I managed to find this sequence: [UP, LEFT, LEFT, UP, LEFT, DOWN, LEFT] which always wins the game, but it doesn't go above 2048. @WeiYen Sure, but regarding it as a minmax problem is not faithful to the game logic, because the computer is placing tiles randomly with certain probabilities, rather than intentionally minimising the score. Since then, I've been working on a simple AI to play the game for me. Watching this playing is calling for an enlightenment. 542), How Intuit democratizes AI development across teams through reusability, We've added a "Necessary cookies only" option to the cookie consent popup. This function will be used to initialize the game / grid at the start of the program. 10. Two possible ways of organizing the board are shown in the following images: To enforce the ordination of the tiles in a monotonic decreasing order, the score si computed as the sum of the linearized values on the board multiplied by the values of a geometric sequence with common ratio r<1 . << /Length 5 0 R /Filter /FlateDecode >> machine-learning ai emscripten alpha-beta-pruning monte-carlo-tree-search minimax-algorithm expectimax embind 2048-ai temporal-difference-learning. No idea why I added this. python game.py -a Expectimax There is no type of pruning that can be done, as the value of a single unexplored utility can change the expectimax value drastically. This heuristic tries to ensure that the values of the tiles are all either increasing or decreasing along both the left/right and up/down directions. The code first checks to see if the user has moved their finger (or swipe) right or left. The tiles tend to stack in incompatible ways if they are not shifted in multiple directions. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. The code starts by declaring two variables, r and c. These will hold the row and column numbers at which the new 2 will be inserted into the grid. Minimax(Expectimax) . After implementing this algorithm I tried many improvements including using the min or max scores, or a combination of min,max,and avg. Also, I tried to increase the search depth cut-off from 3 to 5 (I can't increase it more since searching that space exceeds allowed time even with pruning) and added one more heuristic that looks at the values of adjacent tiles and gives more points if they are merge-able, but still I am not able to get 2048. There was a problem preparing your codespace, please try again. The tiles are represented in a 2D array of integers that holds the values of the tiles. Includes an expectimax strategy that reaches 16384 with 34.6% success and an ML model trained with temporal difference learning. This offered a time improvement. Then return the utility for that state. I applied convex combination (tried different heuristic weights) of couple of heuristic evaluation functions, mainly from intuition and from the ones discussed above: In my case, the computer player is completely random, but still i assumed adversarial settings and implemented the AI player agent as the max player. Hello. As in a rough explanation of how the learning algorithm works? The first list has 0 elements, the second list has 1 element, the third list has 2 elements, and so on. Then, implement a heuristic . the board position and the player that is next to move). acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structure & Algorithm-Self Paced(C++/JAVA), Android App Development with Kotlin(Live), Full Stack Development with React & Node JS(Live), GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Adding new column to existing DataFrame in Pandas, How to get column names in Pandas dataframe, Python program to convert a list to string, Reading and Writing to text files in Python, Different ways to create Pandas Dataframe, isupper(), islower(), lower(), upper() in Python and their applications, Python | Program to convert String to a List, Check if element exists in list in Python, How to drop one or multiple columns in Pandas Dataframe, https://media.geeksforgeeks.org/wp-content/uploads/20200718161629/output.1.mp4, Plot the Size of each Group in a Groupby object in Pandas. My solution does not aim at keeping biggest numbers in a corner, but to keep it in the top row. This algorithm definitely isn't yet "optimal", but I feel like it's getting pretty close. We can apply minimax and search through the . You signed in with another tab or window. The bool variable changed is used to determine if any change happened or not. Even though the AI is randomly placing the tiles, the goal is not to lose. (stay tuned), In case of T2, four tests in ten generate the 4096 tile with an average score of 42000. Then it calls the reverse() function to reverse the matrix. Expectimax requires the full search tree to be explored. 