2048 AI
ov3y.github.io
ov3y.github.io
Minimax assumes that the game/computer which the AI is playing against is playing adversarially - i.e. that the computer will insert the new tile that's the worst possible tile for the player/AI to receive.
But that's not actually what the game is doing. Instead, new tiles are inserted randomly. As a result, minimax probably isn't the best approach here.
I think something like monte carlo rollouts would work better. In other words, rather than evaluating a move by "what's the worst that could happen if I make this move", evaluate a move by "what is stochastically likely to happen if I make this move, weighted by how good/bad that outcome is for me." (Losing the game would have a big negative weight, of course).
Given that the current AI isn't actually winning the game, I guess that some sort of monte carlo rollout strategy would do better.
It's still cool to see how minimax does, though, so kudos to the authors - it'd be really interesting to see a comparison of different methods.
>I think this is for the same reason that all minimax algos assume optimal play by the opponent: if you assume optimal and they play less than so, it can only work in your favor.
That doesn't make sense when talking about a random opponent, though.
Imagine its chess. You are considering moving a pawn into a position where it can be obviously taken with no cost by the other player, but where, if the other player doesn't take the pawn, you'll get a sure checkmate on the next go.
You'd never make that move against an 'intelligent' player (i.e. in a minimax setup). But you'd definitely consider it against an enemy that moves randomly, because the expected value is so high.
This isn't to discount your empirical experience with this game, just to make a more general point.
Further: the potential for gain (trivial mate against a stupid opponent in a handful of moves) is great. If you are thereby playing for "fastest win over time", taking advantage of random's suboptimality seems sane. In 2048, the bottleneck on your score seems best approximated by how long you survive: pulling stunts won't get you to 2048 all that faster as you need to have worked through enough tiles to arrive at that point: I'd imagine the difference would be at best a tiny fraction of the "required" moves.
Meanwhile, the computer has only a few possible moves, increasing the probability of doing something accidentally optimal. As you approach the end, needing over half the board just for unbuilt path up to 1024 and thereby not having as much scratch space, the probability of it hitting a problematic (even if not "devastating", one that suddenly requires you to reorganize things to "clean up the mess") move seems more more of a problem than when playing chess.
In summary: I just don't think comparing this game to chess is leading to useful intuitions.
Mini-max isn't even 'playing safe'.
Consider the following choice of moves, each leading to one of 5 random tile inserts:
Move A, which leads to 5 possible moves with the following game state goodnesses: ['Loss','Win','Win','Win','Win']
Move B, resulting in: ['99%CertainLoss','99%CertainLoss','99%CertainLoss','99%CertainLoss','99%CertainLoss']
Minimax is never going to choose A. Is that really what you want, even if your strategy is 'play it safe'?
Also, we should note that the "value" in "expected value" doesn't have to mean "score." It could be the logarithm of your score, or your chance of winning against a rational adversary, or even the enjoyability of the game to spectators (if you have a precise metric for that).
First off, the chess counter-example was in response to something general ovolve said about 'all minimax algos'.
More broadly, in practice the issue is that you only have computational resources to search a fraction of possible game states. So where do you direct your limited resources?
Some candidate strategies: minimax with AB, exhaustive, MC sampling.
Even with any of these strategies, we can usually only search the game state tree to a given depth. (This is the tree of 'If I go here, then the game goes there, then I go here etc'.) When we reach our depth limit (if we haven't reached a terminal state - i.e. a win or loss), we use a heuristic (maybe 'number of free tiles') which approximates the value of the state to us at that point.
>Meanwhile, the computer has only a few possible moves, increasing the probability of doing something accidentally optimal.
If the computer only has a few possible moves, then, yes, it might randomly choose the best one. But if it only has a few possible moves, chances are that any tree search technique, even stochastic, will start to expand all of them.
The question is, though, how will you know how good each of these moves is when you start examining are? I.e. Which of the moves available to you right now should you make? With Minimax/AB, you'll get a sense the move is optimal, because you'll look at the consequences of the move, (if the game makes the worst response to me (if I make the best move for me (if the game makes the worst response for me ... (heuristic evaluation)))).
