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While Nash equilibria are guaranteed to exist, they may exhibit dense support, making them difficult to understand and execute in some applications. In this paper, we study k-sparse commitments in games where one player is restricted to mixed strategies with support size at most k. Finding k-sparse commitments is known to be computationally hard. We start by showing several structural properties of k-sparse solutions, including that the optimal support may vary dramatically as k increases. These results suggest that naive greedy or double-oracle-based approaches are unlikely to yield practical algorithms. We then develop a simple approach based on mixed integer linear programs (MILPs) for zero-sum games, general-sum Stackelberg games, and various forms of structured sparsity. We also propose practical algorithms for cases where one or both players have large (i.e., practically innumerable) action sets, utilizing a combination of MILPs and incremental strategy generation. We evaluate our methods on synthetic and real-world scenarios based on security applications. In both settings, we observe that even for small support sizes, we can obtain more than 90% of the true Nash value while maintaining a reasonable runtime, demonstrating the significance of our formulation and algorithms.
Present bias, the tendency to overvalue immediate rewards while undervaluing future ones, is a well-known barrier to achieving long-term goals. As artificial intelligence and behavioral economics increasingly focus on this phenomenon, the need for robust mathematical models to predict behavior and guide effective interventions has become crucial. However, existing models are constrained by their reliance on the discreteness of time and limited discount functions. This study introduces a novel continuous-time mathematical model for agents influenced by present bias. Using the variational principle, we model human behavior, where individuals repeatedly act according to a sequence of states that minimize their perceived cost. Our model not only retains analytical tractability but also accommodates various discount functions. Using this model, we consider intervention optimization problems under exponential and hyperbolic discounting and theoretically derive optimal intervention strategies, offering new insights into managing present-biased behavior.
We study the fundamental problem of fairly dividing a set of indivisible items among agents with (general) monotone valuations. The notion of envy-freeness up to any item (EFX) is considered to be one of the most fascinating fairness concepts in this line of work. Unfortunately, despite significant efforts, existence of EFX allocations is a major open problem in fair division, thereby making the study of approximations and relaxations of EFX a natural line of research. Recently, Caragiannis et al. [2023] introduced a promising relaxation of EFX, called epistemic EFX (EEFX). An allocation is EEFX, if for every agent, it is possible to shuffle the items in the remaining bundles so that she becomes ``EFX-satisfied''. Caragiannis et al. [2023] prove existence and polynomial-time computability of EEFX allocations for additive valuations. A natural question asks what happens when we consider valuations more general than additive? We address this important open question and answer it affirmatively by establishing the existence of EEFX allocations for an arbitrary number of agents with general monotone valuations. To the best of our knowledge, besides EF1, EEFX is the only known relaxation of EFX to have such strong existential guarantees. Furthermore, we complement our existential result by proving computational and information-theoretic lower bounds. We prove that even for an arbitrary number of (more than one) agents with identical submodular valuations, it is PLS-hard to compute EEFX allocations and it requires exponentially-many value queries to do so.
We study the problem of computing fair divisions of a set of indivisible goods among agents with additive valuations. For the past many decades, the literature has explored various notions of fairness, that can be primarily seen as either having envy-based or share-based lens. For the discrete setting of resource-allocation problems, envy-free up to any good (EFX) and maximin share (MMS) are widely considered as the flag-bearers of fairness notions in the above two categories, thereby capturing different aspects of fairness herein. Due to lack of existence results of these notions and the fact that a good approximation of EFX or MMS does not imply particularly strong guarantees of the other, it becomes important to understand the compatibility of EFX and MMS allocations with one another. In this work, we identify a novel way to simultaneously achieve MMS guarantees with EFX/EF1 notions of fairness, while beating the best known approximation factors by Chaudhury et al. and Amanatidis et al. Our main contribution is to constructively prove the existence of (i) a partial allocation that is both 2/3-MMS and EFX, and (ii) a complete allocation that is both 2/3-MMS and EF1. Our algorithms run in pseudo-polynomial time if the approximation factor for MMS is relaxed to 2/3 - e for any constant e>0 and in polynomial time if, in addition, the EFX (or EF1) guarantee is relaxed to (1-d)-EFX (or (1-d)-EF1) for any constant d>0. In particular, we improve from the best approximation factor known prior to our work by Chaudhury et al., which computes partial allocations that are 1/2-MMS and EFX in pseudo-polynomial time.
