization of a ratio scale concatenation structure being distributive in a conjoint one, which has been shown to force a multiplicative representation of the latter and the product-of-powers representation of units found in physics? It is established (Theorems 5.1 and 5.2) that for homogeneous structures, the latter two questions are equivalent to it having the property that the set of all translations forms a homogeneous Archimedean ordered group. A sufficient condition for Archimedeaness of the translations is that they form a group, which is equivalent to their being 1-point unique, and the structure be Dedekind complete and order dense (Theorems 2.1 and 2.2). It is suggested that Archimedean order of the translations is, indeed, also the answer to the first question. As a lead into that conclusion, a number of results are reported in Section 3 on Archimedeaness in concatenation structures, including for positive structures sufficient conditions for several different notions of Archimedeaness to be equivalent. The results about idempotent structures are fragmentary.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00337695
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  • 2
    Electronic Resource
    Electronic Resource
    Thurstone's discriminal processes fifty years later (1977)
    Springer
    Psychometrika 42 (1977), S. 461-489 
    Details
    ISSN: 1860-0980
    Keywords: Thurstonian scaling model ; discriminal process ; neural schema for auditory psychophysics ; attention bands ; attention as sampling ; absolute identification ; magnitude estimation ; frequency discrimination
    Source: Springer Online Journal Archives 1860-2000
    Topics: Psychology
    Notes: Abstract Four issues are discussed concerning Thurstone's discriminal processes: the distributions governing the representation, the nature of the response decision rules, the relation of the mean representation to physical characteristics of the stimulus, and factors affecting the variance of the representation. A neural schema underlying the representation is proposed which involves samples in time of pulse trains on individual neural fibers, estimators of parameters of the several pulse trains, samples of neural fibers, and an aggregation of the estimates over the sample. The resulting aggregated estimate is the Thurstonian representation. Two estimators of pulse rate, which is monotonic with signal intensity, are timing and counting ratios and two methods of aggregation are averaging and maximizing. These lead to very different predictions in a speed-accuracy experiment; data indicate that both estimators are available and the aggregation is by averaging. Magnitude estimation data are then used both to illustrate an unusual response rule and to study the psychophysical law. In addition, the pattern of variability and correlation of magnitude estimates on successive trials is interpreted in terms of the sample size over which the aggregation takes place. Neural sample size is equated with selective attention, and is an important factor affecting the variability of the representation. It accounts for the magical number seven phenomenon in absolute identification and predicts the impact of nonuniform distributions of intensities on the absolute identification of two frequencies.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02295975
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  • 3
    Electronic Resource
    Electronic Resource
    On Elements of Chance (2000)
    Springer
    Theory and decision 49 (2000), S. 97-126 
    Details
    ISSN: 1573-7187
    Keywords: Element of chance ; Idempotence ; Kernel equivalent ; Rank-dependent utility ; Utility of chance ; Utility of gambling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Sociology , Economics
    Notes: Abstract One aspect of the utility of gambling may evidence itself in failures of idempotence, i.e., when all chance outcomes give rise to the same consequence the `gamble' may not be indifferent to its common consequence. Under the assumption of segregation, such gambles can be expressed as the joint receipt of the common consequence and what we call `an element of chance', namely, the same gamble with the common consequence replaced by the status quo. Generalizing, any gamble is indifferent to the joint receipt of its element of chance and a certain consequence, which is called the `kernel equivalent' of the gamble. Under idempotence, the kernel equivalent equals the certainty equivalent. Conditions are reported (Theorem 4) that are sufficient for the kernel equivalents to have the kind of utility representation first discussed by Luce and Fishburn (1991), including being idempotent. This utility representation of the kernel equivalents together with the derived form of utility over joint receipts yields a utility representation of the original structure. Possible forms for the utility of an element of chance are developed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1005288012381
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    Paper (German National Licenses)
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