With (sensible) MC search, you'll instead get a sense of which move is optimal by looking at more like what happens (if I make the best move for me (for each of a bunch of random moves the game could make (then if I make the best move for me (for each of a bunch of random moves the game could make ... (heuristic evaluation)))).
My point is that the latter is more suitable for this domain.
>As you approach the end, needing over half the board just for unbuilt path up to 1024 and thereby not having as much scratch space, the probability of it hitting a problematic (even if not "devastating", one that suddenly requires you to reorganize things to "clean up the mess") move seems more more of a problem than when playing chess.
Well, then, if the probability of it randomly hitting a problematic move is high in a certain state, your MC search will be highly likely to come across that move, and thus you'll be likely to avoid moves that bring you to that state. So, no problem.
In summary: 1) If the game is highly likely to make a devastating move by random chance, then you are highly likely to come across that move in your stochastic search, so that's not really an objection. 2) In this game, just as in chess, there are always more states you'd like to search and expand than you have the resources to. Even if the computer only has a handful of 'moves' at any time, in order to tell how good these moves around, you've still got to expand out a lot of states. Choosing to use minimax instead of MC doesn't save you from that.
Also, while I do not have the background you do with search functions, I didn't feel like my comment (which I saw as offering an idea more than a proof: a comment about intuitions based on having wasted way way too much time playing 2048 yesterday and from being in the chess club at a different time in my life) warranted the "let me teach you the basics" paragraph, especially under the "we are working together to figure out why you are wrong" assumption. I am sufficiently confused by these differing approaches to the conversation as to not be certain how to proceed.
Like, "huh, ok, if you had a different idea for why you are wrong, what would it be?" is all I can come up with, but I don't think that fits your side of this interaction. (Maybe, if you simply feel you aren't wrong, you could look at ovolve's code and find something "wrong"/suboptimal with his algorithm? I assumed you had already done this, given the context, but maybe not? Clearly my assumptions are failing here.) I think I will just bow out, actually get some work done, and maybe ask my friends (whom have much more experience in this space than, to my belief, either of us) to explain this to me later ;P.
The utility function is an approximation. So MCMC is aggregating information over an erroneous space, and min/max is also optimizing over an erroneous space too.
Which is the correct thing to do is conditioned on how the utility function behaves. In this scenario I think min/max plays maximumly conservative, which empirically seems to be the best thing to do.
If the utility is minimize free space on the board, the min/max will try to get to a free board but doesn't take risks so gets there in a suboptimal route. The MCMC will take the odd risky move as long as a large proportion of the futures lead to a even emptier board.
Clearly you don't want risky moves, because do enough of them and it ends in disaster (and its a long game). So the utility function should be exponentially weighted against going near risky situations. However, developing such a utility function which combines well in MCMC really requires understanding too much about the future game dynamics.
For MCM to work, the utility function really needs to capture how potentially bad a situation is, and I don;t think that is easy. Min/max is naturally pessimistic in stratergy, which is probably the correct thing to do.
I accept the argument that "minimax is conservative, and conservative is good" might be correct. But I don't think its likely, and, without the time to code my own solution, all I can do is give arguments to that end.
I gave one intuitive argument here: https://news.ycombinator.com/item?id=7381382
Another argument is to remember that minimax, and AB-pruning, is a really strong way of reducing your search space - because of how unfavourable adversary moves are propagated up the tree - which could result in drastic pruning if the minimax assumption is wrong.
One 'bad' state, 6 or 8 ply deep through your branching factor ~10 tree, can result in you pruning entire lines of enquiry using minimax AB; surely that can't be right if the chance of the bad state happening is tiny, and especially if an alternative is chosen which isn't much better.
