In Thor: Ragnarok, why is Thor captured and forced to be a gladiator, but Loki can just walk around and be friends with the Grandmaster?
So Loki falls out of the Bifrost, and lands on Sakaar.
The scavengers come for him; Loki does his projection, "I'm not there, I'm really here, honestly," trick; they don't capture him.
He uses his magic to say hidden & observes what's going on, then cozies himself up to the Grandmaster, telling him something he wants to hear (because it's easier to stab someone in the back when you're close to them and you've gained their trust).
I don't know if this is what actually happened, but it's Loki's modus operandi and it's how I see it going down.
Loki seems like he found a real clears throat awkwardly "sugar daddy" in The Grandmaster. Ok, I could be wrong, but just wait for a second.
When Thor said, "You don't have one of these metal clips on you!" or electrocution devices or whatever they're called, Loki didn't reply. He just gave a wistful look. Perhaps Loki WAS enslaved for like, his first 2 days, but worked from the bottom and got to the top (probably from creating illusions, screaming, "I AM A GOD!" occasionally, killing some of the employees, and just being himself in general).
The Grandmaster said, "I could be millions of years old by now, but here on Sakaar...." He is about to say, "Age is meaningless," when he gives Loki a flirty look, which Loki returns with a nervous look, and Thor is sitting there in confusion. Either the Grandmaster has the hots for Loki, or, something happened during Loki's first week there.
Also following through, the Grandmaster is rich and in charge of Sakaar. Loki loves power. So, if they wanted something from each other, they might've made an arrangement. Loki gets treated like a rular in the planet, and the Grandmaster gets... well, whatever he wants from Loki. Loki is canonlly bi, and the Grandmaster, come on. He plays disco music everywhere and wears blue, yellow, and pink clothes everywhere.
This doesn't mean that they have feelings for each other. Loki wanted hierarchy, and the Grandmaster might've been lonely (a weird thought) and smitten with the looks of this long haired, tight leather pant wearing young man. You've heard of "friends with benefits"? Well, get ready for "aprentinces with benefits". This even makes sense for the scene where Loki says to Thor, "I like it here." Well, he gets treated well there. Who knows in how many ways. And how FAR apart Loki sits with the Grandmaster on that couch during fight matches.
I have to make it clear that I don't ship this. Literally, it sounds like an arrangement between a young god and weird ruler, nothing more. Just a theory.