Question

Prove that with a positive linear transformation of leaf values, the move choice remains unchanged in a game tree with chance nodes.

Answer 1

When we have a positive linear transformation of leaf values in a game tree with chance nodes, the move choice remains unchanged.

Step-by-step explanation:

When we have a positive linear transformation of leaf values in a game tree with chance nodes, the move choice remains unchanged.

This can be proven by considering the nature of the game tree and the effect of the linear transformation on the probabilities of different outcomes.

Let's consider a simple example. Suppose we have a game tree with a chance node that has two child nodes.

The probabilities of reaching each child node are represented by the leaf values.

If we apply a positive linear transformation to the leaf values, increasing them by a constant factor, the relative probabilities of reaching each child node remain the same.

The move choice is determined by the relative probabilities, so it remains unchanged.

Answer 2

Base on research and further investigation, the positive linear transformation of leaf, and the move choices remains unchanged in a game tree with chance nodes is x to ax+b where as a is greater that 0. I hope you are satisfied with my answer and feel free to ask for if you have questions and further clarifications