Leadership in Singleton Congestion Games: What is Hard and What is Easy

TL;DR

This paper investigates the computational complexity of finding Stackelberg equilibria in singleton congestion games, identifying cases where the problem is hard or tractable, and providing algorithms and MILP formulations for practical computation.

Contribution

It characterizes the complexity of computing Stackelberg equilibria in singleton congestion games under various conditions and offers efficient algorithms for the easy cases.

Findings

Computing leader's cost is NP-hard with different action spaces.

Identical action spaces and monotonic costs allow polynomial-time solutions.

MILP formulations enable practical computation in hard cases.

Abstract

We study the problem of computing Stackelberg equilibria Stackelberg games whose underlying structure is in congestion games, focusing on the case where each player can choose a single resource (a.k.a. singleton congestion games) and one of them acts as leader. In particular, we address the cases where the players either have the same action spaces (i.e., the set of resources they can choose is the same for all of them) or different ones, and where their costs are either monotonic functions of the resource congestion or not. We show that, in the case where the players have different action spaces, the cost the leader incurs in a Stackelberg equilibrium cannot be approximated in polynomial time up to within any polynomial factor in the size of the game unless P = NP, independently of the cost functions being monotonic or not. We show that a similar result also holds when the players have nonmonotonic cost functions, even if their action spaces are the same. Differently, we prove that the case with identical action spaces and monotonic cost functions is easy, and propose polynomial-time algorithm for it. We also improve an algorithm for the computation of a socially optimal equilibrium in singleton congestion games with the same action spaces without leadership, and extend it to the computation of a Stackelberg equilibrium for the case where the leader is restricted to pure strategies. For the cases in which the problem of finding an equilibrium is hard, we show how, in the optimistic setting where the followers break ties in favor of the leader, the problem can be formulated via mixed-integer linear programming techniques, which computational experiments show to scale quite well.