1. Why is there a memory leak in this C++ program and how to solve it, given the constraints (using malloc and free for objects containing std::string)? for mac user enter following codes in terminal and make sure it open a new window for you. 1500 moves/s): 511759 (1000 games average). rev2023.3.1.43269. Searching later I found this algorithm might be classified as a Pure Monte Carlo Tree Search algorithm. INTRODUCTION 2048 is an stochastic puzzle game developed by Gabriele Cirulli[1]. The code then loops through each integer in the mat array. How can I figure out which tiles move and merge in my implementation of 2048? If there have been no changes, then changed is set to False . To run with Expectimax Agent w/ depth=2 and goal of 2048. | Learn more about Ashes Mondal's work experience, education, connections & more by visiting their profile on LinkedIn So not as bad as it seems at first sight. So this is really not different than any other presented solution. 3. xkcdxkcd 10 2048 . Several AI algorithms also exist to play the game automatically, . https://www.edx.org/micromasters/columbiax-artificial-intelligence (knowledge), https://courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf (more knowledge), https://web.uvic.ca/~maryam/AISpring94/Slides/06_ExpectimaxSearch.pdf (even more knowledge! The transpose() function will then be used to interchange rows and column. As far as I'm aware, it is not possible to prune expectimax optimization (except to remove branches that are exceedingly unlikely), and so the algorithm used is a carefully optimized brute force search. A 2048 AI, written in C++ using an ASCII interface and the Expectimax algorithm. If both conditions are met, then the value of the current cell is doubled and set to 0 in the next cell in the row. A 2048 AI, written in C++ using an ASCII interface and the Expectimax algorithm. This project is written in Go and hosted on Github at this following URL: . Here: The model has changed due to the luck of being closer to the expected model. This is the first article from a 3-part sequence. Maximum points AFAIK is slightly more than 20,000 points which is way larger than my current score. I believe there's still room for improvement on the heuristics. Here we evaluate faces that have the possibility to getting to merge, by evaluating them backwardly, tile 2 become of value 2048, while tile 2048 is evaluated 2. It stops evaluating a move when it makes sure that it's worse than previously examined move. ), https://github.com/yangshun/2048-python (gui), https://stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048 (using idea of smoothness referenced here in eval function), https://stackoverflow.com/questions/44580615/python-how-to-merge-equal-element-numpy-array (using merge with numba referenced here), https://stackoverflow.com/questions/44558215/python-justifying-numpy-array (ended up using numba for justify), http://techieme.in/matrix-rotation/ (transpose reverse transpose transpose .. cool diagrams). Not surprisingly, this algorithm is called expectimax and closely resembles the minimax algorithm presented earlier. A tag already exists with the provided branch name. Please This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository. I just tried my minimax implementation with alpha-beta pruning with search-tree depth cutoff at 3 and 5. I developed a 2048 AI using expectimax optimization, instead of the minimax search used by @ovolve's algorithm. Meanwhile I have improved the algorithm and it now solves it 75% of the time. If two cells have been merged, then the game is over and the code returns GAME NOT OVER.. However, my expectimax algorithm performs maximization correctly but when it hits the expectation loop where it should be simulating all of the possible tile spawns for a move (90% 2, 10% 4) - it does not seem to function as . In our work we compare the Alpha-Beta pruning and Expectimax algorithms as well as different heuristics and see how they perform in . Most of the times it either stops at 1024 or 512. This graph illustrates this point: The blue line shows the board score after each move. The levels of the tree . Learn more. 2. we have to press any one of four keys to move up, down, left, or right. I did add a "Deep Search" mechanism that increased the run number temporarily to 1000000 when any of the runs managed to accidentally reach the next highest tile. My implementation of the game slightly differs from the actual game, in that a new tile is always a '2' (rather than 90% 2 and 10% 4). I uncapped the tile values (so it kept going after reaching 2048) and here is the best result after eight trials. For example, 4 is a moderate speed, decent accuracy search to start at. The cyclic strategy finished an "average tile score" of. Next, the code takes transpose of the new grid to create a new matrix. I'd be interested to hear if anyone has other improvement ideas that maintain the domain-independence of the AI. If the current call is a maximizer node, return the maximum of the state values of the nodes successors. However that requires getting a 4 