We examine an approval-based model of Liquid Democracy with a budget constraint on voting and delegating costs, aiming to centrally select casting voters ensuring complete representation of the electorate. From a computational complexity perspective, we focus on minimizing overall costs, maintaining short delegation paths, and preventing excessive concentration of voting power. Furthermore, we explore computational aspects of strategic control, specifically, whether external agents can change election components to influence the voting power of certain voters.
In contemporary e-commerce platforms, search result pages display two types of items: ad items and organic items. Ad items are determined through an advertising auction system, while organic items are selected by a recommendation system. These systems have distinct optimization objectives, creating the challenge of effectively merging these two components. Recent research has explored merging mechanisms for e-commerce platforms, but none have simultaneously achieved all desirable properties: incentive compatibility, individual rationality, adaptability to multiple slots, integration of inseparable candidates, and avoidance of repeated exposure for ads and organic items. This paper addresses the design of a merging mechanism that satisfies all these properties. We first provide the necessary conditions for the optimal merging mechanisms. Next, we introduce two simple and effective mechanisms, termed the generalized fix mechanism and the generalized change mechanism. Finally, we theoretically prove that both mechanisms offer guaranteed approximation ratios compared to the optimal mechanism in both simplest and general settings.
We study the fair allocation of indivisible goods among a group of agents, aiming to limit the envy between any two agents. The central open problem in this literature, which has proven to be extremely challenging, is regarding the existence of an EFX allocation, i.e., an allocation such that any envy from some agent i toward another agent j would vanish if we were to remove any single good from the bundle allocated to j. Prior work has shown that when the agents’ valuations are additive, which has been the main focus of prior works, an EFX allocation is guaranteed to exist for all instances involving up to three agents. Subsequent work extended this guarantee to more general valuations, like nice-cancelable and MMS-feasible. However, the existence of EFX allocations for instances involving four agents remains open, even for additive valuations. We contribute to this literature by focusing on EF2X, a relaxation of EFX which requires that any envy toward some agent would vanish if any two of the goods allocated to that agent were to be removed. Our main result shows that EF2X allocations exist for any instance with four agents, even for the class of cancelable valuations, which is more general than additive. Our proof is constructive, proposing an algorithm that computes such an allocation in pseudo-polynomial time. Furthermore, for instances involving three agents we provide an algorithm that computes an EF2X allocation in polynomial time, in contrast to EFX for which the fastest known algorithm for three agents is only pseudo-polynomial.
In this paper, we consider the classic fair division problem of allocating m divisible items to n agents with linear valuations over the items. We define novel notions of fair shares from the perspective of individual agents via the cake-cutting process. These shares generalize the notion of proportionality by taking into account the valuations of other agents via constraints capturing envy. We study what fraction (approximation) of these shares are achievable in the worst case, and present tight and non-trivial approximation bounds as a function of n and m. In particular, we show a tight approximation bound of Θ(√n) for various notions of such shares. We show this bound via a novel application of dual fitting, which may be of independent interest. We also present a bound of O(m^(2/3)) for a strict notion of share, with an almost matching lower bound. We further develop weaker notions of shares whose approximation bounds interpolate smoothly between proportionality and the shares described above. We finally present empirical results showing that our definitions lead to more reasonable shares than the standard fair share notion of proportionality.
The design of multi-item, multi-bidder auctions involves a delicate balancing act of economic objectives, bidder incentives, and real-world complexities. Efficient auctions, that is, auctions that allocate items to maximize total bidder value, are practically desirable since they promote the most economically beneficial use of resources. Arguably the biggest drawback of efficient auctions, however, is their potential to generate very low revenue. In this work, we show how the auction designer can artificially inject competition into the auction to boost revenue while striving to maintain efficiency. First, we invent a new auction family that enables the auction designer to specify competition in a precise, expressive, and interpretable way. We then introduce a new model of bidder behavior and individual rationality to understand how bidders act when prices are too competitive. Next, under our bidder behavior model, we use our new competitive auction class to derive the globally revenue-optimal efficient auction under two different knowledge models for the auction designer: knowledge of full bidder value distributions and knowledge of bidder value quantiles. Finally, we study a third knowledge model for the auction designer: knowledge of historical bidder valuation data. In this setting we present sample and computationally efficient learning algorithms that find high-revenue probably-efficient competitive auctions from bidder data. Our learning algorithms are instance adaptive and can be run in parallel across bidders, unlike most prior approaches to data-driven auction design.
