Cardinal Utility Research Articles

Learning Decision-Making Functions Given Cardinal and Ordinal Consensus Data

Decision-making and reaching consensus are an integral part of everyday life, and studying how individuals reach these decisions is an important problem in psychology, economics, and social choice theory. Our work develops methods and theory for learning the nature of decisions reached upon by individual decision makers or groups of individuals using data. We consider two tasks, where we have access to data on: 1) Cardinal utilities for d individuals with cardinal consensus values that the group or decision maker arrives at, 2) Cardinal utilities for d individuals for pairs of actions, with ordinal information about the consensus, i.e., which action is better according to the consensus. Under some axioms of social choice theory, the set of possible decision functions reduces to the set of weighted power means, M(u, w, p) = (∑ᵢ₌₁ᵈ wᵢ uᵢᵖ)¹ᐟᵖ, where uᵢ indicate the d utilities, w ∈ ∆_{d - 1} denotes the weights assigned to the d individuals, and p ∈ ℝ (Cousins 2023). For instance, p = 1 corresponds to a weighted utilitiarian function, and p = -∞ is the egalitarian welfare function. Our goal is to learn w ∈ ∆_{d - 1} and p ∈ ℝ for the two tasks given data. The first task is analogous to regression, and we show that owing to the monotonicity in w and p (Qi 2000}, learning these parameters given cardinal utilities and social welfare values is a PAC-learnable task. For the second task, we wish to learn w, p such that, given pairs of actions u, v ∈ ℝ₊ᵈ, the preference is given as C((u, v), w, p) = sign(ln(M(u, w, p)) - ln(M(v, w, p))). This is analogous to classification; however, convexity of the loss function in w and p is not guaranteed. We analyze two related cases - one in which the weights w are known, and another in which the weights are unknown. We prove that both cases are PAC-learnable given positive u, v by giving an O(log d) bound on the VC dimension for the known weights case, and an O(d log d) bound for the unknown weights case. We also establish PAC-learnability for noisy data under IID (Natarajan 2013) and logistic noise models for this task. Finally, we demonstrate how simple algorithms can be useful to learn w and p up to moderately high d through experiments on simulated data.

Two-Person Fair Division with Additive Valuations

In the literature, many desirable properties for allocations of indivisible goods have been proposed, including envy-freeness, Pareto optimality, and maximization of either the total welfare of all agents, the welfare of the worst-off agent, or the Nash product of agents’ welfares. In the two-person context, we study relationships among these properties using both analytical models and simulation in a setting where individual preferences are given by additive cardinal utilities. We provide several new theorems linking these criteria and use simulation to study how their values are related to problem characteristics, assuming that utilities are assigned randomly. We draw some conclusions concerning the relation of problem characteristics to the availabilty of allocations with particular properties.

Negotiation strategies for agents with ordinal preferences: Theoretical analysis and human study

Negotiation is a very common interaction between agents. Many common negotiation protocols work with cardinal utilities, even though ordinal preferences, which only rank the outcomes, are easier to elicit from humans. In this work, we focus on negotiation with ordinal preferences over a finite set of outcomes. We study an intuitive protocol for bilateral negotiations, where the two parties make offers alternately. We analyze the negotiation protocol under two settings: First, we consider the full information setting, where each party is fully aware of the other party's preference order. For this case, we provide elegant strategies that specify a sub-game perfect equilibrium. In addition, we show how the studied negotiation protocol almost completely implements a known bargaining rule. Second, we analyze the complementary no-information setting where neither party knows the preference order of the other party. For this case, we provide a Maxmin strategy and show that every pair of Maxmin strategies specifies a robust-optimization equilibrium. Finally, through a human study (N=150), we empirically study the practical relevance of our full information analysis to people engaging in negotiations with each other and/or with an automated agent using the studied protocol. Surprisingly, our results indicate that people tend to arrive at the equilibrium outcomes despite frequently departing from the proposed strategies. In addition, in contrast to commonly held beliefs, we find that an equilibrium-following agent performs very well with people.

