First party is where the console company develops it in their own studio.
Second party is where the company gives the project to a wholly owned subsidiary, i.e. a separate corporate entity that could in theory split and join another company, example: Bungie under MS, Rare, Monolith Soft. For games like Mario Party, where the IP is all first-party and it is also published by the first-party, it is effectively second-party as the real developer is "invisible' as in a true second-party. It's like subcontracting a job: you still own it.
Third party is developed by an external company.
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