We discuss possible improvements in the efficiency of dye-sensitized photovoltaic cells using dyes capable of singlet fission into two triplets, thus producing two electron-hole pairs from a single photon. It is pointed out that, in addition to derivatives of large alternant hydrocarbons, those of biradicals are also likely candidates for a favorable ordering of excited-state energy levels, E(T-2), E(S-1) > 2E(T-1). A large number of potentially favorable structures has been examined by the semiempirical Pariser-Parr-Pople method and some also by the time-dependent density functional theory method. Several likely candidates have been identified for experimental examination.