In voting with ranked ballots each agent submits a strict ranking of the form a > b > c > d over the alternatives, and the voting rule decides on the winner based on these rankings. Although this ballot format has desirable characteristics, there is a question of whether it is expressive enough for the agents. Kahng et. al. address this issue by adding intensities to the rankings. They introduce ranking with intensities ballot format, where agents can use both >> and > in their rankings to express intensive and normal preferences between consecutive alternatives in their rankings. While Kahng et. al. focus on analyzing this ballot format in the utilitarian distortion framework, in this work, we look at the potentials of using this ballot format from the metric distortion view point. We design a class of voting rules coined Positional Scoring Rules, which can be used for different problems in the metric setting, and show that by solving a zero-sum game we can find the optimal member of this class for our problem. This rule takes intensities into account and achieves a lower distortion. In addition, by proving a bound on the price of ignoring intensities, we show that we might lose a great deal in terms of distortion by not taking the intensities into account.

We introduce a model of fair division with market values, where indivisible goods must be partitioned among agents with (additive) subjective valuations, and each good additionally has a market value. The market valuation can be viewed as a separate additive valuation that holds identically across all the agents. We seek allocations that are simultaneously fair with respect to the subjective valuations and under the market valuation. We show that an allocation that satisfies stochastically-dominant envy-freeness up to one good (SD-EF1) with respect to both the subjective valuations and the market valuation does not always exist, but the weaker guarantee of EF1 with respect to the subjective valuations along with SD-EF1 with respect to the market valuation can be guaranteed. We also study a number of other guarantees such as Pareto optimality, EFX, and MMS. In addition, we explore non-additive valuations and extend our model to cake-cutting. Along the way, we identify several tantalizing open questions.

Participatory budgeting (PB) is an increasingly popular tool for democratically allocating limited budgets to public-good projects. In PB, constituents vote on their preferred projects via ballots, and then an aggregation rule selects a set of projects whose total cost fits within the budget. Recent work studies how to design PB ballots and aggregation rules that yield low-distortion outcomes (informally, outcomes with high social welfare). Existing distortion bounds, however, rely on strong assumptions that restrict voters' latent utilities. We prove that low distortion PB outcomes can be achieved by dropping these assumptions and instead leveraging the established idea that voters can be public-spirited: they may consider others' interests alongside their own when voting. Flanigan, Procaccia, and Wang (2023) prove that in public-spirited single-winner voting (the special case of PB where exactly one project can be funded) with ranking ballots, deterministic aggregation rules can achieve constant distortion. Our first contribution is to extend this analysis to PB; there, we prove that the best distortion permitted by deterministic rules with ranking ballots grows linearly in the number of projects. We find that this impossibility --- a problem in practice, where m is often large --- holds for other known ballots as well. Our second contribution is the design of a new PB ballot format that breaks this linear distortion barrier. This ballot asks voters to rank a predetermined set of entire feasible bundles of projects. We design multiple protocols for implementing these ballots, each striking a different trade-off between the number of bundles voters must rank and the distortion: with m bundles, we get sublinear distortion; with polynomial bundles, we get logarithmic distortion; and with pseudopolynomial bundles, we get constant distortion.

We study the distortion of one-sided and two-sided matching problems on the line. In the one-sided case, n agents need to be matched to n items, and each agent's cost in a matching is their distance from the item they were matched to. We propose an algorithm that is provided only with ordinal information regarding the agents' preferences (each agent's ranking of the items from most- to least-preferred) and returns a matching aiming to minimize the social cost with respect to the agents' true (cardinal) costs. We prove that our algorithm simultaneously achieves the best-possible approximation of 3 (known as distortion) with respect to a variety of social cost measures which include the utilitarian and egalitarian social cost. In the two-sided case, where the agents need be matched to n other agents and both sides report their ordinal preferences over each other, we show that it is always possible to compute an optimal matching. In fact, we show that this optimal matching can be achieved using even less information, and we provide bounds regarding the sufficient number of queries.

Abstract We study matching settings in which a set of agents have private utilities over a set of items. Each agent reports a partition of the items into approval sets of different threshold utility levels. Given this limited information on input, the goal is to compute an assignment of the items to the agents (subject to cardinality constraints depending on the application) that (approximately) maximizes the social welfare (the total utility of the agents for their assigned items). We first consider the well-known, simple one-sided matching problem in which each of n agents is to be assigned exactly one of n items. We show that with t threshold utility levels, the distortion of deterministic matching algorithms is $$\Theta (\root t \of {n})$$ while that of randomized algorithms is $$\Theta (\root t+1 \of {n})$$ . We then show that our distortion bounds extend to a more general setting in which there are multiple copies of the items, each agent can be assigned a number of items (even copies of the same one) up to a capacity, and the utility of an agent for an item depends on the number of its copies that the agent is given.

