$\mathcal{NP}$ -Hardness of Pure Nash Equilibrium in Scheduling and Connection Games

Abstract

We prove $\mathcal{NP}$-hardness of pure Nash equilibrium for some problems of scheduling games and connection games. The technique is standard: first, we construct a gadget without the desired property and then embed it to a larger game which encodes a $\mathcal{NP}$-hard problem in order to prove the complexity of the desired property in a game. This technique is very efficient in proving $\mathcal{NP}$-hardness for deciding the existence of Nash equilibria. In the paper, we illustrate the efficiency of the technique in proving the $\mathcal{NP}$-hardness of deciding the existence of pure Nash equilibria of Matrix Scheduling Games and Weighted Connection Games. Moreover, using the technique, we can settle the complexity not only of the existence of equilibrium but also of the existence of good cost-sharing protocol.