So I still think that if you want to tune the search algorithm to be risk adverse, then, yes, do so; but minimax is a drastic way to achieve that. But yes, how big of an effect that decision has in practice depends on complicated things, such as correlations in the search tree. (i.e. If you find a bad state in a section of the game tree, maybe there are likely to be other bad states nearby, so its not such a big deal to prune that whole section using minimax).
so you only need to decide 1 of 4 moves at the beginning
1. min max to a finite horizon using a heuristic utility function (as implemented)
2. Dynamic program/MCMC to a finite horizon and use the heuristic. Good at modelling the opponent behaviour, but could lead to bad results with a bad heuristic. (commented out approach)
3. Sample till the game ends (infinite horizon), pick the first move that lead to the game that went the longest (or won). This avoids developing an ad hoc heuristic.
So now I vote for 3. :p
Sure it does. AB search means you play to maximise your own value and minimise the value for your opponent. If your opponent is using a random strategy they will likely make a poor move and that's extra advantage for you.
Your argument here (and elsewhere in this thread) seems to be: if the opponent is random a stochastic strategy is best. This is simply not true. Stochastic strategies like MC search have an advantage over game-tree search when the game's branching factor limits you to considering just a few ply ahead (e.g. as in Go).
You choose either Die or Coin.
If you chose Die, I will roll a die. If the die is 1, I give you $0; otherwise, $1000.
If you choose the coin, I'll give you $10 for heads, and $20 for tails.
If you use a minimax policy, you will choose the coin, because the worst outcome of that choice is a $10 payoff. The 'best' 'worst' outcome is what you get with minimax, and that's $10.
It is true that, as you say, if you instead get $20, you will be pleasantly surprised. "extra advantage for you", as you put it.
But it should be crystal clear that minimax is nonetheless the wrong decision policy here. If you use a minimax policy when the outcomes are random, you will generally be doing it wrong.
There are exceptions (e.g. you are starving and need $10 or else you'll die; or you are in my casino, and you think I'm using loaded dice, in which case the outcomes aren't random and you have an adversary; etc.) but in general, minimax is just wrong there.
>your argument here (and elsewhere in this thread) seems to be: if the opponent is random a stochastic strategy is best. This is simply not true.
No - my argument is that if an opponent is random a minimax strategy is generally not best.
>Stochastic strategies like MC search have an advantage over game-tree search when the game's branching factor limits you to considering just a few ply ahead (e.g. as in Go).
Yes, that's also true.
There are almost no opportunities to make a 'great' move - since a great move with a significantly better effect on chance of winning than a move which doesn't change anything at all - is possible only if the move fixes a huge earlier mistake that you wouldn't have made in the first place.
For this particular game, advantages are temporary and disadvantages are near-permanent - so it makes sense to play very defensively, which minmax does. Imagine a game of Die or Coin, where if you choose coin, then you get $10 for heads, and $20 for tails; and for dice you throw a hundred-side die and get $25 for values 2-100 but if you roll 1, then you get shot and die.
[edit] what I'm saying is that assuming that [a] all payoffs are either effectively 0 or -infinity; and [b] most moves will (in the near expected future) be either 0 chance of the bad event or >0 chance of the bad event; then minmax would generate equal results to MC search - however, MC search would fail badly if you put overly optimistic payoffs, i.e., give too large rewards for 'good moves' and too little penalties for bad moves; and this is hard to estimate.
Minmax works if your payoff scale is completely wrong by orders of magnitude as long as the preference ordering is correct, MC search doesn't. If you know that position A is better than B but don't know if it is 1.1 times better or a million times better - then you can't implement a good MC search but can implemnt minmax.
Lots of situations. If you had $0, obviously.
Probably if you had less than $833.33 (the expected value of the dice in my game), slightly more subtly.
Maybe even other scenarios, depending on how long you wanted to play for, if there was a time cost to playing, if you could play for as long as you wanted, what your utility/risk preferences were, etc.
The point is that there are appropriate tools for reasoning about such games correctly - utility, decision theory, etc. - and they beat minimax.
> The point is that there are appropriate tools for reasoning about such games correctly - utility, decision theory, etc. - and they beat minimax.
Perhaps this is true but you have not convinced me that this is the case here. Moreover, the OP claims to have empirical results showing that AB-pruning does better than MC-search for the game of 2048.
