First-order Stochastic Dominance Research Articles

Measures of stochastic non-dominance in portfolio optimization

Stochastic dominance rules are well-characterized and widely used. This work aims to describe and better understand the situations when they do not hold by developing measures of stochastic non-dominance. They quantify the error caused by assuming that one random variable dominates another one when it does not. To calculate them, we search for a hypothetical random variable that satisfies the stochastic dominance relationship and is as close to the original one as possible. The Wasserstein distance between the optimal hypothetical random variable and the original one is considered as the measure of stochastic non-dominance. Depending on the conditions imposed on the probability distribution of the hypothetical random variable, we distinguish between general and specific measures of stochastic non-dominance. We derive their exact values for random variables with uniform, normal, and exponential distributions. We present relations to almost first-order stochastic dominance and to tractable almost stochastic dominance. Using monthly returns of twelve assets captured by the German stock index DAX, we solve portfolio optimization problems with the first-order and second-order stochastic dominance constraints. The measures of stochastic non-dominance allow us to compare the optimal portfolios with respect to different orders of stochastic dominance from a new angle. We also defined the closest dominating and closest approximately dominating portfolios. They brought a better understanding of the relationship between the two types of optimal portfolios. Using moving window analysis, the relationship of the in-sample measure of stochastic non-dominance to out-of-sample performance was studied, too.

Distributionally Robust Newsvendor Under Stochastic Dominance with a Feature-Based Application

Problem definition: In this paper, we study the newsvendor problem under some distributional ambiguity sets and explore their relations. Additionally, we explore the benefits of implementing this robust solution in the feature-based newsvendor problem. Methodology and results: We propose a new type of discrepancy-based ambiguity set, the JW ambiguity set, and analyze it within the framework of first-order stochastic dominance. We show that the distributionally robust optimization (DRO) problem with this ambiguity set admits a closed-form solution for the newsvendor loss. This result also implies that the newsvendor problem under the well-known infinity-Wasserstein ambiguity set and Lévy ball ambiguity set admit closed-form inventory levels as a by-product. In the application of feature-based newsvendor, we adopt general kernel methods to estimate the conditional demand distribution and apply our proposed DRO solutions to account for the estimation error. Managerial implications: The closed-form solutions enable an efficient computation of optimal inventory levels. In addition, we explore the property of optimal robust inventory levels with respect to the nonrobust version via concepts of perceived critical ratio and mean repulsion. The results of numerical experiments and the case study indicate that the proposed model outperforms other state-of-the-art approaches, particularly in environments where demand is influenced by covariates and difficult to estimate. Funding: X. Li is supported by the Singapore Ministry of Education [Tier 1 Grant 23-0619-P0001, 24-0500-A0001] and National Natural Science Foundation of China [Grant 72331004]. L. Zhang is partially supported by the National Natural Science Foundation of China [Grants 72171156 and 72231002] and the Hong Kong Research Grants Council [Grant 16212419]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0159 .

Exact Characterization of the Jointly Optimal Restocking and Auditing Policy in Inventory Systems with Record Inaccuracy

We present a continuous-time stochastic model of an inventory system with record inaccuracy. In this formulation, demand is modeled by a point process and is observable only when it leads to sales. In addition to demand that can reduce the stock, an unobservable stochastic loss process can also reduce the stock. The retailer’s goal is to identify the restocking and auditing policy that minimizes the expected discounted cost of carrying a product over an infinite horizon. We analytically characterize the optimal restocking and jointly optimal auditing policy. We prove that the optimal restocking policy is a threshold policy. Our proof of this result is based on a coupling argument that is valid for any demand and loss model. Unlike the optimal restocking policy, the jointly optimal auditing policy is not of threshold type. We show that a complete proof of this statement cannot be obtained by solely resorting to the first-order stochastic dominance property of the Bayesian shelf stock distribution induced by the demand and loss process. Instead, our characterization of the jointly optimal auditing policy is based on proving that the dynamics of the shelf stock distribution constitute a (strictly) sign-regular kernel. To our knowledge, this is the first paper that characterizes the optimal policy of a complex control problem by establishing sign regularity of its underlying Markovian dynamics. Our theoretical results lead to a fast algorithm for computing the exact jointly optimal auditing/restocking policy over the problem’s entire state space. This enables comparative statics analysis, which allows us to determine how inventory record inaccuracy affects the economic significance of various cost drivers. This, in turn, allows us to determine when or, better, under what conditions auditing can be an effective tool for reducing the total cost.

