Consider the game tree for a 2-player turn-taking minimax game shown in the figure. The value of the terminal node represent the utility of the game state if the game end there. There are two player Max & Min. At any particular point state of the game Max prefers to move to a state of maximum value, on the other hand Min prefers to move to a state of min value. Suppose Max starts the game at the root and has three strategies 1, 2, 3. Next, Min plays and also has three strategies 1, 2, 3. The game end there. Both player always take optimal strategies throughout the game. At the root, the best strategy for max is ______(int).

The correct answer is 1.

To determine the optimal choice and game value, we apply the Minimax algorithm, working backward from the terminal leaves at the bottom of the tree up to the root node.

Step 1: Analyze the Min player's optimal moves
The bottom level of the tree shows the terminal utility values after the Min player chooses their strategy (1, 2, or 3). Because Min is a minimizing player, they will always choose the strategy that leads to the lowest utility for each subtree:

If Max chooses Strategy 1, the terminal utilities available to Min are 8, 6, and -1. Min chooses the minimum:
$$min(8,6,-1)=-1$$

If Max chooses Strategy 2, the terminal utilities available to Min are 1, 5, and 7. Min chooses the minimum:
$$min(1,5,7)=1$$

If Max chooses Strategy 3, the terminal utilities available to Min are -4, -3, and -12. Min chooses the minimum:
$$min(-4,-3,-12)=-12$$

Step 2: Analyze the Max player's optimal decision at the root
The root node represents the Max player's turn. Max wants to maximize the utility resulting from Min's optimal response. The values resulting from Max's strategies 1, 2, and 3 are -1, 1, and -12, respectively. Max chooses the maximum of these options:

$$max(-1,1,-12)=1$$

The optimal game value that Max can guarantee is 1 (which corresponds to choosing the middle branch, or index 1 under 0-indexed notation). Therefore, the correct final answer is 1.