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We formally assume that players in a game consider Nash Equilibrium (NE) the behavioral norm. In finite games of perfect information this leads to a refinement of NE: Faithful Nash Equilibrium (FNE). FNE is outcome equivalent to NE of the trimmed game, obtained by restricting the original tree to its NE paths. Thus, it always exists but it need not be unique. Iterating the norm ensures uniqueness of outcome. FNE may violate backward induction when subgame perfection requires play according to the SPE following a deviation from it. We thus provide an alternative view of tenable threats in equilibrium analysis.

Threat is a strategy that can be used to impact decision-making processes in bargaining. Abundant evidence suggests that credible threat and incredible threat both influence the obeisance of others. However, it is not clear whether the decision-making processes under credible threat and incredible threat during bargaining involve differential neurocognitive mechanisms. Here, we employed cathodal transcranial direct current stimulation (tDCS) to deactivate the right dorsolateral prefrontal cortex (rDLPFC) to address this question while subjects allocated and reported the subjective probability of future rejection under incredible threat and credible threat. We found that application of cathodal tDCS over the rDLPFC decreased the proposer's subjective inference of probability of rejection and the offer to the responder under incredible threat. Conversely, the same stimulation did not lead to a significant difference compared to the sham group in subjective probability and offer under credible threat. These results suggested that decision-making processes under the two types of threat during bargaining were associated with different neurocognitive substrates, because the punishment for non-compliance was uncertain under incredible threat, whereas it was certain under credible threat. We decreased activity in the rDLPFC, which is involved in decision-making processes related to bargaining under incredible threats, and observed significantly impacted behavior. The differential neurocognitive bases of subjective probability of rejection under incredible threat and credible threat resulted in different tDCS effects.