Proportional representation plays a crucial role in electoral systems. In ordinal elections, where voters rank candidates based on their preferences, the Single Transferable Vote (STV) is the most widely used proportional voting method. STV is considered proportional because it satisfies an axiom requiring that large enough "solid coalitions" of voters are adequately represented. Using real-world data from local Scottish elections, we observe that solid coalitions of the required size rarely occur in practice. This observation challenges the importance of proportionality axioms and raises the question of how the proportionality of voting methods can be assessed beyond their axiomatic performance. We address these concerns by developing quantitative measures of proportionality. We apply these measures to evaluate the proportionality of voting rules on real-world election data. Besides STV, we consider SNTV, the Expanding Approvals Rule, and Sequential Ranked-Choice Voting. We also study the effects of ballot truncation by artificially completing truncated ballots and comparing the proportionality of outcomes under complete and truncated ballots.
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.
In this article we address the multi-robot task allocation problem, where robots must cooperatively assign themselves to accomplish a set of tasks in their environment. We consider the colony maintenance problem as an example, where a team of robots are tasked with continuously maintaining the energy supply of a central colony. We model this as a global game, where each robot measures the energy level of the colony, and the current number of assigned robots, to determine whether or not to forage for energy sources. The key to our approach is introducing a negative feedback term into the robots' utility, which also eliminates the trivial solution where foraging or not foraging are strictly dominant strategies. We compare our approach qualitatively to existing global games, where a positive positive feedback term admits threshold-based decision making, and encourages many robots to forage simultaneously. We show how positive feedback can lead to a cascading failure in the presence of a human who recruits robots, and we demonstrate the resilience of our approach in simulation.
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.
The predominant setting in classic auction theory considers bidders as utility maximizers (UMs), who aim to maximize quasi-linear utility functions. Recent autobidding strategies in online advertising have sparked interest in auction design with value maximizers (VMs), who aim to maximize the total value obtained. In this work, we investigate revenue-maximizing auction design for selling a single item to a mix of UMs and VMs. Crucially, we assume the UM/VM type is private information of a bidder. This shift to a multi-parameter domain complicates the design of incentive compatible mechanisms. Under this setting, we first characterize the optimal auction structure for auctions with a single bidder. We observe that the optimal auction moves gradually from a first-price auction to a Myerson auction as the probability of the bidder being a UM increases from 0 to 1. We also extend our study to multi-bidder setting and present an algorithm for deriving the optimal lookahead auction with multiple mixed types of bidders.
Today we rely on networks that are created and maintained by smart devices. For such networks, there is no governing central authority but instead the network structure is shaped by the decisions of selfish intelligent agents. A key property of such communication networks is that they should be easy to navigate for routing data. For this, a common approach is greedy routing, where every device simply routes data to a neighbor that is closer to the respective destination. Networks of intelligent agents can be analyzed via a game-theoretic approach and in the last decades many variants of network creation games have been proposed and analyzed. In this paper we present the first game-theoretic network creation model that incorporates greedy routing, i.e., the strategic agents in our model are embedded in some metric space and strive for creating a network among themselves where all-pairs greedy routing is enabled. Besides this, the agents optimize their connection quality within the created network by aiming for greedy routing paths with low stretch. For our model, we analyze the existence of (approximate)-equilibria and the computational hardness in different underlying metric spaces. E.g., we characterize the set of equilibria in 1-2-metrics and tree metrics and show that Nash equilibria always exist. For Euclidean space, the setting which is most relevant in practice, we prove that equilibria are not guaranteed to exist but that the well-known Θ-graph construction yields networks having a low stretch that are game-theoretically almost stable. For general metric spaces, we show that approximate equilibria exist where the approximation factor depends on the cost of maintaining any link.
Imperfect-recall games—in which players may forget previously acquired information—have found many practical applications, ranging from game abstractions to team games and testing AI agents. In this paper, we quantify the utility gain by endowing a player with perfect recall, which we call the value of recall (VoR). While VoR can be unbounded in general, we parameterize it in terms of various game properties, namely the structure of chance nodes and the degree of absentmindedness (the number of successive times a player enters the same information set). Further, we identify several pathologies that arise with VoR, and show how to circumvent them. We also study the complexity of computing VoR, and how to optimally apportion partial recall. Finally, we connect VoR to other previously studied concepts in game theory, including the price of anarchy. We use that connection in conjunction with the celebrated smoothness framework to characterize VoR in a broad class of games.
