We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth–death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665–1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.

我们研究Moran随机过程的固定概率，该过程具有三种或更多类型的个体和频率依赖的适应度。与具有两种类型的个体的人口情况相反，在这种情况下，可以通过精确的公式计算出固定概率，在这里我们必须解决大型线性方程组。我们首先显示该系统始终具有独特的解决方案。其他结果是通过将三种方法的Moran过程与仅用两种策略的生死过程耦合而获得的注视概率的上限和下限。我们也将边界应用于人口中的合作进化问题，该种群的三种类型的个体已经由NúñezRodríguez和Neves在确定性的背景下进行了研究（J Math Biol 73：1665-1690，2016）。