This study introduces an optimal mechanism in a dynamic stochastic knapsack environment. The model features a single seller who has a fixed quantity of a perfectly divisible item. Impatient buyers with a piece-wise linear utility function arrive randomly and they report the two-dimensional private information: marginal value and demanded quantity. We derive a revenue-maximizing dynamic mechanism in a finite discrete time framework that satisfies incentive compatibility, individual rationality, and feasibility conditions. This is achieved by characterizing buyers' utility and utilizing the Bellman equation. Moreover, we establish the essential penalty scheme for incentive compatibility, as well as the allocation and payment policies. Lastly, we propose algorithms to approximate the optimal policy, based on the Monte Carlo simulation-based regression method and reinforcement learning.
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The maximum Nash social welfare (NSW)---which maximizes the geometric mean of agents' utilities---is a fundamental solution concept with remarkable fairness and efficiency guarantees. The computational aspects of NSW have been extensively studied for *one-sided* preferences where a set of agents have preferences over a set of resources. Our work deviates from this trend and studies NSW maximization for *two-sided* preferences, wherein a set of workers and firms, each having a cardinal valuation function, are matched with each other. We provide a systematic study of the computational complexity of maximizing NSW for many-to-one matchings under two-sided preferences. Our main negative result is that maximizing NSW is NP-hard even in a highly restricted setting where each firm has capacity 2, all valuations are in the range {0,1,2}, and each agent positively values at most three other agents. In search of positive results, we develop approximation algorithms as well as parameterized algorithms in terms of natural parameters such as the number of workers, the number of firms, and the firms' capacities. We also provide algorithms for restricted domains such as symmetric binary valuations and bounded degree instances.
We consider spatial voting where candidates are located in the Euclidean d-dimensional space, and each voter ranks candidates based on their distance from the voter's ideal point. We explore the case where information about the location of voters' ideal points is incomplete: for each dimension, we are given an interval of possible values. We study the computational complexity of finding the possible and necessary winners for positional scoring rules. Our results show that we retain tractable cases of the classic model where voters have partial-order preferences. Moreover, we show that there are positional scoring rules under which the possible-winner problem is intractable for partial orders, but tractable in the one-dimensional spatial setting. We also consider approval voting in this setting. We show that for up to two dimensions, the necessary-winner problem is tractable, while the possible-winner problem is hard for any number of dimensions.
The problem of fairly allocating a set of indivisible items is a well-known challenge in the field of (computational) social choice. In this scenario, there is a fundamental incompatibility between notions of fairness (such as envy-freeness and proportionality) and economic efficiency (such as Pareto-optimality). However, in the real world, items are not always allocated once and for all, but often repeatedly. For example, the items may be recurring chores to distribute in a household. Motivated by this, we initiate the study of the repeated fair division of indivisible goods and chores, and propose a formal model for this scenario. In this paper, we show that, if the number of repetitions is a multiple of the number of agents, there always exists a sequence of allocations that is proportional and Pareto-optimal. On the other hand, irrespective of the number of repetitions, an envy-free and Pareto-optimal sequence of allocations may not exist. For the case of two agents, we show that if the number of repetitions is even, it is always possible to find a sequence of allocations that is overall envy-free and Pareto-optimal. We then prove even stronger fairness guarantees, showing that every allocation in such a sequence satisfies some relaxation of envy-freeness. Finally, in case that the number of repetitions can be chosen freely, we show that envy-free and Pareto-optimal allocations are achievable for any number of agents.
In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.
In this paper, we delve into the problem of using monetary incentives to encourage players to shift from an initial Nash equilibrium to a more favorable one within a game. Our main focus revolves around computing the minimum reward required to facilitate this equilibrium transition. The game involves a single row player who possesses m strategies and k column players, each endowed with n strategies. Our findings reveal that determining whether the minimum reward is zero is NP-complete, and computing the minimum reward becomes APX-hard. Nonetheless, we bring some positive news, as this problem can be efficiently handled if either k or n is a fixed constant. Furthermore, we have devised an approximation algorithm with an additive error that runs in polynomial time. Lastly, we explore a specific case wherein the utility functions exhibit single-peaked characteristics, and we successfully demonstrate that the optimal reward can be computed in polynomial time.
