Ordinal Mechanisms Research Articles

Designing information to improve welfare in matching markets

In matching markets, objects are allocated to agents without monetary transfers, based on agents’ preferences. However, agents may not have enough information to determine their preferences over the objects precisely. How should a benevolent planner optimally reveal information to maximize social welfare in this context? I show that when agents are symmetric and there are just two options, letting each agent know his rank in the realized distribution of preferences – but not his actual preferences – always improves social welfare over providing no information. When there are more objects, this rank-based information policy generalizes to the Object Recommendation (OR) Signal, which consists of simply recommending each agent to pick his socially-optimal choice. Under a mild regularity condition, I show that, when agents’ a priori relative preferences over the objects are “not too strong”, the OR Signal, used together with any standard ordinal mechanism, not only maximizes welfare, but achieves the unconstrained social optimum — formalizing the intuition that when people do not have strong opinions over several options, it is easy to sway them.

Hollow self-floating microspheres capture cobalt (Co2+)/nickel (Ni2+) ions from the acidic leachate of spent lithium-ion battery cathodes

The effectiveness of lithium recovery determines the future use of the devices powered by lithium-ion batteries (LIBs) under lithium resource shortages. Selective separation of Co2+/Ni2+ from the cathode leachate of LIBs is a key step to reduce the lithium loss in the recovery progress. This study reported a hollow structured self-floating adsorbent synthesized by co-grafting of (3-Aminopropyl)triethoxysilane (APTES) and ethylenediaminetetraacetic acid (EDTA) on hollow glass microsphere (HGM) for selective capture of Co2+/Ni2+ from LIBs cathode leachate. The introduction of the N atom of APTES was crucial for bridging the inorganic HGM and organic EDTA. The organic component of the synthesized adsorbent accounted for 12.82%, which was responsible for the Co2+/Ni2+ adsorptions. Physic- and chemic-adsorptions were the ordinal mechanisms during the three-stages divided adsorption progress. The negative –COO− groups promoted the electrostatic attraction between metal ions and the adsorbents. The N atoms of EDTA were the effective adsorption sites for Co2+ and Ni2+ ions, resulting in chemical chelation of the metal ions with adsorbents followed by ion exchanges of Na+ and Ca2+. The adsorbents captured 94.18 mg g−1 of Co2+ and 62.56 mg g−1 of Ni2+ with a < 1% lithium loss in the LIBs cathode leachate. The self-flotation process of the adsorbent was non-linearly fitted by our original equation for predicting its movement track for further solid–liquid separation. This research proposed a new strategy for the selective adsorption of Co2+/Ni2+ from LIBs cathode leachate with limited lithium loss.

Peeking behind the ordinal curtain: Improving distortion via cardinal queries

Aggregating the preferences of individuals into a collective decision is the core subject of study of social choice theory. In 2006, Procaccia and Rosenschein considered a utilitarian social choice setting, where the agents have explicit numerical values for the alternatives, yet they only report their linear orderings over them. To compare different aggregation mechanisms, Procaccia and Rosenschein introduced the notion of distortion, which quantifies the inefficiency of using only ordinal information when trying to maximize the social welfare, i.e., the sum of the underlying values of the agents for the chosen outcome. Since then, this research area has flourished and bounds on the distortion have been obtained for a wide variety of fundamental scenarios. However, the vast majority of the existing literature is focused on the case where nothing is known beyond the ordinal preferences of the agents over the alternatives. In this paper, we take a more expressive approach, and consider mechanisms that are allowed to further ask a few cardinal queries in order to gain partial access to the underlying values that the agents have for the alternatives. With this extra power, we design new deterministic mechanisms that achieve significantly improved distortion bounds and, in many cases, outperform the best-known randomized ordinal mechanisms. We paint an almost complete picture of the number of queries required by deterministic mechanisms to achieve specific distortion bounds.

Peeking Behind the Ordinal Curtain: Improving Distortion via Cardinal Queries

The notion of distortion was introduced by Procaccia and Rosenschein (2006) to quantify the inefficiency of using only ordinal information when trying to maximize the social welfare. Since then, this research area has flourished and bounds on the distortion have been obtained for a wide variety of fundamental scenarios. However, the vast majority of the existing literature is focused on the case where nothing is known beyond the ordinal preferences of the agents over the alternatives. In this paper, we take a more expressive approach, and consider mechanisms that are allowed to further ask a few cardinal queries in order to gain partial access to the underlying values that the agents have for the alternatives. With this extra power, we design new deterministic mechanisms that achieve significantly improved distortion bounds and outperform the best-known randomized ordinal mechanisms. We draw an almost complete picture of the number of queries required to achieve specific distortion bounds.