However, I would not dismiss MC-search or expectimax. Whereas alpha-beta should perform well when the current position is in a "good" state, when the current position is in a "bad" or "tricky" state, it almost certainly will perform way worse than search methods that actually take into account the opponent being random.
To consider the extreme example, imagine the position is extremely tenuous and that alpha-beta has proven that if the tiles are placed in the worst possible way in the next several turns, it is guaranteed the position is a loss and a 2048 is impossible. Obviously, it would be better that point to switch to something like expectimax and play so as to minimize the probability of a loss and maximize the probability of recovering, which could still be quite large. Whereas with alpha-beta the only thing you can do is maximize the number of moves until the forced loss, which will often not be the same thing as maximizing the probability of recovery, particularly if most reasonable branches of play lead to a forced loss in the same number of moves.
Similar arguments apply when the position is not a forced loss yet but is still "bad". For example, if the "bad" position is such that your long-term win probability is only 30%, I would expect alpha-beta to pass-up opportunities to fix the position if those opportunities carried any risk of making the position worse, whereas taking those risks is clearly correct if they could fix the position into a probable win and the probability of making things worse was only 1/2 or 1/3.
On the relevant stackoverflow thread, there is another answer much further down that has received far less attention, which is a shame: http://stackoverflow.com/questions/22342854/what-is-the-opti...
I've downloaded and compiled the solver, which uses expectimax rather than alpha-beta, and uses heuristics that are arguably even simpler and dumber, and yet it seems to outperform the OP's solver. It gets 4096 with good consistency and even often gets an 8192, whereas the OP's solver (after a tweak to make it not stop at 2048) can sometimes require a few tries to get a 4096 and only rarely gets an 8192. Granted, there's a large computing power difference, since one runs as native compiled code and one runs as javascript in the browser, but it shows that expectimax, which one would expect to be the most theoretically-sound approach, is indeed at least a comparably good approach, and possibly a superior one.
I was at an early stage, my scoring is just the game score (which apparently is flawed!) so nothing fancy. My approach to the large branching factor with the random tiles was just to ignore them and continue the search with a random one until there are less than N tiles free (I empirically found 4/5 to be good), where I perform a full search and apply a simple mean to the resulting scores. Mine is not even minimax, just a simple depth-first search. It wins between 1/4 and 1/5 of the time in a small sample of 200 games, how does yours perform?
This would be a great AI competition! Too bad it takes so long to run lots of games (mine is ~10 turns a second).
I wonder if web workers would speed it up.
Isn't this guaranteed by the no free lunch theorem?
But 2048 is a highly-structured search space. Even with random opponent, the opponent's choices are constrained by the game's rule structure.
Minimax[1] looks "several" moves ahead to see what may happen. It assumes you always play your best possible move (to MAXimise your score) and your opponent always plays their best possible move (to maximise their score and MINimises yours). Because this creates a huge amount of possibilities, their is a method to reduce the number of moves considered called alpha beta pruning[2]. Some future moves can never lead to a better score for you and so are discarded or ignored.
Both approaches need you to look as many moves ahead as you can - each extra move you consider multiplies the number of possibilities exponentially, so there is another method called iterative deepening[3] which uses the algorithm to look 1 move ahead to find the best move and records that move. Then it repeats looking at 2 moves ahead and if it finds a better score than with 1 move it keeps that instead. Then 3 moves, then 4. The program will continue looking at more and more moves ahead until it reaches a time limit (or a size limit.)
[1] https://en.wikipedia.org/wiki/Minimax#Minimax_algorithm_with...
[2] https://en.wikipedia.org/wiki/Alpha-beta_pruning
[3] https://en.wikipedia.org/wiki/Iterative_deepening_depth-firs...
So minimax is probably right in many cases and close enough to being right in most cases.
It's kinda nice to see how much harder the game becomes if you'd be playing against someone, as even getting to the 128 is hard. The current AI usually only gets to the 32 and sometims the 64 tile.
The most straightforward solution is expectiminimax, which I see this solution has implemented quite nicely. In case someone here isn't familiar with minimax, the OP wrote some very elegant and well-commented code that would be a great primer.
