An Analysis of Zero-Sum Games on Networks with Voter Model Dynamics

Abstract

This paper will begin by introducing several existing opinion dynamics models and assessing their ability to represent empirical observations accurately. We will then exclusively focus on the classic voter model and consider the effects of introducing a small number of stubborn agents that can hold one of two different opinions. Stubborn agents holding the same opinion are considered to belong to the same party and to have directly opposing objectives to those belonging to the other party. Assuming that each party’s objective is to maximize its expected number of supporters as t → ∞, we will explore the optimal strategies for each party in both sequential and simultaneous games. In the sequential case, we will propose the ‘convergence time minimization’ strategy as a tractable proxy for the first mover’s optimal strategy and assess its performance on network structures with varying degrees of homogeneity. In the simultaneous case, we will seek to qualitatively understand the conditions under which a pure strategy Nash equilibrium is expected to arise. Finally, we will explore how the parties’ optimal strategies would change if the objective function were adjusted so as to incorporate variance or if it were evaluated at a finite time horizon. Our analysis and conclusions find applications in the commercial and political fields, where a small number of entities seek to diffuse opposing beliefs through the positioning of advertising resources.