Constrained Phi-Equilibria

TL;DR

This paper introduces constrained Phi-equilibria in normal-form games, explores their computational complexity, and provides algorithms for their approximation and computation under certain conditions.

Contribution

It defines constrained Phi-equilibria, analyzes their computational hardness, and offers polynomial-time algorithms for approximate solutions in specific scenarios.

Findings

Computing constrained Phi-equilibria is generally intractable.

The set of constrained Phi-equilibria may be non-convex.

Polynomial-time algorithms exist for certain constrained equilibrium computations.

Abstract

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.