The less computationally-intensive approach we came up with was to model the game state as a graph G(V, E), where V is the set of active tiles and E is a set of edges connecting adjacent tiles, weighted by the function c(v1, v2), which returns the absolute value of the difference between two tiles. For each decision, the AI picks the move that minimizes the sum of all edge weights in the new game state.
The reasoning behind this is that the only way to progress in the game is to have tiles of equal values adjacent to each other, for which the weight in G would be 0. Thus, the AI should try to minimize total weight. Eventually there will be large numbers on the boards with large edge weights to adjacent tiles, so the AI would try to keep these tiles next to other large tiles to minimize the difference.
Since the game is stochastic the approach I described may not work in a worst case scenario, but it could also be applied to the existing minimax solution as the weight function for each node in the tree.
Check out the eval function, and specifically the function smoothness() in grid.js. It implements the edge weighting you describe!
One more thing - have you looked into storing the game tree? I noticed it is starting the search from the beginning every time. I'd expect you would see a branching factor of around 10, so this would only really make a difference at depths greater than 4.
You started an excellent and inspiring GitHub project - I feel like the AI research into this game has only just begun.
Deleted comment
http://www.youtube.com/watch?v=Sn2o2hb1bi0
Mine is using a minimax variant that replaces the minimum nodes with expectation (given that the choice of next tile is uniformly at random). This is sort of the algorithm used by backgammon solvers. The fun thing is that when expectation factors in, the branching factor is quite wide, but the necessary depth for the algorithm to beat humans is much smaller (with 8-ply on Threes this thing is miles ahead of me, no contest)
Try to keep your cascading layers down to 2, and keep about 2 steps ahead when you get to the 3rd layer. Many people have suggested keeping your highest number in the corner but I've found that doing so makes it easier to have 2's and 4's "invade" that fortress of high numbers you're building close to the wall. The best position is for the highest number to be 2nd or 3rd in the row closest to the wall cushioned by the second highest numbers on either side so that you can build up the numbers to either side of them, and eventually add them in.
Example:
X___X___X___X
X___X___2___4
8___16__32__16
64__256_512_128
EDIT: 5 times in a row now. I think I'm addicted.
EDIT: lost on my attempt for 6 in a row. I had to press up right after the 1024 appeared and was scrambling the rest of the game. I managed to get the 512 and 2 256s to appear as well but couldn't get them together.
That way your large values tend to be clump together and you don't get a small value buried which burns up a square.
Also every time I move in the 4th direction I lose in few more moves (I did it at the begining to try this "theory", and it happened then too).
But I couldn't tell if it was really just for my own cognitive convenience. That is, it's easier for me to reason about moves if I constrain moves to keep my high tiles against an edge.
I watched the AI play once, and it of course does not constrain itself this way. To really rub it in, the AI won the game, which I have not.
I always wanted to re-implement it in node and for a cooler game. Alas! I have too many side projects.
The source is here: https://github.com/mkoryak/Evolutionary-Neural-Net-Checker-A...
and you can play against it here: http://mkoryak.github.io/checkers/nn_checker_ai_demo.html (requires java, might also require a 7 year old computer)
EDIT: This AI is a better player than me.
I've won once.
The 'technique' I used was to keep the numbers on the left hand edge.
Then using left, down and up built up the rest of the tiles with the movement to left hand edge being the preferred move when it is possible.
I've not managed to repeat the victory yet though.
1. "tumbler" until 128 can move to the upper left corner (up right down left, repeat)
2. The highest number on the board is always in the upper left. Make the top row descend left to right.
3. Before combining values on the top row, keep hitting up until one of the lower rows will not combine from a left.
4. Often, the slot you are filling (eg, top right or one below top right) will have a two. Alternate pressing left and right until a two appears, allowing you to combine
5. Keep the second row locked as soon as possible with unique values ASCENDING left to right. This way you can use up, left and right without moving the slot you are filling on the far left of the second row.
6. Never fill the top three rows such that a left or right cannot be used. You will be forced to use a down, messing up the top row.
