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I believe I've figured this out.  The ringmaster gives up when you're one shot away from winning.  (That happens when the 4th light is illuminated and you score another hit.)  Through at least 20 Ringmaster Battle attempts, that's always held true.  I've had it going for 5+ minutes without him surrendering, and I've had him surrender within one minute on nailing that critical second-to-last hit.  Every time he gives up, the next shot always wins the battle.  Drains don't matter, time doesn't matter, progress doesn't matter other than that penultimate hit.

Next question: can a surrendered Ringmaster be turned into TOTAN-style unlimited score milking?  You've got unlimited balls while the ringmaster is in that state, just as long as you can avoid hitting him again.  Points can still be scored - go into the status report and you can see your score during Ringmaster Battle, I did that and checked that it still increased from odd change like slingshots and spinners.  But I'm not sure what you can do on the table that's safe from hitting the Ringmaster again.