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# TicTacToe
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This repository is a simple implementation of the game of TicTacToe and some
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experiments around Reinforcement Learning with it.
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## Structure
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* `game.py` contains the implementation of the game itself.
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* `generate_board_hash_list.py` creates a pickle object containing a list of hash
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for every possible unique non-ending variation of the board. It is useful to
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create the Q-table latter and need to be precomputed.
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* `q_learning.py` contains some experimentation with Q-learning using the
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TicTacToe game as an exemple.
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## Implementation details
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The TicTacToe game is a Python Class. The board is a 3x3 ndarray (numpy) of the
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dtype `int`. Input it taken from 1 to 9 following this scheme:
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```
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+---+---+---+
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| 1 | 2 | 3 |
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+---+---+---+
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| 4 | 5 | 6 |
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+---+---+---+
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| 7 | 8 | 9 |
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+---+---+---+
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```
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It is automatically raveled/unravaled when necessary.
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We only need to check if there is a win above 5 moves because it impossible to
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have a winner below this limit. At 9 moves the board is full and the game is
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considered draw if no one won.
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## Combinatorics
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Without taking into account anything, we can estimate the upper bound of the
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number of possible boards. There is `3**9 = 19683` possibilites.
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There are 8 different symetries possibles (dihedral group of order 8, aka the
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symetry group of the square). This drastically reduce the number of possible
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boards.
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Taking into account the symetries and the impossible boards (more O than X for
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example), we get `765` boards.
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Since we do not need to store the last board in the DAG, this number drops to
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`627` non-ending boards.
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This make our state space size to be `627` and our action space size to be `9`.
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## Reward
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* `+1` for the winning side
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* `-1` for the losign side
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* `±0` in case of draw
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The reward are given only at the end of an episode, when the winner is
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determined. We backtract over all the states and moves to update the Q-table,
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given the appropriate reward for each player.
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Since the learning is episodic it can only be done at the end.
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The learning rate α is set to `1` because the game is fully
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deterministic.
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We use an ε-greedy (expentionnally decreasing) strategy for
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exploration/exploitation.
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The Bellman equation is simplified to the bare minimum for the special case of
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an episodic, deterministic, 2 player game.
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Maybe some reward shaping could be done to get better result and we would also
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try a more complete version of the Bellman equation by considering Q[s+1,a]
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which we do not right now. This would necessitate to handle the special case of
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the winning board, which are not stored in order to reduce the state space
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size.
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