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We propose a simple uncertainty modification for the agent model in normal-form games; at any given strategy profile, the agent can access only a set of “possible profiles” that are within a certain distance from the actual action profile. We investigate the various instantiations in which the agent chooses her strategy using well-known rationales e.g., considering the worst case, or trying to minimize the regret, to cope with such uncertainty. Any such modification in the behavioral model naturally induces a corresponding notion of equilibrium; a distance-based equilibrium. We characterize the relationships between the various equilibria, and also their connections to well-known existing solution concepts such as Trembling-hand perfection. Furthermore, we deliver existence results, and show that for some class of games, such solution concepts can actually lead to better outcomes.

We study a recently introduced class of strategic games that is motivated by and generalizes Schelling's well-known residential segregation model. These games are played on undirected graphs, with the set of agents partitioned into multiple types; each agent either occupies a node of the graph and never moves away or aims to maximize the fraction of her neighbors who are of her own type. We consider a variant of this model that we call swap Schelling games, where the number of agents is equal to the number of nodes of the graph, and agents may swap positions with other agents to increase their utility. We study the existence, computational complexity and quality of equilibrium assignments in these games, both from a social welfare perspective and from a diversity perspective.

We consider a class of coalition formation games that can be succinctly represented by means of hypergraphs and properly generalizes symmetric additively separable hedonic games. More precisely, an instance of hypegraph hedonic game consists of a weighted hypergraph, in which each agent is associated to a distinct node and her utility for being in a given coalition is equal to the sum of the weights of all the hyperedges included in the coalition. We study the performance of stable outcomes in such games, investigating the degradation of their social welfare under two different metrics, the k-Nash price of anarchy and k-core price of anarchy, where k is the maximum size of a deviating coalition. Such prices are defined as the worst-case ratio between the optimal social welfare and the social welfare obtained when the agents reach an outcome satisfying the respective stability criteria. We provide asymptotically tight upper and lower bounds on the values of these metrics for several classes of hypergraph hedonic games, parametrized according to the integer k, the hypergraph arity r and the number of agents n. Furthermore, we show that the problem of computing the exact value of such prices for a given instance is computationally hard, even in case of non-negative hyperedge weights.

We consider settings where agents are evaluated based on observed features, and assume they seek to achieve feature values that bring about good evaluations. Our goal is to craft evaluation mechanisms that incentivize the agents to invest effort in desirable actions; a notable application is the design of course grading schemes. Previous work has studied this problem in the case of a single agent. By contrast, we investigate the general, multi-agent model, and provide a complete characterization of its computational complexity.

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.

Several relaxations of envy-freeness, tailored to fair division in settings with indivisible goods, have been introduced within the last decade. Due to the lack of general existence results for most of these concepts, great attention has been paid to establishing approximation guarantees. In this work, we propose a simple algorithm that is universally fair in the sense that it returns allocations that have good approximation guarantees with respect to four such fairness notions at once. In particular, this is the first algorithm achieving a (φ−1)-approximation of envy-freeness up to any good (EFX) and a 2/φ+2 -approximation of groupwise maximin share fairness (GMMS), where φ is the golden ratio. The best known approximation factor, in polynomial time, for either one of these fairness notions prior to this work was 1/2. Moreover, the returned allocation achieves envy-freeness up to one good (EF1) and a 2/3-approximation of pairwise maximin share fairness (PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our algorithm and improve further the guarantees for GMMS or PMMS. Finally, we show that GMMS—and thus PMMS and EFX—allocations always exist when the number of goods does not exceed the number of agents by more than two.

In this paper we introduce and study all-pay bidding games, a class of two player, zero-sum games on graphs. The game proceeds as follows. We place a token on some vertex in the graph and assign budgets to the two players. Each turn, each player submits a sealed legal bid (non-negative and below their remaining budget), which is deducted from their budget and the highest bidder moves the token onto an adjacent vertex. The game ends once a sink is reached, and Player 1 pays Player 2 the outcome that is associated with the sink. The players attempt to maximize their expected outcome. Our games model settings where effort (of no inherent value) needs to be invested in an ongoing and stateful manner. On the negative side, we show that even in simple games on DAGs, optimal strategies may require a distribution over bids with infinite support. A central quantity in bidding games is the ratio of the players budgets. On the positive side, we show a simple FPTAS for DAGs, that, for each budget ratio, outputs an approximation for the optimal strategy for that ratio. We also implement it, show that it performs well, and suggests interesting properties of these games. Then, given an outcome c, we show an algorithm for finding the necessary and sufficient initial ratio for guaranteeing outcome c with probability 1 and a strategy ensuring such. Finally, while the general case has not previously been studied, solving the specific game in which Player 1 wins iff he wins the first two auctions, has been long stated as an open question, which we solve.