Local vs. global strategy-proofness: A new equivalence result for ordinal mechanisms

We study two incentive compatibility notions of ordinal mechanisms: strategy-proofness and a new notion of local strategy-proofness, called block-adjacent strategy-proofness, which requires no gain by a flip of two adjacent blocks. A condition on the preference domain called path-nestedness is identified as sufficient for the equivalence between these two notions.

A comes before B, like 1 comes before 2. Is the parietal cortex sensitive to ordinal relationships in both numbers and letters? An fMRI-adaptation study.

How are number symbols (e.g., Arabic digits) represented in the brain? Functional resonance imaging adaptation (fMRI‐A) research has indicated that the intraparietal sulcus (IPS) exhibits a decrease in activation with the repeated presentation of the same number, that is followed by a rebound effect with the presentation of a new number. This rebound effect is modulated by the numerical ratio or difference between presented numbers. It has been suggested that this ratio‐dependent rebound effect is reflective of a link between the symbolic numerical representation system and an approximate magnitude system. Experiment 1 used fMRI‐A to investigate an alternative hypothesis: that the rebound effect observed in the IPS is related to the ordinal relationships between symbols (e.g., 3 comes before 4; C after B). In Experiment 1, adult participants exhibited the predicted distance‐dependent parametric rebound effect bilaterally in the IPS for number symbols during a number adaptation task, however, the same effect was not found anywhere in the brain in response to letters. When numbers were contrasted with letters (numbers > letters), the left intraparietal lobule remained significant. Experiment 2 demonstrated that letter stimuli used in Experiment 1 generated a behavioral distance effect during an active ordinality task, despite the lack of a neural distance effect using fMRI‐A. The current study does not support the hypothesis that general ordinal mechanisms underpin the neural parametric recovery effect in the IPS in response to number symbols. Additional research is needed to further our understanding of mechanisms underlying symbolic numerical representation in the brain.

Random Dictators with a Random Referee: Constant Sample Complexity Mechanisms for Social Choice

We study social choice mechanisms in an implicit utilitarian framework with a metric constraint, where the goal is to minimize Distortion, the worst case social cost of an ordinal mechanism relative to underlying cardinal utilities. We consider two additional desiderata: Constant sample complexity and Squared Distortion. Constant sample complexity means that the mechanism (potentially randomized) only uses a constant number of ordinal queries regardless of the number of voters and alternatives. Squared Distortion is a measure of variance of the Distortion of a randomized mechanism.Our primary contribution is the first social choice mechanism with constant sample complexity and constant Squared Distortion (which also implies constant Distortion). We call the mechanism Random Referee, because it uses a random agent to compare two alternatives that are the favorites of two other random agents. We prove that the use of a comparison query is necessary: no mechanism that only elicits the top-k preferred alternatives of voters (for constant k) can have Squared Distortion that is sublinear in the number of alternatives. We also prove that unlike any top-k only mechanism, the Distortion of Random Referee meaningfully improves on benign metric spaces, using the Euclidean plane as a canonical example. Finally, among top-1 only mechanisms, we introduce Random Oligarchy. The mechanism asks just 3 queries and is essentially optimal among the class of such mechanisms with respect to Distortion.In summary, we demonstrate the surprising power of constant sample complexity mechanisms generally, and just three random voters in particular, to provide some of the best known results in the implicit utilitarian framework.

Truthful and Near-Optimal Mechanisms for Welfare Maximization in Multi-Winner Elections

Mechanisms for aggregating the preferences of agents in elections need to balance many different considerations, including efficiency, information elicited from agents, and manipulability. We consider the utilitarian social welfare of mechanisms for preference aggregation, measured by the distortion. We show that for a particular input format called threshold approval voting, where each agent is presented with an independently chosen threshold, there is a mechanism with nearly optimal distortion when the number of voters is large. Threshold mechanisms are potentially manipulable, but place a low informational burden on voters. We then consider truthful mechanisms. For the widely-studied class of ordinal mechanisms which elicit the rankings of candidates from each agent, we show that truthfulness essentially imposes no additional loss of welfare. We give truthful mechanisms with distortion O(√m log m) for k-winner elections, and distortion O(√m log m) when candidates have arbitrary costs, in elections with m candidates. These nearly match known lower bounds for ordinal mechanisms that ignore the strategic behavior. We further tighten these lower bounds and show that for truthful mechanisms our first upper bound is tight. Lastly, when agents decide between two candidates, we give tight bounds on the distortion for truthful mechanisms.

