The shape of incomplete preferences

Incomplete preferences provide the epistemic foundation for models of imprecise subjective probabilities and utilities that are used in robust Bayesian analysis and in theories of bounded rationality. This paper presents a simple axiomatization of incomplete preferences and characterizes the shape of their representing sets of probabilities and utilities. Deletion of the completeness assumption from the axiom system of Anscombe and Aumann yields preferences represented by a convex set of state-dependent expected utilities, of which at least one must be a probability/utility pair. A strengthening of the state-independence axiom is needed to obtain a representation purely in terms of a set of probability/utility pairs.Comment: Published at http://dx.doi.org/10.1214/009053606000000740 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cognition and Wealth: The Importance of Probabilistic Thinking

This paper utilizes a large set of subjective probability questions from the Health and Retirement Survey to construct an index measuring the precision of probabilistic beliefs (PPB) and relates this index to household choices about the riskiness of their portfolios and the rate of growth of their net worth. A theory of uncertainty aversion based on repeated sampling is proposed that resolves the Ellsberg Paradox within a conventional expected utility model. In this theory, uncertainty aversion is implied by risk aversion. This theory is then used to propose a link between an individual’s degree of uncertainty and his propensity to give “focal” answers of “0”, “50_50” or “100” or “exact” answers to survey questions and the validity of this interpretation is tested empirically. Finally, an index of the precision of probabilistic thinking is constructed by calculating the fraction of probability questions to which each HRS respondent gives a non-focal answer. This index is shown to have a statistically and economically significant positive effect on the fraction of risky assets in household portfolios and on the rate of growth of these assets longitudinally. These results suggest that there is systematic variation in the competence of individuals to manage investment accounts that should be considered in designing policies to create individual retirement accounts in the Social Security system.

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

B

2-coherent and 2-convex Conditional Lower Previsions

In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of $n$-coherent and $n$-convex conditional previsions, at the varying of $n$. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) $2$-convex or, if positive homogeneity and conjugacy is needed, $2$-coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalized Bayes Rule and always have a $2$-convex or, respectively, $2$-coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, $n$-convex and $n$-coherent previsions with $n\geq 3$ either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by $2$-convexity, we discuss generalizations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered $2$-convex. In the final part, we determine the rationality requirements of $2$-convexity and $2$-coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.Comment: This is the authors' version of a work that was accepted for publication in the International Journal of Approximate Reasoning, vol. 77, October 2016, pages 66-86, doi:10.1016/j.ijar.2016.06.003, http://www.sciencedirect.com/science/article/pii/S0888613X1630079

The objective Bayesian conceptualisation of proof and reference class problems

The objective Bayesian view of proof (or logical probability, or evidential support) is explained and defended: that the relation of evidence to hypothesis (in legal trials, science etc) is a strictly logical one, comparable to deductive logic. This view is distinguished from the thesis, which had some popularity in law in the 1980s, that legal evidence ought to be evaluated using numerical probabilities and formulas. While numbers are not always useful, a central role is played in uncertain reasoning by the ‘proportional syllogism’, or argument from frequencies, such as ‘nearly all aeroplane flights arrive safely, so my flight is very likely to arrive safely’. Such arguments raise the ‘problem of the reference class’, arising from the fact that an individual case may be a member of many different classes in which frequencies differ. For example, if 15 per cent of swans are black and 60 per cent of fauna in the zoo is black, what should I think about the likelihood of a swan in the zoo being black? The nature of the problem is explained, and legal cases where it arises are given. It is explained how recent work in data mining on the relevance of features for prediction provides a solution to the reference class problem

An Experiment in the Economic Consequences of Additional Disclosure: The Case of the Fair Value of Unlisted Equity Investments

We investigate the economic consequences of additional disclosure about assets with no active market in terms of liquidity, perception of information reliability and relevance. We use an experimental design: 181 MBA students are asked to value 24 investments. We manipulate the level of disclosure on the fair value of assets (Limited versus Full), the perception of expected profit (Gain versus Loss) and the firm's business risk (Low versus High). In the case of Limited (resp. Full) disclosure, participants are given a point estimate for the fair value of the investment (resp. plus a range of possible fair values). We find that in the Full disclosure situation, participants tend to make an offer more frequently, to bid prices lower than the fair value and to earn a lower return on their investments compared to the Limited disclosure case. These consequences vary with the environment (expected profit and risk).fair value; fair value measurement; disclosure; relevance; reliability; risk; SFAS 157; IFRS 7

Computational Determination of Coherence of Financial Risk Measure as a Lower Prevision of Imprecise Probability

This study is about developing some further ideas in imprecise probability models of financial risk measures. A financial risk measure has been interpreted as an upper prevision of imprecise probability, which through the conjugacy relationship can be seen as a lower prevision. The risk measures selected in the study are value-at-risk (VaR) and conditional value-at-risk (CVaR). The notion of coherence of risk measures is explained. Stocks that are traded in the financial markets (the risky assets) are seen as the gambles. The study makes a determination through computation from actual assets data whether the risk measure assessments of gambles (assets) are coherent as an imprecise probability. It is observed that coherence of assessments depends on the asset's returns distribution characteristic

Stochastic multi-period multi-product multi-objective Aggregate Production Planning model in multi-echelon supply chain

In this paper a multi-period multi-product multi-objective aggregate production planning (APP) model is proposed for an uncertain multi-echelon supply chain considering financial risk, customer satisfaction, and human resource training. Three conflictive objective functions and several sets of real constraints are considered concurrently in the proposed APP model. Some parameters of the proposed model are assumed to be uncertain and handled through a two-stage stochastic programming (TSSP) approach. The proposed TSSP is solved using three multi-objective solution procedures, i.e., the goal attainment technique, the modified ε-constraint method, and STEM method. The whole procedure is applied in an automotive resin and oil supply chain as a real case study wherein the efficacy and applicability of the proposed approaches are illustrated in comparison with existing experimental production planning method