We consider the facility location problem in the one-dimensional setting where each facility can serve a limited number of agents from the algorithmic and mechanism design perspectives. From the algorithmic perspective, we prove that the corresponding optimization problem, where the goal is to locate facilities to minimize either the total cost to all agents or the maximum cost of any agent is NP-hard. However, we show that the problem is fixed-parameter tractable, and the optimal solution can be computed in polynomial time whenever the number of facilities is bounded, or when all facilities have identical capacities. We then consider the problem from a mechanism design perspective where the agents are strategic and need not reveal their true locations. We show that several natural mechanisms studied in the uncapacitated setting either lose strategyproofness or a bound on the solution quality %on the returned solution for the total or maximum cost objective. We then propose new mechanisms that are strategyproof and achieve approximation guarantees that almost match the lower bounds.

We study the problem of fair division when the resources contain both divisible and indivisible goods. Classic fairness notions such as envy-freeness (EF) and envy-freeness up to one good (EF1) cannot be directly applied to the mixed goods setting. In this work, we propose a new fairness notion envy-freeness for mixed goods (EFM), which is a direct generalization of both EF and EF1 to the mixed goods setting. We prove that an EFM allocation always exists for any number of agents. We also propose efficient algorithms to compute an EFM allocation for two agents and for n agents with piecewise linear valuations over the divisible goods. Finally, we relax the envy-free requirement, instead asking for ϵ-envy-freeness for mixed goods (ϵ-EFM), and present an algorithm that finds an ϵ-EFM allocation in time polynomial in the number of agents, the number of indivisible goods, and 1/ϵ.

In hedonic diversity games (HDGs), recently introduced by Bredereck, Elkind, and Igarashi (2019), each agent belongs to one of two classes (men and women, vegetarians and meat-eaters, junior and senior researchers), and agents' preferences over coalitions are determined by the fraction of agents from their class in each coalition. Bredereck et al. show that while an HDG may fail to have a Nash stable (NS) or a core stable (CS) outcome, every HDG in which all agents have single-peaked preferences admits an individually stable (IS) outcome, which can be computed in polynomial time. In this work, we extend and strengthen these results in several ways. First, we establish that the problem of deciding if an HDG has an NS outcome is NP-complete, but admits an XP algorithm with respect to the size of the smaller class. Second, we show that, in fact, all HDGs admit IS outcomes that can be computed in polynomial time; our algorithm for finding such outcomes is considerably simpler than that of Bredereck et al. We also consider two ways of generalizing the model of Bredereck et al. to k ≥ 2 classes. We complement our theoretical results by empirical analysis, comparing the IS outcomes found by our algorithm, the algorithm of Bredereck et al. and a natural better-response dynamics.

Adaptivity to changing environments and constraints is key to success in modern society. We address this by proposing “incrementalized versions” of Stable Marriage and Stable Roommates. That is, we try to answer the following question: for both problems, what is the computational cost of adapting an existing stable matching after some of the preferences of the agents have changed. While doing so, we also model the constraint that the new stable matching shall be not too different from the old one. After formalizing these incremental versions, we provide a fairly comprehensive picture of the computational complexity landscape of Incremental Stable Marriage and Incremental Stable Roommates. To this end, we exploit the parameters “degree of change” both in the input (difference between old and new preference profile) and in the output (difference between old and new stable matching). We obtain both hardness and tractability results, in particular showing a fixed-parameter tractability result with respect to the parameter “distance between old and new stable matching”.

We extend the work of Skowron et al. (AIJ, 2016) by considering the parameterized complexity of the following problem. We are given a set of items and a set of agents, where each agent assigns an integer utility value to each item. The goal is to find a set of k items that these agents would collectively use. For each such collective set of items, each agent provides a score that can be described using an OWA (ordered weighted average) operator and we seek a set with the highest total score. We focus on the parameterization by the number of agents and we find numerous fixed-parameter tractability results (however, we also find some W[1]-hardness results). It turns out that most of our algorithms even apply to the setting where each agent has an integer weight.

We introduce successive committees elections. The point is that our new model additionally takes into account that “committee members” shall have a short term of office possibly over a consecutive time period (e.g., to limit the influence of elitist power cartels or to keep the social costs of overloading committees as small as possible) but at the same time overly frequent elections are to be avoided (e.g., for the sake of long-term planning). Thus, given voter preferences over a set of candidates, a desired committee size, a number of committees to be elected, and an upper bound on the number of committees that each candidate can participate in, the goal is to find a “best possible” series of committees representing the electorate. We show a sharp complexity dichotomy between computing series of committees of size at most two (mostly in polynomial time) and of committees of size at least three (mostly NP-hard). Depending on the voting rule, however, even for larger committee sizes we can spot some tractable cases.

