Coordinating Followers to Reach Better Equilibria: End-to-End Gradient Descent for Stackelberg Games

A growing body of work in game theory extends the traditional Stackelberg to settings with one leader and multiple followers who play a Nash equilibrium. Standard approaches for computing equilibria these games reformulate followers' best response as constraints leader's optimization problem. These reformulation can sometimes be effective, but make limiting assumptions on objectives equilibrium reached by followers, e.g., uniqueness, optimism, or pessimism. To overcome limitations, we run gradient descent update strategy differentiating through followers. Our approach generalizes any stochastic selection procedure that chooses from equilibria, where compute back-propagating sampled using solution partial differential equation establish unbiasedness gradient. Using unbiased estimate, implement gradient-based solve three problems consistently outperforms existing baselines achieve higher utility leader.