In the k-committee election problem, we wish to aggregate the preferences of n agents over a set of alternatives and select a committee of k alternatives that minimizes the cost incurred by the agents. While we typically assume that agent preferences are captured by a cardinal utility function, in many contexts we only have access to ordinal information, namely the agents' rankings over the outcomes. As preference rankings are not as expressive as cardinal utilities, a loss of efficiency is inevitable, and is quantified by the notion of distortion. We study the problem of electing a k-committee that minimizes the sum of the \ell-largest costs incurred by the agents, when agents and candidates are embedded in a metric space. This problem is called the \ell-centrum problem and captures both the utilitarian and egalitarian objectives. When k >= 2, it is not possible to compute a bounded-distortion committee using purely ordinal information. We develop the first algorithms (that we call mechanisms) for the \ell-centrum problem (when k >= 2), which achieve O(1)-distortion while eliciting only a very limited amount of cardinal information via value queries. We obtain two types of query-complexity guarantees: O(log k log n) queries per agent, and O(k^2 log^2 n) queries in total (while achieving O(1)-distortion in both cases). En route, we give a simple adaptive-sampling algorithm for the \ell-centrum k-clustering problem.
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Rent division is the well-studied problem of fairly assigning rooms and dividing rent among a set of roommates within a single apartment. A shortcoming of existing solutions is that renters are assumed to be considering apartments in isolation, whereas in reality, renters can choose among multiple apartments. In this paper, we generalize the rent division problem to the multi-apartment setting, where the goal is to both fairly choose an apartment among a set of alternatives and fairly assign rooms and rents within the chosen apartment. Our main contribution is a generalization of envy-freeness called *negotiated envy-freeness*. We show that a solution satisfying negotiated envy-freeness is guaranteed to exist and that it is possible to optimize over all negotiated envy-free solutions in polynomial time. We also define an even stronger fairness notion called *universal envy-freeness* and study its existence when values are drawn randomly.
We formulate the problem of fair and efficient completion of indivisible goods, defined as follows: Given a partial allocation of indivisible goods among agents, does there exist an allocation of the remaining goods (i.e., a completion) that satisfies fairness and economic efficiency guarantees of interest? We study the computational complexity of the completion problem for prominent fairness and efficiency notions such as envy-freeness up to one good (EF1), proportionality up to one good (Prop1), maximin share (MMS), and Pareto optimality (PO), and focus on the class of additive valuations as well as its subclasses such as binary additive and lexicographic valuations. We find that while the completion problem is significantly harder than the standard fair division problem (wherein the initial partial allocation is empty), the consideration of restricted preferences facilitates positive algorithmic results for threshold-based fairness notions (Prop1 and MMS). On the other hand, the completion problem remains computationally intractable for envy-based notions such as EF1 and EF1+PO even under restricted preferences.
Envy-freeness up to any good (EFX) is a popular and important fairness property in the fair allocation of indivisible goods, of which its existence in general is still an open question. In this work, we investigate the problem of determining the minimum number of EFX allocations for a given instance, arguing that this approach may yield valuable insights into the existence and computation of EFX allocations. We focus on restricted instances where the number of goods slightly exceeds the number of agents, and extend our analysis to weighted EFX (WEFX) and a novel variant of EFX for general monotone valuations, termed EFX+. In doing so, we identify the transition threshold for the existence of allocations satisfying these fairness notions. Notably, we resolve open problems regarding WEFX by proving polynomial-time computability under binary additive valuations, and establishing the first constant-factor approximation for two agents.
Tâtonnement is a simple, intuitive market process where prices are iteratively adjusted based on the difference between demand and supply. Many variants under different market assumptions have been studied and shown to converge to a market equilibrium, in some cases at a fast rate. However, the classical case of linear Fisher markets have long eluded the analyses, and it remains unclear whether tâtonnement converges in this case. We show that, for a sufficiently small stepsize, the prices given by the tâtonnement process are guaranteed to converge to equilibrium prices, up to a small approximation radius that depends on the stepsize. To achieve this, we consider the dual Eisenberg-Gale convex program in the price space, view tâtonnement as subgradient descent on this convex program, and utilize novel last-iterate convergence results for subgradient descent under error bound conditions. In doing so, we show that the convex program satisfies a particular error bound condition, the quadratic growth condition, and that the price sequence generated by tâtonnement is bounded above and away from zero. We also show that a similar convergence result holds for tâtonnement in quasi-linear Fisher markets. Numerical experiments are conducted to demonstrate that the theoretical linear convergence aligns with empirical observations.
