Computing the Strategy to Commit to in Polymatrix Games (Extended Version)

TL;DR

This paper investigates the computational complexity of finding optimistic and pessimistic leader-follower equilibria in polymatrix games with multiple followers, providing algorithms for certain classes and highlighting the problem's difficulty.

Contribution

It introduces an exact algorithm for pessimistic equilibria in polymatrix-like games and analyzes the complexity of computing such equilibria under various conditions.

Findings

Exact algorithm for pessimistic equilibria in specific game classes

Computational complexity results for optimistic and pessimistic equilibria

Polymatrix games are computationally hard even with pure strategies

Abstract

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games---polymatrix like---which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies.