We capture contextuality in a simple quantum cognitive model. We find that in a binary voting problem, letting two citizens with alternative thinking frames and opposite voting intentions deliberate under the guidance of a benevolent facilitator allows reaching consensus. Opinion change occurs as the result of citizens' action in terms of "putting themselves in another citizen's shoes". The probability for reaching consensus depends on the correlation between perspectives and on their sophistication (dimensionality). Maximally uncorrelated sophisticated perspectives give the best chance for opinion change. With more than two citizens, multiple deliberation rounds with experts allow reaching consensus with significant probability.

A first central lesson is that with contextual opinions, the diversity of perspectives is beneficial and necessary to overcome initial disagreement. We also learn that well-design procedures managed by a facilitator are needed to increase the probability for consensus. An additional finding is that the richness of a thinking frame helps convergence toward a joint position. The optimal facilitator's strategy entails focusing deliberation on a properly reduced problem.