The computational study of equilibria involving constraints on players' strategies has been largely neglected. However, in real-world applications, players are usually subject to constraints ruling out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps. Computational studies on constrained versions of the Nash equilibrium have lead to some results under very stringent assumptions, while finding constrained versions of the correlated equilibrium (CE) is still unexplored. In this paper, we introduce and computationally characterize constrained Phi-equilibria -- a more general notion than constrained CEs -- in normal-form games. We show that computing such equilibria is in general computationally intractable, and also that the set of the equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium maximizing a given linear function, when either the number of constraints or that of players' actions is fixed. Moreover, in the special case in which a player's constraints do not depend on other players' strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret learning algorithm.

翻译：对包含策略约束的均衡问题的计算研究长期被忽视。然而在实际应用中，参与者通常受到安全性要求与预算上限等约束，导致其部分策略不可行。关于纳什均衡受约束版本的计算研究仅在极为严苛的假设下取得若干成果，而受约束关联均衡的计算问题仍未被探索。本文针对标准式博弈，提出并计算刻画了受约束Phi均衡——一种比受约束关联均衡更广义的概念。研究表明：计算此类均衡通常是计算不可行的，且均衡集合可能不具备凸性，这与无约束关联均衡形成鲜明对比。尽管如此，我们仍提出了多项式时间算法，可在约束数量或参与者行动数量固定时，计算最大化给定线性函数的受约束（近似）Phi均衡。进一步地，当参与者的约束条件独立于其他参与者策略时，我们证明可在多项式时间内计算出精确的函数最大化均衡，并通过高效的去中心化无遗憾学习算法求得一个（近似）均衡。