Stackelberg Solutions in an Opinion Dynamics Game with Stubborn Agents

The paper examines an opinion dynamics game in a social group with two active agents (influencers) based on the Friedkin–Johnsen model. In the game, we assume sequential announcements of influence efforts by the active agents on the opinions of other (passive) agents of the group. We characterize the Stackelberg solutions as proper solution concepts under sequential play. We then analyze the solutions with a number of measures that quantify them in different aspects: (i) the role of the information structure, i.e., open-loop vs. feedback, (ii) the advantage of sequential over simultaneous moves, and (iii) whether being a leader in the game is more cost-effective than being a follower. Finally, we perform numerical simulations for Zachary’s karate club network to understand how the Stackelberg solutions are sensitive to a change in a parameter characterizing the stubbornness of agents to their initial opinions. The results indicate that the information structure has minimal effect; however, the greatest advantage of the open-loop policy could be achieved with a fully conforming society. In such a society, the efforts of influencers become more efficient, reducing the spread of opinions. Additionally, we observe that the follower has an advantage in the game, which forces each influencer to delay their action until the other one acts.