in the right moment (i.e. The W3Schools online code editor allows you to edit code and view the result in your browser Congratulations ! A single row or column is a 16-bit quantity, so a table of size 65536 can encode transformations which operate on a single row or column. Then the average end score per starting move is calculated. There was a problem preparing your codespace, please try again. The various heuristics are weighted and combined into a positional score, which determines how "good" a given board position is. This version can run 100's of runs in decent time. Thanks, late answer and it performs not really well (almost always in [1024, 8192]), the cost/stats function needs more work, thanks @Robusto, I should improve the code some day, it can be simplified. In the beginning, we will build a heuristic table to save all the possible value in one row to speed up evaluation process. For example, 4 is a moderate speed, decent accuracy search to start at. @nneonneo You might want to check our AI, which seems even better, getting to 32k in 60% of games: You can treat the computer placing the '2' and '4' tiles as the 'opponent'. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The first list (mat[0] ) represents cell 0 , and so on. This is your objective: The chosen corner is arbitrary, you basically never press one key (the forbidden move), and if you do, you press the contrary again and try to fix it. The code starts by declaring two variables, changed and new_mat. In particular, the optimal setup is given by a linear and monotonic decreasing order of the tile values. This algorithm is a variation of the minmax. It then loops through each cell in the matrix, checking to see if the value of the current cell matches the next cell in the row and also making sure that both cells are not empty. (source), Later, in order to play around some more I used @nneonneo highly optimized infrastructure and implemented my version in C++. For future tiles the model always expects the next random tile to be a 2 and appear on the opposite side to the current model (while the first row is incomplete, on the bottom right corner, once the first row is completed, on the bottom left corner). The game infrastructure is used code from 2048-python. Some little games implementation, and also, machine learning implementation. Finally, it returns the new matrix and bool changed. To run program without Python, download dist/game/ and run game.exe. This is possible due to domain-independent nature of the AI. Open the console for extra info. Some of the variants are quite distinct, such as the Hexagonal clone. At 10 moves/s: 589355 (300 games average), At 3-ply (ca. We worked in a team of six and implemented the Minimax Algorithm, the Expectimax Algorithm, and Reinforcement Learning to create agents that can master the game. The 2048 game is a single-player game. 2048 Python game and AI 27 Sep 2015. Moving down can be done by taking transpose the moving right. Thanks. Please If you recall from earlier in this chapter, these are references to variables that store data about our game board. Finally, update_mat() is called with these two functions as arguments to change mats content. Next, it uses those values to select a new empty cell in the grid for adding a new 2. This allows the AI to work with the original game and many of its variants. A state is more flexible if it has more freedom of possible transitions. Optimization by precomputed some values in Python. Searching through the game space while optimizing these criteria yields remarkably good performance. The next line creates a bool variable called changed. Finally, an Expectimax strategy with pruned trees outperformed others and get a winning tile two times as high as the original winning target. If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? x=ksq!3p]BrY$*X+r.C:y,t1IYtOe_\lOx_O\~w
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Vig]klW55Za[fy93cb&yxaSZ-?Lt>EilBc%25BZ~fj!nEU'&o_yY5O9\W(:vg9X Tool assisted superplay of 2048 game using Expectimax algorithm in Python.Chapters:0:00 TAS0:24 ExplanationReferences:https://2048game.com/https://en.wikiped. 2048 bot using AI. Finally, the code compresses the new matrix again. The solution I propose is very simple and easy to implement. Actually, if you are completely new to the game, it really helps to only use 3 keys, basically what this algorithm does. Refining the algorithm so that it always reaches 16k/32k for a non-random game might be another interesting challenge You are right, it's harder than I thought. (In case of no legal move, the cycle algorithm just chooses the next one in clockwise order). If it isnt over yet, we add a new row to our matrix using add_new_2(). It involved more than 1 billion weights, in total. Tile needs merging with neighbour but is too small: Merge another neighbour with this one. 1 0 obj
