Learning in general-sum games can be unstable and often leads to socially undesirable, Pareto-dominated outcomes. To mitigate this, Learning with Opponent-Learning Awareness (LOLA) introduced opponent shaping to this setting, by accounting for the agent's influence on the anticipated learning steps of other agents. However, the original LOLA formulation (and follow-up work) is inconsistent because LOLA models other agents as naive learners rather than LOLA agents. In previous work, this inconsistency was suggested as a cause of LOLA's failure to preserve stable fixed points (SFPs). First, we formalize consistency and show that higher-order LOLA (HOLA) solves LOLA's inconsistency problem if it converges. Second, we correct a claim made in the literature, by proving that, contrary to Sch\"afer and Anandkumar (2019), Competitive Gradient Descent (CGD) does not recover HOLA as a series expansion. Hence, CGD also does not solve the consistency problem. Third, we propose a new method called Consistent LOLA (COLA), which learns update functions that are consistent under mutual opponent shaping. It requires no more than second-order derivatives and learns consistent update functions even when HOLA fails to converge. However, we also prove that even consistent update functions do not preserve SFPs, contradicting the hypothesis that this shortcoming is caused by LOLA's inconsistency. Finally, in an empirical evaluation on a set of general-sum games, we find that COLA finds prosocial solutions and that it converges under a wider range of learning rates than HOLA and LOLA. We support the latter finding with a theoretical result for a simple game.

通过在LOLA算法中引入一种方法称为Consistent LOLA，其中学习更新功能在彼此影响时保持一致，作者在广义和游戏模型中进行了一系列实验，发现这种方法比HOLA和LOLA更容易收敛，并能够找到更加符合社会期望的解决方案。

COLA: 具有对手感知的一致学习