High-quality information set abstraction remains a core challenge in solving large-scale imperfect-information extensive-form games (IIEFGs)--such as no-limit Texas Hold’em--where the finite nature of spatial resources hinders solving strategies for the full game. State-of-the-art AI methods rely on pre-trained discrete clustering for abstraction, yet their hard classification irreversibly discards critical information: specifically, the quantifiable subtle differences between information sets--vital for strategy solving--thus compromising the quality of such solving. Inspired by the word embedding paradigm in natural language processing, this paper proposes the Embedding CFR algorithm, a novel approach for solving strategies in IIEFGs within an embedding space. The algorithm pre-trains and embeds the features of individual information sets into an interconnected low-dimensional continuous space, where the resulting vectors more precisely capture both the distinctions and connections between information sets. Embedding CFR introduces a strategy-solving process driven by regret accumulation and strategy updates in this embedding space, with supporting theoretical analysis verifying its ability to reduce cumulative regret. Experiments on poker show that with the same spatial overhead, Embedding CFR achieves significantly faster exploitability convergence compared to cluster-based abstraction algorithms, confirming its effectiveness. Furthermore, to our knowledge, it is the first algorithm in poker AI that pre-trains information set abstractions via low-dimensional embedding for strategy solving.
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This paper investigates the problem of shared resource allocation, where a set of agents must be assigned to heterogeneous resources, with each agent allocated exactly one resource and each resource potentially shared by multiple agents. An agent’s utility for a given resource is jointly determined by the resource's type and the number of agents sharing it. We focus on two fundamental classes of monotone valuations: monotone nondecreasing and monotone nonincreasing, where an agent’s utility respectively increases or decreases with the number of agents sharing the resource. Within this shared resource framework, we examine classical notions of fairness and stability, including maximin-share fairness, envy-freeness, Nash stability, and two epistemic relaxations—epistemic envy-freeness and epistemic Nash stability—as well as swap stability. We propose formal definitions adapted to this setting and systematically analyze the relationships among these concepts. The primary contributions of this work consist of establishing existence and computational complexity results for each notion under both monotonicity assumptions and developing polynomial-time algorithms in cases where fair or stable allocations are guaranteed to exist.
A perfect clone in an ordinal election (i.e., an election where the voters rank the candidates in a strict linear order) is a set of candidates that each voter ranks consecutively. We consider different relaxations of this notion: *independent* or *subelection clones* are sets of candidates that only some of the voters recognize as a perfect clone, whereas *approximate clones* are sets of candidates such that every voter ranks their members close to each other, but not necessarily consecutively. We establish the complexity of identifying such imperfect clones, and of partitioning the candidates into families of imperfect clones. We also study the parameterized complexity of these problems with respect to a set of natural parameters such as the number of voters, the size or the number of imperfect clones we are searching for, or their level of imperfection.
In the k-Kemeny problem, we are given an ordinal election, i.e., a collection of votes ranking the candidates from best to worst, and we seek the smallest number of swaps of adjacent candidates that ensure that the election has at most k different rankings. We study this problem for a number of structured domains, including the single-peaked, single-crossing, group-separable, and Euclidean ones. We obtain two kinds of results: (1) We show that k-Kemeny remains intractable under most of these domains, even for k=2, and (2) we use k-Kemeny to rank these domains in terms of their diversity.
We study strategic candidate nomination by parties in elections decided by Plurality voting. Each party selects a nominee before the election, and the winner is chosen from the nominated candidates based on the voters' preferences. We introduce a new restriction on these preferences, which we call party-aligned single-peakedness: all voters agree on a common ordering of the parties along an ideological axis, but may differ in their perceptions of the positions of individual candidates within each party. The preferences of each voter are single-peaked with respect to their own axis over the candidates, which is consistent with the global ordering of the parties. We present a polynomial-time algorithm for recognizing whether a preference profile satisfies party-aligned single-peakedness. In this domain, we give polynomial-time algorithms for deciding whether a given party can become the winner under some (or all) nominations, and whether this can occur in some pure Nash equilibrium. We also prove a tight result about the guaranteed existence of pure strategy Nash equilibria for elections with up to three parties for single-peaked and party-aligned single-peaked preference profiles.