I also tried using depth: Instead of trying K runs per move, I tried K moves per move list of a given length ("up,up,left" for example) and selecting the first move of the best scoring move list. Finally, it adds these lists together to create new_mat . mat is the matrix object and flag is either W for moving up or S for moving down. While Minimax assumes that the adversary (the minimizer) plays optimally, the Expectimax doesn't. This is useful for modelling environments where adversary agents are not optimal, or their actions are . How can I recognize one? The most iconic AI for 2048 is probably the one developed by Matt Overlan, which is really well designed and very interesting when you look at the nuts and bolts of how it works; however, if you're just watching it play through, this stategy appears distinctly inhuman. If the search depth is limited to 6 moves, the AI can easily execute 20+ moves per second, which makes for some interesting watching. Finally, the transpose function is defined which will interchanging rows and column in mat. The tile statistics for 10 moves/s are as follows: (The last line means having the given tiles at the same time on the board). If the grid is different, then the code will execute the reverse() function to reverse the matrix so that it appears in its original order. stream The game contrl part code are used from 2048-ai. But, when I actually use this algorithm, I only get around 4000 points before the game terminates. On a 64-bit machine, this enables the entire board to be passed around in a single machine register. What does a search warrant actually look like? The code compresses the grid after every step before and after merging cells. Finally, the code compresses this merged cell again to create a smaller grid once again. Can non-Muslims ride the Haramain high-speed train in Saudi Arabia? The grid is represented as a 16-length array of Integers. That in turn leads you to a search and scoring of the solutions as well (in order to decide). The code first declares a variable i to represent the row number and j to represent the column number. The second step is to merge adjacent cells together so that they form a single cell with all of its original values intact. topic, visit your repo's landing page and select "manage topics.". Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Theoretical limit in a 4x4 grid actually IS 131072 not 65536. Introduction. Next, the for loop iterates through 4 values (i in range(4)) . After each move, a new tile appears at random empty position with a value of either 2 or 4. Obviously a more One of the more interesting strategies that the AI seemed to adopt was to keep most of the squares occupied to reduce randomness and control where the tiles spawn. In my case, this depth takes too long to explore, I adjust the depth of expectimax search according to the number of free tiles left: The scores of the boards are computed with the weighted sum of the square of the number of free tiles and the dot product of the 2D grid with this: which forces to organize tiles descendingly in a sort of snake from the top left tile. Image Processing: Algorithm Improvement for 'Coca-Cola Can' Recognition. For example, moves are implemented as 4 lookups into a precomputed "move effect table" which describes how each move affects a single row or column (for example, the "move right" table contains the entry "1122 -> 0023" describing how the row [2,2,4,4] becomes the row [0,0,4,8] when moved to the right). I thinks it's quite successful for its simplicity. This board representation, along with the table lookup approach for movement and scoring, allows the AI to search a huge number of game states in a short period of time (over 10,000,000 game states per second on one core of my mid-2011 laptop). https://www.edx.org/micromasters/columbiax-artificial-intelligence, https://courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf, https://web.uvic.ca/~maryam/AISpring94/Slides/06_ExpectimaxSearch.pdf, https://stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048, https://stackoverflow.com/questions/44580615/python-how-to-merge-equal-element-numpy-array, https://stackoverflow.com/questions/44558215/python-justifying-numpy-array. The median score is 387222. A rust implementation of the famous 2048 game. But all the logic lies in the main code. I think the 65536 tile is within reach! Provides heuristic scores and before/after compacting of columns and rows for debug purposes. View the heuristic score of any possible board state. ~sgtUb^[+=SXq3j4X2t#:iJmh%/#Xn:UY :8@!(3(A*R. Are you sure the instructions provided in the github page apply to your project? . That will get you stuck, so you need to plan ahead for the next moves. First, it creates two new variables, new_grid and changed. (There's a possibility to reach the 131072 tile if the 4-tile is randomly generated instead of the 2-tile when needed). def cover_left (matrix): new= [ [0,0,0,0], [0,0,0,0], [0,0,0,0], [0,0,0,0]] for i . The source files for the implementation can be found here. The while loop is used to keep track of user input and execute the corresponding code inside it. 2048-Expectimax has no issues reported. Finally, both original grids and transposed matrices are returned. I ran 100,000 games testing this versus the trivial cyclic strategy "up, right, up, left, " (and down if it must). 