Search and recommendation ecosystems exhibit competition among content creators. This competition has been tackled in a variety of game-theoretic frameworks. Content creators generate documents with the aim of being recommended by a content ranker for various information needs. In order for the ecosystem, modeled as a content ranking game, to be effective and maximize user welfare, it should guarantee stability, where stability is associated with the existence of pure Nash equilibrium in the corresponding game. Moreover, if the contents' ranking algorithm possesses a game in which any best-response learning dynamics of the content creators converge to equilibrium of high welfare, the system is considered highly attractive. However, as classical content ranking algorithms, employed by search and recommendation systems, rank documents by their distance to information needs, it has been shown that they fail to provide such stability properties. As a result, novel content ranking algorithms have been devised. In this work, we offer an alternative approach: corpus enrichment with a small set of fixed dummy documents. It turns out that, with the right design, such enrichment can lead to pure Nash equilibrium and even to the convergence of any best-response dynamics to a high welfare result, where we still employ the classical/current content ranking approach. We show two such corpus enrichment techniques with tight bounds on the number of documents needed to obtain the desired results. Interestingly, our study is a novel extension of Borel's Colonel Blotto game.
In forecasting competitions, the traditional mechanism scores the predictions of each contestant against the outcome of each event, and the contestant with the highest total score wins. While it is well-known that this traditional mechanism can suffer from incentive issues, it is folklore that contestants will still be roughly truthful as the number of events grows. Yet thus far the literature lacks a formal analysis of this traditional mechanism. This paper gives the first such analysis. We first demonstrate that the "long-run truthfulness" folklore is false: even for arbitrary numbers of events, the best forecaster can have an incentive to hedge, reporting more moderate beliefs to increase their win probability. On the positive side, however, we show that two contestants will be approximately truthful when they have sufficient uncertainty over the relative quality of their opponent and the outcomes of the events, a case which may arise in practice.
We propose a decentralized market model in which agents can negotiate bilateral contracts. This builds on a similar, but centralized, model of trading networks introduced by Hatfield et al. in 2013. Prior work has established that fully-substitutable preferences guarantee the existence of competitive equilibria which can be centrally computed. Our motivation comes from the fact that prices in markets such as over-the-counter markets and used car markets arise from decentralized negotiation among agents, which has left open an important question as to whether equilibrium prices can emerge from agent-to-agent bilateral negotiations. We design a best response dynamic intended to capture such negotiations between market participants. We assume fully substitutable preferences for market participants. In this setting, we provide proofs of convergence for sparse markets (covering many real world markets of interest), and experimental results for more general cases, demonstrating that prices indeed reach equilibrium, quickly, via bilateral negotiations. Our best response dynamic, and its convergence behavior, forms an important first step in understanding how decentralized markets reach, and retain, equilibrium.
Consider public health officials aiming to spread awareness about a new vaccine in a community interconnected by a social network. How can they distribute information with minimal resources, so as to avoid polarization and ensure community-wide convergence of opinion? To tackle such challenges, we initiate the study of sample complexity of opinion formation in networks. Our framework is built on the recognized opinion formation game, where we regard each agent’s opinion as a data-derived model, unlike previous works that treat opinions as data-independent scalars. The opinion model for every agent is initially learned from its local samples and evolves game-theoretically as all agents communicate with neighbors and revise their models towards an equilibrium. Our focus is on the sample complexity needed to ensure that the opinions converge to an equilibrium such that every agent’s final model has low generalization error. Our paper has two main technical results. First, we present a novel polynomial time optimization framework to quantify the total sample complexity for arbitrary networks, when the underlying learning problem is (generalized) linear regression. Second, we leverage this optimization to study the network gain which measures the improvement of sample complexity when learning over a network compared to that in isolation. Towards this end, we derive network gain bounds for various network classes including cliques, star graphs, and random regular graphs. Additionally, our framework provides a method to study sample distribution within the network, suggesting that it is sufficient to allocate samples inversely to the degree. Empirical results on both synthetic and real-world networks strongly support our theoretical findings.
