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

arXiv: https://arxiv.org/abs/1903.00950 · HTML: https://mlsanity.com/paper/1903.00950/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Mar 2019.

## Built on by (most cited first, 6 of 6)

- [Contextual Games: Multi-Agent Learning with Side Information](https://mlsanity.com/paper/2107.06327/) (2021)
- [Fast First-Order Methods for Monotone Strongly DR-Submodular Maximization](https://mlsanity.com/paper/2111.07990/) (2021)
- [Optimistic Games for Combinatorial Bayesian Optimization with Application to Protein Design](https://mlsanity.com/paper/2409.18582/) (2024)
- [Online SuBmodular + SuPermodular (BP) Maximization with Bandit Feedback](https://mlsanity.com/paper/2207.03091/) (2022)
- [How Bad is Selfish Driving? Bounding the Inefficiency of Equilibria in Urban Driving Games](https://mlsanity.com/paper/2210.13064/) (2022)
- [Cordial Learning: Distributed Training with Correlated Data](https://mlsanity.com/paper/2610.03330/) (2026)

## Abstract (arXiv)

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.