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I remark that whatever the level of the Lichess' Stockfish I choose it plays in average at the same speed. How could I tune my Stockfish (the one I use in Fritz or Arena) to achieve the same with the same performance ?

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The settings used for the Stockfish levels on lichess can be found at https://github.com/niklasf/fishnet/blob/53e218bd184554265748513f85d19e601f18c310/fishnet.py#L117-L119:

LVL_SKILL = [0, 3, 6, 10, 14, 16, 18, 20]
LVL_MOVETIMES = [50, 100, 150, 200, 300, 400, 500, 1000]
LVL_DEPTHS = [1, 1, 2, 3, 5, 8, 13, 22]

where LVL_SKILL refers to the Skill Level UCI option in Stockfish, LVL_MOVETIMES to the maximum move time in milliseconds, and LVL_DEPTHS to the maximum depth for each of the eight levels.

The move time and depth might not be easy to set in a local GUI, but you can simply use the Skill Level option to adjust playing strength where 20 means best play and 0 is the lowest available level for Stockfish.

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