We adopt a parametric approach to analyze the worst-case degradation in social welfare when the allocation of indivisible goods is constrained to be fair. Specifically, we are concerned with cardinality-constrained allocations, which require that each agent has at most k items in their allocated bundle. We propose the notion of the price of cardinality, which captures the worst-case multiplicative loss of utilitarian or egalitarian social welfare resulting from imposing the cardinality constraint. We then characterize tight or almost-tight bounds on the price of cardinality as exact functions of the instance parameters, demonstrating how the social welfare improves as k is increased. In particular, one of our main results refines and generalizes the existing asymptotic bound of Θ(√n) on the price of balancedness. We also further extend our analysis to the problem where the items are partitioned into disjoint categories, and each category has its own cardinality constraint. Through a parametric study of the price of cardinality, we provide a framework which aids decision makers in choosing an ideal level of cardinality-based fairness, using their knowledge of the potential loss of utilitarian and egalitarian social welfare.
In perpetual voting, multiple decisions are made at different moments in time. Taking the history of previous decisions into account allows us to satisfy properties such as proportionality over periods of time. In this paper, we consider the following question: is there a perpetual approval voting method that guarantees that no voter is dissatisfied too many times? We identify a sufficient condition on voter behavior ---which we call 'bounded conflicts' condition---under which a sublinear growth of dissatisfaction is possible. We provide a tight upper bound on the growth of dissatisfaction under bounded conflicts, using techniques from Kolmogorov complexity. We also observe that the approval voting with binary choices mimics the machine learning setting of prediction with expert advice. This allows us to present a voting method with sublinear guarantees on dissatisfaction under bounded conflicts, based on the standard techniques from prediction with expert advice.
This paper studies a generalized variant of the Colonel Blotto game, referred to as the Colonel Blotto game with costs. Unlike the classic Colonel Blotto game, which imposes the use-it-or-lose-it budget assumption, the Colonel Blotto game with costs captures the strategic importance of costs related both to obtaining resources and assigning them across battlefields. We show that every instance of the Colonel Blotto game with costs is strategically equivalent to an instance of the zero-sum Colonel Blotto game with one additional battlefield. This enables the computation of Nash equilibria of the Colonel Blotto game with costs in polynomial time with respect to the game parameters: the number of battlefields plus the number of resources available to the players.
We consider correlated equilibria in an adversarial environment, where an adversary can compromise the public signal used by the players for choosing their strategies, while players aim at detecting a potential attack as soon as possible to avoid loss of utility. We model the interaction between the adversary and the players as a zero-sum game and we derive the maxmin strategies for both the defender and the attacker using the framework of quickest change detection. We define a class of adversarial strategies that achieve the optimal trade-off between the impact and the detectability of the attack for the adversary and show that a generalized CUSUM scheme is asymptotically optimal for their detection. Our numerical results on the Sioux-Falls benchmark traffic routing game show that the proposed detection scheme can effectively limit the utility loss by a potential adversary.
We provide mechanisms and new metric distortion bounds for line-up elections. In such elections, a set of n voters, k candidates, and ell positions are all located in a metric space. The goal is to choose a set of candidates and assign them to different positions, so as to minimize the total cost of the voters. The cost of each voter consists of the distances from itself to the chosen candidates (measuring how much the voter likes the chosen candidates, or how similar it is to them), as well as the distances from the candidates to the positions they are assigned to (measuring the fitness of the candidates for their positions). Our mechanisms, however, do not know the exact distances, and instead produce good outcomes while only using a smaller amount of information, resulting in small distortion. We consider several different types of information: ordinal voter preferences, ordinal position preferences, and knowing the exact locations of candidates and positions, but not those of voters. In each of these cases, we provide constant distortion bounds, thus showing that only a small amount of information is enough to form outcomes close to optimum in line-up elections.
The correlation of values commonly exists in auctions, which can be further exploited to improve revenue. However, the complex correlation structure makes it hard to manually design the optimal auction mechanism. Data-driven auction mechanisms, powered by machine learning, enable to design auctions directly from historical auction data, without relying on specific value distributions. In this work, we synthesize the learning-based auction and the characteristics of strategy-proofness in the correlated value setting, and propose a new auction mechanism, namely Conditional Auction Net (CAN). The CAN can encode the correlation of values into the rank score of each bidder, and further adjust the allocation rule to approach the optimal revenue. The property of strategy-proofness is guaranteed by encoding the game theoretical condition into the neural network structure. Furthermore, all operations in the designed auctions are differentiable to enable an end-to-end training paradigm. We also present CAN can provide a large solution space to adequately encode the correlation of values. Experimental results demonstrate that the proposed auction mechanism can represent almost any strategy-proof auction mechanism, and outperforms the auction mechanisms wildly used in the correlated value settings.