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.

Tournament solutions are frequently used to select winners from a set of alternatives based on pairwise comparisons between alternatives. Prior work has shown that several common tournament solutions tend to select large winner sets and therefore have low discriminative power. In this paper, we propose a general framework for refining tournament solutions. In order to distinguish between winning alternatives, and also between non-winning ones, we introduce the notion of margin of victory (MoV) for tournament solutions. MoV is a robustness measure for individual alternatives: For winners, the MoV captures the distance from dropping out of the winner set, and for non-winners, the distance from entering the set. In each case, distance is measured in terms of which pairwise comparisons would have to be reversed in order to achieve the desired outcome. For common tournament solutions, including the top cycle, the uncovered set, and the Banks set, we determine the complexity of computing the MoV and provide worst-case bounds on the MoV for both winners and non-winners. Our results can also be viewed from the perspective of bribery and manipulation.

We focus on the following natural question: is it possible to influence the outcome of a voting process through the strategic provision of information to voters who update their beliefs rationally? We investigate whether it is computationally tractable to design a signaling scheme maximizing the probability with which the sender's preferred candidate is elected. We resort to the model recently introduced by Arieli and Babichenko (2019) (i.e., without inter-agent externalities), and focus on, as illustrative examples, k-voting rules and plurality voting. There is a sharp contrast between the case in which private signals are allowed and the more restrictive setting in which only public signals are allowed. In the former, we show that an optimal signaling scheme can be computed efficiently both under a k-voting rule and plurality voting. In establishing these results, we provide two contributions applicable to general settings beyond voting. Specifically, we extend a well-known result by Dughmi and Xu (2017) to more general settings and prove that, when the sender's utility function is anonymous, computing an optimal signaling scheme is fixed-parameter tractable in the number of receivers' actions. In the public signaling case, we show that the sender's optimal expected return cannot be approximated to within any factor under a k-voting rule. This negative result easily extends to plurality voting and problems where utility functions are anonymous.

We focus on the scenario in which messages pro and/or against one or multiple candidates are spread through a social network in order to affect the votes of the receivers. Several results are known in the literature when the manipulator can make seeding by buying influencers. In this paper, instead, we assume the set of influencers and their messages to be given, and we ask whether a manipulator (e.g., the platform) can alter the outcome of the election by adding or removing edges in the social network. We study a wide range of cases distinguishing for the number of candidates or for the kind of messages spread over the network. We provide a positive result, showing that, except for trivial cases, manipulation is not affordable, the optimization problem being hard even if the manipulator has an unlimited budget (i.e., he can add or remove as many edges as desired). Furthermore, we prove that our hardness results still hold in a reoptimization variant, where the manipulator already knows an optimal solution to the problem and needs to compute a new solution once a local modification occurs (e.g., in bandit scenarios where estimations related to random variables change over time).

We study an information-structure design problem (a.k.a. a persuasion problem) with a single sender and multiple receivers with actions of a priori unknown types, independently drawn from action-specific marginal probability distributions. As in the standard Bayesian persuasion model, the sender has access to additional information regarding the action types, which she can exploit when committing to a (noisy) signaling scheme through which she sends a private signal to each receiver. The novelty of our model is in considering the much more expressive case in which the receivers interact in a sequential game with imperfect information, with utilities depending on the game outcome and the realized action types. After formalizing the notions of ex ante and ex interim persuasiveness (which differ by the time at which the receivers commit to following the sender's signaling scheme), we investigate the continuous optimization problem of computing a signaling scheme which maximizes the sender's expected revenue. We show that computing an optimal ex ante persuasive signaling scheme is NP-hard when there are three or more receivers. Instead, in contrast with previous hardness results for ex interim persuasion, we show that, for games with two receivers, an optimal ex ante persuasive signaling scheme can be computed in polynomial time thanks to the novel algorithm we propose, based on the ellipsoid method.

We study single-candidate voting embedded in a metric space, where both voters and candidates are points in the space, and the distances between voters and candidates specify the voters' preferences over candidates. In the voting, each voter is asked to submit her favorite candidate. Given the collection of favorite candidates, a mechanism for eliminating the least popular candidate finds a committee containing all candidates but the one to be eliminated. Each committee is associated with a social value that is the sum of the costs (utilities) it imposes (provides) to the voters. We design mechanisms for finding a committee to optimize the social value. We measure the quality of a mechanism by its distortion, defined as the worst-case ratio between the social value of the committee found by the mechanism and the optimal one. We establish new upper and lower bounds on the distortion of mechanisms in this single-candidate voting, for both general metrics and well-motivated special cases.

