Ordinal Preferences Research Articles

Two-player fair division of indivisible items: Comparison of algorithms

We study algorithms for allocating a set of indivisible items to two players who rank them differently. We compare eleven such algorithms, mostly taken from the literature, in a computational study, evaluating them according to fairness and efficiency criteria that are based on ordinal preferences as well as Borda counts. Our study is exhaustive in that, for every possible instance of up to twelve items, we compare the output of each algorithm to all possible allocations. We thus can search for “good” allocations that no algorithm finds. Overall, the algorithms do very well on ordinal properties but fall short on Borda properties. We also discuss the similarity of algorithms and suggest how they can be usefully combined.

Compromises and Rewards: stable and non-manipulable probabilistic matching

Can we reconcile stability with non-manipulability in two-sided matching problems by selecting lotteries over matchings? We parameterize, through sets of utility functions, how ordinal preferences induce preferences over lotteries and develop corresponding notions of ex-ante stability and non-manipulability. For most sets, the properties are incompatible. However, for the set of utility functions with increasing differences, stability and non-manipulability characterize Compromises and Rewards. This novel rule is fundamentally different from the one that has attracted most attention in the literature, Deferred Acceptance. We then derive complementary negative results that show that increasing differences essentially is a necessary condition for the properties to be compatible.

Investigating the characteristics of one-sided matching mechanisms under various preferences and risk attitudes

One-sided matching mechanisms are fundamental for assigning a set of indivisible objects to a set of self-interested agents when monetary transfers are not allowed. Two widely-studied randomized mechanisms in multiagent settings are the Random Serial Dictatorship (RSD) and the Probabilistic Serial Rule (PS). Both mechanisms require only that agents specify ordinal preferences and have a number of desirable economic and computational properties. However, the induced outcomes of the mechanisms are often incomparable and thus there are challenges when it comes to deciding which mechanism to adopt in practice. In this paper, we first consider the space of general ordinal preferences and provide empirical results on the (in)comparability of RSD and PS. We analyze their respective economic properties under general and lexicographic preferences. We then instantiate utility functions with the goal of gaining insights on the manipulability, efficiency, and envyfreeness of the mechanisms under different risk-attitude models. Our results hold under various preference distribution models, which further confirm the broad use of RSD in most practical applications.

Utilitarians Without Utilities: Maximizing Social Welfare for Graph Problems Using Only Ordinal Preferences

We consider ordinal approximation algorithms for a broad class of utility maximization problems for multi-agent systems. In these problems, agents have utilities for connecting to each other, and the goal is to compute a maximum-utility solution subject to a set of constraints. We represent these as a class of graph optimization problems, including matching, spanning tree problems, TSP, maximum weight planar subgraph, and many others. We study these problems in the ordinal setting: latent numerical utilities exist, but we only have access to ordinal preference information, i.e., every agent specifies an ordering over the other agents by preference. We prove that for the large class of graph problems we identify, ordinal information is enough to compute solutions which are close to optimal, thus demonstrating there is no need to know the underlying numerical utilities. For example, for problems in this class with bounded degree b a simple ordinal greedy algorithm always produces a (b + 1)-approximation; we also quantify how the quality of ordinal approximation depends on the sparsity of the resulting graphs. In particular, our results imply that ordinal information is enough to obtain a 2-approximation for Maximum Spanning Tree; a 4-approximation for Max Weight Planar Subgraph; a 2-approximation for Max-TSP; and a 2- approximation for various Matching problems.

A new ex-ante efficiency criterion and implications for the probabilistic serial mechanism

We introduce and analyze an efficiency criterion for probabilistic assignment of objects, when only ordinal preference information is available. This efficiency criterion is based on the following domination relation: a probabilistic assignment dominates another assignment if it is ex-ante efficient for a strictly larger set of utility profiles consistent with the ordinal preferences. We provide a simple characterization of this domination relation. We revisit an extensively studied assignment mechanism, the Probabilistic Serial mechanism (Bogomolnaia and Moulin, 2001), which always chooses a “fair” assignment. We show that the Probabilistic Serial assignment may be dominated by another fair assignment. We provide conditions under which the serial assignment is undominated among fair assignments.

