Competitions for shareable and limited resources have long been studied with
strategic agents. In reality, agents often have to learn and maximize the
rewards of the resources at the same time. To design an individualized
competing policy, we model the competition between agents in a novel
multi-player multi-armed bandit (MPMAB) setting where players are selfish and
aim to maximize their own rewards. In addition, when several players pull the
same arm, we assume that these players averagely share the arms' rewards by
expectation. Under this setting, we first analyze the Nash equilibrium when
arms' rewards are known. Subsequently, we propose a novel SelfishMPMAB with
Averaging Allocation (SMAA) approach based on the equilibrium. We theoretically
demonstrate that SMAA could achieve a good regret guarantee for each player
when all players follow the algorithm. Additionally, we establish that no
single selfish player can significantly increase their rewards through
deviation, nor can they detrimentally affect other players' rewards without
incurring substantial losses for themselves. We finally validate the
effectiveness of the method in extensive synthetic experiments.

在多人多臂老虎机（MPMAB）竞争模型中，我们提出了基于纳什均衡的自私 MPMAB 和平均分配（SMAA）策略，有效解决了自私玩家学习和最大化奖励的问题，并在理论和实验中证明其有效性和稳健性。