Voting problems with a continuum of voters and finitely many alternatives are considered. Since the gibbard–satterthwaite theorem persists in this model, we relax the non-manipulability requirement as follows: are there social choice functions (scfs) such that for every profile of preferences there exists a strong nash equilibrium resulting in the alternative assigned by the scf? such scfs are called exactly and strongly consistent. The paper extends the work of peleg (econometrica 46:153–161, 1978a) and others. Specifically, a class of anonymous scfs with the required property is characterized through blocking coefficients of alternatives and through associated effectivity functions.