One of you was never briefed. When the game starts, everyone is secretly told the same location (say, The Beach) and given a job there, like Lifeguard or Ice Cream Seller. One player gets no briefing at all. That player is the spy, and nobody knows who it is.
The question chain. Everyone gives one opening line, then the questions start. One player asks at a time, and whoever answers asks the next question — you just cannot fire straight back at the person who asked you. Each round has a shared pool of questions; the counter shows how many are left. Vague answers look suspicious. Too-precise answers tell the spy exactly where you are.
Accuse, defend, then vote. When the questions run out, everyone in turn gets one open accusation: name your suspect — they defend themselves on the spot — or pass. Then the secret vote: accuse or pass, sealed until everyone has voted. If most players accuse the same person, that person is revealed.
How it ends. Everyone who got the briefing is an insider. Catch the spy with a majority and the insiders win. Pile onto an innocent player and the spy wins. The spy has one more trick: guess any time from round 1; it ends the game. Right, spy wins on the spot. Wrong, the insiders win.
3 rounds, no eliminations. The game lasts at most 3 rounds. If no majority ever forms, the spy slips away and wins. Nobody is ever eliminated: everyone plays to the very end.