The electronic structures of ${\rm AgNi}_{1-{\rm x}}{\rm Co}_{\rm x}{\rm O}_{2}$ $(0\leq{\rm x}\leq 1)$ delafossite oxides have been investigated by employing photoemission spectroscopy (PES) and X-ray absorption spectroscopy (XAS) in the soft X-ray regime of synchrotron radiation. The Ag 3d core-level PES spectra shift toward the higher binding energy with increasing x, in agreement with the metallic nature for low values of x and the insulating nature for high values of x in ${\rm AgNi}_{1-{\rm x}}{\rm Co}_{\rm x}{\rm O}_{2}$ . Further, the Ag 3d core-level PES spectra become more asymmetric with decreasing x, which reflects the larger number of electron-hole pairs in the Ni-rich regime. Valence-band PES spectra reveal that the band width of the occupied Ni 3d states is broader than that of the Co 3d states, implying the larger Ni 3d-O 2p hybridization than the Co 3d-O 2p hybridization. The Co and Ni 2p XAS spectra show that Co ions are in the low-spin ${\rm Co}^{3+}$ states for all x, while Ni ions are in the ${\rm Ni}^{2+}-{\rm Ni}^{3+}$ mixed-valent states. The ${\rm Ni}^{3+}$ component increases with x.