A diffusion approximation for a randomized 2 × 2-matrix game in a large finite population is ascertained in the case of random payoffs whose expected values, variances and covariances are of order given by the inverse of the population size $N$. Applying the approximation to a Randomized Prisoner’s Dilemma (RPD) with independent payoffs for cooperation and defection in random pairwise interactions, conditions on the variances of the payoffs for selection to favor the evolution of cooperation, favor more the evolution of cooperation than the evolution of defection, and disfavor the evolution of defection are deduced. All these are obtained from probabilities of ultimate fixation of a single mutant. It is shown that the conditions are lessened with an increase in the variances of the payoffs for defection against cooperation and defection and a decrease in the variances of the payoffs for cooperation against cooperation and defection. A RPD game with independent payoffs whose expected values are additive is studied in detail to support the conclusions. Randomized matrix games with non-independent payoffs, namely the RPD game with additive payoffs for cooperation and defection based on random cost and benefit for cooperation and the repeated RPD game with Tit-for-Tat and Always-Defect as strategies in pairwise interactions with a random number of rounds, are studied under the assumption that the population-scaled expected values, variances and covariances of the payoffs are all of the same small enough order. In the first model, the conditions in favor of the evolution of cooperation hold only if the covariance between the cost and the benefit is large enough, while the analysis of the second model extends the results on the effects of the variances of the payoffs for cooperation and defection found for the one-round RPD game.

在随机收益的情况下，确定了一个随机的2×2矩阵博弈在较大有限总体中的扩散近似，其预期值，方差和协方差由总体大小的倒数给出 $ñ$。将近似值应用于随机成对互动中具有合作和叛逃的独立收益的随机囚徒困境（RPD），选择收益的方差条件有利于合作的演变，而不是叛变的演变，更有利于合作的发展，并推论了不利的演变。所有这些都是从单个突变体的最终固定概率中获得的。结果表明，随着针对合作与叛变的叛变收益的方差的增加以及针对合作与叛变的合作收益的方差的减少，条件得到了减轻。详细研究了具有独立收益且期望值可加的RPD游戏，以支持结论。具有非独立收益的随机矩阵博弈，即具有基于随机成本和合作收益的用于合作与叛逃的附加收益的RPD游戏，以及以Tit-for-Tat和Always-Defect为策略的重复RPD博弈，与双方互动在假设总体规模的期望值，收益的方差和协方差都足够小的前提下，对随机回合数进行了研究。在第一个模型中，只有当成本和收益之间的协方差足够大时，有利于合作发展的条件才成立，而对第二个模型的分析则将结果扩展到合作收益的方差影响上一轮RPD游戏发现叛逃。分别研究了基于随机成本和合作收益的具有合作与叛逃的附加收益的RPD博弈，以及以Tit-for-Tat和Always-Defect为策略的随机RPD博弈，该策略具有随机轮次的成对互动。假设总体规模的期望值，收益的方差和协方差都足够小。在第一个模型中，只有当成本和收益之间的协方差足够大时，有利于合作发展的条件才成立，而对第二个模型的分析则将结果扩展到合作收益的方差影响上一轮RPD游戏发现叛逃。分别研究了基于随机成本和合作收益的具有合作与叛逃的附加收益的RPD博弈，以及以Tit-for-Tat和Always-Defect为策略的随机RPD博弈，该策略具有随机轮次的成对互动。假设总体规模的期望值，收益的方差和协方差都足够小。在第一个模型中，只有当成本和收益之间的协方差足够大时，有利于合作发展的条件才成立，而对第二个模型的分析则将结果扩展到合作收益的方差影响上一轮RPD游戏发现叛逃。