Stable target opinion through power law bias in information exchange

We study a model of binary decision making when a certain population of agents is initially seeded with two different opinions, `$+$' and `$-$', with fractions $p_1$ and $p_2$ respectively, $p_1+p_2=1$. Individuals can reverse their initial opinion only once based on this information exchange. We study this model on a completely connected network, where any pair of agents can exchange information, and a two-dimensional square lattice with periodic boundary conditions, where information exchange is possible only between the nearest neighbors. We propose a model in which each agent maintains two counters of opposite opinions and accepts opinions of other agents with a power law bias until a threshold is reached, when they fix their final opinion. Our model is inspired by the study of negativity bias and positive-negative asymmetry known in the psychology literature for a long time. Our model can achieve stable intermediate mix of positive and negative opinions in a population. In particular, we show that it is possible to achieve close to any fraction $p_3, 0\leq p_3\leq 1$, of `$-$' opinion starting from an initial fraction $p_1$ of `$-$' opinion by applying a bias through adjusting the power law exponent of $p_3$.