An Unexpected Stochastic Dominance: Pareto Distributions, Dependence, and Diversification

Diversification is generally regarded as an efficient tool to reduce portfolio risks. In “An unexpected stochastic dominance: Pareto distributions, dependence, and diversification,” Chen, Embrechts, and Wang showed that the weighted average of independent and identically distributed (i.i.d.) Pareto random variables with infinite mean is larger than one such random variable in the sense of first-order stochastic dominance, and thus diversification is, surprisingly, worse than no diversification. The relation implies superadditivity of value-at-risk, a regulatory risk measure used in the finance and insurance sectors. The obtained relation also holds under some form of negative dependence.

Mixture independence foundations for expected utility

An alternative preference foundation for expected utility is provided. Our segregated approach considers four logically independent implications of the classic von Neumann–Morgenstern independence axiom. The monotonicity principle is, for a transitive relation, equivalent to monotonicity with respect to first-order stochastic dominance. Rank-dependent separability is similar to the comonotonic sure-thing principle used in the ambiguity literature. The remaining two properties are weak formulations of the independence principle which invoke the latter only for probability mixtures with the extreme, that is the best, respectively, the worst outcome. These four implications of independence, together with completeness, transitivity and continuity of a preference relation, characterize expected utility. Furthermore, if rank-dependent separability is dropped, expected utility still holds on each subset of three-outcomes lotteries that give positive probability to both the best and the worst outcomes.

Violations of first-order stochastic dominance

AbstractI find necessary and sufficient conditions for first-order stochastic dominance (FOSD) violations for choices from a budget line of Arrow securities. Applying this characterization to existing data, I compare FOSD violation rates across a broad set of risk preference elicitation tasks.

Risk Classification in Insurance Markets with Risk and Preference Heterogeneity

Abstract This paper studies a competitive model of insurance markets in which consumers are privately informed about their risk and risk preferences. We provide a characterization of the equilibria, which depend non-trivially on consumers’ type distribution, a desirable feature for policy analysis. The use of consumer characteristics for risk classification is modeled as the disclosure of a public informative signal. A novel property of signals, monotonicity, is shown to be necessary and sufficient for their release to be welfare improving for almost all consumer types. We also study the effect of changes to the risk distribution in the population as the result of demographic changes or policy interventions. We show that an increase in the risk distribution, according to the monotone likelihood ratio ordering of distribution, leads to lower utility for almost all consumer types. In contrast, the effect is ambiguous when considering the first-order stochastic dominance ordering.

Extreme Points and First-Order Stochastic Dominance: Theory and Applications

Extreme Points and First-Order Stochastic Dominance: Theory and Applications

First-Order Stochastic Dominance Violations, Risk Preferences and Framing Effects

First-Order Stochastic Dominance Violations, Risk Preferences and Framing Effects

Contests with an uncertain number of prizes with a fixed total value

We study multiple-prize contests where the number of prizes to be awarded is a random variable. We identify conditions under which a unique symmetric Nash equilibrium exists. We compare the equilibrium efforts according to different probability distributions of the number of prizes. Considering multi-prize contests proposed so far in the literature, we show that each player’s effort decreases with the average number of prizes (first-order stochastic dominance) and may increase or decrease with the risk in the number of prizes (second-order stochastic dominance) depending on the contest technology adopted.

Almost first-order stochastic dominance by distorted expectations

Almost stochastic dominance has been receiving a great amount of attention in the financial and economic literatures. In this paper, we characterize the properties of almost first-order stochastic dominance (AFSD) via distorted expectations and investigate the conditions under which AFSD is preserved under a distortion transform. The main results are also applied to establish stochastic comparisons of order statistics and receiver operating characteristic curves via AFSD.