Ordinal utility differences

It is widely held that under ordinal utility, utility differences are ill-defined. Allegedly, for these to be well-defined (without turning to choice under risk or the like), one should adopt as a new kind of primitive quaternary relations, instead of the traditional binary relations underlying ordinal utility functions. Correlatively, it is also widely held that the key structural properties of quaternary relations are entirely arbitrary from an ordinal point of view. These properties would be, in a nutshell, the hallmark of cardinal utility. While much is obviously true in these two tenets, this note explains why, as stated, they should be abandoned. Any ordinal utility function induces a rich quaternary relation. There is such a thing as ordinal utility differences. Furthermore, this induced quaternary relation respects, apart from completeness, the most standard structural properties of quaternary relations. These properties are, from an ordinal point of view, anything but arbitrary; from a quaternary perspective only completeness should be considered the hallmark—if any—of cardinal utility. These facts are explained to be especially relevant to the critical appreciation of the ordinalist methodology.

Probability Spaces Identifying Ordinal and Cardinal Utilities in Problems of an Economic Nature: New Issues and Perspectives

Prevision bundles identifying expected returns on risky assets are established. A probability space associated with risky assets is defined. In this research work, the optimization principle is based on the notion of distance. This is because problems of an economic nature are not handled in an axiomatic or intrinsic way, but they are investigated with regard to a given coordinate system. The latter is shown to be invariant. The notion of mathematical expectation applied to summarizing both monetary values and utilities is treated. Such a notion is extended to study portfolios of financial assets. Objective conditions of coherence connected with the notion of mathematical expectation are extended. Rational behaviors towards risk are based on them. A model representing diagrams considered inside the same coordinate system is shown. Such a model identifies as many optimal choices as pair comparisons it is possible to take into account in order to obtain a multilinear measure. The latter is the expected return on a specific portfolio of financial assets.

Random Serial Dictatorship for horizontal collaboration in logistics

In this paper, we propose the application of matching mechanisms to handle horizontal collaboration in logistics. Based on the requirements of a real-life setting, we introduce a new variation of matching under preferences. We provide a Random Serial Dictatorship (RSD) mechanism for finding a solution that incorporates Pareto optimality, incentive compatibility and fairness, which are desired properties for enabling collaboration between antagonistic participants. Further, we present an extensive experimental analysis that is twofold. We compare the proposed RSD mechanism to two well-known matching mechanisms, namely Maximising Cardinal Utilities (MAXCU) and Bundled Probabilistic Serial (BPS). We compare these mechanisms with respect to cardinal efficiency and envy, and highlight the complementarity between them for handling different settings. Most importantly, we show that RSD is the most appropriate choice for settings similar to the one motivating our study. Moreover, we examine the effect of randomisation of RSD to fairness. Finally, we discuss how our matching mechanism was applied in the real-life context of four 3rd Party Logistics providers. Results show that the proposed approach creates significant potential for synergies and may be used to support horizontal collaboration in logistics.

On the Coexistence Of Stability and Incentive Compatibility in Fractional Matchings

Stable matchings have been studied extensively in social choice literature. The focus has been mostly on integral matchings, in which the nodes on the two sides are wholly matched. A fractional matching, which is a convex combination of integral matchings, is a natural extension of integral matchings. The topic of stability of fractional matchings has started receiving attention only very recently. Further, incentive compatibility in the context of fractional matchings has received very little attention. With this as the backdrop, our paper studies the important topic of incentive compatibility of mechanisms to find stable fractional matchings. We work with preferences expressed in the form of cardinal utilities. Our first result is an impossibility result that there are matching instances for which no mechanism that produces a stable fractional matching can be incentive compatible or even approximately incentive compatible. This provides the motivation to seek special classes of matching instances for which there exist incentive compatible mechanisms that produce stable fractional matchings. Our study leads to a class of matching instances that admit unique stable fractional matchings. We first show that a unique stable fractional matching for a matching instance exists if and only if the given matching instance satisfies the conditional mutual first preference (CMFP) property. To this end, we provide a polynomial-time algorithm that makes ingenious use of envy-graphs to find a non-integral stable matching whenever the preferences are strict and the given instance is not a CMFP matching instance. For this class of CMFP matching instances, we prove that every mechanism that produces the unique stable fractional matching is (a) incentive compatible and further (b) resistant to coalitional manipulations.