This paper investigates the impact of perturbations on the best-response-based algorithms approximating Nash equilibria in zero-sum games, namely Double Oracle and Fictitious Play. More precisely, we assume that the oracle computing the best responses perturbs the utilities before selecting the best response. We show that using such an oracle reduces the number of iterations for both algorithms. For some cases, suitable perturbations ensure the expected number of iterations is logarithmic. Although the utility perturbation is computationally demanding as it requires iterating through all pure strategies, we demonstrate that one can efficiently perturb the utilities in games where pure strategies have further inner structure.
Communication networks are essential for our economy and our everyday lives. This makes them lucrative targets for attacks. Today, we see an ongoing battle between criminals that try to disrupt our key communication networks and security professionals that try to mitigate these attacks. However, today's networks, like the Internet or peer-to-peer networks among smart devices, are not controlled by a single authority, but instead consist of many independently administrated entities that are interconnected. Thus, both the decisions of how to interconnect and how to secure against potential attacks are taken in a decentralized way by selfish agents. This strategic setting, with agents that want to interconnect and potential attackers that want to disrupt the network, was captured via an influential game-theoretic model by Goyal, Jabbari, Kearns, Khanna, and Morgenstern (WINE 2016). We revisit this model and show improved tight bounds on the achieved robustness of networks created by selfish agents. As our main result, we show that such networks can resist attacks of a large class of potential attackers, i.e., these networks maintain asymptotically optimal welfare post attack. This improves several bounds and resolves an open problem. Along the way, we show the counter-intuitive result, that attackers that aim at minimizing the social welfare post attack do not actually inflict the greatest possible damage.
We introduce River, a novel Condorcet-consistent voting method that is based on pairwise majority margins and can be seen as a simplified variation of Tideman's Ranked Pairs method. River is simple to explain, simple to compute even "by hand," and gives rise to an easy-to-interpret certificate in the form of a directed tree. Like Ranked Pairs and Schulze's Beat Path method, River is a refinement of the Split Cycle method and shares with those many desirable properties, including independence of clones. Unlike the other three methods, River satisfies a strong form of resistance to agenda-manipulation that is known as independence of Pareto-dominated alternatives.
Recently, the River Method was introduced as novel refinement of the Split Cycle voting rule. The decision-making process of River is closely related to the well established Ranked Pairs Method. Both methods consider a margin graph computed from the voters' preferences and eliminate majority cycles in that graph to choose a winner. As ties can occur in the margin graph, a tiebreaker is required along with the preferences. While such a tiebreaker makes the computation efficient, it compromises the fundamental property of neutrality: the voting rule should not favor alternatives in advance. One way to reintroduce neutrality is to use Parallel-Universe Tiebreaking (PUT), where each alternative is a winner if it wins according to any possible tiebreaker. Unfortunately, computing the winners selected by Ranked Pairs with PUT is NP-complete. Given the similarity of River to Ranked Pairs, one might expect River to suffer from the same complexity. Surprisingly, we show the opposite: We present a polynomial-time algorithm for computing River winners with PUT, highlighting significant structural advantages of River over Ranked Pairs. Our Fused-Universe (FUN) algorithm simulates River for every possible tiebreaking in one pass. From the resulting FUN diagram one can then directly read off both the set of winners and, for each winner, a certificate that explains how this alternative dominates the others.
An important desideratum in approval-based multiwinner voting is proportionality. We study the problem of reconfiguring proportional committees: given two proportional committees, is there a transition path that consists only of proportional committees, where each transition involves replacing one candidate with another candidate? We show that the set of committees satisfying the proportionality axiom of justified representation (JR) is not always connected, and it is PSPACE-complete to decide whether two such committees are connected. On the other hand, we prove that any two JR committees can be connected by committees satisfying a 2-approximation of JR. We also obtain similar results for the stronger axiom of extended justified representation (EJR). In addition, we demonstrate that the committees produced by several well-known voting rules are connected or at least not isolated, and investigate the reconfiguration problem in restricted preference domains.
