Question

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).

Hint:

Always apply the minimax algorithm from the leaf nodes upwards. Label each level with "Max" or "Min" to keep track of which operation (maximum or minimum) to apply at that level. The value of the root node is the final value of the game.

Answer 1

Correct Answer: 1

Step 1: Understanding the Question:
The question asks for the outcome or value of the game for the Max player assuming both players play optimally, according to the minimax algorithm. The phrasing "the best strategy for max is ___" is ambiguous and, given the integer answer of 1, likely refers to the *value* of the game when Max plays the best strategy, rather than the index of the strategy (which would be 2).
Step 2: Key Formula or Approach:
The minimax algorithm determines the optimal move by propagating values up the game tree from the terminal nodes. - At nodes where it's Min's turn to play, the node's value is the minimum of its children's values. - At nodes where it's Max's turn to play, the node's value is the maximum of its children's values.
Step 3: Detailed Explanation:
We work from the bottom of the tree upwards.
1. Min's Turn (Layer 1): We calculate the value for each of the three nodes where Min makes a choice. - Left Node (from Max's move 1): Min chooses between terminal nodes with values \{8, 6, -1\}. Min will choose the smallest value. \[ \min(8, 6, -1) = -1 \] - Middle Node (from Max's move 2): Min chooses between terminal nodes with values \{1, 5, 7\}. Min will choose the smallest value. \[ \min(1, 5, 7) = 1 \] - Right Node (from Max's move 3): Min chooses between terminal nodes with values \{-4, -3, -12\}. Min will choose the smallest value. \[ \min(-4, -3, -12) = -12 \] 2. Max's Turn (Root): Now, the root Max player has three choices, which lead to outcomes with values -1, 1, and -12. Max will choose the move that leads to the largest value. \[ \max(-1, 1, -12) = 1 \] Step 4: Final Answer:
The maximum value that the Max player can guarantee is 1. This is achieved by choosing strategy 2. The value of the game at the root is 1. Interpreting the question as asking for this value, the answer is 1.