Determination of Break-Even Point through Consumer Preference Relation

Social choice theory beliefs about how the consumers function to chose their interested goods and services. Preference relation with affine indifference curves that has a concave representation has a linear utility representation. This study asks how individual preference relations might be combined to give a single ordering which captures the overall wishes of the group of individuals. There are certain properties that one would like such a utility rule, utility have thus become a more abstract concept that is not necessarily solely based on the satisfaction or pleasure received. Concept of cardinal utility is studied in three different situations Debreu (1958) gave quite different approach. This study maintains link between mathematical theory and financial concept to determine break-even point through the consumers’ preference relation.

A Rehabilitation of the Law of Diminishing Marginal Utility: An Ordinal Marginal Utility Approach

Abstract The model in which an individual maximizes his ordinal or cardinal total utility has long been the paradigm of individual choice theory. However, the two mainstream utility theories, the ordinal and cardinal total utility theories, have caused a dilemma, i.e. one has to sacrifice one of the following two: the good property of utility ordinality, or common-sense notions such as the law of diminishing marginal utility. Ordinal theory keeps the former but gives up the latter, while cardinal theory keeps the latter but sacrifices the former. We propose an ordinal marginal utility approach aiming to solve this dilemma by changing the very first assumption regarding individual choice.

Fair Influence Maximization: a Welfare Optimization Approach

Several behavioral, social, and public health interventions, such as suicide/HIV prevention or community preparedness against natural disasters, leverage social network information to maximize outreach. Algorithmic influence maximization techniques have been proposed to aid with the choice of ``peer leaders'' or ``influencers'' in such interventions. Yet, traditional algorithms for influence maximization have not been designed with these interventions in mind. As a result, they may disproportionately exclude minority communities from the benefits of the intervention. This has motivated research on fair influence maximization. Existing techniques come with two major drawbacks. First, they require committing to a single fairness measure. Second, these measures are typically imposed as strict constraints leading to undesirable properties such as wastage of resources. To address these shortcomings, we provide a principled characterization of the properties that a fair influence maximization algorithm should satisfy. In particular, we propose a framework based on social welfare theory, wherein the cardinal utilities derived by each community are aggregated using the isoelastic social welfare functions. Under this framework, the trade-off between fairness and efficiency can be controlled by a single inequality aversion design parameter. We then show under what circumstances our proposed principles can be satisfied by a welfare function. The resulting optimization problem is monotone and submodular and can be solved efficiently with optimality guarantees. Our framework encompasses as special cases leximin and proportional fairness. Extensive experiments on synthetic and real world datasets including a case study on landslide risk management demonstrate the efficacy of the proposed framework.

On continuity and general equilibrium: Pareto, Cassel, and the foundations of neoclassical economics

This article explores the application of the principle of continuity to general equilibrium theory by Vilfredo Pareto, Gustav Cassel, and Knut Wicksell. It begins by recapping Pareto’s early works based on cardinal utility and continuity. After that, Cassel’s proposal of an elementary framing for general equilibrium with discontinuous functions is presented. The next section covers some unpublished letters between Pareto, Maffeo Pantaleoni, and Cassel related to the issue. It is followed by a review of the reception of Cassel’s ideas advanced in his 1899 Grundriss. In the end, some reflections are offered on the scientific implications of the principle of continuity to neoclassical economics.

R&D innovation efficiency of Chinese high-tech industries: Three-stage network approach with fairness consideration

Recently, innovation evaluation has played an increasingly important role in the development of high-tech industry and the enhancement of international competitiveness. Data envelopment analysis (DEA) has been viewed as an effective data-driven method in evaluating the performance of decision-making units (DMUs). However, the traditional DEA models treat the DMUs as “black box.” As an extension of traditional DEA, a three-stage network evaluation transforming from a “black box” to a “gray box” structure has appeared. Unfortunately, the problem of intermediate elements has reduced the usefulness of the extended method. Although some gaming approaches have been introduced in the DEA model to reduce internal conflicts, the issue of fairness in the three stages, which have a significant impact on the behavior of the decision-makers, has still not been considered. To fill this gap, Neumann-Morgenstern cardinal utility is adopted to express the basic utility of a certain stage by considering itself efficiency and additional fairness utility from the other’s efficiency. Afterward, several non-cooperative DEA models have been developed by considering the efficiency of the leader stage and fairness disutility from other stages. Finally, a numerical example is presented and an empirical application is given to verify the proposed method.

