It is well known that if f is a Holder continuous function from a mixing shift of finite type X to R, then there exists a unique equilibrium state which is an invariant Gibbs measure having f as a potential function. This result has been generalized to wider classes, such as when X is a subshift with the specification property and f is a function in the Bowen class. Recently Baker and Ghenciu showed that there exists a (non-invariant) Gibbs measures for the zero potential if and only if X is (right-)balanced. We extend this result and show that a necessary and sufficient condition for the existence of invariant Gibbs measures on X for the potential 0 is the bi-balanced condition for X. We define a new condition, called f-balanced condition for the pair (X,f) and present a similar result for the existence of Gibbs measure with respect to f. Using this result, we construct a class of shift spaces which have a Gibbs measure but do not have invariant Gibbs measures for the potential 0, or equivalently, which are one-sided balanced but not bi-balanced, answering a question raised by Baker and Ghenciu.