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• Figure 2: Game tree for the symmetry-pruned tic-tac-toe and the corresponding Boolean formulas.
• Tic-tac-toe
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Without caching, playing a single game takes about 1.

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Caching takes that down to about 0. A full search will encounter the same position over again many times. Caching allows us to speed this process up considerably: If we've seen a position before, we don't need to re-explore that part of the game tree.

If we're at the end of a game, we return the game result as the value for the position. Then we either get the minimum or the maximum of all of those values, depending on who's turn it is:.

Going back to caching for a moment, the caching code also takes into account positions that are equivalent. That includes rotations as well as horizontal and vertical reflections. Without taking into account rotation and reflection, a single game takes approximately 0.

If you're interested in having a closer look, the github repo with all of the code for this project is available at:. Below are the most recent demo results. The current qtable agent plays near-perfect games as O against itself, minimax, and random. Getting good result for the X player was pretty straightforward, but for O it took quite a bit of fiddling with the hyperparameters.

Below are the winning percentages for the different combinations of minimax and random players, with games played for each combination:. We can see that if both players play perfectly, only a draw is possible, but X is more likely to win if both players play at random. Go to the "misc" section of your settings and select night theme. Navigation menu.

The diagram below shows an example of minimax applied to a board position: If the position is an end-of-game state, then the value of that position is the result of that game - that's the terminating condition for the recursive calls. Demo project for different approaches for playing tic-tac-toe. Code requires python 3, numpy, and pytest. View on GitHub.

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Share to Reddit. Share to Hacker News. Share to Facebook. Sore eyes? Unfortunately, this is far too long for the very first moves as the algorithm completly unfolds all the possible games. In memory, this is not a big issue as we unfold on request but in time A significant speed-up can be obtained making use of AlphaBeta prunning. If the current node is the play of the opponent if you already know that one of the sons is winning for him, then you know the value of the current node as well. These are the alpha and beta cuts.

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Now, you just need to manually unroll these recursive functions to decide whether you stop or go on evaluating the sons. This might give you something like the following:. You can now embed everything in a function to select an action and play the game. The full solution is given in preliminaries. This game doesn't allow one to win all the time and a significant proportion of games played results in a draw. This study is aimed at evolving of no-loss strategies in the game using decision tree algorithm and comparing them with existing methodologies, mainly focused on the implementation of the game using the minimax algorithm.

The minimax algorithm does provide an optimal no-loss strategy by assuming that both players play optimally. So the question that comes out is what happens when the opponent plays un-optimally, in these cases the minimax proved to play non optimal moves, even though it wins at the next state rather than the expected state.

Thus this paper provides a clear study of those trivial states and provides an optimal game play using an decision tree algorithm independent of the opponent's game strategy.