  Empirical studies suggest a deep intertwining between opinion formation and
decision-making processes, but these have been treated as separate problems in
the study of dynamical models for social networks. In this paper, we bridge the
gap in the literature by proposing a novel coevolutionary model, in which each
individual selects an action from a binary set and has an opinion on which
action they prefer. Actions and opinions coevolve on a two-layer network. For
homogeneous parameters, undirected networks, and under reasonable assumptions
on the asynchronous updating mechanics, we prove that the coevolutionary
dynamics is an ordinal potential game, enabling analysis via potential game
theory. Specifically, we establish global convergence to the Nash equilibria of
the game, proving that actions converge in a finite number of time steps, while
opinions converge asymptotically. Next, we provide sufficient conditions for
the existence of, and convergence to, polarized equilibria, whereby the
population splits into two communities, each selecting and supporting one of
the actions. Finally, we use simulations to examine the social psychological
phenomenon of pluralistic ignorance.