We present a study on a repeated delegated choice problem, which is the first to consider an online learning variant of Kleinberg and Kleinberg, EC'18. In this model, a principal interacts repeatedly with an agent who possesses an exogenous set of solutions to search for efficient ones. Each solution can yield varying utility for both the principal and the agent, and the agent may propose a solution to maximize its own utility in a selfish manner. To mitigate this behavior, the principal announces an eligible set which screens out a certain set of solutions. The principal, however, does not have any information on the distribution of solutions nor the number of solutions in advance. Therefore, the principal dynamically announces various eligible sets to efficiently learn the distribution. The principal's objective is to minimize cumulative regret compared to the optimal eligible set in hindsight. We explore two dimensions of the problem setup, whether the agent behaves myopically or strategizes across the rounds, and whether the solutions yield deterministic or stochastic utility. We obtain sublinear regret upper bounds in various regimes, and derive corresponding lower bounds which implies the tightness of the results. Overall, we bridge a well-known problem in economics to the evolving area of online learning, and present a comprehensive study in this problem.
Single-elimination (SE) tournaments are a popular format used in competitive environments and decision making. Algorithms for SE tournament manipulation have been an active topic of research in recent years. In this paper, we initiate the algorithmic study of a novel variant of SE tournament manipulation that aims to model the fact that certain matchups are highly desired in a sporting context, incentivizing an organizer to manipulate the bracket to make such matchups take place. We obtain both hardness and tractability results. We show that while the problem of computing a bracket enforcing a given set of matches in an SE tournament is NP-hard, there are natural restrictions that lead to polynomial-time solvability. In particular, we show polynomial-time solvability if there is a linear ordering on the ability of players with only a constant number of exceptions where a player with lower ability beats a player with higher ability.
We study worst-case VCG redistribution mechanism design for the public project problem. The mechanism design task comes down to designing a payment function that maximizes the worst-case allocative efficiency ratio. We use a multilayer perceptron (MLP) with ReLU activation to model the payment function and use mixed integer programming (MIP) to solve for the worst-case type profiles that maximally violate the mechanism design constraints. We collect these worst-case type profiles and use them as training samples to train toward better worst-case mechanisms. In practice, we require a tiny neural network structure for the above approach to scale. The Lottery Ticket Hypothesis states that a large network is likely to contain a "winning ticket" -- a much smaller subnetwork that "won the initialization lottery", which makes its training particularly effective. Motivated by this hypothesis, we train a large network and prune it into a tiny subnetwork. We run MIP-based worst-case training on the drawn subnetwork and evaluate the resulting mechanism's worst-case performance. If the subnetwork does not achieve good worst-case performance, then we record the type profiles that cause the current draw to be bad. To draw again, we restore the large network to its initial weights and prune using recorded type profiles from earlier draws, therefore avoiding drawing the same ticket twice. We expect to eventually encounter a tiny subnetwork that leads to effective training for our worst-case mechanism design task. Lastly, a by-product of multiple ticket draws is an ensemble of mechanisms with different worst cases, which improves the worst-case performance further. Using our approach, we find previously unknown optimal mechanisms for up to 5 agents. Our results confirm the tightness of existing theoretical upper bounds. For up to 20 agents, we derive significantly improved worst-case mechanisms, surpassing a long list of existing manual results.
This work studies the behaviors of two large-population teams competing in a discrete environment. The team-level interactions are modeled as a zero-sum game while the agent dynamics within each team is formulated as a collaborative mean-field team problem. Drawing inspiration from the mean-field literature, we first approximate the large-population team game with its infinite-population limit. Subsequently, we construct a fictitious centralized system and transform the infinite-population game to an equivalent zero-sum game between two coordinators. Via a novel reachability analysis, we study the optimality of coordination strategies, which induce decentralized strategies under the original information structure. The optimality of the resulting strategies is established in the original finite-population game, and the theoretical guarantees are verified by numerical examples.
