In order to maintain a stable arc and avoid wear of the projections of electrode tip ends even if the lamp is lighted with a lamp power of at most 70% of the nominal electric power, a high pressure discharge lamp light source device is provided with a power supply device for supplying alternating current to the high pressure discharge lamp in a low electric power lighting mode having a defined current value in the range of 40 to 70% of a nominal electric power consumption, and wherein the power supply device, in the low electric power lighting mode supplies an alternating current with a low frequency which is lower than the steady-state lighting frequency, and supplies a boost current at least two times during a half cycle of the alternating current with a low frequency which is larger than the alternating current with a low frequency.