Due to the sensitivity of data, Federated Learning (FL) is employed to enable distributed machine learning while safeguarding data privacy and accommodating the requirements of various devices. However, in the context of semidecentralized FL, clients’ communication and training states are dynamic. This variability arises from local training fluctuations, heterogeneous data distributions, and intermittent client participation. Most existing studies primarily focus on stable client states, neglecting the dynamic challenges inherent in real-world scenarios. To tackle this issue, we propose a TRust-Aware clIent scheduLing mechanism called TRAIL, which assesses client states and contributions, enhancing model training efficiency through selective client participation. We focus on a semi-decentralized FL framework where edge servers and clients train a shared global model using unreliable intra-cluster model aggregation and inter-cluster model consensus. First, we propose an adaptive hidden semi-Markov model to estimate clients’ communication states and contributions. Next, we address a client-server association optimization problem to minimize global training loss. Using convergence analysis, we propose a greedy client scheduling algorithm. Finally, our experiments conducted on real-world datasets demonstrate that TRAIL outperforms state-of-the-art baselines, achieving an improvement of 8.7% in test accuracy and a reduction of 15.3% in training loss.
We initiate the study of matching roommates and rooms wherein the preferences of agents over other agents and rooms are complementary and represented by Leontief utilities. In this setting, 2n agents must be paired up and assigned to n rooms. Each agent has cardinal valuations over the rooms as well as compatibility values over all other agents. Under Leontief preferences, an agent’s utility for a matching is the minimum of the two values. We focus on the tradeoff between maximizing utilitarian social welfare and strategyproofness. Our main result shows that—in a stark contrast to the additive case— under binary Leontief utilities, there exist strategyproof mechanisms that maximize the social welfare. We further devise a strategyproof mechanism that implements such a welfare maximizing algorithm and is parameterized by the number of agents. Along the way, we highlight several possibility and impossibility results, and give upper bounds and lower bounds for welfare with or without strategyproofness.
By classic results in social choice theory, any reasonable preferential voting method sometimes gives individuals an incentive to report an insincere preference. The extent to which different voting methods are more or less resistant to such strategic manipulation has become a key consideration for comparing voting methods. Here we measure resistance to manipulation by whether neural networks of varying sizes can learn to profitably manipulate a given voting method in expectation, given different types of limited information about how other voters will vote. We trained over 100,000 neural networks of 26 sizes to manipulate against 8 different voting methods, under 6 types of limited information, in committee-sized elections with 5-21 voters and 3-6 candidates. We find that some voting methods, such as Borda, are highly manipulable by networks with limited information, while others, such as Instant Runoff, are not, despite being quite profitably manipulated by an ideal manipulator with full information. For the three probability models for elections that we use, the overall least manipulable of the 8 methods we study are Condorcet methods, namely Minimax and Split Cycle.
Two prominent objectives in social choice are utilitarian - maximizing the sum of agents' utilities, and leximin - maximizing the smallest agent's utility, then the second-smallest, etc. Utilitarianism is typically computationally easier to attain but is generally viewed as less fair. This paper presents a general reduction scheme that, given a utilitarian solver, produces a distribution over states (deterministic outcomes) that is leximin in expectation. Importantly, the scheme is robust in the sense that, given an approximate utilitarian solver, it produces a lottery that is approximately-leximin (in expectation) - with the same approximation factor. We apply our scheme to several social choice problems: stochastic allocations of indivisible goods, giveaway lotteries, and fair lotteries for participatory budgeting.
A *citizens' assembly* is a group of people who are randomly selected to represent a larger population in a deliberation. While this approach has successfully strengthened democracy, it has certain limitations that suggest the need for assemblies to form and associate more organically. In response, we propose *federated assemblies*, where assemblies are interconnected, and each parent assembly is selected from members of its child assemblies. The main technical challenge is to develop random selection algorithms that meet new representation constraints inherent in this hierarchical structure. We design and analyze several algorithms that provide different representation guarantees under various assumptions on the structure of the underlying graph.