Computing Stackelberg Equilibria of Large General-Sum Games

TL;DR

This paper investigates the computational complexity of finding Stackelberg Equilibria in large general-sum games, revealing that efficient computation in zero-sum games does not extend straightforwardly to general-sum scenarios.

Contribution

It demonstrates that structural properties enabling efficient zero-sum Nash equilibrium computation are insufficient for general-sum Stackelberg equilibrium computation.

Findings

Efficient algorithms exist for zero-sum games with polynomial strategies.

Structural properties do not guarantee efficient solutions in general-sum games.

Computing Stackelberg equilibria in general-sum games is computationally complex.

Abstract

We study the computational complexity of finding Stackelberg Equilibria in general-sum games, where the set of pure strategies of the leader and the followers are exponentially large in a natrual representation of the problem. In \emph{zero-sum} games, the notion of a Stackelberg equilibrium coincides with the notion of a \emph{Nash Equilibrium}~\cite{korzhyk2011stackelberg}. Finding these equilibrium concepts in zero-sum games can be efficiently done when the players have polynomially many pure strategies or when (in additional to some structural properties) a best-response oracle is available~\cite{ahmadinejad2016duels, DHL+17, KV05}. Despite such advancements in the case of zero-sum games, little is known for general-sum games. In light of the above, we examine the computational complexity of computing a Stackelberg equilibrium in large general-sum games. We show that while there are natural large general-sum games where the Stackelberg Equilibria can be computed efficiently if the Nash equilibrium in its zero-sum form could be computed efficiently, in general, structural properties that allow for efficient computation of Nash equilibrium in zero-sum games are not sufficient for computing Stackelberg equilibria in general-sum games.