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By a safe advantage, I mean it is easy for the side with the advantage to keep it. Winning the game is extremely likely under normal conditions. Example: KQ vs KN is a very safe advantage.

By an unsafe advantage, I mean it is a complicated position and the side with the advantage can only keep it with very accurate play. If they make even one mistake/inaccurate move, the advantage may be lost and they may even lose the game. Many complicated middle game positions are "unsafe" in this sense.

Does there exist terminologies that describe a "safe advantage" and "unsafe advantage"? Can an engine give a "safety index" of an advantage?

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What you are looking for is the term "sharp". A position that is sharp requires accurate play to maintain the evaluation; there may only be one good move for either side. From there defining this sharpness is hard to do from a first principles basis as what is sharp for you or I may not be for Magnus. That aside here is a free tool you may want to play with. The fritz engine also claims to have its own sharpness evaluator.

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