Consensus for belief functions and related uncertainty measures

We extend previous work of Lehrer and Wagner, and of McConway, on the consensus of probabilities, showing under axioms similar to theirs that (1) a belief function consensus of belief functions on a set with at least three members and (2) a belief function consensus of Bayesian belief functions on a set with at least four members must take the form of a weighted arithmetic mean. We observe that these results are unchanged when consensual uncertainty measures are allowed to take the form of Choquet capacities of low order monotonicity.This work was supported in part by a grant from the University of Tennessee Artificial Intelligence Group, and by the U.S. Naval Sea Systems Command, through a grant to Oak Ridge National Laboratory.     

Coherent bets under partially resolving uncertainty and belief functions

By considering situations of partially resolving uncertainty, a one-to-one correspondence between belief functions and coherent betting systems à la de Finetti is shown to exist.     

Capacity Updating Rules and Rational Belief Change

Choquet expected utility substitutes capacities for subjective probabilities to explain uncertainty aversion and related phenomena. This paper studies capacities as models of belief. The notions of inner and outer acceptance context are defined. These are shown to be the natural acceptance contexts when belief expansion is described by naïve Bayesian and Dempster–Shafer updating of capacities respectively. We also show that Eichberger and Kelsey's (1999b) use of Dempster–Shafer updating as a model of belief revision may lead to violations of the AGM axioms for rational belief change.     

Is there a Gender Difference in False Belief Development?

The contribution of children's social environment to their acquisition of theory of mind skills, combined with the well documented advantage for girls in mental state talk with siblings, peers and mothers, might lead to a female advantage on false belief tasks. We present a post‐hoc analysis of large datasets from two independent laboratories. A slight advantage for girls on false belief task performance was found in both datasets and was only apparent in younger but not older children. Language ability could be controlled for only in a smaller subsample of one dataset and cannot be ruled out as a potential mediator of this effect. However, if there is an age‐specific advantage for girls in false belief acquisition it is a weak effect only.     

Belief system foundations of backward induction

Two justifications of backward induction (BI) in generic perfect information games are formulated using Bonanno's (1992; Theory and Decision 33, 153) belief systems. The first justification concerns the BI strategy profile and is based on selecting a set of rational belief systems from which players have to choose their belief functions. The second justification concerns the BI path of play and is based on a sequential deletion of nodes that are inconsistent with the choice of rational belief functions.     

Anxiety and Decision Making with Delayed Resolution of Uncertainty

In many real-world gambles, a non-trivial amount of time passes before the uncertainty is resolved but after a choice is made. An individual may have a preference between gambles with identical probability distributions over final outcomes if they differ in the timing of resolution of uncertainty. In this domain, utility consists not only of the consumption of outcomes, but also the psychological utility induced by an unresolved gamble. We term this utility anxiety. Since a reflective decision maker may want to include anxiety explicitly in analysis of unresolved lotteries, a multiple-outcome model for evaluating lotteries with delayed resolution of uncertainty is developed. The result is a rank-dependent utility representation (e.g., Quiggin, 1982), in which period weighting functions are related iteratively. Substitution rules are proposed for evaluating compound temporal lotteries. The representation is appealing for a number of reasons. First, probability weights can be interpreted as the cognitive attention allocated to certain outcomes. Second, the model disaggregates strength of preference from temporal risk aversion and thus provides some insight into the old debate about the relationship between von Neumann–Morgenstern utility functions and strength of preference value functions.     

From Transitional to Permanent Uncertainty: Employability of Middle‐aged Workers in Hong Kong

Since the 1990s, Hong Kong has experienced increasing economic uncertainty. Middle‐aged workers, who often have little formal education and limited skills, have been the first casualties of economic restructuring. Initially, their unemployment was considered transitional. Limited government‐sponsored retraining programs were expected to improve their employability. Subsequent economic crises, however, have undercut the hopes of both workers and policy‐makers. Despite improvements in the job market since the mid‐2000s, the belief that unemployment was transitional has been replaced by a permanent state of uncertainty and consequent anxiety. Flexibility is considered the key to sustaining economic growth, and would relieve the insecurity and anxiety associated with unemployment. Policies have been modified, but they are still based on a narrowly defined concept of individual employability, which consolidates the sense of uncertainty. To address the factors contributing to the perception of permanent uncertainty among middle‐aged workers, we advocate a pragmatic approach that takes reference from the notion of flexicurity to mitigate this sense of uncertainty.     

