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In this paper, we consider simple cooperative games with winning coalitions that satisfy three natural assumptions: monotonicity, non-emptiness, and inverse consistency. We investigate the implementability of the weak core in two cases: when there are at least two agents under strong Nash equilibria, and when there are exactly two agents under Nash equilibria. To achieve this, we use a strong version of Maskin monotonicity, introduced by Yi [2012], which we call Y-monotonicity. We demonstrate that, when combined with unanimity, this property is sufficient for implementation in both cases. Based on this result, we extend the findings of Shinotsuka and Takamiya [2003] on the implementability of the weak core in our setting, moving from Nash equilibrium to strong Nash equilibrium for at least three agents, while also addressing the case with two agents. Finally, we relate our results to the existing literature on the existence of the weak core.