We study a novel approach to information design in the standard traffic model of network congestion games. It captures the natural condition that the demand is unknown to the users of the network. A principal (e.g., a mobility service) commits to a signaling strategy, observes the realized demand and sends a (public) signal to agents (i.e., users of the network). Based on the induced belief about the demand, the users then form an equilibrium. We consider the algorithmic goal of the principal: Compute a signaling scheme that minimizes the expected total cost of the induced equilibrium. We concentrate on single-commodity networks and affine cost functions, for which we obtain the following results. First, we devise a fully polynomial-time approximation scheme (FPTAS) for the case that the demand can only take two values. It relies on several structural properties of the cost of the induced equilibrium as a function of the updated belief about the distribution of demands. We show that this function is piecewise linear for any number of demands, and monotonic for two demands. Second, we give a complete characterization of the graph structures for which it is optimal to fully reveal the information about the realized demand. This signaling scheme turns out to be optimal for all cost functions and probability distributions over demands if and only if the graph is series-parallel. Third, we propose an algorithm that computes the optimal signaling scheme for any number of demands whose time complexity is polynomial in the number of supports that occur in a Wardrop equilibrium for some demand. Finally, we conduct a computational study that tests this algorithm on real-world instances.
Fair allocation of indivisible goods presents intriguing challenges from both a social choice perspective and an algorithmic standpoint. Due to the indivisibility of goods, it is common for one agent to envy the bundle of goods assigned to another agent and, indeed, envy-free solutions do not exist in general. In line with the classical game-theoretic concept of Nucleolus in coalitional games, we propose that a fair allocation should minimize the agents’ dissatisfaction profile in a lexicographic manner, where the dissatisfaction of an agent is defined as her maximum envy towards other agents. Therefore, we seek allocations that minimize the maximum envy. In cases where multiple solutions have an equal maximum value, we minimize the second-worst value, and so on. Additionally, as is customary in fair division problems, we also consider an efficiency requirement: among the allocations with the best agents’ dissatisfaction profile, we prioritize those that maximize the sum of agents’ utilities, known as maximum social welfare. Such allocations, referred to as maxileximin allocations, always exist. In this study, we analyze the computational properties of maxileximin allocations in the context of fair allocation problems with constraints. Specifically, we focus on the Connected Fair Division problem, where goods correspond to the nodes of a graph, and a bundle of goods is allowed if the subgraph formed by those goods is connected. We demonstrate that the problem is F∆P2 -complete, even for instances with simple graphical structures such as path and star graphs. However, we identify islands of tractability for instances with more intricate graphs, such as those having bounded treewidth, provided that the number of agents is bounded by a fixed number and utility functions use small values.
We study the budget aggregation problem in which a set of strategic voters must split a finite divisible resource (such as money or time) among a set of competing projects. Our goal is twofold: We seek truthful mechanisms that provide fairness guarantees to the projects. For the first objective, we focus on the class of moving phantom mechanisms, which are -- to this day -- essentially the only known truthful mechanisms in this setting. For project fairness, we consider the mean division as a fair baseline, and bound the maximum difference between the funding received by any project and this baseline. We propose a novel and simple moving phantom mechanism that provides optimal project fairness guarantees. As a corollary of our results, we show that our new mechanism minimizes the L1 distance to the mean for three projects and gives the first non-trivial bounds on this quantity for more than three projects.
Among the recent work on designing algorithms for selecting citizens' assembly participants, one key property of these algorithms has not yet been studied: their manipulability. Strategic manipulation is a concern because these algorithms must satisfy representation constraints according to volunteers' self-reported features; misreporting these features could thereby increase a volunteer's chance of being selected, decrease someone else's chance, and/or increase the expected number of seats given to their group. Strikingly, we show that Leximin — an algorithm that is widely used for its fairness — is highly manipulable in this way. We then introduce a new class of selection algorithms that use Lp norms as objective functions. We show that the manipulability of the Lp-based algorithm decreases in O(1/n^(1-1/p)) as the number of volunteers n grows, approaching the optimal rate of O(1/n) as p approaches infinity. These theoretical results are confirmed via experiments in eight real-world datasets.