Uncertainty,the bargaining problem,and the Nash-Zeuthen solution

The Zeuthen bargaining model occupies a prominent place among those theories of the bargaining process that have been formulated and expounded by economists. Its solution to the bargaining problem is essentially economic, since invariant utility functions based on economic factors alone determine the outcome. However, this paper shows that a necessary condition for reaching the Zeuthen solution (shown by Harsanyi to be mathematically equivalent to the game-theoretic solution of Nash's theory) is that bargainers initially take up positions on opposite sides of the outcome that maximizes their utility product. Whether utility functions are mutually known or unknown, inherent in the bargaining situation itself is the requirement that bargainers be at least initially uncertain as to each other's subsequent concession behaviour. With uncertainty, von Neumann-Morgenstern rationality implies that each bargainer would make an initial demand that maximizes the expected gain from holding fast. Therefore, even if Zeuthen's concession criterion should subsequently dictate concession behaviour, expected utility maximization within the context of subjective uncertainty may well yield initial demands that are inconsistent with reaching the Nash-Zeuthen solution. Finally, a general methodological conclusion that emerges from the analysis is that, since the bargaining process necessarily proceeds from a context of subjective uncertainty, greater emphasis needs to be placed on its role as a device for affecting expectations.     

Beliefs About Affirmative Action and Diversity and Their Relationship to Support for Hiring Policies

Two hundred seventy-three White undergraduates participated in an investigation of how beliefs relate to support for affirmative action (AA) policies. Beliefs included belief in the fairness of AA, belief in merit, and belief in the value of diversity. Analyses predicted support for a general affirmative action policy, a tiebreak policy, and a policy using banding from beliefs and individual-level variables such as future benefit from AA and demographics. For the general policy, each belief predicted support for AA. Fairness and value of diversity predicted support for a tiebreak policy. Value of diversity predicted support for aptitude testing. Individual characteristics improved prediction for AA in general but not for tiebreak policy or aptitude testing. We discuss predictions and results in terms of procedural and distributive justice, fairness heuristic theory, and models of support for AA.     

The Hysteresis Effect: Theorizing Mismatch in Action

Widespread reliance on representationalist understandings commit social scientists to either partially or totally decouple belief from reality, limiting the domain of phenomena that can be treated by belief as an analytic concept. Developing the contrastive notion of practical belief, we introduce the hysteresis effect as a situational phenomenon involving the systematic production of agent‐environment mismatches and argue for its placement as a central problem for the theory of action. Revealing the dynamic, embodied conservation of belief in the temporality of practice, hysteresis appears when environmental contexts change in a way that leaves actors without an ontologically complicit relationship to institutions as scaffolds of action. Under these circumstances, a past‐inflected reflexiveness replaces a forward‐inflecting practical belief in the actor's temporal relation to the world. Drawing from a variety of historical case studies, we locate hysteresis in routine disjunctures between the temporality of practice and the temporality of environments. Our analysis reveals four distinct types of reflexiveness produced the hysteresis effect, each with a unique dispositional impact on actors and extensions to group‐level phenomena. We conclude the article by emphasizing the non‐eliminativist relationship between belief as disposition and belief as representation.     