In a public goods game, every player chooses whether or not to buy a good that all neighboring players will have access to. We consider a setting in which the good is indivisible, neighboring players are out-neighbors in a directed graph, and there is a capacity constraint on their number, k, that can benefit from the good. This means that each player makes a two-pronged decision: decide whether or not to buy and, conditional on buying, choose which k out-neighbors to share access. We examine both pure and mixed Nash equilibria in the model from the perspective of existence, computation, and efficiency. We perform a comprehensive study for these three dimensions with respect to both sharing capacity (k) and the network structure (the underlying directed graph), and establish sharp complexity dichotomies for each.
Dividing Indivisible Items for the Benefit of All: It Is Hard to Be Fair Without Social Awareness
PDF ↗In standard fair division models, we assume that all agents are selfish. However, in many scenarios, division of resources has a direct impact on the whole group or even society. Therefore, we study fair allocations of indivisible items that, at the same time, maximize social impact. In this model, each agent is associated with two additive functions that define their value and social impact for each item. The goal is to allocate items so that the social impact is maximized while maintaining some fairness criterion. We reveal that the complexity of the problem heavily depends on whether the agents are socially aware, i.e., they take into consideration the social impact functions. For socially unaware agents, we prove that the problem is NP-hard for a variety of fairness notions, and that it is tractable only for very restricted cases, e.g., if for every agent valuation equals social impact and it is binary. On the other hand, social awareness allows for fair allocations that maximize social impact, and such allocations can be computed in polynomial time. Interestingly, the problem becomes again intractable as soon as the definition of social awareness is relaxed.
Breaking Barriers, Finding Boundaries: Not Obviously Manipulable Budget-Feasible Mechanism Design
PDF ↗Strategyproofness has been the holy grail in mechanism design for decades, providing strong incentive compatibility guarantees under the assumption of perfectly rational agents. However, this assumption is questionable when agents exhibit bounded rationality. Moreover, strategyproofness often imposes strong impossibility results that prevent mechanisms from surpassing certain approximation barriers. We study this tension in budget-feasible mechanism design, where a designer wants to procure services of maximum value from agents subject to a budget constraint. Here, strategyproofness imposes approximation barriers of 2.41 and 2 for deterministic and randomized mechanisms, respectively. We investigate how much we can potentially gain under bounded rationality. We adopt the weaker notion of not obviously manipulable (NOM), which only prevents "obvious" strategic deviations. We fully resolve the achievable approximation guarantees under NOM: We derive a deterministic 2-approximate NOM mechanism under the general class of monotone subadditive valuations. We also show that this bound is tight (even for additive valuations). Additionally, we provide a simple randomized NOM mechanism that is approximately optimal. These results demonstrate a clear separation between strategyproof and NOM mechanisms. Our mechanisms use Golden Tickets and Wooden Spoons as natural design primitives, arising from our characterization of NOM mechanisms.
We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuations. We focus on envy-freeness up to any good (EFX) — an important fairness notion in fair division of indivisible goods. A central open question in this field is whether EFX allocations always exist for any number of agents. While recent results have established EFX existence for settings with at most three distinct valuations and for two types of goods, the general case remains unresolved. In this paper, we extend the existent knowledge by proving that EFX allocations satisfying Pareto optimality (PO) always exist and can be computed in quasiliniear time when there are two types of goods, given that the valuations are positive. Our findings demonstrate a fairly simple and efficient algorithm constructing an EFX+PO allocation.
Fraud can pose a challenge in many resource allocation domains, including social service delivery and credit provision. For example, agents may misreport private information in order to gain benefits or access to credit. To mitigate this, a principal can design strategic audits to verify claims and penalize misreporting. In this paper, we introduce a general model of audit policy design as a principal-agent game with multiple agents, where the principal commits to an audit policy, and agents collectively choose an equilibrium that minimizes the principal’s utility. We examine both adaptive and non-adaptive settings, depending on whether the principal's policy can be responsive to the distribution of agent reports. Our work provides efficient algorithms for computing optimal audit policies in both settings and extends these results to a setting with limited audit budgets.
