Organizations: LIDS, EECS, Massachusetts Institute of Technology
Abstract
Most familiar equilibrium concepts, such as Nash and correlated equilibrium, guarantee only that no single player can improve their utility by deviating unilaterally. They offer no guarantees against profitable coordinated deviations by coalitions. Although the literature proposes solution concepts that provide stability against multilateral deviations (\emph{e.g.}, strong Nash and coalition-proof equilibrium), these generally fail to exist. In this paper, we study an alternative solution concept that minimizes coalitional deviation incentives, rather than requiring them to vanish, and is therefore guaranteed to exist. Specifically, we focus on minimizing the average gain of a deviating coalition, and extend the framework to weighted-average and maximum-within-coalition gains. In contrast, the minimum-gain analogue is shown to be computationally intractable. For the average-gain and maximum-gain objectives, we prove a lower bound on the complexity of computing such an equilibrium and present an algorithm that matches this bound. Finally, we use our framework to solve the \emph{Exploitability Welfare Frontier} (EWF), the maximum attainable social welfare subject to a given exploitability (the maximum gain over all unilateral deviations).
Recent work in fair division has focused on either simultaneously satisfying closely related fairness notions or achieving a single notion across the ex-ante and ex-post worlds. We study the compatibility of two fundamentally different fairness notions: envy-freeness and equitability. For indivisible goods-only and chores-only settings, we study the existence and complexity of simultaneously satisfying their relaxations, revealing sharp contrasts between the two settings. We show that EF1+EQ1 may fail to exist even for normalized binary goods: we construct an instance with 113 agents and 341 goods in which every agent approves exactly 165 goods, but no complete allocation satisfies both notions. Our main algorithmic result computes an EF1+EQ1 allocation for every normalized binary goods instance with at most seven agents. Thus, the smallest number of agents admitting a counterexample lies between 8 and 113, leaving the cases from 8 through 112 unresolved. In sharp contrast, binary chores admit the stronger EFX+EQX guarantee for any number of agents, even without normalization. We further initiate the study of cross-notion ex-ante and ex-post guarantees, asking whether randomized allocations can provide ex-ante guarantees for one notion while preserving ex-post guarantees for another.
Many important games have more than two players and imperfect information. Existing approaches for computing Nash equilibrium, the central game-theoretic solution concept, in such games either lack scalability or obtain poor performance. In this paper we introduce a new algorithm called projected exploitability descent (PED) for approximating Nash equilibria in multiplayer games of imperfect information. The algorithm works by running projected subgradient descent minimizing a proxy for the multiplayer generalized exploitability function. The objective is nonconvex and nonsmooth, but can be represented as the sum of the maxima of linear functions, for which a subgradient can easily be computed and projected to the polytope of feasible sequence-form strategies. We explore performance of PED on a generalized version of the well-studied benchmark game three-player Kuhn poker. No prior exact algorithms scale to the version of the game with deck size larger than 4, and we compare performance to the popular algorithms of fictitious play (FP) and counterfactual regret minimization (CFR). We find that PED obtains a consistent near-monotonic improvement throughout all runs, though both FP and CFR perform significantly better in the initial iterations. This inspires a hybrid algorithm FP-PED that runs FP for an initial burn-in period before switching to PED for stable long-run refinement. We can alternatively view this as a multi-step algorithm that runs FP as a pre-processing step to obtain a strong initialization for PED.
To verify the robustness of a program or protocol, it is common in the computer science community to rely on the theoretical framework of game theory. In particular, if one seeks to enforce a desired property, or specification, despite an unpredictable environment, a useful abstraction is to model the situation as a two-player zero-sum game. The goal is then to find a strategy for the system that guarantees the specification against any strategy of the environment. However, to model more complex situations, such as multiple systems with different objectives or an environment composed of various agents, the richer framework of multiplayer games must be considered. In this setting, a natural question is to identify equilibria, i.e., strategy profiles that are robust in the sense that no player has an incentive to deviate. The most well-known equilibrium concept is the Nash equilibrium, but several alternatives exist. We study five such notions and, for each of them, we provide complexity results for the constrained existence problem, which consists of deciding whether a given game contains an equilibrium that ensures each player a payoff within a specified interval.