On the existence of Pareto Efficient and envy-free allocations

Envy-freeness and Pareto Efficiency are two major goals in welfare economics. The existence of an allocation that satisfies both conditions has been studied for a long time. Whether items are indivisible or divisible, it is impossible to achieve envy-freeness and Pareto Efficiency ex post even in the case of two people and two items. In contrast, in this work, we prove that, for any cardinal utility functions (including complementary utilities for example) and for any number of items and players, there always exists an ex ante mixed allocation which is envy-free and Pareto Efficient, assuming the allowable assignments satisfy an anonymity property. The problem remains open in the divisible case.We also investigate the communication complexity for finding a Pareto Efficient and envy-free allocation.

Interpreting the Will of the People: A Positive Analysis of Ordinal Preference Aggregation

We investigate how individuals think groups should aggregate members’ ordinal preferences – that is, how they interpret “the will of the people.” In an experiment, we elicit revealed attitudes toward ordinal preference aggregation and classify subjects according to the rules they apparently deploy. Majoritarianism is rare. Instead, people employ rules that place greater weight on compromise options. The classification’s fit is excellent, and clustering analysis reveals that it does not omit important rules. We ask whether rules are stable across domains, whether people impute cardinal utility from ordinal ranks, and whether attitudes toward aggregation differ across countries with divergent traditions.

To Integrated Management and Financial Coverage for Health Care Infrastructure of Zaporizhchyna Districts

Introduction. Health care is characterized by the fact that it belongs to the major state functions and the main kinds of economic activity at the same time, as well as the fact that in contemporary conditions it provides dual-use products – use for both conventional and defense against the latest biothreats. In the course of reforming this state function in Ukraine, the main financing is provided through the National Health Service of Ukraine, where management changes relatively frequently. The purpose. Protection against biohazards, health care, health insurance requires systemic resilience and integrated management based on modern information and communication technologies. Such technologies for social insurance have been successfully developed and implemented by the V.M.Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine. Results. A specific example of Zaporizhia region shows which health facilities are without proper government support and how to anticipate and manage distributed networks on big data. In all the above issues of protection against biohazards, health care, health insurance, government institutions cannot make rational decisions without comprehensive and accurate assessment of future gains (and losses) caused by the implementation of a particular project, as well as a comparison of such gains with the present value of costs associated with this project. It is important for decision makers to measure gains and costs in the same units applying the known principle «Who canʼt measure cannot manage». Since project costs are usually measured in monetary terms, it makes sense to measure all gains in monetary terms as well. Different approaches to economic assessment of health status compare the benefits from a medical intervention with the costs of that intervention. Conclusions. Gains from medical intervention can be measured in physical units on a one-dimensional scale, monetary units, units of cardinal utility function, reflecting the multidimensional concept of health via the scalar index or key performance indicator. Nowadays multiple dimensions mentioned are gradually developing into big data for each node and link of the health care grid. Keywords: biothreats, system resilience, social insurance, health insurance, big data, distributed networks.

Incorporating Mortality in Health Utility Measures.

The creation of multiattribute health utility systems requires design choices that have profound effects on the utility model, many of which have been documented and studied in the literature. Here we describe one design choice that has, to the best of our knowledge, been unrecognized and therefore ignored. It can emerge in any multiattribute decision analysis in which one or more essential outcomes cannot be described in terms of the multiattribute space. In health applications, the state of being dead is such an outcome. When the remaining health is conceptualized as a multidimensional space, determining the utility of the state of being dead requires using the interval-scale properties of cardinal utility, combined with elicited utilities for the state of being dead and the all-worst state, to produce a utility function in which the state of being dead has a utility of 0 and full health has a utility of 1 (i.e., the quality-adjusted life-year scale). Although previously unrecognized, there are two approaches to accomplish that step, and they produce different results in almost all cases. As a corollary, the choice of approach determines the proportion of states rated as worse than dead by the system. For example, in the Health Utility Index 3 (HUI3), the method used classifies 78% of the 972,000 unique health states in the classification system as worse than dead, and that proportion increases to 85% when the HUI3 is recalculated using the alternative approach. Studies of populations with significant morbidity are the most likely to be sensitive to the design choice. Those who design utility measures should be aware that they are using a researcher degree of freedom when they decide how to scale the state of being dead.

