51–55 of 55 posts
Wow. This one was insanely stressful compared to every other variant I've seen so far— which is weird, because (unlike the original 2048) I was actually able to win this one. Either way, really cool.
Earlier quoted context omitted.
No, because the problem is deciding whether, for all possible moves you make, there is a move that the opponent can make, such that for all possible moves you make, there is a move the opponent can make, ... that will force a win or lose for one player. This makes it smell like a PSPACE-hard problem (if you make the board size arbitrary). [EDIT] Now I see what you mean, that you could start in a position where you ca…
What I meant is that no matter how the computer plays, you can always serve it a sequence of 2s and 4s that will never reach 2048. Easiest is to make sure there's exactly one 2 on the board, then spawn 4s in the corners, always picking the corner farthest away from the other tiles. From time to time spawn a 2 if you can guarantee it won't align with the other 2 in a couple of moves. Near the endgame it is trivial to…
So let me be formal. I'm posing the following decision problem: given a target score n and some initial board configuration of size (k * k), can one player force a score of at least n? I conjecture this problem is PSPACE-hard.
Your claim is that if n >= 2048 and k = 4 then the answer is always no (though I don't buy your justification). My question is more general, and it's clear that there is not a constant answer (e.g. if n is 2 the answer is always yes, but for some sufficiently large n the answer is always no)
As a computer scientist I can't help but now ask: what is the complexity of this problem? If it's in P then this game is not very interesting (in that I can't hope to fool the AI).
Take chess as an example. It's an EXP-TIME complete game with a branching factor of 35, and AI techniques win pretty much all of the time. In this case you're looking at a much simpler game (if you treat it as an expectiminimax problem, you alternate between turns with a branching factor of less than or equal to 4 and ones with a branching factor of less than or equal to 32). The game is much less complicated than chess. The game tree is really small, meaning that it should be tractable to most AI techniques.
Even though it is reasonable to 'solve' the problem, there are board setups that are unwinnable. Since the player has a lot more power over the game in placing tiles than the agent does in moving them, it is easier for the player to force the game to one of the unwinnable states than it is for the agent to prevent these things. Even if the agent is acting optimally, the player should still be able to win.
As a computer scientist I can't help but now ask: what is the complexity of this problem? If it's in P then this game is not very interesting (in that I can't hope to fool the AI).
Stop me if I'm wrong, but it seems misleading to ask whether or not the problem is in P. It seems to me that it is better to ask more generally whether the problem is tractable. There are a lot of games that are solvable in exponential time (not in P) that are still tractable because the branching factor is low enough that simple methods like minimax searches work well. Take chess as an example. It's an EXP-TIME comp…
You say this game is less complicated than chess, but what measure of complexity are you talking about? Might it be computational complexity? Computer scientists of all people know that simple rules can provide arbitrary complexity.