We study a mean field game in continuous time over a finite horizon, T, where the state of each agent is binary and where players base their strategic decisions on two, possibly competing, factors: the willingness to align with the majority (conformism) and the aspiration of sticking with the own type (stubbornness). We also consider a quadratic cost related to the rate at which a change in the state happens: changing opinion may be a costly operation. Depending on the parameters of the model, the game may have more than one Nash equilibrium, even though the corresponding N-player game does not. Moreover, it exhibits a very rich phase diagram, where polarized/unpolarized, coherent/incoherent equilibria may coexist, except for T small, where the equilibrium is always unique. We fully describe such phase diagram in closed form and provide a detailed numerical analysis of the N-player counterpart of the mean field game. In this finite dimensional setting, the equilibrium selected by the population of players is always coherent (favoring the subpopulation whose type is aligned with the initial condition), but it does not necessarily minimize the cost functional. Rather, it seems that, among the coherent ones, the equilibrium prevailing is the one that most benefits the underdog subpopulation forced to change opinion.

翻译：在一定的地平线上,我们不断研究一种平均的野外游戏,T, 每一个代理人的状态是二进制的,而球员的战略决定以两个可能相互竞争的因素为基础: 是否愿意与多数(符合)和坚持自己类型(天生)的愿望相一致。 我们还考虑一种与国家变化发生速度有关的四进制成本: 改变观点可能是一个代价高昂的操作。 根据模型参数, 游戏可能有一个以上的纳什平衡, 即使相应的N- 玩家游戏并不如此。 此外, 它展示了一个非常丰富的阶段图, 极化/ 极化、 一致/ 不一致的 Equilibria 可能同时存在, 除了小的, 平衡总是独一无二的。 我们完全以封闭的形式描述这种阶段的图表, 并详细分析中场游戏的N- 玩家对应方的数值。 在这个有限的维度设置中, 玩家群体选择的平衡总是一致的( 与初始状态一致的亚人口组一致 ) 。但是它不一定将成本的功能最大化最小化。 相反,, 一种是, 稳定的 。