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If I trade, say, Pikachu, for my friend's Pikachu.

Will the chance of getting a Lucky Pokemon increase when trading two of the same Pokemon than two different Pokemon?

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No. Trading the same Pokemon does not affect the chance.

However, if one of the Pokemon was obtained more than a year ago, then the chance of a Lucky Pokemon increases as per this previous answer.

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    While there is no hard statistical evidence and probably just extremely lucky RNG, I’ve actually more luck with getting lucky Pokémon with same-species trade. I got six lucky Snorlaxes during the “Snoozing with Snorlax” event last year; two shiny Blazikens after community day; back-to-back Alolan Geodudes; and a few others. Again, probably just extremely lucky RNG, but I’m curious if the OPs question has some merit
    – Wondercricket
    Mar 30, 2020 at 11:39

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