Stable and Pareto optimal group activity selection from ordinal preferences

In several situations agents need to be assigned to activities on basis of their preferences, and each agent can take part in at most one activity. Often, the preferences of the agents do not depend only on the activity itself but also on the number of participants in the respective activity. In the setting we consider, the agents hence have preferences over pairs “(activity, group size)” including the possibility “do nothing”; in this work, these preferences are assumed to be strict orders. The task will be to find stable assignments of agents to activities, for different concepts of stability such as Nash or core stability, and Pareto optimal assignments respectively. In this respect, particular focus is laid on two natural special cases of agents’ preferences inherent in the considered model, namely increasing and decreasing preferences, where agents want to share an activity with as many (as few, respectively) agents as possible.

The Random Assignment Problem with Submodular Constraints on Goods

Problems of allocating indivisible goods to agents in an efficient and fair manner without money have long been investigated in literature. The random assignment problem is one of them, where we are given a fixed feasible (available) set of indivisible goods and a profile of ordinal preferences over the goods, one for each agent. Then, using lotteries, we determine an assignment of goods to agents in a randomized way. A seminal paper of Bogomolnaia and Moulin (2001) shows a probabilistic serial (PS) mechanism to give an ordinally efficient and envy-free solution to the assignment problem. In this article, we consider an extension of the random assignment problem to submodular constraints on goods. We show that the approach of the PS mechanism by Bogomolnaia and Moulin is powerful enough to solve the random assignment problem with submodular (matroidal and polymatroidal) constraints. Under the agents’ ordinal preferences over goods we show the following: (1) The obtained PS solution for the problem with unit demands and matroidal constraints is ordinally efficient, envy-free, and weakly strategy-proof with respect to the associated stochastic dominance relation. (2)For the multiunit demand and polymatroidal constraint problem, the PS solution is ordinally efficient and envy-free but is not strategy-proof in general. However, we show that under a mild condition (that is likely to be satisfied in practice) the PS solution is a weak Nash equilibrium.

Benchmarking container port security risks by applying a FIS methodology

This paper presents a fuzzy inference approach to estimate the security level of a port in a manner that it provides essential information to the stakeholders when evaluating security risks under uncertainty. A fuzzy inference system (FIS) methodology is developed on account to four predefined security factors. A team of experts is used to rank and survey potential port security risks whereas the experts' ordinal preferences were combined using the Cook and Seiford method to come up with a consensus risks' ranking. To validate the model, results are compared with those from an established fuzzy evidential reasoning approach given the same security risk inputs. The verified FIS can provide useful insights for security analysts to conduct security risk quantification under high uncertainty in data in the maritime sector as well as a wider range of other industries (e.g., aerospace and process) facing high terrorism threats with appropriate tailor and adaptation.

Benchmarking container port security risks by applying a FIS methodology

This paper presents a fuzzy inference approach to estimate the security level of a port in a manner that it provides essential information to the stakeholders when evaluating security risks under uncertainty. A fuzzy inference system (FIS) methodology is developed on account to four predefined security factors. A team of experts is used to rank and survey potential port security risks whereas the experts' ordinal preferences were combined using the Cook and Seiford method to come up with a consensus risks' ranking. To validate the model, results are compared with those from an established fuzzy evidential reasoning approach given the same security risk inputs. The verified FIS can provide useful insights for security analysts to conduct security risk quantification under high uncertainty in data in the maritime sector as well as a wider range of other industries (e.g., aerospace and process) facing high terrorism threats with appropriate tailor and adaptation.