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.

Incentive properties for ordinal mechanisms

We study three incentive properties for ordinal mechanisms: (i) strategy-proofness, which requires that no agent gain by misrepresenting his preferences; (ii) adjacent strategy-proofness, which requires that no agent gain by switching the rankings of two adjacent alternatives; and (iii) mistake monotonicity, which requires that the welfare of each agent weakly decrease as he reports increasingly bigger mistakes. Each of these properties has three versions, depending on whether preferences over sure alternatives are extended to preferences over lotteries by the stochastic dominance, downward lexicographic, or upward lexicographic extension. We identify conditions on the preference domain that guarantee the equivalence of these properties. The universal domain and the domains of single-dipped and single-peaked preferences satisfy our conditions.

Optimal Allocation Without Money: An Engineering Approach

We study the allocation of heterogeneous services to agents without monetary transfers under incomplete information. Agents have private, multi-dimensional utilities over services, drawn from commonly known priors, which may depend on agents' observable information. The social planner's goal is to maximize a public objective. Potential applications include the allocation of seats to public schools, spaces in college dorms or courses, or spots in subsidized housing.For tractability, we take an engineering'' approach, in which we first solve a simpler large market approximation, and show using real data that a corresponding finite market mechanism still yields good results. In the large market model, there are finitely many types'' of agents, and a continuum of agents of each type. (Here, type'' corresponds to agents' observation information.) For both cardinal and ordinal mechanisms, we characterize all mechanisms that are incentive compatible and Pareto optimality within type. Such characterizations reduce the mechanism design problem to a well-defined optimization problem with a few continuous variables.We apply this framework to real data from Boston 2012-2013 school choice reform. The goal is to design an optimal ordinal mechanism to assign students to public schools to maximize a linear combination of utilitarian and max-min welfare, subject to capacity and transportation constraints. We show how to optimally solve a large market model with over 868 types of students and 77 schools, translate the solution into the finite market setup and yield a feasible finite market mechanism, which significantly outperforms the baseline plan chosen by the city in terms of efficiency, equity, and predictability.

Representation of time in time-place learning.

Ordinal, interval, and circadian mechanisms of solving a time-place task were tested. Rats searched for food twice in the morning and once in the afternoon (Group AB-C, n = 5) or once in the morning and twice in the afternoon (Group A-BC, n = 5) in a box with four food troughs. The location of the food depended on the time of day in a 12:12-h light:dark cycle. Acquisition was documented by food-site inspections at the correct locations prior to food availability. On nonrewarded probes, the time of the middle search (B) was shifted late (for Group AB-C) or early (for Group A-BC). The rats visited Location B at chance, contrary to an ordinal mechanism. When the posttesting meal and light-dark transitions were omitted, the rats visited correct locations with impaired performance but at above-chance levels on nonrewarded probes. The results are consistent with interval and circadian representations of time.

Magnetosheath electrons in anomalously low density solar wind observed by Geotail

We present an overview of the magnetosheath electron observations above 60 eV in the anomalously low density solar wind on 11 May 1999 observed by Geotail. We observed anisotropic ‘strahl’ component not only in the solar wind but also in the magnetosheath. Strahl is observed during the interval of 6–21h UT and most enhanced in the interval surrounding the solar wind density minimum in 18–19h UT. Its temperature seems positively correlated to its flux. We also evaluate the density and temperature of low energy isotropic component. The variation of density is associated with proton density. Its temperature is continuous, and suddenly decreased after the density minimum in ∼18–19h UT. We also compared our downstream data with upstream one observed by the ACE spacecraft. We conclude that the electron acceleration and heating at the bow shock is executed in the ordinal mechanisms even in the low density solar wind.

An experimental examination into the design of decentralized methods to solve the assignment problem with and without money

An experimental examination of the assignment problem, matching individuals to positions or slots, is conducted in which various assignment mechanisms are analyzed. Generalized versions of both the Vickrey and English auctions are designed to solve the assignment problem along with oridinal ranking mechanisms (serial dictator and “funny” money system). The generalized auctions result in efficient allocations. In contrast, the ordinal ranking mechanisms, which require no monetary transfers, are significantly less efficient in their assignments. However, the efficient allocations obtained from the competitive bidding processes are at the expense of consumers' surplus since demanders retain significantly larger profits with the ordinal ranking mechanisms.