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I play around with Monte Carlo tree search and tic-tac-toe. For now I have followed the Wikipedia article. There is one place, where I am stuck, the selection phase. The given procedure is the following:

  1. Selection: Start from the root and choose child nodes based on their number of $w$, number of playouts $n$ and the number of playouts of the parent $N$ such that one picks the child which maximizes $$w/n + c \sqrt{\frac{\ln(N)}{n}} \,.$$

    Repeat that process until one has found a leaf.

  2. Expansion. Unless that leaf is a terminal node in the game, create one or multiple child nodes. Choose one of them.

  3. Simulation. From the chosen child, play the game until a terminal node is reached.

  4. Backpropagation. Propagate the number of playouts and wins to the root of the tree.

It all sounds fine. I start with a tree consisting of just the root, and that happens to be a non-terminal root. So I just add all the child states that my game offers. Then I uniformly sample from them (random policy) and continue down until the game is over. Backpropagation goes back to the root.

The problem is in the next step where I now have expanded children from the previous expansions. These have $n=0$ playouts because they were not chosen. This makes the given condition undefined. There are multiple ways to work around that, but none make any sense to me.

  • I could just initialize them with $n = 1$. But then the sums of $n$ would not match the parent playout count $N$. This doesn't seem right.
  • If I just insert one of the children into a leaf, it won't be a leaf in the next round. Therefore I will not do an expansion there. This happens on all the levels. This means that my tree will never broaden, but I only play one random trajectory again and again.
  • If I do expand all of them, but only sample from those with $n = 0$ until all have $n \ge 1$, I will expand all nodes on the first level before taking any of them again. This happens again for every level. I don't exactly explore the whole tree, but it feels like I would be exploring a bit too much.
  • If there are children with $n = 0$ and others with $n \ge 1$, I could just choose the argmax from the explored ones with say 90 % chance and randomly sample the others with say 10 %. This sounds like it might work, but I cannot take that from the explanations that I've read so far.

So how do I treat children which I have expanded but still have $n = 0$?

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