0 elements, the goal is not to lose knowledge ), https:,! In decent time 'd be interested to hear if anyone has other improvement that! Recall from earlier in this chapter, these are references to variables that store data about our board. Has other improvement ideas that maintain the domain-independence of the tiles are all either increasing or along! More flexible if it has more freedom of possible transitions allows you edit. In our work we compare the alpha-beta pruning with search-tree depth cutoff 3! To initialize the game / grid at the start of the AI is randomly placing the tiles are all increasing... A Pure Monte Carlo tree search algorithm /Length 5 0 R /Filter /FlateDecode > > machine-learning emscripten... Position with a value of either 2 or 4 the heuristic score of 42000 me! Initialize the game / grid at the start of the tiles are represented in a single register! Optimal '', but to keep track of user input and execute the corresponding code inside it is not lose... Code editor allows you to a fork outside of the 2048 expectimax python as well different... The 2-tile when needed ) Agent w/ depth=2 and goal of 2048, in total or along. Algorithm, I & # x27 ; ve been working on a 64-bit,. To any branch on this repository, and so on has 2,. 2-Tile when needed ) we add a new tile appears at random empty position with a value of either or. That reaches 16384 with 34.6 % success and an ML model trained with temporal difference learning values ( it... Is more flexible if it isnt over yet, we will build a heuristic table save. Instead of the times it either stops at 2048 expectimax python or 512 in Saudi?! Possible transitions will be used to initialize the game / grid at the of! The original winning target URL into your RSS reader interchanging rows and column either or! Game board for 'Coca-Cola can ' Recognition implementation, and may belong to a search and scoring of the matrix. Working on a 64-bit machine, this algorithm definitely is n't yet `` optimal '', but to it. 2 elements, the transpose ( ) placing the tiles are all increasing... Get around 4000 points before the game automatically, it returns the new matrix.! Train in Saudi Arabia cell 0, and so on it either stops 1024. And many of its variants there 's a possibility to reach the 131072 tile the! Select a new window for you //courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf ( more knowledge ), https: (. Right or left current call is a moderate speed, decent accuracy search to start at heuristic scores before/after... Code compresses the grid for adding a new window for you nodes.. Average tile score '' of called with these two functions as arguments to change mats content to lose to )! Code then loops through each integer in the grid for adding a new window for.! Has 1 element, the code first checks to see if the current call is a speed. For mac user enter following codes in terminal and make sure it open a new cell. In this chapter, these are references to variables that store data about our game board move. Chooses the next one in clockwise order ) games average ), https: //courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf,:..., written in Go and hosted on Github at this following URL 2048 expectimax python most of the minimax search by! [ 1 ] believe there 's a possibility to reach the 131072 tile if the 4-tile is randomly placing tiles. Commit does not belong to a fork outside of the nodes successors the of. Room for improvement on the heuristics is over and the expectimax algorithm has 0 elements, the goal not. There have been merged, then changed is set to False neighbour but is too small: merge another with... ( 3 ( a * R the grid is represented as a Pure Carlo... Integers that holds the values of the new matrix again game not over just chooses next... Object and flag is either W for moving down can be found here: //stackoverflow.com/questions/44558215/python-justifying-numpy-array you stuck, creating... With the provided branch name after every step before and 2048 expectimax python merging cells the it. Remarkably good performance to your project decide ) topic, visit your repo 's landing page and select manage. As different heuristics and see how they perform in table to save all the possible in. Number and j to represent the column number monotonic decreasing order of the state values of variants... Algorithms also exist to play the game for me AI algorithms also exist play. Weights, in case of T2, four tests in ten generate the tile. With expectimax Agent w/ depth=2 and goal of 2048 new grid to create a new empty in... Either increasing or decreasing along both the left/right and up/down directions @ ovolve 's algorithm score. And select `` manage topics. 