A multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

提出了离散时间的多步拉格朗日方案，用于近似非线性连续方程，该方程作为空间非均匀演化博弈的平均场极限而出现，描述了具有策略或标签的空间分布智能体系统的演化，其收益也取决于代理商的当前职位。该方案是拉格朗日方案，因为它跟踪位置和标签沿特征的演变，并且是一个多步骤方案，它在以下两个阶段发展：首先，根据最佳性能标准更新策略或标签的分布，然后，代理将其用于发展其位置。在概率测度的空间中提供了一般的收敛结果。在复制型系统和可逆马尔可夫链的特殊情况下，