Is perfect play always possible in a game with perfect information and more than 2 players? So I was interested in this game called Sprouts, invented by Conway and Paterson. It's a game with perfect information, so no information is hidden, like in chess and unlike stratego or card games. I understand that perfect play is possible in such games. But happens when you have more than 2 players? In sprouts, it's perfectly possible to play it with more than 2 players, you'd just have to make it so that the player who moves last, would win. And a draw can never be reached. So a player must win. So I guess perfect play is possible too? It's just confusing. You could play the game of sprouts with 10 players, or a hundred players. But there should always be one player who can force a win, no matter what the other 9 or 99 other players do. Or not?
I think it's quite paradoxical.
EDIT:
I also imagined 3-player chess, which exists, and how it would work for a computer program.  And it's weird because if you'd be some pieces up, compared to the other two players, it may be the best thing for the other two players to cooperate against you, since they might loose otherwise. But then it might not be a good idea to be several pieces up. It's weird to me. It makes me wonder if it's still predetermined or not.
 A: In the case of two-player games, especially if we know that the game must end in finitely many steps, we can imagine the whole game tree and backtrack from the leaves in such a way that at each node that player that moves will choose the move that is most beneficial for them, assuming that their opponent does the same. This is essentially the minimax algorithm.
Now, in the case of multiple players something is radically different: in the case when one of the players has no winning move, they still need to choose a move, but their choice can determine which of the other players wins. In such a case it is not obvious to define what perfect play would be.
One way around this is can be the following: each player has their own ranking of all the players which says who they want to have as winner of the game. Obviously, each player will put themselves at the top of this ranking. Now, when they choose their move in a minimax-type approach, they will aim to choose a move that favors a player as high as possible in their ranking. With this, we can talk again about optimal play, but we expect that the results might change when players change their rankings.
Update: The discussion above is somewhat abstract, so here's also a made-up game to go with it. Consider $n$ players placed in a circle, taking turns clockwise, where the only action a player can do is to choose one of the other players and award them $1$ point. The first player to collect $k$ points wins (assume $k > 0$). For $n = 2$ this game is trivial because the players have no choice and always award a point to their opponent, so the second player will always win. For $n \geq 3$ none of the players has a winning strategy, because their own points always depend on the actions of the others. Yet, the game will still always end, in at most $(k - 1)n + 1$ turns, and will have exactly one winner.
