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Chess engines such as StockFish provide centi-pawn evaluation of positions. Clearly the longer the engine runs the deeper it can search and thus provide more accurate centi-pawn evaluation.

Basically, given a random position (say of equal material), I seek to know how the centipawn evaluation changes as a function of the depth parameter. When the depth is very low (say 2-5 moves) the evaluations expected to be erratic and change dramatically, but how does it look like on higher depth (say 20-30 or even higher) - do we tend to see convergence to a value? In other words, is it the case that we learn less and less about the position the deeper we go?

Interestingly, in both cases having the engine run long enough is somewhat futile (unless we are looking for a forcing mate), for (1) if depth 30 is good enough and we learn less and less the deeper we go, why bother to reach depth 40? Or (2) if the other case is true: namely, that there could be some significant difference between depth 30 and depth 40 evaluation, why stop at 40? who promise us that at depth 50 the evaluation would be closer to the 40 value than of the 30?

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  • No idea about sigma, but convergence will ultimately be to 0, 0.5 or 1, and that's also what's to learn in any given positions eval.
    – Pit
    Jan 20 at 18:28
  • @Pit, That's right of course. But I have no idea what depth is needed to solve chess. I think I've read somewhere that depth is not actual moves. I guess we are very very far from this. Basically I've asked about maximal depth we actually using today in engines. say should we really let the engine think for another hour after it already thought for an hour?
    – user34075
    Jan 20 at 19:23

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Not possible to predict

An extra node (i.e., position searched) can always change the evaluation, potentially by a lot. Regardless of how far ahead you search, as long as you don't search the entire game tree (not realistic for chess), it's always possible there is something just "beyond the horizon" that you didn't find.

See example here. When going from d = 28 to d = 29, Stockfish's evaluation crashed ("moobed" in the lingo of engine chess enthusiasts). That's because it missed a tactic that required d = 29 to find.

You can find more examples of such positions where the engine's eval changes dramatically after a certain period of thinking time here.

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  • thanks! guess my follow-up question would be then: how many tactics are to be found in each depth. Still it seems the chances to come with a novel tactic requiring 20 depth is higher than the chances to come up with a novel tactic requiring 30 depth.
    – user34075
    Jan 21 at 10:05
  • Presumably a sufficiently-deep tactic ceases to be a tactic (or opponent will spot it and avoid it), but the gist remains. If you don't see the tactic now, when you see it later, it might stop your best move and therefore the position is worse for you than you thought. Alternatively, maybe the tactic prevents what you thought was your opponent's best move, so your position is actually better for you than you thought.
    – Allure
    Jan 21 at 10:26

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