Security and potential level preferences with thresholds

The security level models of Gilboa [1988. A combination of expected utility and maxmin decision criteria. Journal of Mathematical Psychology, 32, 405-420] and of Jaffray [1988. Choice under risk and the security factor: An axiomatic model. Theory and Decision, 24, 169-200] as well as the security and potential level model of Cohen [1992. Security level, potential level, expected utility: A three-criteria decision model under risk. Theory and Decision, 33, 101-104] and Essid [1997. Choice under risk with certainty and potential effects: A general axiomatic model. Mathematical Social Sciences, 34, 223-247] successfully accommodate classical Allais paradoxes while they offer an interesting explanation for their occurrence. However, experimental data suggest a systematic violation of these models when lotteries with low probabilities of bad or good outcomes are involved. In our opinion, one promising candidate for the explanation of these violations is the assumption of thresholds in the perception of security and potential levels. The present paper develops an axiomatic model that allows for such thresholds, so that the derived representation of preferences can accommodate the observed violations of the original security and potential level models.     

Reference dependence in cumulative prospect theory

A central feature of (cumulative) prospect theory is reference dependence, i.e. the value function depends on gains and losses relative to a status quo and not on final wealth positions as in expected utility theory. Until now, all theoretical work on prospect theory only considered a fixed status quo. In this case, however, reference dependence and the dependence of the value function on final wealth positions are equivalent concepts, at least if the status quo is known. In order to identify the differences between both concepts, an axiomatization of cumulative prospect theory is developed which allows for a variable status quo. In this framework a preference condition can be formulated which discriminates between reference dependence and the dependence on final wealth positions. Furthermore, this framework allows one to analyze in which way the shape of the value and weighting functions change with increasing status quo. With respect to this question several hypotheses are defined and characterized in terms of preference conditions.     

‘Additive difference’ models without additivity and subtractivity

This paper studies conjoint measurement models tolerating intransitivities that closely resemble Tversky's additive difference model while replacing additivity and subtractivity by mere decomposability requirements. We offer a complete axiomatic characterisation of these models without having recourse to unnecessary structural assumptions on the set of objects. This shows the pure consequences of several cancellation conditions that have often been used in the analysis of more traditional conjoint measurement models. Our models contain as particular cases many aggregation rules that have been proposed in the literature.     

A simple derivation of Prelec's probability weighting function

Since Kahneman and Tversky [(1979). Prospect theory: An analysis of decision under risk. Econometrica, 47, 263-291], it has been generally recognized that decision makers overweight low probabilities. Of the several weighting functions that have been proposed, that of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528] has the attractions that it is parsimonious, consistent with much of the available empirical evidence and has an axiomatic foundation. Luce [(2001). Reduction invariance and Prelec's weighting functions. Journal of Mathematical Psychology, 45, 167-179] provided a simpler derivation based on reduction invariance, rather than compound invariance of Prelec [(1998). The probability weighting function. Econometrica, 60, 497-528]. This note introduces a behavioral assumption that we call power invariance and provides a simple derivation of Prelec's function. Thus, we have three a priori different behavioral assumptions all leading to Prelec's function.     

Maximum likelihood additivity analysis

A maximum likelihood estimation procedure was developed to fit unweighted and weighted additive models to conjoint data obtained by the categorical rating, the pair comparison or the directional ranking method. The scoring algorithm used to fit the models was found to be both reliable and efficient, and the program MAXADD is capable of handling up to 300 parameters to be estimated. Practical uses of the procedure are reported to demonstrate various advantages of the procedure as a statistical method.The research reported here was supported by Grant A6394 to the author from the Natural Sciences and Engineering Research Council of Canada. Portions of this research were presented at the Psychometric Society meeting in Iowa City, Iowa, in May, 1980.Thanks are due to Jim Ramsay, Justine Sergent and anonymous reviewers for their helpful comments.Two MAXADD programs which perform the computations discussed in this paper may be obtained from the author.     

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:

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

;

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

;

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

.     

On additivity of duration reproduction functions

In the present paper properties of theoretical response functions in the duration reproduction task (drt) are studied. If the drt response is to be interpreted as the measure of a temporal duration elapsed in the past, the reproduction function (rf) must be additively measuring. The additivity property is defined for bivariate rfs, taking into account the delay between retention and reproduction. The requirement that an rf be additively measuring leads to a system of two functional equations, which are solved by (i) linear functions of time, or (ii) functions of the form yielded by the ‘dual klepsydra model’ of duration reproduction [Wackermann, J., & Ehm, W. (2006). The klepsydra model of internal time representation and time reproduction. Journal of Theoretical Biology, 239, 482-493]. This result supports the outstanding rôle of the ‘klepsydraic reproduction function’ (krf) in models of internal representation and reproduction of temporal durations.     

