Generalising the pari-mutuel model

We introduce two models for imprecise probabilities which generalise the Pari-Mutuel Model while retaining its simple structure. Their consistency properties are investigated, as well as their capability of formalising an assessor\u2019s different attitudes. It turns out that one model is always coherent, while the other is (occasionally coherent but) generally only 2-coherent, and may elicit a conflicting attitude towards risk

Distortion models for estimating human error probabilities

Human Reliability Analysis aims at identifying, quantifying and proposing solutions to human factors causing hazardous consequences. Quantifying the influence of the human factors gives rise to human error probabilities, whose estimation is a cumbersome problem. Since these human factors are usually related to other organisational or technological factors, it has been proposed to apply probabilistic graphical models, such as Bayesian or credal networks. However, these can be problematic when conditional probabilities on missing data are involved. While the solutions proposed so far combine frequentist and subjective approaches and are in general not robust to small modifications in the dataset, in this paper we propose an alternative based on distortion models, which are a type of imprecise probabilities. We perform a comparative analysis, showing that our proposal is consistent with the previous studies while giving rise to robust estimations

Quantifying Degrees of E-admissibility in Decision Making with Imprecise Probabilities

This paper is concerned with decision making using imprecise probabilities. In the first part, we introduce a new decision criterion that allows for explicitly modeling how far decisions that are optimal in terms of Walley’s maximality are accepted to deviate from being optimal in the sense of Levi’s E-admissibility. For this criterion, we also provide an efficient and simple algorithm based on linear programming theory. In the second part of the paper, we propose two new measures for quantifying the extent of E-admissibility of an E-admissible act, i.e. the size of the set of measures for which the corresponding act maximizes expected utility. The first measure is the maximal diameter of this set, while the second one relates to the maximal barycentric cube that can be inscribed into it. Also here, for both measures, we give linear programming algorithms capable to deal with them. Finally, we discuss some ideas in the context of ordinal decision theory. The paper concludes with a stylized application examples illustrating all introduced concepts

Quantifying Degrees of E-admissibility in Decicion Making with Imprecise Probabilities

This paper is concerned with decision making using imprecise probabilities. In the first part, we introduce a new decision criterion that allows for explicitly modeling how far decisions that are optimal in terms of Walley’s maximality are accepted to deviate from being optimal in the sense of Levi’s E-admissibility. For this criterion, we also provide an efficient and simple algorithm based on linear programming theory. In the second part of the paper, we propose two new measures for quantifying the extent of E-admissibility of an E-admissible act, i.e. the size of the set of measures for which the corresponding act maximizes expected utility. The first measure is the maximal diameter of this set, while the second one relates to the maximal barycentric cube that can be inscribed into it. Also here, for both measures, we give linear programming algorithms capable to deal with them. Finally, we discuss some ideas in the context of ordinal decision theory. The paper concludes with a stylized application examples illustrating all introduced concepts

Concepts for Decision Making under Severe Uncertainty with Partial Ordinal and Partial Cardinal Preferences

We introduce three different approaches for decision making under uncertainty if (I) there is only partial (both cardinally and ordinally scaled) information on an agent’s preferences and (II) the uncertainty about the states of nature is described by a credal set (or some other imprecise probabilistic model). Particularly, situation (I) is modeled by a pair of binary relations, one specifying the partial rank order of the alternatives and the other modeling partial information on the strength of preference. Our first approach relies on decision criteria constructing complete rankings of the available acts that are based on generalized expectation intervals. Subsequently, we introduce different concepts of global admissibility that construct partial orders between the available acts by comparing them all simultaneously. Finally, we define criteria induced by suitable binary relations on the set of acts and, therefore, can be understood as concepts of local admissibility. For certain criteria, we provide linear programming based algorithms for checking optimality/admissibility of acts. Additionally, the paper includes a discussion of a prototypical situation by means of a toy example

Lucky Stores, Gambling, and Addiction: Empirical Evidence from State Lottery Sales

There is a large body of literature in both psychology and economics documenting mistaken perceptions of randomness. In this paper we demonstrate that people appear to believe that "lightning will strike twice" when it comes to lottery jackpots. First, we show that in the week following the sale of a winning ticket, retailers that sell a winning jackpot ticket experience relative increases in game-specific ticket sales of between 12 and 38 percent, with the sales response increasing in the size of the jackpot. In addition, the increase in sales experienced by the winning vendor increases with the proportion of the local population comprised of high school dropouts, elderly adults, and households receiving public assistance. We further show that this increase in retail-game sales initially reflects an increase in total sales at the retail and zip code level. Second, we show that the increase in sales is persistent at the winning retailer. However, the data no not provide clear evidence that the increase in sales at the zip code level is persistent. It thus appears that in the long run, consumers are persistent in their habit of buying lottery tickets at the "lucky" store; however, as the shock to total gambling dissipates, there is no evidence that lottery gambling itself is habit forming or addictive.

Bivariate p-boxes

A p-box is a simple generalization of a distribution function, useful to study a random number in the presence of imprecision. We propose an extension of p-boxes to cover imprecise evaluations of pairs of random numbers and term them bivariate p-boxes. We analyze their rather weak consistency properties, since they are at best (but generally not) equivalent to 2-coherence. We therefore focus on the relevant subclass of coherent p-boxes, corresponding to coherent lower probabilities on special domains. Several properties of coherent p-boxes are investigated and compared with those of (one-dimensional) p-boxes or of bivariate distribution functions

Weak Dutch Books with Imprecise Previsions

Uncertainty assessments for imprecise previsions based on coherence and related concepts require that the suprema of certain random numbers (interpreted as gains) are non-negative. The extreme situation that a supremum is zero represents what is called a Weak Dutch Book (WDB) in a betting interpretation language. While most of the previous dedicated literature focused on WDBs for de Finetti's coherence with precise probabilities, in this paper we analyse the properties of WDBs with imprecise previsions, notably for conditional (Williams') coherent lower previsions. We show that WDB assessments ensure a certain `local precision' property and imply, in the agent's evaluation, some kind of `protection' against real losses. Further, these properties vary with the consistency notion we adopt, tending to vanish with weaker ones. A generalisation of the classical strict coherence and other alternative approaches to WDBs are also discussed