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Rorys approach was different, taking a leaf out of one of his favourite games. The game’s most memorable lesson for Rory was blackmail.
Is the apostrophe in this case correct?
Yes, it is a genitive. Whose most memorable lesson? The game's.
Whose can be considered to work with things, too.
Thanks to @MichaelHarvey:
Also at the beginning, there should be an apostrophe. Whose approach? Rory's.