  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.