On mechanisms eliciting ordinal preferences

When is a mechanism designer justified in only asking for ordinal information about preferences? Simple examples show that, even if the planner's goal (expressed by a social choice correspondence, or SCC) depends only on ordinal information, eliciting cardinal information may help with incentives. However, if agents may be uncertain about their own cardinal preferences, then a strong robustness requirement can justify the focus on ordinal mechanisms. Specifically, when agents' preferences over pure outcomes are strict, if a planner is able to implement an SCC (in ex-post equilibrium) using a mechanism that is robust to interdependence of arbitrary form in cardinal preferences, then there must exist such a mechanism that elicits only ordinal preferences. The strictness assumption can be dropped if we further allow the possibility of non-expected-utility preferences.

Heterogeneity in preferences and behavior in threshold models

A coordination game is repeatedly played on a graph by players (vertices) who have heterogeneous cardinal preferences and whose strategy choice is governed by the individualistic asynchronous logit dynamic. The idea of potential driven autonomy of sets of players is used to derive results on the possibility of heterogeneous preferences leading to heterogeneous behavior. In particular, a class of graphs is identified such that for large enough graphs in this class, diversity in ordinal preferences will nearly always lead to heterogeneity in behavior, regardless of the cardinal strength of the preferences. These results have implications for network design problems, such as when a social planner wishes to induce homogeneous/heterogeneous behavior in a population.

How voters use grade scales in evaluative voting

During the first round of the 2012 French presidential election, participants in an in situ experiment were invited to vote according to “evaluative voting”, which involves rating the candidates using a numerical scale. Various scales were used: (0,1), (-1,0,1), (0,1,2), and (0,1,…,20). The paper studies scale calibration effects, i.e., how individual voters adapt to the scale, leading to possibly different election outcomes. The data show that scales are not linearly equivalent, even if individual ordinal preferences are not inconsistent. Scale matters, notably because of the symbolic power of negative grades, which does not affect all candidates uniformly.

An axiomatic approach to the measurement of envy

We characterize a class of envy-as-inequity measures. There are three key axioms. Decomposability requires that overall envy is the sum of the envy within and between subgroups. The other two axioms deal with the two-individual setting and specify how the envy measure should react to simple changes in the individuals’ commodity bundles. The characterized class measures how much one individual envies another individual by the relative utility difference (using the envious’ utility function) between the bundle of the envied and the bundle of the envious, where the utility function that must be used to represent the ordinal preferences is the ‘ray’ utility function. The class measures overall envy by the sum of these (transformed) relative utility differences. We discuss our results in the light of previous contributions to envy measurement and multidimensional inequality measurement.

Strategy-proofness of scoring allocation correspondences for indivisible goods

We study strategy-proofness in a model of resource allocation due to Brams and King (Ration Soc 17:387–421, 2005) and Brams et al. (Theory Decis 55:147–180, 2003), further developed by Baumeister et al. (J Auton Agents Multi Agent Syst 31(3):628–655, 2017). We assume resources to be indivisible and nonshareable and that agents have responsive preferences over the power set of the resources, but only submit ordinal preferences over single resources to the social planner. Using scoring vectors, these ordinal preferences induce additive utility functions. We then focus on allocation correspondences that maximize utilitarian social welfare, and we use extension principles (from social choice theory, such as the Kelly and the Gardenfors extension) for preferences to study manipulation of allocation correspondences. We characterize strategy-proofness of the utilitarian allocation correspondence: It is Gardenfors/Kelly-strategy-proof if and only if the number of different values in the scoring vector is at most two or the number of occurrences of the greatest value in the scoring vector is larger than half the number of resources.

Election outcomes under different ways to announce preferences: an analysis of the 2015 parliament election in the Austrian federal state of Styria

We use preference data from the 2015 parliament election in the Austrian federal state of Styria to analyze different voting rules. An exit poll right after the election collected data on ordinal and cardinal preferences from approximately 1000 actual voters. Our analysis is threefold. First, we determine the hypothetical social outcomes under different voting rules; second, we investigate the stability of the outcomes under those rules. Finally, we provide a categorization of different types of parties and analyze the impact of certain voting rules (Plurality Rule, Plurality Run Off, Hare System, Condorcet Method, Approval Voting, Borda Rule, Evaluative Voting, and Majority Judgment) on the performances of parties in those scenarios.