In Hotelling's model of spatial competition, a unit mass of voters is distributed in the interval [0,1] (with their location corresponding to their political persuasion), and each of m candidates selects as a strategy their distinct position in this interval. Each voter votes for the nearest candidate, and candidates choose their strategy to maximize their votes. It is known that if there are more than two candidates, equilibria may not exist in this model. It was unknown, however, how close to an equilibrium one could get. Our work studies approximate equilibria in this model, where a strategy profile is an (additive) ϵ-equilibria if no candidate can increase their votes by ϵ, and provides tight or nearly-tight bounds on the approximation ϵ achievable. We show that for 3 candidates, for any distribution of the voters, ϵ ≥ 1/12. Thus, somewhat surprisingly, for any distribution of the voters and any strategy profile of the candidates, at least 1/12th of the total votes is always left ``on the table.'' Extending this, we show that in the worst case, there exist voter distributions for which ϵ ≥ 1/6, and this is tight: one can always compute a 1/6-approximate equilibria in polynomial time. We then study the general case of m candidates, and show that as m grows large, we get closer to an exact equilibrium: one can always obtain a 1/(m+1)-approximate equilibria in polynomial time. We show this bound is asymptotically tight, by giving voter distributions for which ϵ ≥ 1/(m+3).
Equilibrium problems in Bayesian auction games can be described as systems of differential equations. Depending on the model assumptions, these equations might be such that we do not have a rigorous mathematical solution theory. The lack of analytical or numerical techniques with guaranteed convergence for the equilibrium problem has plagued the field and limited equilibrium analysis to rather simple auction models such as single-object auctions. Recent advances in equilibrium learning led to algorithms that find equilibrium under a wide variety of model assumptions. Monotonicity and the Minty condition are the known sufficient conditions for learning algorithms to converge to an equilibrium in games. Not much is known about convergence of learning algorithms beyond these conditions. We analyze first- and second-price auctions where simple learning algorithms consistently converge to an equilibrium. The analysis is challenging, because these properties need to be shown in infinite dimensions. Interestingly, we show that neither monotonicity nor pseudo- or quasi-monotonicity holds for the respective variational inequalities (VIs). The second-price auction's equilibrium is a Minty-type solution, but the first-price auction is not. However, the analysis via infinite-dimensional VIs allows us to get ex-post guarantees for gradient-based algorithms. We show that the Bayes--Nash equilibrium is the unique solution to the VI within the class of uniformly increasing bid functions, which ensures that gradient-based algorithms attain the equilibrium in case of convergence, as also observed in numerical experiments.
In two-player cooperative games, agents can play together effectively when they have accurate assumptions about how their teammate will behave, but may perform poorly when these assumptions are inaccurate. In language games, failure may be due to disagreement in the understanding of either the semantics or pragmatics of an utterance. We model coarse uncertainty in semantics using a prior distribution of language models and uncertainty in pragmatics using the cognitive hierarchy, combining the two aspects into a single prior distribution over possible partner types. Fine-grained uncertainty in semantics is modeled using noise that is added to the embeddings of words in the language. To handle all forms of uncertainty we construct agents that learn the behavior of their partner using Bayesian inference and use this information to maximize the expected value of a heuristic function. We test this approach by constructing Bayesian agents for the game of Codenames, and show that they perform better in experiments where semantics is uncertain.
An important - but very demanding - property in collective decision-making is strategyproofness, which requires that voters cannot benefit from submitting insincere preferences. Gibbard (1977) has shown that only rather unattractive rules are strategyproof, even when allowing for randomization. However, Gibbard's theorem is based on a rather strong interpretation of strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for at least one utility function consistent with his ordinal preferences. In this paper, we study weak strategyproofness, which deems a manipulation successful if it increases the voter's expected utility for all utility functions consistent with his ordinal preferences. We show how to systematically design attractive, weakly strategyproof social decision schemes (SDSs) and explore their limitations for both strict and weak preferences. In particular, for strict preferences, we show that there are weakly strategyproof SDSs that are either ex post efficient or Condorcet-consistent, while neither even-chance SDSs nor pairwise SDSs satisfy both properties and weak strategyproofness at the same time. By contrast, for the case of weak preferences, we discuss two sweeping impossibility results that preclude the existence of appealing weakly strategyproof SDSs.
