Static equilibria and dynamic evolution in noisy binary choice (Ising) games on graphs are considered. Equations defining static quantal response equilibria (QRE) for Ising games on graphs with arbitrary topology and noise distribution are written. It is shown that in the special cases of complete graph and arbitrary noise distribution, and circular and star topology and logistic noise distribution the resulting equations can be cast in the form coinciding with that derived in the earlier literature. Explicit equations for non-directed random graphs in the annealed approximation are derived. It is shown that the resulting effect on the phase transition is the same as found in the literature on phase transition in the mean-field versions of the Ising model on graphs . Evolutionary Ising game having the earlier described QRE as its stationary equilibria in the mean field approximation is constructed using the formalism of master equation for the complete, star, circular and random annealed graphs and the formalism of population games for random annealed graphs.

翻译：在图表上,可以考虑到静态平衡和噪音分布的动态演进,在图表上,在噪音分布和任意分布的图表上,为Ising游戏的静态二次响应平衡(QRE)定義了静态二次响应平衡(QRE)的方程式。可以显示,在完整图形和任意噪音分布的特殊情况下,以及圆形和恒星地形及后勤噪音分布中,所产生的方程式可以以与先前文献中生成的方程式相匹配的形式投出。还得出了Anneald近似中非定向随机图的清晰方程式。它表明,对阶段过渡产生的效果与图中平均版Ising模型中关于阶段过渡的文献中发现的效果相同。将QRE早先描述为平均场近似中的固定线的演进性方程式,是利用全方程式、恒星、圆形和随机Anneald 图形的主方程式以及随机面图形的人口游戏形式来构建的演进式图。