Three ancient problems solved by using the game theory logic based on the Shapley value

The ancient problems of bankruptcy, contested garment, and rights arbitration have generated many studies, debates, and controversy. The objective of this paper is to show that the Shapley value from game theory, measuring the power of each player in a game, may be consistently applied for getting the general one-step solution of all these three problems viewed as n-person games. The decision making is based on the same tool, namely the game theory logic based on the use of the Shapley value, but the specific games involved are slightly different in each problem. The kind of claims of the players, the relationship between the given claims and the given resources available, and the particular way of calculating the generalized characteristic function of the game determine the specific type of game which has to be solved in each of the three ancient problems mentioned. The iterative use of the Shapley value may also justify the well-known Aumann–Maschler step-by-step procedure for solving the bankruptcy problem.

     

The bias function of the maximum likelihood estimate of ability for the dichotomous response level

Samejima has recently given an approximation for the bias function for the maximum likelihood estimate of the latent trait in the general case where item responses are discrete, generalizing Lord's bias function in the three-parameter logistic model for the dichotomous response level. In the present paper, observations are made about the behavior of this bias function for the dichotomous response level in general, and also with respect to several widely used mathematical models. Some empirical examples are given.     

Oblique factor analytic solutions by orthogonal transformations

A general framework for obtaining all possible factor analytic solutions, orthogonal and oblique, for a given common factor space is developed in detail. Interestingly, and seemingly paradoxically, any one of these solutions may be obtained by orthogonal transformations of selected matrices; thus an oblique solution may be determined by orthogonal transformations. Within the possible oblique solutions, two distinct categories of solutions emerge, a special case of the simpler of which apparently provides a definitive solution to the problem of independent, but correlated, clusters. Possible further specializations of the general approach to specific problems are discussed.     

Pattern variants on a square field

A quantitative approach to the psychology of pattern recognition requires knowledge of the number of possible variants of any particular pattern. The general solution for the number [p/m ²] of pattern variants thatp counters can form on a square network ofm ² positions is obtained by elementary group theory. The exact solution is given in terms of the different types (symmetric, asymmetric, etc.) of patterns possible and an approximate formula for the total number of patterns is also developed.The author is indebted to Mr. J. Sandiford for his invaluable assistance on the theoretical aspects of the problem and to Mr. J. L. Griffith for his helpful criticism.     

Language and cognition-Kurt Goldstein's theory of semantics

Kurt Goldstein is regarded as one of the major proponents of the holistic movement at the beginning of the 20th century. He rejected the strong localization hypothesis in the field of aphasiology and attempted to link language disturbances to an underlying general intellectual impairment. Goldstein's criticism was based on his subtle symptomatology, his organismic biology, and his philosophical reflections. In his concept of abstract attitude Goldstein searched for a general psychological function that might explain a variety of aphasic symptoms. Abstract attitude bridges the gap between cognitive and linguistic structures. According to Goldstein, it is the basis for words to have a meaning, to be employed in a categorical sense. Since amnesic aphasics are confined to a concrete attitude, their words have lost their representational function. Although Goldstein's concept of abstract attitude is no longer used in scientific discourse, it is analyzed for its heuristic value. It led Goldstein to questions about the relation between cognition and language and to fragments of a semantic theory.     

