By Max Pumperla and Kevin Ferguson
This article, from Deep Learning and the Game of Go, discusses tree search algorithms and their relevance to various types of games.
What games does tree search apply to?
Tree search algorithms are mainly relevant to games where you take turns, and there’s a discrete set of options on each turn. Many board and card games fit this description. On the other hand, tree search won’t help a computer play basketball, charades, or World of Warcraft. We can further classify board and card games according to two useful properties.
- Deterministic vs non-deterministic. In a deterministic game, the course of the game depends only on the players’ decisions. In a non-deterministic game, there’s some element of randomness involved, like rolling dice or shuffling cards.
- Perfect information vs hidden information. In perfect information games, both players can see the full game state at all time; the whole board is visible, or everyone’s cards are on the table. In hidden information games, each player can only see part of the game state. This is common in card games, where each player is dealt a few cards and can’t see what the other players are holding. Part of the appeal of hidden information games is guessing what the other players know based on their game decisions.
Table 1. Classifying board and card games
In this article, we primarily focus deterministic perfect information games. In particular, the minimax algorithm generates perfect play for any such games.
Anticipating your opponent with minimax search
How can we program a computer to decide what move to make next in a game? To start, we can think about how humans would make the same decision. Let’s start with the simplest deterministic perfect information game: tic-tac-toe. The technical name for the strategy we’ll describe is minimaxing (a contraction of “minimizing and maximizing”). You can sum it up in one sentence: assume your opponent is as smart as you are. Let’s see how minimaxing works in practice.
Figure 1. What move should X make next? This is an easy one: playing in the lower right corner wins the game.
Take a look at figure 1. What move should X make next? No tricks here; taking the lower right corner wins the game. We can make that into a general rule: take any move that immediately wins the game. This plan can’tgo wrong. We could implement this rule in code with something like this:
def find_winning_move(game_state, next_player): for candidate_move in game_state.legal_moves(next_player): ❶ next_state = game_state.apply_move(candidate_move) ❷ if next_state.is_over() and next_state.winner == next_player: return candidate_move ❸ return None ❹
❶ Loop over all legal moves
❷ Calculates what the board would look like if we pick this move
❸ This is a winning move! No need to look further
❹ Can’t win on this turn
Figure 2 illustrates the hypothetical board positions this function examines. This structure, where a board position points to possible follow-ups, is called a game tree.
Figure 2. An illustration of an algorithm to find the winning move. We start with the position at the top. We loop over every possible move and calculate the game state that would result if we played that move. Then we check if that hypothetical game state is a winning position for X.
Let’s back up a bit. How did we get in this position? Perhaps the previous position looked like figure 3. The O player naively hoped to make three in a row across the bottom. But that assumes that X cooperates with the plan. This gives a corollary to our previous rule: don’t choose any move that gives our opponent a winning move.
Figure 3. What move should O make next? If O plays in the lower left, we must assume that X will follow up in the lower right to win the game. O must find the only move that prevents this.
def eliminate_losing_moves(game_state, next_player): opponent = next_player.other() possible_moves =  ❶ for candidate_move in game_state.legal_moves(next_player): ❷ next_state = game_state.apply_move(candidate_move) ❸ opponent_winning_move = find_winning_move(next_state, opponent) ❹ if opponent_winning_move is None: ❹ possible_moves.append(candidate_move) ❹ return possible_moves
❶ possible_moves will become a list of all moves worth considering
❷ Loops over all legal moves
❸ Calculates what the board would look like if we play this move
❹ Does this give our opponent a winning move? If not, this move is plausible
Figure 4. What move should X make? If X plays in the center, there are two ways to complete three-in-a-row: top middle and lower right. O can only block one of them, and X is guaranteed a win.
Now, we know that we must block our opponent from getting into a winning position. Therefore, we should assume that our opponent’s going to do the same to us. With that in mind, how can we play to win? Take a look at the board in 4. If we play in the center, we have two ways to complete three in a row: top middle or lower right. The opponent can’t block them both. We can describe this general principle as: look for a move where our opponent can’t block from setting up a winning move. Sounds complicated, but it’s easy to build this logic on top of the functions we’ve already written:
def find_two_step_win(game_state, next_player): opponent = next_player.other() for candidate_move in game_state.legal_moves(next_player): ❶ next_state = game_state.apply_move(candidate_move) ❷ good_responses = eliminate_losing_moves(next_state, opponent) ❸ if not good_responses: ❸ return candidate_move ❸ return None ❹
❶ Loop over all legal moves
❷ Calculates what the board would look like if we play this move
❸ Does our opponent have a good defense? If not, pick this move
❹ No matter what move we pick, our opponent can prevent a win
Our opponent will anticipate that we’ll try to do this, and also try to block such a play. We can start to see a general strategy forming:
- First, see if we can win on the next move. If yes, play that move.
- If not, see if our opponent can win on the next move. If yes, block that.
- If not, see if we can force a win in two moves. If yes, play to set that up.
