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A Mexican standoff is a three person duel. In a two person duel the person who shoots first has the advantage. In a Mexican duel it is different. If person A shoots B, then person C has the advantage as they can shoot person A while they are busy. So in a Mexican standoff nobody wants to be the first one to shoot. As well as this, nobody can take the risk of backing down. The term is often used to describe a confrontation between two equals, where neither side will gain a winning advantage by being the first to shoot, but neither side can risk backing down. The Cold War between the USA and the Soviet Union is often described as being a Mexican standoff, with neither country prepared to be the first to risk using nuclear weapons.
The idea of a Mexican standoff is often used as part of the plot in movies.
This page mentioned, but misquoted, in the last scene of The Big Bang Theory episode #506.