MCTS Backpropagation Phase

You are executing the BACKPROPAGATION phase of Monte Carlo Tree Search.

Backpropagation Algorithm

1. •Start from the simulated node
2. •Traverse up to root:
   • •For each node on the path:
     • •Increment visit count: N = N + 1
     • •Add reward to value: Q = Q + reward
3. •Record the update for analysis

Using MCP Tools

Call mcts_backpropagate with:

• •node_id: The leaf node where simulation ended
• •reward: The reward from simulation
• •path: (optional) Explicit path to update

The tool returns:

• •nodes_updated: List of updated node IDs
• •new_statistics: Updated Q and N for each node
• •tree_depth: Current maximum depth

Statistics Update

For each node in the path from leaf to root:

node.N += 1
node.Q += reward
node.avg_reward = node.Q / node.N

Backpropagation Strategy

For the current context: $ARGUMENTS

Standard Update

• •Each node gets the same reward
• •Simple and effective for most problems

Discounted Update (optional)

• •Apply discount factor γ as you go up
• •Nodes closer to outcome get more credit
• •node.Q += reward * (γ ^ depth_from_leaf)

Observation Recording

After backpropagation:

1. •Record any new insights as observations
2. •Update beliefs if the result was surprising
3. •Note if any branch is now clearly best/worst

Convergence Check

After updating, check:

1. •Best path stability: Has the best path changed?
2. •Value convergence: Are top nodes' values stabilizing?
3. •Sufficient exploration: Have all branches been tried?

Output

After backpropagation, report:

1. •Nodes updated with new statistics
2. •Current best path and its average reward
3. •Exploration coverage (% of nodes visited)
4. •Whether to continue or extract solution

If continuing, return to SELECTION phase. If converged or budget exhausted, extract the solution.