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Abstract We show that each of the properties of equilibrium problems considered in several recent Nash equilibrium existence results satisfy the following condition: All the existence problems that satisfy the property on a compact subset $$K$$ K of a fixed strategy space have, on $$K$$ K , an equilibrium set that contains the equilibrium set of an existence problem that is well behaved on $$K$$ K . The properties that satisfy this condition are called reducible equilibrium properties. This notion is used to clarify recent existence results and the relationship of some of the properties considered in those results.