There are a few other pattern recognition tricks that you'll pick up to aid in filling a slot for higher values. Other misc tips:
* try to keep high number squares close together, and merge up to the top row as soon as possible. Otherwise, they will just close off a slot
* You may get a 2 trapped on the top row blocked by a higher value below it. Unfreeze the second row by combining squares & hope that a two appears in the new opening
* only 2's or 4's will appear. They (always?) appear in the space left behind by the previous movement
Whenever two sets of tiles combine in a single move, only one of their scores is counted. For example, let's say that a pair of 8s and a pair of 4s are about to be combined into a new 16 tile and a new 8 tile. The game should be giving us 24 points total for this move because we are creating a 16 tile (16 pts) and an 8 tile (8 pts) where 8+16=24.
However, this does not happen. Only one or the other combination will actually be counted, it seems. In a more egregious case, I combined two 512 tiles to form a new 1024 tile and should have gotten those corresponding points. However, there was also a pair of 2s which combined to make a 4-tile. I only received 4 points for the entire move.
The game should be counting the score from all combined tiles!!! This is why my calculated minimum theoretical score of 20480 (assuming only twos are generated) was completely blown out of the water by winning scores of 12k - because in many cases, the score is incorrectly calculated!
If this is fixed.......
The minimum possible score to win the game (I think...) is 18432, although right now, with the bug, scores can be much lower. Here's how I get that. If we assume only 2s can be generated, then the minimum score is 10*2048 = 20480. However, sometimes a 4 is generated rather than a 2. Apparently, this happens 10% of the time. In theory, it is possible for someone to be given only 4s and also have a perfectly efficient game. In this case, the scoring contribution to get all the 4s from the 2s in the first example is eliminated. The total score of any tier is 2048, so we're remove 2048 from 20480, yielding 18432.
The minimum possible score to reach a winning 2048 tile, once this bug is fixed, should be 18432.
1. Up
2. Right
3. Down
4. Left
5. Go to 1.
That loop scores better than my trying.
- Default to placing tiles in one corner, so, for example, using Up & Left primarily so that high numbers will accumulate in the upper left corner.
- Don't automatically consolidate tiles, but rather, maximally fill out the diagonal half-matrix with 2-4-8-etc... before moving to force a consolidation. Example:
https://www.dropbox.com/s/aargiwvoyg1shdp/Screen%20Shot%2020...
- When no Up or Left move is available, move right. Sometimes, this allows for a general consolidation chain reaction to occur, which generally radically empties the board.
- repeat - sometimes you'll need to switch corners to the upper right.
What's important to realize that after being forced to go right you will occasionally get a small tile trapped either in the corner or directly below it. You need to focus all your energy into merging other tiles into this small corner until it is the largest tile on the top row. Learning how to deal with these small tiles correctly was the key to my victory.
left, down, right, down, (repeat)
My new simple algo is:
1. Down until you cannot go down
2. Left
3. Down until you cannot go down
4. Right
5. Go to 1.
When I began playing I was moving around the board in an unorganized fashion, but am now finding that moving the larger numbers to one corner (bottom-right for me) is the best method. By having your block consolidation take place in one particular area, the odds that you'll have matching large numbers goes up dramatically.
Yup, this one works very well.
Another interesting test was to always play a clockwise or counter-clockwise pattern. The consolidation happens in the middle of the board. This method seemed to result in higher scores than the right/down/right/down pattern.
Something that occurred to me is that "score" and "winning" can have different optimizations. Score is based upon combining blocks, where as winning is based upon reaching the 2048 block. This means the game can be optimzed in two different ways: 1) To maximize score, wherein your goal is to delay reaching 2048 until the last possible moment to allow yourself time to rack up score, and 2) to reach 2048 in the optimal number of moves, which means a lower score.
You're scored on what you combine in a move. So, if you combine two "4" tiles to yield and "8" tile, you'll get 8 points, and so on. To maximize score, the idea would be to basically waste space on the board building up tiles you don't need, while avoiding getting to 2048. In theory, one could build up many 1024 tiles, and maybe even combine several at once to yield multiple 2048 tiles.
