Both the nature of avalanche ionization (AI) and the role of multi-photon ionization (MPI) in the studies of laser-induced damage have remained controversial up to now. According to the model proposed by Stuart et al., we study the role of MPI and AI in laser-induced damage in two dielectric films, fused silica (FS) and barium aluminum borosilicate (BBS), irradiated by 780-nm laser pulse with the pulse width range of 0.01-5 ps. The effects of MPI and initial electron density on seed electron generation are numerically analyzed. For FS, laser-induced damage is dominated by AI for the entire pulse width regime due to the wider band-gap. While for BBS, MPI becomes the leading power in damage for the pulse width \tau less than about 0.03 ps. MPI may result in a sharp rise of threshold fluence Fth on \tau, and AI may lead to a mild increase or even a constant value of Fth on \tau. MPI serves the production of seed electrons for AI when the electron density for AI is approached or exceeded before the end of MPI. This also means that the effect of initial electron can be neglected when MPI dominates the seed electron generation. The threshold fluence Fth decreases with the increasing initial electron density when the latter exceeds a certain critical value.