Repulsion in controversial debate drives public opinion into fifty-fifty stalemate.
Opinion formation is a process with strong implications for public policy. In controversial debates with large consequences, the public opinion is often trapped in a fifty-fifty stalemate, jeopardizing broadly accepted political decisions. Emergent effects from millions of private discussions make it hard to understand or influence this kind of opinion dynamics. Here we demonstrate that repulsion from opinions favors fifty-fifty stalemates. We study a voter model where agents can have two opinions or an undecided state in between, and where we allow for repulsion of opinions and for doubt: in pairwise discussions, undecided agents can be not only convinced, but also repelled from the opinion expressed by another agent, and decided agents may return to the undecided state. As a result, we observe that, if an agent is repelled instead of being convinced in at least one out of four interactions, as in controversial debates, the frequencies of both opinions equalize. This voter model attractor reproduces the phenomenology of repeated Brexit poll data well and provides a mechanism solely based on local interactions between agents that may explain stalemate polarization in controversial opinion formation.