To minimize moves in order to reach the fastest would basically be a game of golf. You'd need to reach 2048, but the lowest score in doing so would, by default, mean you've completed the game more efficiently. There's probably some absolute minimum score, but I'm too lazy to figure that out right now...
http://i.imgur.com/4rqdhQQ.jpg
3/10 games were won. Of these three, the lowest score was 12268, with only 8, 4, and 2 tiles left on the board (aside from the 2048 tile, obviously). The highest score of all was also the first game to be won with a score of 13404. This was also the least "efficient" game to be won, with 10 other tiles left on the board aside from 2048.
Intuitively, I want to say that the optimal score is more like 10x2048/2, but I haven't been able to prove that yet (at least not in between work today :) )
Add the logged value of the numbers on the corners to the score and the higher numbers will tend to 'stick' to them. This also serves as a mechanism to guarantee that the new numbers appear away from the large numbers, which tend to block them from combining.
I also turned down the compute time to 20ms and it still runs well.
The basic idea is minimax search. Googling that will get you started, but basically the algorithm plays out the game and keeps a score of the position after every move. Then it just makes the move that leads to the best score.
The "score" here is basically a count of how many free squares there are (with a little extra to keep things aligned if possible).
One major thing with these search algorithms is that the game tree grows exponentially as you move forward. To combat that, implemented alpha-beta pruning and a heuristic to only search the nastier computer moves rather than all of the possibilities.
This basically means he's running an optimized version of minimax -- essentially, depth first search of game states looking for states that minimize tiles and keep them in line.
At each step, he looks several steps into the future, assigns them each a numerical score, then picks the move that leads to the best outcome.
Iterative deepening means that he evaluates all 1 move options, then all two move options, then all 3 moves options and so on. It increases the chances of finding a good move within a time constraint.
Alpha beta is a "lossless" optimization that allows you to more aggressively prune the tree when you're searching.
If there is a guaranteed win, the algorithm will find it. However, if there isn't any, how does it pick a path when it thinks it will lose every time? Are all losses equal to each other? Are some better than the others? Could the elgorithm take the path with best chances where the most of the plays end up winning while only a few of them end up losing?
The gameplay is much more interesting but it seems that getting from 1024 to 2048 is much harder this way, probably because the randomness accumulates over time making the depth search unreliable. The more structured solution accumulates much less risk so has some margin to deal with this.
AI.prototype.translate = function(move) {
return {
0: 'up',
1: 'right',
2: 'down',
3: 'left'
}[move];
}
I find it frustrating to watch it hit (where I have not managed to get to) where you have one block 128, 256, 512, and a 1024. Moving around only makes it harder to join things together.
I am rather convinced this game is more by luck than actual good-play.
It was a little upsetting when I had actually been putting thought into my play and he was doing better.
It's just a wicked fast board implementation with a simple depth first search, as of now. But the idea of "making up" an opponent to make it possible to do alpha beta pruning is a cool idea. I might try to implementet it myself.
I regularly get scores above 50k with AI_DEPTH=5-6 and NUM_TRIES=20-30. My record so far is a score of 220k :-).
I can't speak on statistics but I had best luck when following a set of heuristic rules that minimized the chances of locking valuable tiles in an unreachable location. If you stack the higher value blocks to a corner you're less likely to trap them with lower value blocks that are almost impossible to reach.
Ultimately I think the game is a better comparison to minesweeper where there are rules that will help you progress but at some point you're forced to guess and hope the outcome is in your favor.
edit: I used 'look' 3 times in the last 20 words, what's wrong with me
I have finished solitaire a lot of times.
Colossus by Dennis Feltham Jones
( don't read the wikipedia entry as it contains spoilers ( whole plot ) )
The way I won myself rather quickly was using bottom right priority and ignoring up key(which was suggested by HN).
That is put highest scoring tiles bottom and sorted to the right if possible.
So it is mostly down, right, with some lefts, but no ups. This way is really easy to get 1024, I think I reached 2048 on the 3rd try with this strategy(score was 20k something).
Tada! I got my 2048 tile :) So happy
https://drive.google.com/file/d/0Bz9_OO8kXRkfY0RWdzZ6cDA4RDQ...