Serial dictatorship is a simple mechanism for coordinating agents in solving combinatorial optimization problems according to their preferences. The most representative such problem is one-sided matching, in which a set of n agents have values for a set of n items, and the objective is to compute a matching of the agents to the items of maximum total value (a.k.a., social welfare). Following the recent framework of Caragiannis and Rathi (2023), we consider a model in which the agent-item values are not available upfront but become known by querying agent sequences. In particular, when the agents are asked to act in a sequence, they respond by picking their favorite item that has not been picked by agents who acted before and reveal their value for it. Can we compute an agent sequence that induces a social welfare-optimal matching? We answer this question affirmatively and present an algorithm that uses polynomial number (specifically, O(n^5) of queries). This solves the main open problem stated by Caragiannis and Rathi (2023). Our analysis uses a potential function argument that measures progress towards learning the underlying edge-weight information. Furthermore, the algorithm has a truthful implementation by adapting the paradigm of VCG payments.
We study the exploration-exploitation trade-off for large multiplayer coordination games where players strategise via Q-Learning, a common learning framework in multi-agent reinforcement learning. Q-Learning is known to have two shortcomings, namely non-convergence and potential equilibrium selection problems, when there are multiple fixed points, called Quantal Response Equilibria (QRE). Furthermore, whilst QRE have full support for finite games, it is not clear how Q-Learning behaves as the game becomes large. In this paper, we characterise the critical exploration rate that guarantees convergence to a unique fixed point, addressing the two shortcomings above. Using a generating-functional method, we show that this rate increases with the number of players and the alignment of their payoffs. For many-player coordination games with perfectly aligned payoffs, this exploration rate is roughly twice that of p-player zero-sum games. As for large games, we provide a structural result for QRE, which suggests that as the game size increases, Q-Learning converges to a QRE near the boundary of the simplex of the action space, a phenomenon we term asymptotic extinction, where a constant fraction of the actions are played with zero probability at a rate o(1/N) for an N -action game.
Man-made and natural disruptions such as planned constructions on roads, suspensions of bridges, and blocked roads by trees/mudslides/floods can often create obstacles that separate two connected regions. As a result, the traveling and reachability of agents from their respective regions to other regions can be affected. To minimize the impact of the obstacles and maintain agent accessibility, we initiate the problem of constructing a new pathway (e.g., a detour or new bridge) connecting the regions disconnected by obstacles from the mechanism design perspective. In the problem, each agent in their region has a private location and is required to access the other region. The cost of an agent is the distance from their location to the other region via the pathway. Our goal is to design strategyproof mechanisms that elicit truthful locations from the agents and approximately optimize the social or maximum cost of agents by determining locations in the regions for building a pathway. We provide a characterization of all strategyproof and anonymous mechanisms. For the social and maximum costs, we provide upper and lower bounds on the approximation ratios of strategyproof mechanisms.
Knockout tournaments, also known as single-elimination or cup tournaments, are a popular form of sports competitions. In the standard probabilistic setting, for each pairing of players, one of the players wins the game with a certain (a priory known) probability. Due to their competitive nature, tournaments are prone to manipulation. We investigate the computational problem of determining whether, for a given tournament, a coalition has a manipulation strategy that increases the winning probability of a designated player above a given threshold. More precisely, in every round of the tournament, coalition players can strategically decide which games to throw based on the advancement of other players to the current round. We call this setting adaptive constructive coalition manipulation. To the best of our knowledge, while coalition manipulation has been studied in the literature, this is the first work to introduce adaptiveness to this context. We show that the above problem is hard for every complexity class in the polynomial hierarchy. On the algorithmic side, we show that the problem is solvable in polynomial time when the coalition size is a constant. Furthermore, we show that the problem is fixed-parameter tractable when parameterized by the coalition size and the size of a minimum player set that must include at least one player from each non-deterministic game. Lastly, we investigate a generalized setting where the tournament tree can be imbalanced.
Coalition formation concerns autonomous agents that strategically interact to form self-organized coalitions. When agents lack initial sufficient information to evaluate their preferences before interacting with others, they learn them online through repeated feedback while iteratively forming coalitions. In this work, we introduce online learning in coalition formation from a non-cooperative perspective, studying the impact of collective data utilization where selfish agents aim to accelerate their learning by leveraging a shared data platform. Thus, the efficiency and dynamics of the learning process are affected by each agent's local feedbacks, motivating us to explore the tension between semi-bandit and bandit feedback, which differ in the granularity of utility information observed by each agent. Under our non-cooperative viewpoint, we evaluate the system by means of Nash stability, where no agent can improve her utility by unilaterally deviating. Our main result is a sample-efficient algorithm for selfish agents that aims to minimize their Nash regret under both semi-bandit and bandit feedback, implying approximately Nash stable outcomes. Under both feedback settings, our algorithm enjoys Nash regret and sample complexity bounds that are optimal up to logarithmic factors.