Satisfied two-sided matching: a method considering elation and disappointment of agents

In practical two-sided matching problems, every agent usually cares about the matching result. If the result reaches or exceeds his/her expectation, he/she will experience elation; otherwise, he/she will experience disappointment. This is the psychological behavior of agents in two-sided matching, and the satisfaction degrees to the potential matching result of agents are closely related to the psychological behavior. However, the psychological behavior of agents is missing in the existing two-sided matching methods. The purpose of this paper is to develop a method for the two-sided matching problem considering psychological behavior of agents on both sides. First, the expected preference ordinals of each agent toward opposite agents are calculated based on the uncertain preference ordinals provided by agents. Then, the preference utility function is constructed, and the expected preference ordinals are transformed into preference utility values using the preference utility function. Next, based on the disappointment theory, the modified preference utility values are determined by calculating the disappointment values and elation values of each agent to the possible matching results. Furthermore, to maximize the sum of modified preference utility values of all agents on each side, a bi-objective optimization model is constructed, and the satisfied matching result can be obtained by solving the optimization model. Finally, a numerical example is used to illustrate the use of the proposed method.

K-maxitive Sugeno integrals as aggregation models for ordinal preferences

We consider an order variant of k-additivity, so-called k-maxitivity, and present an axiomatization of the class of k-maxitive Sugeno integrals over distributive lattices. To this goal, we characterize the class of lattice polynomial functions with degree at most k and show that k-maxitive Sugeno integrals coincide exactly with idempotent lattice polynomial functions whose degree is at most k. We also discuss the use of this parametrized notion in preference aggregation and learning. In particular, we address the question of determining optimal values of k through a case study on empirical data.

Epsilon-stability in school choice

In many school choice practices, scores, instead of ordinal rankings, are used to indicate students’ qualification. We study school choice problems where students have ordinal preference over schools while their priorities at schools are in the form of cardinal scores. The cardinality of scores allows us to measure the intensity of priority violations and hence relax stability by proposing epsilon-stability. We also propose the epsilon-EADA mechanism to select the constrained efficient matching under epsilon-stability.

Fairness and efficiency in strategy-proof object allocation mechanisms

I consider the problem of allocating N indivisible objects among N agents according to their preferences when transfers are absent and an outside option may exist. I study the tradeoff between fairness and efficiency in the class of strategy-proof mechanisms. The main finding is that for strategy-proof mechanisms the following efficiency and fairness criteria are mutually incompatible: (1) ex-post efficiency and envy-freeness, (2) ordinal efficiency and weak envy-freeness, and (3) ordinal efficiency and equal division lower bound. Result 1 is the first impossibility result for this setting that uses ex-post efficiency; results 2 and 3 are more practical than similar results in the literature. In addition, for N=3, I give two characterizations of the celebrated random serial dictatorship mechanism: it is the unique strategy-proof, ex-post efficient mechanism that (4) provides agents that have the same ordinal preferences with assignments not dominated by each other (weak envy-freeness among equals), or (5) provides agents that have the same cardinal preferences with assignments of equal expected utility (symmetry). These results strengthen the characterization by Bogomolnaia and Moulin (2001); result 5 implies the impossibility result by Zhou (1990).

Randomized Social Choice Functions Under Metric Preferences

We determine the quality of randomized social choice algorithms in a setting in which the agents have metric preferences: every agent has a cost for each alternative, and these costs form a metric. We assume that these costs are unknown to the algorithms (and possibly even to the agents themselves), which means we cannot simply select the optimal alternative, i.e. the alternative that minimizes the total agent cost (or median agent cost). However, we do assume that the agents know their ordinal preferences that are induced by the metric space. We examine randomized social choice functions that require only this ordinal information and select an alternative that is good in expectation with respect to the costs from the metric. To quantify how good a randomized social choice function is, we bound the distortion, which is the worst-case ratio between the expected cost of the alternative selected and the cost of the optimal alternative. We provide new distortion bounds for a variety of randomized algorithms, for both general metrics and for important special cases. Our results show a sizable improvement in distortion over deterministic algorithms.