Second time around was pretty close http://i.imgur.com/psdSwjM.png
Now, since the board is 16 tiles, and that you need two 2^n can make a 2^(n+1), the max game that can be played is 2^16=65536.
Can a 65536 game can be reallistically won, can it be by OP's AI?
http://cl.ly/image/0H1o1P2Q0A0u/Image%202014-03-12%20at%2012...
I feel bad about my 1024 now, bested by a minmax.
now... for a who can make the optimal AI competition... :P
Heuristics rely on simplified rules which don't accurately model the system on which they're acting 100% of the time. Good heuristics can come close to 100%, however.
But why? Glad you asked!
A 100% correct solution would be to write an algorithm which enumerates all of the possible moves as a decision tree, and walks the decision tree to find the correct answer.
However, given that there are four possible moves the user can make (up, down, left, right) and an upper bound of 32 possible moves the computer can make (computer places "2" or "4" anywhere in a 4x4 grid), each level of the tree could require up to 128 times the number of computations that it took to compute the previous level of the tree.
For example, calculating the first turn is on the order of 128 calculations. Calculating the second move is on the order of 128^2 calculations. Calculating the third is 128^3, an so on. By order of magnitude, how many moves do you think it takes on average to get to 2048? 10^3? Even if we were being really optimistic, maybe it's 10^2. In that case, you'd have to perform somewhere around 128^100 computations in order to solve the game perfectly every time.
Incidentally, python tells me that'd be
5260135901548373507240989882880128665550339802823173859498280903068732154297080822113666536277588451226982968856178217713019432250183803863127814770651880849955223671128444598191663757884322717271293251735781376
Or for fun, this is 128^1000:
16216967556622020264666650854783770951911124303637432562359820841515270231627023529870802378794460004651996019099530984538652557892546513204107022110253564658647431585227076599373340842842722420012281878260072931082617043194484266392077784125099996860169436006660011209817579296678781962552377006552947572566780558092938446272186402161088626008160971328747492043520874011018626908423275017246052311293955235059054544214554772509509096507889478094683592939574112569473438619121529684847434440674120417402088754037186942170155022073539838122429925874353753616104159343594557666561701790904172597025336526662682021808493892812699709528570890696375575414344876088248369941993802415197514510125127043829087280919538476302857811854024099958895964192277601255360491156240349994714416090573084242931396211995367937301294479560024833357073899839202991032234659803895306904298017400980173252106913079712420169633972302183530075897845195258485537108858195631737000743805167411189134617501484521767984296782842287373127422122022517597535994839257029877907706355334790244935435386660512591079567291431216297788784818552292819654176600980398997991681404749384215743515802603811510682864067897304838292203460427757655073776567547507027144662263487685709621261074762705203049488907208978593689047063428548531668665657327174660658185609066484950801276175461457216176955575199211750751406777510449672859082255854777144724233490076402632176089211355256124119453870268029904400183858505767193696897593661213568888386800238409325673807775018914703049621509969838539752071549396339237202875920415172949370790977853625108320092839604807237954887069546621688044652112493076290091990717742355039135117441532973747930089955830518884135334798464113680004999403737245600354288112326328218661131064550772899229969469156018580839820741704606832124388152026099584696588161375826382921029547343888832163627122302921229795384868355483535710603407789177417026363656202726955437517780741313455101810009468809407811220573803353711246329589162370895804762245950918253016369092362406714116443316561598280583720783439888562390892028440902553829376
It's been too long since I've taken any hardcore discrete math for me to reliably reason about the bounds on the number of moves required to win. All I can do is make estimates based on simplifications.
How I arrived at my estimates for the order of magnitude of the minimum number of moves required to win:
At most, you can merge 4 tiles in one move. Assuming you were doing 4 tiles every move, and the game just didn't produce twos, it'd take just 128 moves. Order of magnitude: 10^2.
Assuming exactly one merge per move, and that the game only produces twos, it'd take 1024 moves. Order of magnitude: 10^3.
i KNEW the AI would have been coming sooner or later [no this fast though], GREAT JOB!