ElementaryNet: A Non-Strategic Neural Network for Predicting Human Behavior in Normal-Form Games
PDF ↗Behavioral game theory models serve two purposes: yielding insights into how human decision-making works, and predicting how people would behave in novel strategic settings. A system called GameNet represents the state of the art for predicting human behavior in the setting of unrepeated simultaneous-move games, combining a simple "level-k" model of strategic reasoning with a complex neural network model of non-strategic "level-0" behavior. Although this reliance on well-established ideas from cognitive science ought to make GameNet interpretable, the flexibility of its level-0 model raises the possibility that it is able to emulate strategic reasoning. In this work, we prove that GameNet's level-0 model is indeed too general. We then introduce ElementaryNet, a novel neural network that is provably incapable of expressing strategic behavior. We show that these additional restrictions are empirically harmless, with ElementaryNet and GameNet having statistically indistinguishable performance. We then show how it is possible to derive insights about human behavior by varying ElementaryNet's features and interpreting its parameters, finding evidence of iterative reasoning, learning about the depth of this reasoning process, and showing the value of a rich level-0 specification.
Tournaments are widely used models to represent pairwise dominance between candidates, alternatives, or teams. We study the problem of providing certified explanations for why a candidate appears among the winners under various tournament rules. To this end, we identify minimal supports—minimal sub-tournaments in which the candidate is guaranteed to win regardless of how the rest of the tournament is completed (that is, the candidate is a necessary winner of the sub-tournament). This notion corresponds to an abductive explanation for the question,"Why does the winner win the tournament?"—a central concept in formal explainable AI. We focus on common tournament solutions: the top cycle, the uncovered set, the Copeland rule, the Borda rule, the maximin rule, and the weighted uncovered set. For each rule we determine the size of the smallest minimal supports, and we present polynomial-time algorithms to compute them for all solutions except for the weighted uncovered set, for which the problem is NP-complete. Finally, we show how minimal supports can serve to produce compact, certified, and intuitive explanations for tournament solutions.
We study the problem of (approximate) maximin share (MMS) allocation of indivisible items among a set of agents. We focus on the graphical valuation model, in which the input is given by a graph where edges correspond to items, and vertices correspond to agents. An edge may have non-zero marginal value only for its incident vertices. We study additive, XOS and subadditive valuations and we present positive and negative results for (approximate) MMS fairness, and also for (approximate) pair-wise maximin share (PMMS) fairness.
Equilibria of realistic multiplayer games constitute a key solution concept both in practical applications, such as online advertising auctions and electricity markets, and in analytical frameworks used to study strategic voting in elections or assess policy impacts in integrated assessment models. However, efficiently computing these equilibria requires games to have a carefully designed structure and satisfy numerous restrictions; otherwise, the computational complexity becomes prohibitive. In particular, finding even approximate Nash equilibria in general normal-form games with three or more players is known to be PPAD-complete. Current state-of-the-art algorithms for computing Nash equilibria in multiplayer normal-form games either suffer from poor scalability due to their reliance on non-convex optimization solvers, or lack guarantees of convergence to a true equilibrium. In this paper, we propose a novel reformulation of the Nash equilibrium computation problem and develop a complete and sound spatial branch-and-bound algorithm based on this reformulation. We provide a qualitative analysis arguing why one should expect our approach to perform better than conventional formulation, and show the relationship between approximate solution to our reformulation and that of computing an approximate Nash equilibrium. Empirical evaluations demonstrate that our algorithm substantially outperforms existing complete methods.
Committee-selection problems arise in many contexts and applications, and there has been increasing interest within the social choice research community on identifying which properties are satisfied by different multi-winner voting rules. In this work, we propose a data-driven framework to evaluate how frequently voting rules violate axioms across diverse preference distributions in practice, shifting away from the binary perspective of axiom satisfaction given by worst-case analysis. Using this framework, we analyze the relationship between multi-winner voting rules and their axiomatic performance under several preference distributions, and propose a methodology for systematically minimizing axioms violations. Our results suggest that data-driven approaches to social choice can inform the design of new voting systems and support the continuation of data-driven research in social choice.