In this paper, we delve into the problem of using monetary incentives to encourage players to shift from an initial Nash equilibrium to a more favorable one within a game. Our main focus revolves around computing the minimum reward required to facilitate this equilibrium transition. The game involves a single row player who possesses $m$ strategies and $k$ column players, each endowed with $n$ strategies. Our findings reveal that determining whether the minimum reward is zero is NP-complete, and computing the minimum reward becomes APX-hard. Nonetheless, we bring some positive news, as this problem can be efficiently handled if either $k$ or $n$ is a fixed constant. Furthermore, we have devised an approximation algorithm with an additive error that runs in polynomial time. Lastly, we explore a specific case wherein the utility functions exhibit single-peaked characteristics, and we successfully demonstrate that the optimal reward can be computed in polynomial time.

翻译：本文深入探讨了在博弈中通过经济激励促使参与者从初始纳什均衡转向更优均衡的问题。我们主要致力于计算实现这一均衡转移所需的最小奖励金额。该博弈涉及一个拥有$m$种策略的行参与者与$k$个列参与者（每个列参与者拥有$n$种策略）。研究结果表明：确定最小奖励是否为零是NP完全问题，而计算最小奖励本身则属于APX困难问题。但我们带来了积极发现：当$k$或$n$为固定常量时，该问题可被高效求解。此外，我们设计了一种具有加性误差的多项式时间近似算法。最后，我们探讨了效用函数呈现单峰特性的特殊情形，并成功证明在该情形下最优奖励可在多项式时间内计算得出。