Chance theory: A separation of riskless and risky utility

In a temporal context, sure outcomes may yield higher utility than risky ones as they are available for the execution of plans before the resolution of uncertainty. By observing a disproportionate preference for certainty, empirical research points to a fundamental difference between riskless and risky utility. Chance Theory (CT) accounts for this difference and, in contrast to earlier approaches to separate risky and riskless utility, does not violate basic rationality principles like first-order stochastic dominance or transitivity. CT evaluates the lowest outcome of an act with the riskless utility v and the increments over that outcome, called chances, by subjective expected utility (EU) with a risky utility u. As a consequence of treating sure outcomes differently to risky ones, CT is able to explain the EU-paradoxes of Allais (Econometrica, 21(4): 503–546, 1953) that rely on the certainty effect, and also the critique to EU put forward by Rabin (Econometrica, 68(5): 1281–1292, 2000). Moreover, CT separates risk attitudes in the strong sense, captured entirely by u, from attitude towards wealth reflected solely through the curvature ofv.

Multiple-switching behavior in choice-list elicitation of risk preference

This study examines multiple-switching behavior (MSB) in choice-list elicitation of risk preference from the perspectives of stochastic choice. We distinguish between “regular” and “irregular” MSB, and find that subjects with more irregular MSB are more likely to violate first-order stochastic dominance. In contrast, subjects with more regular MSB are more likely to concurrently exhibit non-expected utility behavior and reduce compound lottery, and to deliberately randomize in repeated choice. Our results suggest the need to diagnose the quality of MSB when applying choice-list elicitations, and distinguish stochastic choice models including random utility and deliberate randomization.

Expected scalarised returns dominance: a new solution concept for multi-objective decision making

AbstractIn many real-world scenarios, the utility of a user is derived from a single execution of a policy. In this case, to apply multi-objective reinforcement learning, the expected utility of the returns must be optimised. Various scenarios exist where a user’s preferences over objectives (also known as the utility function) are unknown or difficult to specify. In such scenarios, a set of optimal policies must be learned. However, settings where the expected utility must be maximised have been largely overlooked by the multi-objective reinforcement learning community and, as a consequence, a set of optimal solutions has yet to be defined. In this work, we propose first-order stochastic dominance as a criterion to build solution sets to maximise expected utility. We also define a new dominance criterion, known as expected scalarised returns (ESR) dominance, that extends first-order stochastic dominance to allow a set of optimal policies to be learned in practice. Additionally, we define a new solution concept called the ESR set, which is a set of policies that are ESR dominant. Finally, we present a new multi-objective tabular distributional reinforcement learning (MOTDRL) algorithm to learn the ESR set in multi-objective multi-armed bandit settings.

The impact of risk aversion and ambiguity aversion on annuity and saving choices

We analyze the impact of risk aversion and ambiguity aversion on the competing demands for annuities and bequeathable savings using a lifecycle recursive utility model. Our main finding is that risk aversion and ambiguity aversion have similar effects: an increase in either of the two reduces annuity demand and enhances bond holdings. We obtain this unequivocal result in the flexible intertemporal framework of Hayashi and Miao (2011) by assuming that the agent’s preferences are monotone with respect to first-order stochastic dominance. Our contribution is then twofold. First, from a decision-theoretic point of view, we show that monotonicity allows one to obtain clear-cut results about the respective roles of risk and ambiguity aversion. Second, from the insurance point of view, our result that the demand for annuities decreases with risk and ambiguity aversion stands in contrast with what is usually found with other insurance products. As such, it may help explain the low annuitization level observed in the data.

Social preferences or sacred values? Theory and evidence of deontological motivations.

Recent advances in economic theory, largely motivated by experimental findings, have led to the adoption of models of human behavior where decision-makers take into consideration not only their own payoff but also others’ payoffs and any potential consequences of these payoffs. Investigations of deontological motivations, where decision-makers make their choice based on not only the consequences of a decision but also the decision per se, have been rare. We provide a formal interpretation of major moral philosophies and a revealed preference method to distinguish the presence of deontological motivations from a purely consequentialist decision-maker whose preferences satisfy first-order stochastic dominance.