Some neglected problems in IRT

The paper addresses three neglected questions from IRT. In section 1, the properties of the “measurement” of ability or trait parameters and item difficulty parameters in the Rasch model are discussed. It is shown that the solution to this problem is rather complex and depends both on general assumptions about properties of the item response functions and on assumptions about the available item universe. Section 2 deals with the measurement of individual change or “modifiability” based on a Rasch test. A conditional likelihood approach is presented that yields (a) an ML estimator of modifiability for given item parameters, (b) allows one to test hypotheses about change by means of a Clopper-Pearson confidence interval for the modifiability parameter, or (c) to estimate modifiability jointly with the item parameters. Uniqueness results for all three methods are also presented. In section 3, the Mantel-Haenszel method for detecting DIF is discussed under a novel perspective: What is the most general framework within which the Mantel-Haenszel method correctly detects DIF of a studied item? The answer is that this is a 2PL model where, however, all discrimination parameters are known and the studied item has the same discrimination in both populations. Since these requirements would hardly be satisfied in practical applications, the case of constant discrimination parameters, that is, the Rasch model, is the only realistic framework. A simple Pearsonx ² test for DIF of one studied item is proposed as an alternative to the Mantel-Haenszel test; moreover, this test is generalized to the case of two items simultaneously studied for DIF.     

Distance and Similarity Measures in Generalised Quantum Theory

A summary of recent experimental results shows that entanglement can be generated more easily than before, and that there are improved chances for its persistence. An eminent finding of Generalised Quantum Theory is the insight that the notion of entanglement can be extended, such that, e.g., psychological or psychophysical problem areas can be included, too. First, a general condition for entanglement to occur is given by the term ‘common prearranged context’. A formalised treatment requires a quantitative definition of the similarity or dissimilarity between two complex structures which takes their internal structures into account. After some specific remarks on distance, metrics, and semi-metrics in mathematics, a procedure is described for setting up a similarity function with the required properties. This procedure is in analogy with the two-step character of measurement and with the well-known properties of perspective notions. A general methodology can be derived for handling perspective notions. Finally, these concepts supply heuristic clues towards a formalised treatment of the notions of ‘meaning’ and ‘interpretation’.     

A generalization of Verhelst's solution for a constrained regression problem in alscal and related MDS-algorithms

Verhelst derived a solution for a constrained regression problem which occurs in the interval measurement application of ALSCAL and related MDS-algorithms. In the present paper it is shown that Verhelst's solution is based on an implicit nonsingularity assumption. A general solution, which contains Verhelst's solution as a special case, is derived by a simple completing-the-squares type approach instead of partial differentiation with a Lagrange multiplier. In addition, this approach permits the identification of a small interval which uniquely contains the optimal value of a parameter needed to solve the special case where Verhelst's solution is valid.The author is obliged to Dirk Knol and Klaas Nevels for helpful comments.     

Using dual-process theory and analogical transfer to explain facilitation on a hypothetico-deductive reasoning task

Using the analogical transfer paradigm, the present study investigated the competing explanations of Girotto and Legrenzi (Psychological Research 51: 129–135, 1993) and Griggs, Platt, Newstead, and Jackson (Thinking and Reasoning 4: 1–14, 1998) for facilitation on the SARS version of the THOG problem, a hypothetico-deductive reasoning task. Girotto and Legrenzi argue that facilitation is based on logical analysis of the task [System 2 reasoning in Evans’s (Trends in Cognitive Sciences 7: 454–459, 2003) dual-process account of reasoning] while Griggs et al. maintain that facilitation is due to an attentional heuristic produced by the wording of the problem (System 1 reasoning). If Girotto and Legrenzi are correct, then System 2 reasoning, which is volitional and responsible for deductive reasoning, should be elicited, and participants should comprehend the solution principle of the THOG task and exhibit analogical transfer. However, if Griggs et al. are correct, then System 1 reasoning, which is responsible for heuristic problem solving strategies such as an attentional heuristic, should occur, and participants should not abstract the solution principle and transfer should not occur. Significant facilitation (68 and 82% correct) was only observed for the two SARS source problems, but significant analogical transfer did not occur. This lack of transfer suggests that System 1 reasoning was responsible for the facilitation observed in the SARS problem, supporting Griggs et al.’s attentional heuristic explanation. The present results also underscore the explanatory value of using analogical transfer rather than facilitation as the criterion for problem understanding.     

