GAME TREE — By Harsh

A game tree is basically a directed graph where the nodes (vertices of the graph) are used to represent positions in the game. Additionally, the edges (links between the vertices in the graph) are used as “moves” from one position to another. In essence, the game tree is used in games that analyze the moves of players to determine the winner.

Game trees begin at the root node. In games which employ AI-controlled opponents of the player, the game tree is completely searched and pruned using rules, so as to find the best move that can be made, in response to the player’s own.

Basic games with a low total number of moves, such as tic-tac-toe, can easily have their tree searched, to find the right move. On the other hand, games that are much larger, such as chess, have trees that are too large to be searched efficiently. To deal with this, the tree is only partially searched: only the sub-tree from the player’s current position is completely searched to find a suitable move to be made.

In most cases, the higher the extent of searching is done, the better are the chances of finding the right move to be used against the player.

Alpha-Beta Pruning

Alpha: Alpha is the highest value that we have found at any instance along the path of Maximiser. The initial value for alpha is — ∞.

Beta: Beta is the lowest value that we have found at any instance along the path of Minimizer. The initial value for alpha is + ∞.

The condition for Alpha-beta Pruning is that α >= β.

Each node has to keep track of its alpha and beta values. Alpha can be updated only when it’s MAX’s turn and, similarly, beta can be updated only when it’s MIN’s chance. MAX will update only alpha values and MIN player will update only beta values.

Working :

Step 1: At the first step the, Max player will start first move from node A where α=-∞ and β= +∞, these value of alpha and beta passed down to node B where again α= -∞ and β= +∞, and Node B passes the same value to its child D.

Step 2: At Node D, the value of α will be calculated as its turn for Max. The value of α is compared with firstly 2 and then 3, and the max (2, 3) = 3 will be the value of α at node D and node value will also 3.

Step 3: Now algorithm backtrack to node B, where the value of β will change as this is a turn of Min, Now β= +∞, will compare with the available subsequent nodes value, i.e. min (∞, 3) = 3, hence at node B now α= -∞, and β= 3.

In the next step, algorithm traverse the next successor of Node B which is node E, and the values of α= -∞, and β= 3 will also be passed.

Step 4: At node E, Max will take its turn, and the value of alpha will change. The current value of alpha will be compared with 5, so max (-∞, 5) = 5, hence at node E α= 5 and β= 3, where α>=β, so the right successor of E will be pruned, and algorithm will not traverse it, and the value at node E will be 5.

Step 5: At next step, algorithm again backtrack the tree, from node B to node A. At node A, the value of alpha will be changed the maximum available value is 3 as max (-∞, 3)= 3, and β= +∞, these two values now passes to right successor of A which is Node C.

At node C, α=3 and β= +∞, and the same values will be passed on to node F.

Step 6: At node F, again the value of α will be compared with left child which is 0, and max(3,0)= 3, and then compared with right child which is 1, and max(3,1)= 3 still α remains 3, but the node value of F will become 1.

Step 7: Node F returns the node value 1 to node C, at C α= 3 and β= +∞, here the value of beta will be changed, it will compare with 1 so min (∞, 1) = 1. Now at C, α=3 and β= 1, and again it satisfies the condition α>=β, so the next child of C which is G will be pruned, and the algorithm will not compute the entire sub-tree G.

Step 8: C now returns the value of 1 to A here the best value for A is max (3, 1) =3. Following is the final game tree which is the showing the nodes which are computed and nodes which has never computed. Hence the optimal value for the maximizer is 3 for this example.

Advantages of Alpha Beta Pruning

The biggest benefit of Alpha Beta pruning lies in the fact that the branches of the tree can be eliminated thus making the Mini- Max algorithm a bit fast without affecting the final decision of the tree.

The optimization reduces the effective depth to slightly more than half that of simple minimax.

Hope this article is useful.