Non-monetary coordination mechanisms for time slot allocation in warehouse delivery

Recent empirical evidence suggests that delivery to retail warehouses suffers from a lack of coordination. While carriers try to optimize their routes, they often experience very long waiting times at loading docks, which renders their individual planning useless. To reduce such inefficiencies, carriers need to coordinate. This problem has received considerable attention in practice, but the design of coordination mechanisms is challenging for several reasons: First, the underlying package assignment problem is NP-hard. Second, efficiency, incentive-compatibility, and fairness are important design desiderata, but in most economic environments they are conflicting. Third, the market for logistics services is competitive and price-based mechanisms where carriers might have to pay for time slots suffer from low acceptance. We draw on recent advances in market design, more specifically randomized matching mechanisms, which set incentives for carriers to share information truthfully such that a central entity can coordinate their plans in a fair and approximately efficient way. We use and adapt the existing maximizing cardinal utilities (MAXCU) framework to a retail logistics problem, which yields a new and powerful approach for coordination. We report numerical experiments based on field data from a real-world logistics network to analyze the average reduction in waiting times and the computation times required and compare to first-come, first-served and an auction mechanism. Our results show that randomized matching mechanisms provide an effective means to reduce waiting times at warehouses without requiring monetary transfers by the carriers. They run in polynomial time and provide a practical solution to wide-spread coordination problems.

Fair Allocation with Diminishing Differences

Ranking alternatives is a natural way for humans to explain their preferences. It is used in many settings, such as school choice, course allocations and residency matches. Without having any information on the underlying cardinal utilities, arguing about the fairness of allocations requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation whenever it exists. Using simulations, we compare the various fairness criteria in terms of their probability of existence, and their probability of being fair by the underlying cardinal valuations. We find that necessary-DD-proportionality fares well in both measures. We also consider envy-freeness and Pareto optimality under diminishing-differences, as well as chore allocation under the analogous condition --- increasing-differences.

Modeling agent's conditional preferences under objective ambiguity in Dempster-Shafer theory

We manage decisions under “objective” ambiguity by considering generalized Anscombe-Aumann acts, mapping states of the world to generalized lotteries on a set of consequences. A generalized lottery is modeled through a belief function on consequences, interpreted as a partially specified randomizing device. Preference relations on these acts are given by a decision maker focusing on different scenarios (conditioning events). We provide a system of axioms which are necessary and sufficient for the representability of these “conditional preferences” through a conditional functional parametrized by a unique full conditional probability P on the algebra of events and a cardinal utility function u on consequences. The model is able to manage also “unexpected” (i.e., “null”) conditioning events and distinguishes between a systematically pessimistic or optimistic behavior, either referring to “objective” belief functions or their dual plausibility functions. Finally, an elicitation procedure is provided, reducing to a Quadratically Constrained Linear Program (QCLP).

Heuristic Voting as Ordinal Dominance Strategies

Decision making under uncertainty is a key component of many AI settings, and in particular of voting scenarios where strategic agents are trying to reach a joint decision. The common approach to handle uncertainty is by maximizing expected utility, which requires a cardinal utility function as well as detailed probabilistic information. However, often such probabilities are not easy to estimate or apply.To this end, we present a framework that allows for “shades of gray” of likelihood without probabilities. Specifically, we create a hierarchy of sets of world states based on a prospective poll, with inner sets contain more likely outcomes. This hierarchy of likelihoods allows us to define what we term ordinally-dominated strategies. We use this approach to justify various known voting heuristics as bounded-rational strategies.