  From a self-centered perspective, it can be assumed that people only hold
opinions that can benefit them. If opinions have no intrinsic value, and
acquire their value when held by the majority of individuals in a discussion
group, then we have a situation that can be modeled as an $N$-player game. Here
we explore the dynamics of (binary) opinion formation using a game-theoretic
framework to study an $N$-player game version of Galam's local majority-vote
model. The opinion dynamics is modeled by a stochastic imitation dynamics in
which the individuals copy the opinion of more successful peers. In the
infinite population limit, this dynamics is described by the classical
replicator equation of evolutionary game theory. The equilibrium solution shows
a threshold separating the initial frequencies that lead to the fixation of one
opinion or the other. A comparison with Galam's deterministic model reveals
contrasting results, especially in the presence of inflexible individuals, who
never change their opinions. In particular, the $N$-player game predicts a
polarized equilibrium consisting only of extremists. Using finite-size scaling
analysis, we evaluate the critical exponents that determine the population size
dependence of the opinion's fixation probability and mean fixation times near
the threshold. The results underscore the usefulness of combining evolutionary
game theory with opinion dynamics and the importance of statistical physics
tools to summarize the results of Monte Carlo simulations.
