Optimal Opinion Control: The Campaign Problem

Abstract

Opinion dynamics is nowadays a very common field of research. In this article we formulate and then study a novel, namely strategic perspective on such dynamics: There are the usual 'normal' agents that update their opinions, for instance according the well-known bounded confidence mechanism. But, additionally, there is at least one strategic agent. That agent uses opinions as freely selectable strategies to get control on the dynamics: The strategic agent of our benchmark problem tries, during a campaign of a certain length, to influence the ongoing dynamics among normal agents with strategically placed opinions (one per period) in such a way, that, by the end of the campaign, as much as possible normals end up with opinions in a certain interval of the opinion space. Structurally, such a problem is an optimal control problem. That type of problem is ubiquitous. Resorting to advanced and partly nonstandard methods for computing optimal controls, we solve some instances of the campaign problem. But even for a very small number of normal agents, just one strategic agent, and a ten-period campaign length, the problem turns out to be extremely difficult. Explicitly we discuss moral and political concerns that immediately arise, if someone starts to analyze the possibilities of an "optimal opinion control".

Optimal control of the freezing time in the Hegselmann–Krause dynamics

“…It would be very interesting to see optimized strategies and/or an improved analysis further reducing the upper bound of the optimal freezing time for these two specific examples for m ¼ 1. Given a concrete HK system and a number m of strategic agents, the optimal freezing time might be determined using an exact algorithm based on Journal of Difference Equations and Applications 645 ILP, similar to those ILP formulations presented in [11,16]. However, we do not go into details here and propose the development of (practically) efficient exact algorithms for the optimal control problem of minimizing the freezing time in the HK model as a research challenge.…”

“…It can indeed be easily shown, see, e.g. [11], that one strategic agent suffices to bring any configuration of starting positions to a consensus in a finite number of time steps, i.e. the opinions of all non-strategic agents coincide after some rounds.…”

“…Recently researchers started to look at the problem of controlling or steering an instance of an opinion dynamics system towards a desired state, see [1,4,11,25]. Exogenous interventions into the system's dynamics were, e.g.…”

“…studied in [6,7,21]. The desired state in [11] is one in which as many as possible of the opinions lie in some specified subset of the opinion space, called a conviction interval. The practical problem that one can have in mind is that of a political or commercial campaign.…”

“…Via media channels, speeches or personal communications the opinion dynamics can be influenced. Formally such a controllable exogenous influence was modelled by introducing strategic agents in [11]. Here we will use strategic agents to accelerate the freezing time in the HK model, i.e.…”