- If not, see if our opponent could set up a two-move win on their next move…
Notice that all three of our functions have a similar structure. Each function loops over all valid moves and examines the hypothetical board position that we’d get after playing that move. Furthermore, each function builds on the previous function to simulate what our opponent would do in response. If we generalize this concept, we get an algorithm that can always identify the best possible move.
Solving tic-tac-toe: a minimax example
In the previous section, we examined how to anticipate your opponent’s play one or two moves ahead. In this section, we show how to generalize that strategy to pick perfect moves in tic-tac-toe. The core idea is exactly the same, but we need the flexibility to look an arbitrary number of moves in the future.
First let’s define an enum that represents the three possible outcomes of game: win, loss, or draw. These possibilities are defined relative to particular player: a loss for one player is a win for the other.
Listing 1. An enum to represent the outcome of a game.
class GameResult(enum.Enum): loss = 1 draw = 2 win = 3
Imagine we have a function
best_result that takes a game state and tells us the best outcome that a player can achieve from that state. If that player can guarantee a win — by any sequence, no matter how complicated — the
best_result function returns
GameResult.win. If that player can force a draw, it returns
GameResult.draw. Otherwise, it returns
GameResult.loss. If we assume that function already exists, it’s easy to write a function to pick a move: we loop over all possible moves, call
best_result, and pick the move that leads to the best result for us. There may be multiple moves that lead the equal results; we can pick randomly from them in that case. Listing 2 shows how to implement this.
Listing 2. A game-playing agent that implements minimax search.
class MinimaxAgent(Agent): def select_move(self, game_state): winning_moves =  draw_moves =  losing_moves =  for possible_move in game_state.legal_moves(): ❶ next_state = game_state.apply_move(possible_move) ❷ opponent_best_outcome = best_result(next_state) ❸ our_best_outcome = reverse_game_result(opponent_best_outcome) ❸ if our_best_outcome == GameResult.win: ❹ winning_moves.append(possible_move) ❹ elif our_best_outcome == GameResult.draw: ❹ draw_moves.append(possible_move) ❹ else: ❹ losing_moves.append(possible_move) ❹ if winning_moves: ❺ return random.choice(winning_moves) ❺ if draw_moves: ❺ return random.choice(draw_moves) ❺ return random.choice(losing_moves) ❺
❶ Loops over all legal moves
❷ Calculates the game state if we select this move
❸ Since our opponent plays next, figure out their best possible outcome from there. Our outcome is the opposite of that
❹ Categorizes this move according to its outcome
❺ Picks a move that leads to our best outcome
Now the question’s how to implement
best_result. As in the previous section, we can start from the end of the game and work backward. Listing 3 shows the easy case: if the game is already over, there’s only one possible result. We return it.
Listing 3. First of the minimax search algorithm. If the game is already over, we already know the result.
def best_result(game_state): """Find the best result that next_player can get from this game state. Returns: GameResult.win if next_player can guarantee a win GameResult.draw if next_player can guarantee a draw GameResult.loss if, no matter what next_player chooses, the opponent can still force a win """ if game_state.is_over(): if game_state.winner() == game_state.next_player: return GameResult.win elif game_state.winner() is None: return GameResult.draw else: return GameResult.loss
If we’re somewhere in the middle of the game, we need to search ahead. By now, the pattern should be familiar. We start by looping over all possible moves and calculating the next game state. Then we must assume our opponent will do their best to counter our hypothetical move. To do this we can call
best_result from this new position. That tells us the result our opponent can get from the new position; we invert it to find out our result. Out of all the moves we consider, we select the one that leads to the best result for us. Listing 4 shows how to implement this logic, which makes up the second half of
best_result. Figure 5 illustrates the board positions this function considers for a particular tic-tac-toe board.
Figure 5. A tic-tac-toe game tree. In the top position, it’s X’s turn. If X plays in the top center, then O can guarantee a win. If X plays in the left center, X will win. If X plays right center, then O can force a draw. Therefore, X will choose to play in the left center.
Listing 4. Implementing
best_result_so_far = GameResult.loss opponent = game_state.next_player.other for candidate_move in game_state.legal_moves(): ❶ next_state = game_state.apply_move(candidate_move) ❷ opponent_best_result = best_result(next_state) ❸ our_result = reverse_game_result(opponent_best_result) ❹ ifour_result.value>best_result_so_far.value: best_result_so_far = our_result returnbest_result_so_far
❶ See what the board would look like if we play this move.
❷ Find out our opponent’s best move.
❸ Whatever our opponent wants, we want the opposite.
❹ See if this result is better than the best we’ve seen so far.
If we apply this algorithm to a simple game, such as tic-tac-toe, we get an unbeatable opponent. You can play against it and see for yourself: try the
play_ttt.py example on GitHub (https://github.com/maxpumperla/deep_learning_and_the_game_of_go). In theory, this algorithm also works for chess, Go, or any other deterministic perfect information game. In reality, it’s far too slow for any of those games.
That’s all for this article. To learn about enhancing tree search with deep learning to make game-playing machines, read the first chapter of Deep Learning and the Game of Go here and see this slide deck.