  Group polarization, the phenomenon where individuals become more extreme
after interacting, has been gaining attention, especially with the rise of
social media shaping people's opinions. Recent interest has emerged in formal
reasoning about group polarization using logical systems. In this work we
consider the modal logic PNL that captures the notion of agents agreeing or
disagreeing on a given topic. Our contribution involves enhancing PNL with
advanced formal reasoning techniques, instead of relying on axiomatic systems
for analyzing group polarization. To achieve this, we introduce a semantic game
tailored for (hybrid) extensions of PNL. This game fosters dynamic reasoning
about concrete network models, aligning with our goal of strengthening PNL's
effectiveness in studying group polarization. We show how this semantic game
leads to a provability game by systemically exploring the truth in all models.
This leads to the first cut-free sequent systems for some variants of PNL.
Using polarization of formulas, the proposed calculi can be modularly adapted
to consider different frame properties of the underlying model.
