To approach perfect chess, I took drawn late endgame positions which Stockfish plays almost perfectly (although the scores are still mostly heuristic). Zurichess Appenzeller (1800 ELO CCRL) plays them pretty miserably in self-games, getting only 30% draws. So, on this sort of "openings" we can talk of Zurichess as "regular engine" and Stockfish "almost perfect engine". In this case the pathological behavior is obvious:hgm wrote:For the purpose of this discussion a blunder is just a move in the far tail of the distribution. Classifying moves as blunder or not in the example, each with a different distribution, was just a way to get an overall distribution that is not a pure Gaussian, but has longer tails.
It is true that score loss per move is a bit ill defined, as game theoretical the only scores are mate scores. So in this model we are selling ourselves to a heuristic evaluation. Large score losses will be mostly caused by tactical errors; at some point the engine will unwittingly do a move that unavoidably results in the loss of a piece, while another move could have saved it. Deeper search at that position would have solved the problem, and engines that are better because of deeper search thus make these errors less frequently. Because distant unavoidable losses are more rare in the game tree than close ones. OTOH, in positionally poor positions (low mobility, poor center control) unavoidable material losses of any kind get more frequent in the tree.
Perhaps the model of adding independent per-move errors is entirely wrong, and most games are won by slowly outperforming the opponent positionally in small steps, to steer him to a part of the tree where blunder possibilities are more frequent, so that he has a higher probability to fall for one. The density of blunders will not go completely to zero in equal or even better positions, though.
So, all those nice tails I produced earlier are due to the arbiter (Stockfish) being itself far from perfect in regular chess.