New Math Reveals What Happens When Competing Influencers Battle for Your Mind
Your opinions are being shaped by competing algorithms — this explains the fallout.
This paper studies competitive one-step-ahead control of Friedkin-Johnsen networks with overlapping player influence. The one-step interaction is an exact potential game with a unique Nash equilibrium obtained from a symmetric positive-definite system.
Sequential best-response sweeps converge for every frozen network state, parallel sweeps obey an exact Jacobi condition (and always converge with two players), and one-sweep implementations require an augmented state-action stability test. For marginal networks, a signed left-right damping condition is sufficient for exact-equilibrium stability and becomes a sharp first-order instability test when its sign is reversed.
A resolvent identity clarifies the feedback geometry, while a control-aware centrality identifies the goal conflicts that matter most. We characterize attainable equilibria under unconstrained, convex, and sparse goal restrictions and give a closed form for the same-state welfare loss caused by competition. Numerical examples verify the stability thresholds, geometry, and welfare predictions.
- Competing influencers (like campaigns or platforms) that try to steer public opinion can create a stable outcome, not chaos.
- Competition has a measurable cost: rivals' efforts cancel each other out, wasting influence.
- The math identifies key people in a network whose opinions matter most — useful for spotting where manipulation might hit hardest.
Why It Matters
This research helps explain and predict how online battles over our opinions actually play out, informing policy on misinformation.