Gaussian model‐based partitioning using iterated local search

The emergence of Gaussian model‐based partitioning as a viable alternative to K‐means clustering fosters a need for discrete optimization methods that can be efficiently implemented using model‐based criteria. A variety of alternative partitioning criteria have been proposed for more general data conditions that permit elliptical clusters, different spatial orientations for the clusters, and unequal cluster sizes. Unfortunately, many of these partitioning criteria are computationally demanding, which makes the multiple‐restart (multistart) approach commonly used for K‐means partitioning less effective as a heuristic solution strategy. As an alternative, we propose an approach based on iterated local search (ILS), which has proved effective in previous combinatorial data analysis contexts. We compared multistart, ILS and hybrid multistart–ILS procedures for minimizing a very general model‐based criterion that assumes no restrictions on cluster size or within‐group covariance structure. This comparison, which used 23 data sets from the classification literature, revealed that the ILS and hybrid heuristics generally provided better criterion function values than the multistart approach when all three methods were constrained to the same 10‐min time limit. In many instances, these differences in criterion function values reflected profound differences in the partitions obtained.     

An approximation for the bias function of the maximum likelihood estimate of a latent variable for the general case where the item responses are discrete

Lord developed an approximation for the bias function for the maximum likelihood estimate in the context of the three-parameter logistic model. Using Taylor's expansion of the likelihood equation, he obtained an equation that includes the conditional expectation, given true ability, of the discrepancy between the maximum likelihood estimate and true ability. All terms of orders higher thann ^?1 are ignored wheren indicates the number of items. Lord assumed that all item and individual parameters are bounded, all item parameters are known or well-estimated, and the number of items is reasonably large. In the present paper, an approximation for the bias function of the maximum likelihood estimate of the latent trait, or ability, will be developed using the same assumptions for the more general case where item responses are discrete. This will include the dichotomous response level, for which the three-parameter logistic model has been discussed, the graded response level and the nominal response level. Some observations will be made for both dichotomous and graded response levels.     

Empirical evaluation of a model of global psychophysical judgments: IV. Forms for the weighting function

Understanding the psychological interpretation of numerals is of both practical and theoretical interest. In classical magnitude estimation, respondents match numerals to sensations and in magnitude production they select sensations that stand in a prescribed numerical ratio to a given standard. The present work focusses on evaluating several possible, and related, forms for the function W formulating the distortion of numerals. The main form, of which a power function is a special case, is the Prelec exponential/power representation. Behavioral equivalents to power and to Prelec functions are formulated, tested, and rejected. It is argued that either the mathematical form or the assumption W(1)=1 is wrong. Whereas, the axiomatic literature has focussed exclusively on the former inference, we explore the alternate that W(1)≠1. Behavioral axioms are formulated in each case and experimentally tested. We conclude that most respondents satisfy a general power function and that those who do not, satisfy the general Prelec function.     

Time-distance records and the power function

World record performances in running, walking, swimming, speed skating, and horse racing are closely described by a function of the form y=ax^b. In addition, all the functions have approximately the same exponent! These data are discussed in relation to S. S. Stevens’s sensory power law as a type of complex case of cross-modality matching, and it is suggested that the power function describes a more general behavioral process which necessarily includes sensory discriminations.     

One articulation of the structural family therapy model: a biobehavioral family model of chronic illness in children

This article presents the Biobehavioral Family Model (BBFM), which, as a general systems model, has the advantage of being conceptually neutral with respect to health versus pathology, and promotes integration of individual, family and social levels of theory. The BBFM provides a heuristic theory of pathways by which family patterns of interaction and individual family member physiological function influence one another. The BBFM also can be extrapolated to the clinical domain, providing a developmental biopsychosocial approach to assessment and intervention in childhood chronic illness or disability. A case illustration demonstrates clinical application.     