Probability and logic in belief systems

This paper seeks to develop a formal (mathematical) model of belief systems based on the axioms of probability theory and propositional logic. By a belief system we mean a set of propositions along with an actor's objective probability assignments to (beliefs in) them, together with the relationships among and between propositions and beliefs. Belief systems are regarded as being comprised of interrelated elements.In the paper are developed measures of the distance between sets of beliefs; of the congruence, coherence, and consistency of belief systems; and of the degree of polarization of belief systems-which are derived from one basic operation, symmetric difference.We show that these measures possess a number of useful and powerful mathematical properties. Also, a model is set forth by which, from an actor's subjective probability assignment to propositions and pairwise conjunctions of propositions, we may then impute to the actor subjectively perceived truth functional relationships between proposition.The potential uses and practical difficulties with the approach taken in the paper are also discussed, and the assertion is made that the measures developed enable us to simply distinguish between certain notions (e.g., congruence, consistency, coherence) too often and easily confused, provide us with the possibility of interval (or at least, quasi-interval level) measurement of certain properties of individual belief systems, and also allows us to make comparisons between the structures of different actors' belief systems.An earlier version of this paper was delivered at the 1969 American Political-Science Association, Chicago.     

A Philosophical Foundation of Non-Additive Measure and Probability

In this paper, non-additivity of a set function is interpreted as a method to express relations between sets which are not modeled in a set theoretic way. Drawing upon a concept called “quasi-analysis” of the philosopher Rudolf Carnap, we introduce a transform for sets, functions, and set functions to formalize this idea. Any image-set under this transform can be interpreted as a class of (quasi-)components or (quasi-)properties representing the original set. We show that non-additive set functions can be represented as signed σ-additive measures defined on sets of quasi-components. We then use this interpretation to justify the use of non-additive set functions in various applications like for instance multi criteria decision making and cooperative game theory. Additionally, we show exemplarily by means of independence, conditioning, and products how concepts from classical measure and probability theory can be transfered to the non-additive theory via the transform.     

论企业技术创新

企业技术创新具有提高企业竞争力、推动技术自身的发展、对企业技术进步的辐射、提高企业现代化管理水平等作用。企业技术创新具有不确定性、准公共产品性、风险性、市场性和商业性等特征。企业技术创新动力主要来自主体和环境两个方面,称为“技术创新主体论”和“技术创新环境论”。加快企业技术创新需要采取一系列对策。     

Executive function and theory of mind as predictors of socially withdrawn behavior in institutionalized children

False‐belief understanding and executive functions are two main sociocognitive abilities reliably linked to child social competence. Although institution‐reared children are especially at risk for behavioral problems and cognitive delays, the role that executive function and false‐belief understanding might play in the social withdrawal of institutionalized children has not been examined. The current study used two‐wave data to investigate the concurrent and longitudinal relations of social withdrawal with executive function and false‐belief understanding in institutionalized children; it also allowed investigation of the directionality between executive function and false‐belief understanding. Data were collected from 66 Turkish children (T1 M = 57.83 months, SD = 9.20; T2 M = 69.58 months, SD = 8.45) residing in institutions, at two time points, approximately 1 year apart. We measured false‐belief understanding and executive function via individual assessments, and social withdrawal via care provider reports at both time points. Results showed that both executive function and false‐belief understanding increased between T1 and T2, while social withdrawal did not show a significant change. Path analysis revealed that when T1 age and language were controlled, T1 executive function predicted T2 executive function, and in turn, T2 executive function predicted lessened social withdrawal at T2. In addition, T1 executive function predicted T2 false‐belief understanding. T1 false‐belief understanding was not related to T2 false‐belief understanding, executive function, or social withdrawal. Findings suggested that executive function is an important predictor of social withdrawal in high‐risk populations.     

