Satisfaction and Regret in Stackelberg Games

TL;DR

This paper explores the concept of follower satisfaction in Stackelberg games, demonstrating that satisfaction can improve leader utility and analyzing regret-based algorithms' convergence behavior in this context.

Contribution

It introduces follower satisfaction in Stackelberg games, compares it with standard models, and investigates regret algorithms' convergence, highlighting new insights into strategic behavior.

Findings

Follower satisfaction increases leader utility in Stackelberg games.

Regret matching algorithms may not always converge to no-regret solutions in this setting.

Conditional regret matching can approximate Stackelberg solutions under certain conditions.

Abstract

This paper introduces the new concept of (follower) satisfaction in Stackelberg games and compares the standard Stackelberg game with its satisfaction version. Simulation results are presented which suggest that the follower adopting satisfaction generally increases leader utility. This important new result is proven for the case where leader strategies to commit to are restricted to be deterministic (pure strategies). The paper then addresses the application of regret based algorithms to the Stackelberg problem. Although it is known that the follower adopts a no-regret position in a Stackelberg solution, this is not generally the case for the leader. The report examines the convergence behaviour of unconditional and conditional regret matching (RM) algorithms in the Stackelberg setting. The paper shows that, in the examples considered, that these algorithms either converge to any pure Nash equilibria for the simultaneous move game, or to some mixed strategies which do not have the "no-regret" property. In one case, convergence of the conditional RM algorithm over both players to a solution "close" to the Stackelberg case was observed. The paper argues that further research in this area, in particular when applied in the satisfaction setting could be fruitful.