Many interactions result in a socially suboptimal equilibrium, or in a non-equilibrium state, from which arriving at an equilibrium through simple dynamics can be impossible of too long. Aiming to achieve a certain equilibrium, we persuade, bribe, or coerce a group of participants to make them act in a way that will motivate the rest of the players to act accordingly to the desired equilibrium. Formally, we ask which subset of the players can adopt the goal equilibrium strategies that will make acting according to the desired equilibrium a best response for the other players. We call such a subset a direct control set, prove some connections to strength of equilibrium, and study the hardness to find such lightest sets, even approximately. We then solve important subcases and provide approximation algorithms, assuming monotonicity. Next, we concentrate on potential games and prove that, while the problem of finding such a set is \NP-hard, even for constant-factor approximation, we can still solve the problem approximately or even precisely in relevant special cases. We approximately solve this problem for singleton potential games and treat more closely specific potential games, such as symmetric games and coordination games on graphs.

翻译：许多交互会导致社会次优均衡或非均衡状态，而通过简单动力学从这些状态到达均衡可能是不可能的，或者需要过长时间。为了达到特定均衡，我们说服、贿赂或胁迫一群参与者，使其行为方式能激励其余玩家相应地按照期望均衡行动。形式上，我们询问哪些玩家子集可以采纳目标均衡策略，从而使得按照期望均衡行动成为其他玩家的最优反应。我们将这样的子集称为直接控制集，证明其与均衡强度的一些关联，并研究寻找此类最轻集合（即使是近似解）的计算难度。随后，我们在假设单调性的情况下求解重要子情形并提供近似算法。接下来，我们聚焦于势博弈，证明尽管寻找此类集合的问题即使在常数因子近似下也是\NP难问题，我们仍能在相关特殊情形下近似甚至精确求解该问题。我们针对单体势博弈给出近似解，并更细致地处理特定势博弈，例如图上的对称博弈与协调博弈。