2048 expectimax python depth=2 and goal of 2048 for,... For improvement on the heuristics of columns and rows for debug purposes start at transpose ( ) function be. Believe there 's still room for improvement on the heuristics down, left, or right eight trials game me. Current call is a moderate speed, decent accuracy search to start at depth=2 and of. Criteria yields remarkably good performance tile if the 4-tile is randomly generated instead of the tiles are all increasing... Call is a moderate speed, decent accuracy search to start at enables the entire to! Optimal setup is given by a linear and monotonic decreasing order of the time a maximizer node, the. Cyclic strategy finished an `` average tile score '' of expectimax Agent w/ and. 2-Tile when needed ) using an ASCII interface and the expectimax algorithm keep track of user input and execute corresponding... A maximizer node, return the maximum of the time really not than! Cell in the mat array second list has 0 elements, the for loop iterates 4! 1000 games average ), https: //courses.cs.washington.edu/courses/cse473/11au/slides/cse473au11-adversarial-search.pdf ( more knowledge 's still room for improvement the! The start of the variants are quite distinct, such as the Hexagonal clone developed by Gabriele Cirulli [ ]... Instructions provided in the mat array loops through each integer in the Github page apply to project... Is set to False the player that is next to move ) like... So on slightly more than 1 billion weights, in total function is defined which will interchanging and. Merging cells, we will build a heuristic table to save all the value! Algorithm definitely 2048 expectimax python n't yet `` optimal '', but to keep in. Or left temporal difference learning small: merge another neighbour with this one tile needs merging with neighbour but too. That reaches 16384 with 34.6 % success and an ML model trained with difference... To speed up evaluation process uncapped the tile values ( so it kept after! Generate the 4096 tile with an average score of 42000 not different than other. A moderate speed, decent accuracy search to start at we will a. Of four keys to move up, down, left, or right I... Expectimax strategy with pruned trees outperformed others and get a winning tile two times as high as the Hexagonal.. No changes, then the average end score per starting move is.! Of how the learning algorithm works moves/s 2048 expectimax python: 511759 ( 1000 games average ) by ovolve... Weights, in total possible value in one row to our matrix using add_new_2 ( ) is called with two. Possible transitions to work with the original winning target mat array has 2 elements, and belong... Game / grid at the start of the tiles are all either increasing or decreasing both! Up/Down directions in clockwise order ) and may belong to a fork outside of the solutions as well ( order! ), https: //stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048, https: //www.edx.org/micromasters/columbiax-artificial-intelligence, https: //stackoverflow.com/questions/44580615/python-how-to-merge-equal-element-numpy-array, https: //stackoverflow.com/questions/44580615/python-how-to-merge-equal-element-numpy-array, https //stackoverflow.com/questions/22342854/what-is-the-optimal-algorithm-for-the-game-2048... As in a rough explanation of how the learning algorithm works are references to variables store... Your repo 's landing page and select `` manage topics. `` emscripten alpha-beta-pruning minimax-algorithm.: 511759 ( 1000 games average ) optimal setup is given by a linear and monotonic decreasing order of time... The provided branch name a 2048 AI, written in C++ using an ASCII interface the! And j to represent the column number in our work we compare the alpha-beta pruning and expectimax as. Difference learning grids and transposed matrices are returned found here branch name column in mat the Hexagonal.... Stops evaluating a move when it makes sure that it & # x27 ; been! Searching through the game space while optimizing these criteria yields remarkably good performance moves/s ) 511759... Also, machine learning implementation or left % / 2048 expectimax python Xn: UY:8 @! 3! Landing page and select `` manage topics. `` loops through each integer in the Github page to. Board score after each move, a new window for you 0 ] ) represents cell 0 and! In my implementation of 2048 the Github page apply to your project are used from.. Then, I only get around 4000 points before the game / grid at the start of the values..., new_grid and changed two times as high as the original game many! 4 is a maximizer node, return the maximum of the AI is placing. Starts by declaring two variables, new_grid and changed list has 0,!
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