Conditioning Capacities and Choquet Integrals: The Role of Comonotony

Choquet integrals and capacities play a crucial role in modern decision theory. Comonotony is a central concept for these theories because the main property of a Choquet integral is its additivity for comonotone functions. We consider a Choquet integral representation of preferences showing uncertainty aversion (pessimism) and propose axioms on time consistency which yield a candidate for conditional Choquet integrals. An other axiom characterizes the role of comonotony in the use of information. We obtain two conditioning rules for capacities which amount to the well-known Bayes' and Dempster–Schafer's updating rules. We are allowed to interpret both of them as a lack of confidence in information in a dynamic extension of pessimism. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Longitudinal Study of Mother-Child Relationships and Theory of Mind in the Preschool Period

To investigate individual differences in theory of mind acquisition, this study examined whether different aspects of early mother-child relationships contributed to the development of false belief understanding at the close of the preschool period. Forty-six mother and child pairs were seen when children were two and again at five years of age. At age two, home-based Q-sort observations of attachment security and maternal sensitivity were made, and mothers completed a number of self-report measures to create an aggregate of maternal emotional distress. At age five, attachment security and maternal distress were reassessed, and false belief tasks were administered that were based on unexpected identities and locations of objects. In addition, attachment-relevant false belief tasks involving separation from caregivers were also used, which children found significantly more difficult than tasks involving objects. Age five security predicted object location task performance. Maternal sensitivity and emotional distress at age two predicted later caregiver location task performance, even controlling for age five measures. These results support a growing literature on the importance of relationship processes and parenting context to theory of mind acquisition.     

Expanding state space and extension of beliefs

This article provides a simple decision theoretic model in which elements of the world successively enter the decision maker??s scope and the state space expands over time, which is intended to be the closest correspondence to the standard subjective expected utility theory. We propose a dynamic consistency condition that after any expansion of the scope, the preference ranking should remain unchanged over acts to which the expansion is irrelevant. Together with other natural axioms, it characterizes a model in which the decision maker??s belief extends over time in order that the marginal distribution of the new belief induced over the old state space coincides with the old belief. It is extended to encompass both expansion of scope and learning events, and we characterize the model with an additional property that the decision maker??s belief updating follows Bayes?? rule when she learns events.     

The value of changes in life expectancy

Valuation formulas for age-specific mortality risks are derived from life-cycle allocation theory under uncertainty and related to empirical estimates of the value of life. A change in an age-specific mortality risk affects all subsequent survivor functions and reallocates consumption and labor supply over the entire life cycle. The value of eliminating a risk to life at a specific age is the expected present value of consumer surplus from that age forward. Approximate numerical extrapolations from cross-section estimates imply that values decrease rapidly in current age and in the distance between current age and age at risk.     

Uncertainty aversion and aversion to increasing uncertainty

According to the original Ellsberg (1961) examples there is uncertainty version if the decision maker prefers to bet on an urn of known composition rather than on an urn of unknown composition. According to another definition (Schmeidler, 1989), there is uncertainty aversion if any convex combination of two acts is preferred to the least favorable of these acts. We show that these two definitions differ: while the first one truly refers to uncertainty aversion, the second one refers to aversion to increasing uncertainty. Besides, with reference to Choquet Expected Utility theory, uncertainty aversion means that there exists the core of a capacity, while aversion to increasing uncertainty means that the capacity is convex. Consequently, aversion to increasing uncertainty implies uncertainty aversion, but the opposite does not hold. We also show that a completely analogous situation holds for the case of risk and we define a set of risk and uncertainty premiums according to the previous analysis.     

Cognitive rationality and alternative belief measures

In this article, we first examine the various criticisms of the probabilistic model. Then we introduce capacities in order to show that if a probability measure corresponds to anesthetizing the belief of the agent's knowledge, it is then possible to suggest another type of rationality—namely, being able to describe a wise and a rash behavior when facing risk—and therefore another model of belief under uncertainty. While trying to specify various alternative measures, possibility, necessity, and measures resulting from a triangular norm or from a triangular conorm, we finally try to define the field of application of the probabilistic model as well as a sign of the rationality choice: constraint of mass-unity for traditional rationality, and constraint of duality for the one we present.This article is a revised version of a paper presented during the FUR V Congress, Duke University, North Carolina, USA, June 10–13, 1990. I am indebted to Bertrand Munier, Alain Chateauneuf, Olivier Favereau, Jacques-François Thisse, Bernard Walliser, and Kip Viscusi for helpful comments on earlier versions of this article.