A HEURISTIC SCIENCE‐BASED NATURALISM AS A PARTNER FOR THEOLOGICAL REFLECTIONS ON THE NATURAL WORLD

After a few general observations on scientific activity, the author briefly comments on different versions of naturalism. Subsequently, he suggests that the birth of evolutionary biology and its successive developments may show how the natural world comes to be differently conceived as scientific advancements are accomplished. Then the main thesis is outlined by introducing the principles of a heuristic science‐based naturalism not conclusively defining the real and the knowable. From the epistemological perspective, heuristic naturalism is meant to be framed in critical realism, whereas from the ontological standpoint it may be framed in emergent monism, given that the latter can also underpin recent trends in investigation addressing human specificity. Finally, attention is turned to some implications of heuristically guided scientific activity with regard to the issues of divine action and of imago Dei.     

Fast and frugal heuristics: rationality and the limits of naturalism

Gerd Gigerenzer and Thomas Sturm have recently proposed a modest form of what they describe as a normative, ecological and limited naturalism. The basic move in their argument is to infer that certain heuristics we tend to use should be used in the right ecological setting. To address this argument, we first consider the case of a concrete heuristic called Take the Best (TTB). There are at least two variants of the heuristic which we study by making explicit the choice functions they induce, extending these variants of TTB beyond binary choice. We argue that the naturalistic argument can be applied to only one of the two variants of the heuristic; we also argue that the argument for the extension requires paying attention to other “rational” virtues of heuristics aside from efficacy, speed, and frugality. This notwithstanding, we show that there is a way of extending the right variant of TTB to obtain a very well behaved heuristic that could be used to offer a stronger case for the naturalistic argument (in the sense that if this heuristic is used, it is also a heuristic that we should use). The second part of the article considers attempts to extending the naturalistic argument from algorithms dealing with inference to heuristics dealing with choice. Our focus is the so-called Priority Heuristic, which we extend from risk to uncertainty. In this setting, the naturalist argument seems more difficult to formulate, if it remains feasible at all. Normativity seems in this case extrinsic to the heuristic, whose main virtue seems to be its ability to describe actual patterns of choice. But it seems that a new version of the naturalistic argument used with partial success in the case of inference is unavailable to solve the normative problem of whether we should exhibit the patterns of choice that we actually display.     

Counterfactual processing and the correspondence between events and outcomes: Normality versus value

Kahneman and Tversky (1982) have proposed a simulation heuristic such that perceivers tend to substitute ‘normal’ antecedent events for exceptional ones in psychologically ‘undoing’ a given outcome. Recently Gavanski and Wells (1989) have demonstrated that exceptional outcomes tend to be perceived as caused by exceptional events and normal outcomes by normal events, a finding more in line with the representativeness heuristic than this ‘normalization’ principle. We argue that representativeness may be determined by the evaluative tone of events as well as by probability—namely that positive events are assumed to underlie positive outcomes and negative events, negative outcomes. Both normality and value were independently manipulated in order to test the relative effects of each of these factors. In contrast to Gavanski and Wells our data indicate that preference was given to the similarity of value between events and outcome for undoing both positive and negative and normal and exceptional outcomes. Some implications of these findings for counterfactual processing are discussed.     

Experts,Novices, and the St. Petersburg paradox: Is one solution enough?

Although the controversy over the correct solution to the St. Petersburg paradox continues in the decision making literature, few of the solutions have been empirically evaluated. Via the development of alternative versions of the St. Petersburg game, we were able to empirically test some of these solutions. Experts and novices behaved in accordance with Treisman's expectation heuristic when bidding for the right to play the various versions of the St. Petersburg game. When subjects were asked their preferences among the game versions. novices continued to behave in accordance with the expectation heuristic but a plurality of experts seemed to follow another strategy. This preference reversal and its implications and possible causes are thoroughly discussed. An alternative theory which mimicks the expectation heuristic is considered, and generalizations of the expectation heuristic and the St. Petersburg Paradox for z-sided 'coins' (where z is any integer greater than or equal to 2) are presented. It appears that no one solution is yet rich enough for the St. Petersburg paradox.