Game tree

In the world of combinatorial game theory, a game tree is a special diagram that shows every possible situation in a sequential game where players know all the information. Games like chess, checkers, Go, and tic-tac-toe are examples of such games.

A game tree helps us understand how complicated a game is by showing all the different ways it can progress. Because some games, like chess, have very large game trees, computers use smaller parts of these trees to make playing the game possible. There are many ways to work with game trees. If we can create the whole tree, certain careful steps can help find the best move. When that's not possible, other clever methods help computers make good decisions in these games.

Understanding the game tree

A game tree is a way to show all the possible moves and results in a game where players take turns, like chess or tic-tac-toe. Imagine each spot on the tree as a different position of the game pieces, and each branch as a possible move a player can make.

Game trees help us understand how games can unfold and are used in computer programs that play games. For simpler games like tic-tac-toe, we can look at the whole tree. But for bigger games like chess, computers can only look at parts of the tree, depending on how much time they have. This helps them choose better moves by exploring more options.

Solving game trees

With a complete game tree, it is possible to "solve" the game — meaning to find a sequence of moves that either the first or second player can follow to get the best result, like winning or tying. One way to do this is called the deterministic algorithm, which includes steps like coloring the end results of the game (wins for player 1, wins for player 2, and ties) and then working backwards to decide the best moves.

Randomized algorithms can also be used to solve game trees. These are faster and more practical because the order of solving is random, making it harder for an opponent to predict the moves. The algorithm uses a method called "short-circuiting" to decide the outcome based on whether the node is an "OR" or "AND" operator.