An algorithm for finite conjoint additivity

A complete solution to the problem of finite conjoint additivity is provided, not only to answer the question of whether a particular order is conjointly additive, but also to detail the possible value assignments which demonstrate conjoint additivity for the order. In the process, a complete solution is given to the problem of solving a finite number of homogeneous linear inequalities in a finite number of variables. While others have reduced the problem to the solution of systems of linear inequalities, the algorithm presented here is based on “basic inequalities,” and it provides considerable computational simplicity over previous algorithms.     

Monotonic weighted power transformations to additivity

A class of monotonic transformations which generalize the power transformation is fit to the independent and dependent variables in multiple regression so that the resulting additive relationship is optimized. This is achieved by minimizing a quadratic fitting criterion with linear inequality constraints on the parameters. A quadratic programming technique which works reliably and quickly in this application is outlined. Some examples of the analysis of artificial and real data are offered.This work was supported by National Research Council of Canada Grant APA 320 to the author.     

Bicartesian Coherence

Coherence is demonstrated for categories with binary products and sums, but without the terminal and the initial object, and without distribution. This coherence amounts to the existence of a faithful functor from a free category with binary products and sums to the category of relations on finite ordinals. This result is obtained with the help of proof-theoretic normalizing techniques. When the terminal object is present, coherence may still be proved if of binary sums we keep just their bifunctorial properties. It is found that with the simplest understanding of coherence this is the best one can hope for in bicartesian categories. The coherence for categories with binary products and sums provides an easy decision procedure for equality of arrows. It is also used to demonstrate that the categories in question are maximal, in the sense that in any such category that is not a preorder all the equations between arrows involving only binary products and sums are the same. This shows that the usual notion of equivalence of proofs in nondistributive conjunctive-disjunctive logic is optimally defined: further assumptions would make this notion collapse into triviality.     

Failure of additivity in bisection of length

Subjects were intructed to select one rod to lie halfway in length between two given rods. These bisection instructions imply an additive model in the subjective metric. However, the data were inherently nonadditive; the length of the bisector could be an increasing or a decreasing function of the length of one given rod, depending on the length of the other given rod. A convexity analysis and a nonmetric analysis both showed that no monotone transformation could make the data additive. The bisection problem is used to contrast the axiomatic and functional approaches to measurement theory.     

Loss aversion and perceptual risk aversion

We prove that, in cumulative prospect theory, the weak loss aversion for S-shaped value functions is equivalent to a notion of risk aversion that we define from the perceptual point of view. No additional assumption or condition on the probability distortion is needed. It is demonstrated that a power S-shaped value function does not satisfy weak loss aversion, i.e., a decision maker is risk seeking with respect to some mixed sign lotteries.     

Subjective sampling distributions and the additivity of estimates

Abstract.— Previous studies of sampling distributions have been conducted almost exclusively under the assumption that persons behave in accordance with the "fundamental convention" of probability, i.e. that the sum of all probability estimates will equal 1. When this assumption was tested by asking subjects to give "unrestricted" probability estimates of all possible outcomes of samples from a given population, a general tendency of overestimation made the sum of all probabilities exceed 1 to a considerable extent. The subjective sampling distributions appeared to be unaffected by sample size ( N=5 or 10) and number of outcomes, and were flatter than the corresponding "objective" sampling distributions.     

Coherence and disagreement

I wish to thank Lars Bergström, Carl Ginet, Joseph Moore, Christian Munthe, Torbjörn Tännsjö, and an anonymous referee for Philosophical Studies for valuable comments on earlier drafts of this paper. The remaining errors are of course my own.     

Coherence and Explanations

The article advocates a particular coherence theory of justification that emphasizes the significance of explanatory relations. It is shown that other approaches to coherence have failed because they underestimate the importance of explanatory theories in forming a system of beliefs. Additionally, a conception of explanation as a unifying substantial embedding of models is sketched that closely conforms with the proposed theory of coherence.     

Coherence and coreference

Coherence in conversations and in texts can be partially characterized by a set of coherence relations, motivated ultimately by the speaker's or writer's need to be understood. In this paper, formal definitions are given for several coherence relations, based on the operations of an inference system; that is, the relations between successive portions of a discourse are characterized in terms of the inferences that can be drawn from each. In analyzing a discourse, it is frequently the case that we would recognize it as coherent, in that it would satisfy the formal definition of some coherence relation, if only we could assume certain noun phrases to be coreferential. In such cases, we will simply assume the identity of the entities referred to, in what might be called a “petty conversational implicature,” thereby solving the coherence and coreference problems simultaneously. Three examples of different kinds of reference problems are presented. In each, it is shown how the coherence of the discourse can be recognized, and how the reference problems are solved, almost as a by-product, by means of these petty conversational implicatures.