Smooth Games of Configuration in the Linear-Quadratic Setting

TL;DR

This paper introduces a novel framework for strategic configuration in differential games, enabling fine-tuning of multi-agent interactions through a two-stage game approach with gradient-based solution methods.

Contribution

It formalizes the concept of a game of configuration within affine-quadratic differential games and provides computational methods for finding equilibrium configurations.

Findings

Effective gradient-based search for local solutions

Applicable to zero-sum and general-sum AQ systems

Provides necessary conditions for equilibrium configurations

Abstract

Dynamic game theory offers a toolbox for formalizing and solving for both cooperative and non-cooperative strategies in multi-agent scenarios. However, the optimal configuration of such games remains largely unexplored. While there is existing literature on the parametrization of dynamic games, little research examines this parametrization from a strategic perspective where each agent's configuration choice is influenced by the decisions of others. In this work, we introduce the concept of a game of configuration, providing a framework for the strategic fine-tuning of differential games. We define a game of configuration as a two-stage game within the setting of finite-horizon, affine-quadratic, AQ, differential games. In the first stage, each player chooses their corresponding configuration parameter, which will impact their dynamics and costs in the second stage. We provide the subgame perfect solution concept and a method for computing first stage cost gradients over the configuration space. This then allows us to formulate a gradient-based method for searching for local solutions to the configuration game, as well as provide necessary conditions for equilibrium configurations over their downstream (second stage) trajectories. We conclude by demonstrating the effectiveness of our approach in example AQ systems, both zero-sum and general-sum.