We consider three distinct discrete-time models of learning and evolution in games: a biological model based on intra-species selective pressure, the dynamics induced by pairwise proportional imitation, and the exponential / multiplicative weights (EW) algorithm for online learning. Even though these models share the same continuous-time limit - the replicator dynamics - we show that second-order effects play a crucial role and may lead to drastically different behaviors in each model, even in very simple, symmetric $2\times2$ games. Specifically, we study the resulting discrete-time dynamics in a class of parametrized congestion games, and we show that (i) in the biological model of intra-species competition, the dynamics remain convergent for any parameter value; (ii) the dynamics of pairwise proportional imitation exhibit an entire range of behaviors for larger time steps and different equilibrium configurations (stability, instability, and even Li-Yorke chaos); while (iii) in the EW algorithm, increasing the time step (almost) inevitably leads to chaos (again, in the formal, Li-Yorke sense). This divergence of behaviors comes in stark contrast to the globally convergent behavior of the replicator dynamics, and serves to delineate the extent to which the replicator dynamics provide a useful predictor for the long-run behavior of their discrete-time origins.

翻译：我们研究了博弈中学习与演化的三种不同离散时间模型：基于种内选择压力的生物模型、成对比例模仿所诱导的动力学，以及在线学习的指数/乘法权重（EW）算法。尽管这些模型共享相同的连续时间极限——复制子动力学，但我们表明，二阶效应起着至关重要的作用，并可能导致每种模型中截然不同的行为，即使在非常简单的对称$2\times2$博弈中也是如此。具体而言，我们在参数化拥堵博弈的一类模型中研究了由此产生的离散时间动力学，并表明：(i) 在种内竞争的生物模型中，动力学对于任何参数值都保持收敛；(ii) 成对比例模仿的动力学在较大时间步长及不同均衡配置（稳定性、不稳定性，甚至李-约克混沌）下展现出完整的行为范围；而(iii) 在EW算法中，增加时间步长（几乎）不可避免地导致混沌（同样是在形式上的李-约克意义下）。这些行为的分歧与复制子动力学的全局收敛行为形成鲜明对比，并界定了复制子动力学在何种程度上可作为其离散时间起源长期行为的有效预测器。