@Brian:
The sputtering is result of high energy ions bombarding the electrode and it happen during whole lamp operation. But it's rate depend on ion's amount (obviously) and energy. The amount is given by the lamp current: higher current = more ions, but there you might do nothing about. Cation energy is given by the fill ionisation energy and is usually very small, but energy of anions come directly from cathode fall (
described on Figure 31 ): The potential difference between end of the anode column accelerate them towards the cathode and when their energy is high enough, they kick away an atom from the electrode.
The cathode fall depend mainly on it's emission capability: If the cathode would emit electrons freely (is hot), the cathode fall would be very small (few V), but if it does not (cold), the cathode fall might be 100's of V. At the same time the cathode fall causes power dissipation, what in fact heat the cathode during normal operation and if this is the only cathode heating mechanism (instant start, switch or starter preheat ballasts, HID's), it settle to the temperature, when the cathode fall causes just enough power dissipation to keep the electrode on such temperature (colder => larger fall => larger dissipation => heat up), what yield typical ~10V fall during operation. Note, then the cathode fall might increase, if the current overload it, e.g. by capacitor discharge current upon arc (re)ignition. This is the reason, why capacitive lead-phase ballasts yield shorter lamp life (more accurately their very high current crest factor).
Of course there is other phenomenon responsible for electrode loss: Evaporation. And this is faster, when the electrode is hotter, but this phenomenon is not as steep (towards the temperature) as the sputtering, so quite significant margin exist. So there is overall temperature optimum for longest electrode life.
And here come the influence of frequent starting: If the used starting method heat the cathode to the operating temperature before and there is no current spike upon ignition, there is no extra electrode wear. If the starting sequence overheat electrodes, they evaporate. If it does not heat them enough, they sputter away, before the cathode fall heat them to correct temperature.
You are very wrong with statement, then sputtering does not influence HID life (LPS i don't count as HID here).
You are right, then electrodes have huge excess of material to not make the limiting factor, but when sputtered away, the material does not disappear. It settle on arctube wall and cause it's blackening, on MH's this is only slowed down by the selfcleaning halogen cycle.
On all lamps the blackening absorb the radiation output, so reduce the lumen output, while at the same time it increase the arctube temperature, mainly in the electrode region (it convert light to heat). This temperature rise has very minor influence on MV (as these use unsaturated vapors), but it does increase the pressure in both HPS and MH's (they have usually "cold spot" behind the electrode). This pressure increase cause the arc voltage to rise, what increase the power delivered to the lamp (the extent depend on the exact ballast characteristic), what increase the arctube temperature even more (and on HPS this power increase compensate the lumen output for the blackening, yielding good overall lumen maintenance over life). After some point this yield to thermal runaway, what result in lamp cycling (voltage rise so high, the ballast is not able to keep the arc), arctube leakage or (on QMH) even explosion ("normal" end of life of these lamps).