In this paper, we present a unified theory for solutions to games of fair division, which are ordinal in nature and appear as non-symmetric variants of the equal income market equilibrium solution. We characterize the entire family of such solutions using consistency, converse-consistency, local-independence, individual rationality and a weak efficiency condition. This is all done in a variable population framework. In the fixed population framework, we obtain an axiomatic characterization for the same family using monotonicity, individual rationality, local-independence, non-discrimination and another weak efficiency property.