-The Riddler, Batman: The Long Halloween
The answer, according to Batman: Arkham Asylum, is when the killer's sole purpose is to provide the player with an opportunity to acquire achievements and fulfill completionist desires. The Riddler does not torment Batman, provide clues to a larger mystery, or care about anything else happening in the Asylum. He is omniscient ("I can't believe you found it!" the moment you pick up a trophy in a secluded vent) and entirely detached from the game's main plot. He is an achievement unlocked pop-up made diegetic, in the loosest sense of the word. Here's another riddle: I appear in games under a variety of guises, but what I tell you is always opposed to what I ask of you. Who am I? Answer: It's that old rogue ludonarrative dissonance!
The main plot of the game's narrative involves hurrying to stop Joker. We need to hurry because we are Batman and our Batmobile is being torn up, or Commissioner Gordon has been kidnapped, or the warden has been kidnapped, or we need to find Dr. Young's research papers before Joker's men, or Ivy's plants are about to destroy the island and then Gotham City, or you get the point.
The riddles are a significant ludic portion of the game. By rewarding the player with XP that can be used to purchase ability upgrades, they become a system inextricably tied to the core skill progression and combat experience. Being rewarded for finding the scattered Riddler trophies and other items suggest, as Clint Hocking puts it, a contract between designer and player. This contract says "Take your time exploring every nook and cranny and you will be rewarded with the skills you need to battle Joker's men." The narrative contract says something like "Time is of the essence! If you hurry I promise to make you feel like you ARE Batman." The narrative is never given a fair chance to so much as fake a sense of urgency in the player that mimics that felt by Batman, because the very existence of the ludic riddle contract discredits it. The moment players engage with the riddles, they disengage from being Batman.
Could this dissonance have been avoided? Of course. If the level of abstraction of the Riddler's demands was reduced they could have been made to support the ludonarrative rather than work against it. Okami is one of the few games that manages to pull this off successfully. Whether it's feeding wild animals or restoring wilted trees, each secondary task that pushes the player towards that 100% completion also contributes to the restoration of nature. It is almost impossible to "abuse" the reward and sidequest system without engaging with the narrative of saving a corrupted land. It is thematic and ludonarrative consonance.
Arkham Asylum could have similarly imbued the riddles with narrative meaning. Maybe you wouldn't seek trophies, but clues to the whereabouts of The Riddler, who has masterminded some elaborate plan to mess with Batman. Perhaps his victim is an unaware Oracle, the other character Batman regularly interacts with through audio alone. The idea is to make it so even players who are just in it for the XP and achievements are nonetheless participating in a narrative that does not contradict the main plot with Joker. The choice to pursue the riddles is then framed as a choice to spend a few minutes working towards saving Oracle before getting back to Joker and his schemes. This would be a choice we could believe Batman would make within the context of the story.
When the hand-to-hand combat and use of gadgets and atmosphere all come together the game shines and we think "I am Batman". I wonder, when crawling through a dead-end vent for the thirteenth time in search of a trophy is Batman thinking "I am a gamer"?