A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs between two well-studied measures of fairness and efficiency --- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency --- by saying, for given constants α and β, whether there exists an α-EFX allocation that guarantees a β-fraction of the maximum Nash welfare (β-MNW). For additive valuations, we show that for any α ∈ [0,1], there exists a partial allocation that is α-EFX and 1/(α+1)-MNW. This tradeoff turns out to be tight (for every α) as demonstrated by an impossibility result that we give. We also show that for α ∈ [0, φ-1 ≃ 0.618] these partial allocations can be turned into complete allocations where all items are assigned. Furthermore, for any α ∈ [0, 1/2], we show that the tight tradeoff of α-EFX and 1/(α+1)-MNW with complete allocations holds for the more general setting of subadditive valuations. Our results improve upon the current state of the art, for both additive and subadditive valuations, and match the best-known approximations of EFX under complete allocations, regardless of Nash welfare guarantees. Notably, our constructions for additive valuations also provide EF1 and constant approximations for maximin share guarantees.
In representative democracy, a redistricting map is chosen to partition an electorate into districts which each elects a representative. A valid redistricting map must satisfy a collection of constraints such as being compact, contiguous, and of almost-equal population. However, these constraints are loose enough to enable an enormous ensemble of valid redistricting maps. This enables a partisan legislature to gerrymander by choosing a map which unfairly favors it. In this paper, we introduce an interpretable and tractable distance measure over redistricting maps which does not use election results and study its implications over the ensemble of redistricting maps. Specifically, we define a central map which may be considered "most typical" and give a rigorous justification for it by showing that it mirrors the Kemeny ranking in a scenario where we have a committee voting over a collection of redistricting maps to be drawn. We include runnning time and sample complexity analysis for our algorithms, including some negative results which hold using any algorithm. We further study outlier detection based on this distance measure and show that our framework can detect some gerrymandered maps. More precisely, we show some maps that are widely considered to be gerrymandered that lie very far away from our central maps in comparison to a large ensemble of valid redistricting maps. Since our distance measure does not rely on election results, this gives a significant advantage in gerrymandering detection which is lacking in all previous methods.
In approval-based committee (ABC) elections, the goal is to select a fixed-size subset of the candidates, a so-called committee, based on the voters' approval ballots over the candidates. One of the most popular classes of ABC voting rules are ABC scoring rules, for which voters give points to each committee and the committees with maximal total points are chosen. While the set of ABC scoring rules has recently been characterized in a model where the output is a ranking of all committees, no full characterization of these rules exists in the standard model where a set of winning committees is returned. We address this issue by characterizing two important subclasses of ABC scoring rules in the standard ABC election model, thereby both extending the result for ABC ranking rules to the standard setting and refining it to subclasses. In more detail, by relying on a consistency axiom for variable electorates, we characterize (i) the prominent class of Thiele rules and (ii) a new class of ABC voting rules called ballot size weighted approval voting. Based on these theorems, we also infer characterizations of three well-known ABC voting rules, namely multi-winner approval voting, proportional approval voting, and satisfaction approval voting.
We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a fully polynomial-time approximation scheme to compute an approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.
We study the computational complexity of fairly allocating a set of indivisible items under externalities. In this recently-proposed setting, in addition to the utility the agent gets from their bundle, they also receive utility from items allocated to other agents. We focus on the extended definitions of envy-freeness up to one item (EF1) and of envy-freeness up to any item (EFX), and we provide the landscape of their complexity for several different scenarios. We prove that it is NP-complete to decide whether there exists an EFX allocation, even when there are only three agents, or even when there are only six different values for the items. We complement these negative results by showing that when both the number of agents and the number of different values for items are bounded by a parameter the problem becomes fixed-parameter tractable. Furthermore, we prove that two-valued and binary-valued instances are equivalent and that EFX and EF1 allocations coincide for this class of instances. Finally, motivated from real-life scenarios, we focus on a class of structured valuation functions, which we term agent/item-correlated. We prove their equivalence to the "standard" setting without externalities. Therefore, all previous results for EF1 and EFX apply immediately for these valuations.
We give a quantitative analysis of the independence of irrelevant alternatives (IIA) axiom. IIA says that the society's preference between x and y should depend only on individual preferences between x and y: we show that, in several contexts, if the individuals express their preferences about additional (``irrelevant'') alternatives, this information helps to estimate better which of x and y has higher social welfare. Our contribution is threefold: (1) we provide a new tool to measure the impact of IIA on social welfare (pairwise distortion), based on the well-established notion of voting distortion, (2) we study the average impact of IIA in both general and metric settings, with experiments on synthetic and real data and (3) we study the worst-case impact of IIA in the 1D-Euclidean metric space.