Gerrymandering is a practice of manipulating district boundaries and locations in order to achieve a political advantage for a particular party. Lewenberg, Lev, and Rosenschein [AAMAS 2017] initiated the algorithmic study of a geographically-based manipulation problem, where voters must vote at the ballot box closest to them. In this variant of gerrymandering, for a given set of possible locations of ballot boxes and known political preferences of n voters, the task is to identify locations for k boxes out of m possible locations to guarantee victory of a certain party in at least ℓ districts. Here integers k and ℓ are some selected parameter. It is known that the problem is NP-complete already for 4 political parties and prior to our work only heuristic algorithms for this problem were developed. We initiate the rigorous study of the gerrymandering problem from the perspectives of parameterized and fine-grained complexity and provide asymptotically matching lower and upper bounds on its computational complexity. We prove that the problem is W[1]-hard parameterized by k + n and that it does not admit an f(n,k) · mo(√k) algorithm for any function f of k and n only, unless the Exponential Time Hypothesis (ETH) fails. Our lower bounds hold already for 2 parties. On the other hand, we give an algorithm that solves the problem for a constant number of parties in time (m+n)O(√k).

Obraztsova et al. (2013) have recently proposed an intriguing convexity axiom for voting rules. This axiom imposes conditions on the shape of the sets of elections with a given candidate as a winner. However, this new axiom is both too weak and too strong: it is too weak because it defines a set to be convex if for any two elements of the set some shortest path between them lies within the set, whereas the standard definition of convexity requires all shortest paths between two elements to lie within the set, and it is too strong because common voting rules do not satisfy this axiom. In this paper, we (1) propose several families of voting rules that are convex in the sense of Obraztsova et al.; (2) put forward a weaker notion of convexity that is satisfied by most common voting rules; (3) prove impossibility results for a variant of this definition that considers all, rather than some shortest paths.

We develop a powerful approach that makes modern SAT solving techniques available as a tool to support the axiomatic analysis of economic matching mechanisms. Our central result is a preservation theorem, establishing sufficient conditions under which the possibility of designing a matching mechanism meeting certain axiomatic requirements for a given number of agents carries over to all scenarios with strictly fewer agents. This allows us to obtain general results about matching by verifying claims for specific instances using a SAT solver. We use our approach to automatically derive elementary proofs for two new impossibility theorems: (i) a strong form of Roth's classical result regarding the impossibility of designing mechanisms that are both stable and strategyproof and (ii) a result establishing the impossibility of guaranteeing stability while also respecting a basic notion of cross-group fairness (so-called gender-indifference).

Liquid democracy is a collective decision making paradigm which lies between direct and representative democracy. One main feature of liquid democracy is that voters can delegate their votes in a transitive manner so that: A delegates to B and B delegates to C leads to A delegates to C. Unfortunately, because voters' preferences over delegates may be conflicting, this process may not converge. There may not even exist a stable state (also called equilibrium). In this paper, we investigate the stability of the delegation process in liquid democracy when voters have restricted types of preference on the agent representing them (e.g., single-peaked preferences). We show that various natural structures of preference guarantee the existence of an equilibrium and we obtain both tractability and hardness results for the problem of computing several equilibria with some desirable properties.

Coarse correlation models strategic interactions of rational agents complemented by a correlation device which is a mediator that can recommend behavior but not enforce it. Despite being a classical concept in the theory of normal-form games since 1978, not much is known about the merits of coarse correlation in extensive-form settings. In this paper, we consider two instantiations of the idea of coarse correlation in extensive-form games: normal-form coarse-correlated equilibrium (NFCCE), already defined in the literature, and extensive-form coarse-correlated equilibrium (EFCCE), a new solution concept that we introduce. We show that EFCCEs are a subset of NFCCEs and a superset of the related extensive-form correlated equilibria. We also show that, in n-player extensive-form games, social-welfare-maximizing EFCCEs and NFCCEs are bilinear saddle points, and give new efficient algorithms for the special case of two-player games with no chance moves. Experimentally, our proposed algorithm for NFCCE is two to four orders of magnitude faster than the prior state of the art.

It is widely observed that individuals prefer to interact with others who are more similar to them (this phenomenon is termed homophily). This similarity manifests itself in various ways such as beliefs, values and education. Thus, it should not come as a surprise that when people make hiring choices, for example, their similarity to the candidate plays a role in their choice. In this paper, we suggest that putting the decision in the hands of a committee instead of a single person can reduce this bias. We study a novel model of voting in which a committee of experts is constructed to reduce the biases of its members. We first present voting rules that optimally reduce the biases of a given committee. Our main results include the design of committees, for several settings, that are able to reach a nearly optimal (unbiased) choice. We also provide a thorough analysis of the trade-offs between the committee size and the obtained error. Our model is inherently different from the well-studied models of voting that focus on aggregation of preferences or on aggregation of information due to the introduction of similarity biases.