When an agent votes, she typically ranks the set of available alternatives. Occasionally, she may also wish to report the intensity of her preferences by indicating adjacent pairs of alternatives in her ranking between which her preference is acutely decisive; for instance, she may suggest that she likes alternative a more than b, but b much more than c. We design near-optimal voting rules which aggregate such preference rankings with intensities using the recently-popular distortion framework. We also show that traditional voting rules, which aggregate preference rankings while ignoring (or not eliciting) intensities, can incur significant welfare loss.

With a rapid growth in the deployment of AI tools for making critical decisions (or aiding humans in doing so), there is a growing demand to be able to explain to the stakeholders how these tools arrive at a decision. Consequently, voting is frequently used to make such decisions due to its inherent explainability. Recent work suggests that using randomized (as opposed to deterministic) voting rules can lead to significant efficiency gains measured via the distortion framework. However, rules that use intricate randomization can often become too complex to explain to the stakeholders; losing explainability can eliminate the key advantage of voting over black-box AI tools, which may outweigh the efficiency gains. We study the efficiency gains which can be unlocked by using voting rules that add a simple randomization step to a deterministic rule, thereby retaining explainability. We focus on two such families of rules, randomized positional scoring rules and random committee member rules, and show, theoretically and empirically, that they indeed achieve explainability and efficiency simultaneously to some extent.

We study the problem of designing voting rules that take as input the ordinal preferences of n agents over a set of n alternatives and output a single alternative, aiming to optimize the overall happiness of the agents. The input to the voting rule is each agent's ranking of the alternatives from most to least preferred, yet the agents have more refined (cardinal) preferences that capture the intensity with which they prefer one alternative over another. To quantify the extent to which voting rules can optimize over the cardinal preferences given access only to the ordinal ones, prior work has used the distortion measure, i.e., the worst-case approximation ratio between a voting rule's performance and the best performance achievable given the cardinal preferences.

A fundamental question in social choice and multi-agent systems is aggregating ordinal preferences expressed by agents into a measurably prudent collective choice. A promising line of recent work views ordinal preferences as a proxy for underlying cardinal preferences. It aims to optimize distortion, the worst-case approximation ratio of the (utilitarian) social welfare. When agents rank the set of alternatives, prior work identifies near-optimal voting rules for selecting one or more alternatives. However, ranking all the alternatives is prohibitive when there are many alternatives. In this work, we consider the setting where each agent ranks only her t favorite alternatives and identify almost tight bounds on the best possible distortion when selecting a single alternative or a committee of alternatives of a given size k. Our results also extend to approximating higher moments of social welfare. Along the way, we close a gap left open in prior work by identifying asymptotically tight distortion bounds for committee selection given full rankings.

Envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is (φ − 1) ≃ 0.61 by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time 0.73-approximation allocation algorithm. For our algorithm we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness.

In Spatial Voting Theory, distortion is a measure of how good the winner is. It has been proved that no deterministic voting mechanism can guarantee a distortion better than 3, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election? We consider an election with two candidates and suggest an abstention model, which is a general form of the abstention model proposed by Kirchgässner. Our results characterize the distortion ¨ value and provide a rather complete picture of the model.

We study the extent to which decentralized cost-sharing protocols can achieve good price of anarchy (PoA) bounds in network cost-sharing games with nagents. We focus on the model of resource-aware protocols, where the designer has prior access to the network structure and can also increase the total cost of an edge (overcharging), and we study classes of games with concave or convex cost functions. We first consider concave cost functions and our main result is a cost-sharing protocol for symmetric games on directed acyclic graphs that achieves a PoA of 2+ε for some arbitrary small positive ε, which improves to 1+ε for games with at least two players. We also achieve a PoA of 1 for series-parallel graphs and show that no protocol can achieve a PoA better than Ω(√n) for multicast games. We then also consider convex cost functions and prove analogous results for series-parallel networks and multicast games, as well as a lower bound of Ω(√n) for the PoA on directed acyclic graphs without the use of overcharging.

In Spatial Voting Theory, distortion is a measure of how good the winner is. It is proved that no deterministic voting mechanism can guarantee a distortion better than 3, even for simple metrics such as a line. In this study, we wish to answer the following question: how does the distortion value change if we allow less motivated agents to abstain from the election?We consider an election with two candidates and suggest an abstention model, which is a more general form of the abstention model proposed by Kirchgässner (2003). We define the¨ concepts of the expected winner and the expected distortion to evaluate the distortion of an election in our model. Our results fully characterize the distortion value and provide a rather complete picture of the model.