Histogram Distortion Bias in Consumer Choices

Existing research on word-of-mouth considers various descriptive statistics of rating distributions, such as the mean, variance, skewness, kurtosis, and even entropy and the Herfindahl-Hirschman index. But real-world consumer decisions are often derived from visual assessment of displayed rating distributions in the form of histograms. In this study, we argue that such distribution charts may inadvertently lead to a consumer-choice bias that we call the histogram distortion bias (HDB). We propose that salient features of distributions in visual decision making may mislead consumers and result in inferior decision making. In an illustrative model, we derive a measure of the HDB. We show that with the HDB, consumers may make choices that violate well-accepted decision rules. In a series of experiments, subjects are observed to prefer products with a higher HDB despite a lower average rating. They could also violate widely accepted modeling assumptions, such as branch independence and first-order stochastic dominance.This paper was accepted by Chris Forman, information systems.Funding: This work was supported by the National Natural Science Foundation of China [Grants 72131001 and 92146003] and the Research Grants Council, University Grants Committee [GRF 14500521, GRF 14501320, GRF 14503818, and TRS:T31-604/18-N].Supplemental Material: Data and the online appendices are available at https://doi.org/10.1287/mnsc.2022.4306 .

A flexible and scalable model to improve decision quality in shale plays

Shale asset investment decisions are difficult to model because they are portfolios of options under complex and time-evolving uncertainties. Sequential development decisions must balance short-term cash flow with long-term value creation. Typical assets consist of thousands of locations, making exact analysis impossible. Consequently, decision support tools are very simple, usually consisting of decision trees, fixed price decks, and no midstream constraints.We develop a heuristic that maps the current information of an asset, such as inventory, prices, and estimated production, to a well schedule. We combine the heuristic with Monte Carlo simulation and decision trees to form a shale field development model that includes all relevant features while being scalable to realistic problem sizes. To demonstrate our method's performance, we create a sample shale asset with 370 wells, price and production uncertainty, midstream constraints, and 30 decision periods. On the sample problem, our method exhibits first-order stochastic dominance over the decision tree approach and improves expected value by 407%. Using our modeling approach will improve overall decision quality in shale asset management.

Further properties of fractional stochastic dominance

AbstractA continuum of stochastic dominance rules, also referred to as fractional stochastic dominance (SD), was introduced by Müller, Scarsini, Tsetlin, and Winkler (2017) to cover preferences from first- to second-order SD. Fractional SD can be used to explain many individual behaviors in economics. In this paper we introduce the concept of fractional pure SD, a special case of fractional SD. We investigate further properties of fractional SD, for example the generating processes of fractional pure SD via $\gamma$ -transfers of probability, Yaari’s dual characterization by utilizing the special class of distortion functions, the separation theorem in terms of first-order SD and fractional pure SD, Strassen’s representation, and bivariate characterization. We also establish several closure properties of fractional SD under quantile truncation, under comonotonic sums, and under distortion, as well as its equivalence characterization. Examples of distributions ordered in the sense of fractional SD are provided.

Robustness of the Poverty Measures: Evidence from Farm Households in Akwa Ibom State, Nigeria

The use of a plethora of poverty indexes is sometimes fraught with difficulties. The purpose of this research was to quantitatively assess poverty and to examine the robustness of the poverty metrics. Selecting representative farm homes required a multistage sample technique, which was implemented. A total of 150 rural homes were surveyed using questionnaires. Stochastic dominance and the weighted poverty measures of Foster, Greer and Thorbecke were used in this work to examine the weighted poverty measures' resilience and sensitivity to changes in the poverty line. According to the findings, as people become older and their families get larger, the likelihood, severity, and depth of poverty increases. An asymptotic sampling distribution was utilized to infer whether poverty was larger across a variety of hypothetical poverty lines by stochastic dominance analysis. First-order stochastic dominance was found, with the Cumulative Distribution Function (CDF) of households headed by people over 60 years old lying totally above the other distribution functions (CDFs). The CDF of single families was lower than the CDF of married households, according to the findings. At any poverty level, the CDF of families with more than 10 household members stochastically dominated those with fewer family members. Many households will be lifted out of poverty if poverty-reduction initiatives are targeted at those over 60 and those with big families.