# Bounding Inefficiency of Equilibria in Continuous Actions Games using Submodularity and Curvature

## TL;DR

This paper establishes efficiency bounds for coarse correlated equilibria in continuous strategy games with social functions that are monotone DR-submodular, extending to approximate submodularity and providing improved guarantees for certain optimization problems.

## Contribution

It introduces a framework for analyzing the efficiency of equilibria in continuous games using submodularity and curvature, offering new bounds and design strategies.

## Key findings

- Efficiency bounds depend on social function curvature.
- Bounds extend to approximately submodular functions.
- Games can be designed for near-optimal solutions with guarantees.

## Abstract

Games with continuous strategy sets arise in several machine learning problems (e.g. adversarial learning). For such games, simple no-regret learning algorithms exist in several cases and ensure convergence to coarse correlated equilibria (CCE). The efficiency of such equilibria with respect to a social function, however, is not well understood. In this paper, we define the class of valid utility games with continuous strategies and provide efficiency bounds for their CCEs. Our bounds rely on the social function being a monotone DR-submodular function. We further refine our bounds based on the curvature of the social function. Furthermore, we extend our efficiency bounds to a class of non-submodular functions that satisfy approximate submodularity properties. Finally, we show that valid utility games with continuous strategies can be designed to maximize monotone DR-submodular functions subject to disjoint constraints with approximation guarantees. The approximation guarantees we derive are based on the efficiency of the equilibria of such games and can improve the existing ones in the literature. We illustrate and validate our results on a budget allocation game and a sensor coverage problem.