| Part of the problem described by Medved can be overcome in HPS lamps by using a neon-argon filled Penning-start arc tube. That solves the initial ignition problem but due to the higher thermal conductivity of neon vs xenon, the arc cools even faster on each half-cycle and the reignition peaks are higher than for other discharge lamps.
The easiest way to reliably get around that is to set the HPS arc voltage so low that even at the point of arc extinction, there is just enough mains voltage available to reignite the arc. However the low arc voltage then means that a proportionally higher voltage filament ballast is needed. The result is the combination of a very low power HPS arc tube with a high power incandescent filament, and that is so inefficient that the product is not commercially viable for general lighting (even though technically possible).
Philips developed a 100W self-ballasted SON in the early 80s based on a slight improvement of this principle, with a capacitor to displace the voltage and current waveforms in the opposite direction than on an inductive ballast, thereby making a higher voltage available for reignition and if I remember rightly, that lamp had about 30% of the power in the arc and 70% in the filament - similar to a self-ballasted mercury. But the efficacy was still too low to be commercially viable, and I can imagine that capacitor overheating / short life may have been a problem. Another problem is the slower run-up of HPS lamps vs mercury, and until the arc voltage rises the ballast filament is severely overloaded - this necessitates designing the filament for lower colour temperature, which again reduces efficacy. Incidentally the Philips lamp was not intended to be sold for the reason of being self-ballasted, but for improved colour rendering.
Many years ago when work was more relaxed I also developed a self-ballsted SON at Sylvania. It was based on a 110W SHX neon-argon arc tube with reduced arc gap for moderately lower voltage and gasfilled outer envelope to use the heat convected from the filament to quickly run up the arc. It was made in a one-piece PAR38 reflector bulb with aluminised coating to further heat the arc tube. We found a balance that was possible to achieve a stable burning lamp rated about 230V 250W but it would extinguish too easily with mains voltage dips, and due to the gasfilled bulb plus strong shortwave infrared radiation of the arc, cap temperatures exceeded the allowed limits for E27. It was not very efficient in lumens per watt, but the photosynthetic photon efficacy for plants was rather good and the increased far red radiation from the filament was also beneficial for this application. At the time many of the Sylvania HID lamps had names like BriteArc, BriteSpot, BriteLux, BriteLine, BriteScreen, BriteBlue etc. This one was known internally as the BriteWeed lamp, so you can guess the application :-) But it was one of the many lab ideas that never made the commercial transition into production due to some unresolved problems - it would extinguish too easily with mains voltage dips, and it was not possible to keep the E27 cap temperature within the allowed limits.
More recently the Chinese have made self-ballasted SON with a large full electronic ballast in a base compartment taken from CFLi lamps, but I think only up to 50W. The heat of high power HID lamps is not really compatible with electronic ballasts. Aside from this, I don’t think there are any other commercialised self-ballasted HPS lamps.
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