It’s interesting to note Wikipedia’s definition:
A Mexican standoff is most precisely a confrontation among three opponents armed with guns. The tactics for such a confrontation are substantially different than for a duel, where the first to shoot has the advantage. In a confrontation among three mutually hostile participants, the first to shoot is at a tactical disadvantage. If opponent A shoots opponent B, then while so occupied, opponent C can shoot A, thus winning the conflict. Since it is the second opponent to shoot that has the advantage, no one wants to go first.