Core stability is a natural and well-studied notion for group fairness in multi-winner voting, where the task is to select a committee from a pool of candidates. We study the setting where voters either approve or disapprove of each candidate; here, it remains a major open problem whether a core-stable committee always exists. In this work, we develop an approach based on mixed-integer linear programming for deciding whether and when core-stable committees are guaranteed to exist. In contrast to SAT-based approaches popular in computational social choice, our method can produce proofs for a specific number of candidates independent of the number of voters. In addition to these computational gains, our program lends itself to a novel duality-based reformulation of the core stability problem, from which we obtain new existence results in special cases. Further, we use our framework to reveal previously unknown relationships between core stability and other desirable properties, such as notions of priceability.

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.

*Variational inequalities (VIs)* encompass many fundamental problems in diverse areas ranging from engineering to economics and machine learning. However, their considerable expressivity comes at the cost of computational intractability. In this paper, we introduce and analyze a natural relaxation—which we refer to as *expected variational inequalities (EVIs)*—where the goal is to find a distribution that satisfies the VI constraint in expectation. By adapting recent techniques from game theory, we show that, unlike VIs, EVIs can be solved in polynomial time under general (nonmonotone) operators. EVIs capture the seminal notion of correlated equilibria, but enjoy a greater reach beyond games. We also employ our framework to capture and generalize several existing disparate results, including from settings such as smooth games, and games with coupled constraints or nonconcave utilities.

Infinitely repeated games support equilibrium concepts beyond those present in one-shot games (e.g., cooperation in the prisoner's dilemma). Nonetheless, repeated games fail to capture our real-world intuition for settings with many anonymous agents interacting in pairs. Repeated games with restarts, introduced by Berker and Conitzer, address this concern by giving players the option to restart the game with someone new whenever their partner deviates from an agreed-upon sequence of actions. In their work, they studied symmetric games with symmetric strategies. We significantly extend these results, introducing and analyzing more general notions of equilibria in asymmetric games with restarts. We characterize which goal strategies players can be incentivized to play in equilibrium, and we consider the computational problem of finding such sequences of actions with minimal cost for the agents. We show that this problem is NP-hard in general. However, when the goal sequence maximizes social welfare, we give a pseudo-polynomial time algorithm.

We study two axioms for social choice functions that capture the impact of similar candidates: independence of clones (IoC) and composition consistency (CC). We clarify the relationship between these axioms by observing that CC is strictly more demanding than IoC, and investigate whether common voting rules that are known to be independent of clones (such as Single Transferable Vote, Ranked Pairs, Schulze Method, and Split Cycle) are composition-consistent. While for most of these rules the answer is negative, we identify a variant of Ranked Pairs that satisfies CC. Further, we show how to efficiently modify any (neutral) social choice function so that it satisfies CC, while maintaining its other desirable properties. Our transformation relies on the hierarchical representation of clone structures via PQ-trees. We extend our analysis to social preference functions. Finally, we interpret IoC and CC as measures of robustness against strategic manipulation by candidates, with IoC corresponding to strategy-proofness and CC corresponding to obvious strategy-proofness.

Φ-equilibria—and the associated notion of Φ-regret—are a powerful and flexible framework at the heart of online learning and game theory, whereby enriching the set of deviations Φ begets stronger notions of rationality. Recently, Daskalakis, Farina, Fishelson, Pipis, and Schneider (STOC '24) settled the existence of efficient algorithms when Φ contains only linear maps under a general, d-dimensional convex constraint set X. In this paper, we significantly extend their work by resolving the case where Φ is k-dimensional; degree-ℓ polynomials constitute a canonical such example with k = dO(ℓ). In particular, we obtain two main positive results: i) a poly(n, d, k, log(1/ϵ))-time algorithm for computing ϵ-approximate Φ-equilibria in n-player multilinear games, and ii) an efficient online algorithm that incurs average Φ-regret at most ϵ using poly(d, k)/ϵ2 rounds.