A mediator observes no-regret learners playing an extensive-form game repeatedly across $T$ rounds. The mediator attempts to steer players toward some desirable predetermined equilibrium by giving (nonnegative) payments to players. We call this the steering problem. The steering problem captures problems several problems of interest, among them equilibrium selection and information design (persuasion). If the mediator's budget is unbounded, steering is trivial because the mediator can simply pay the players to play desirable actions. We study two bounds on the mediator's payments: a total budget and a per-round budget. If the mediator's total budget does not grow with $T$, we show that steering is impossible. However, we show that it is enough for the total budget to grow sublinearly with $T$, that is, for the average payment to vanish. When players' full strategies are observed at each round, we show that constant per-round budgets permit steering. In the more challenging setting where only trajectories through the game tree are observable, we show that steering is impossible with constant per-round budgets in general extensive-form games, but possible in normal-form games or if the per-round budget may itself depend on $T$. We also show how our results can be generalized to the case when the equilibrium is being computed online while steering is happening. We supplement our theoretical positive results with experiments highlighting the efficacy of steering in large games.

翻译：一名中介观察无遗憾学习者在$T$轮重复博弈中参与扩展式博弈。中介通过向玩家提供（非负）支付，试图引导他们走向某个预定的期望均衡。我们将此称为引导问题。该问题涵盖若干有趣的问题，包括均衡选择与信息设计（劝说）。若中介的预算不受限制，引导问题将变得平凡，因为中介可以直接支付玩家以采取期望行动。我们研究了中介支付的两个约束：总预算与每轮预算。若中介的总预算不随$T$增长，我们证明引导是不可能的。然而，我们表明总预算随$T$次线性增长即足够，这意味着平均支付趋于零。当每轮中玩家的完整策略可被观测时，我们证明恒定的每轮预算允许引导。在更具挑战性的场景中，即仅能观测博弈树路径时，我们证明对于一般扩展式博弈，恒定每轮预算无法实现引导，但在正规式博弈中或当每轮预算可随$T$变化时则是可能的。我们还展示了当均衡在引导过程中在线计算时，我们的结果如何推广。我们通过实验补充理论正面结果，凸显引导在大型博弈中的有效性。