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Can anyone please help me to figure out a method in which a type of users can assign quests to a group of players.
They also determine rewards that must be given to the players that successfully complete the quest.

Hello,

If you have your users in a database, you could have a 'permission_type' or whatever column that then interacts with your code differently..

For example,

User: Administrator has Permission_Type 'Admin' > can do everything.
User: GameMaster has permission_type 'Gamemaster' > can assign quests to players/player groups/and give rewards.
User: Player has permisison_type 'Player' > Is treated as a player, without now modificational permissions other than what is required of the game design.

If you require more detailed answer in code level, please let me know what language/storage you are planning to use.

Hope this helps!

It'd essentially be the same as a developer made quest, given by an NPC.

Except here one player would presumably choose from a selection of quest which one to present, then they would send some message to the other player which asks that player if they want to accept the quest.

Which is just slightly more complicated than a quest giving NPC.