Initial: \[\begin{equation*} q_i^{(0)} = p_i^{(0)} \left( p_i^{(0)} + 2 p_{(i+1) \mod 3 + 1} \right) \end{equation*}\]

Then, for the first of our alternating steps (i.e., when two children are from the LB), we have something like

\[\begin{equation*} q_i^{(0.5)} = q_i^{(0)} \left( q_i^{(0)} + 2q_{i \mod 3 + 1}^{(0)} \right) \end{equation*}\]

For convenience (and roughly speaking), let \(z_i^{(k)}\) denote the probability that the loser at height \(k\) of the UB is from component \(i\) (e.g., \(z_1^{(0)}\) is the probability that a loser in the first round is from component \(1\)).

On the next of the alternating steps (one child from the LB, one from the UB), we have something like \[\begin{align*} q_i^{(1)} &= q_i^{(0.5)} \left( z_i^{(1)} + z_{i \mod 3 + 1}^{(1)} \right) \\ &\quad + q_{i \mod 3 + 1}^{(0.5)} \left( z_i^{(1)} \right) \end{align*}\]

Then, note that \[\begin{equation*} z_i^{(k)} = p_i^{(k)} \left( p_i^{(k)} + 2 p_{(i + 1) \mod 3 + 1}^{(k)} \right) \end{equation*}\]

So, we can expand the second step to be \[\begin{align*} q_i^{(1)} &= q_i^{(0.5)} \left( p_i^{(1)} \left( p_i^{(1)} + 2 p_{(i + 1) \mod 3 + 1}^{(1)} \right) + p_{i \mod 3 + 1}^{(1)} \left( p_{i \mod 3 + 1}^{(1)} + 2 p_{i}^{(1)} \right) \right) \\ &\quad + q_{i \mod 3 + 1}^{(0.5)} \left( p_i^{(1)} \left( p_i^{(1)} + 2 p_{(i + 1) \mod 3 + 1}^{(1)} \right) \right) \end{align*}\]

Now, we can expand the intermediary step

\[\begin{align*} q_i^{(1)} &= \left( q_i^{(0)} \left( q_i^{(0)} + 2q_{i \mod 3 + 1}^{(0)} \right) \right) \left( p_i^{(1)} \left( p_i^{(1)} + 2 p_{(i + 1) \mod 3 + 1}^{(1)} \right) + p_{i \mod 3 + 1}^{(1)} \left( p_{i \mod 3 + 1}^{(1)} + 2 p_{i}^{(1)} \right) \right) \\ &\quad + \left( q_{i \mod 3 + 1}^{(0)} \left( q_{i \mod 3 + 1}^{(0)} + 2q_{i + 1 \mod 3 + 1}^{(0)} \right) \right) \left( p_i^{(1)} \left( p_i^{(1)} + 2 p_{(i + 1) \mod 3 + 1}^{(1)} \right) \right) \end{align*}\]

So, the general form for \(q_i^{(k + 1)}\) is

\[\begin{align*} q_i^{(k + 1)} &= \left( q_i^{(k)} \left( q_i^{(k)} + 2q_{i \mod 3 + 1}^{(k)} \right) \right) \left( p_i^{(k + 1)} \left( p_i^{(k + 1)} + 2 p_{(i + 1) \mod 3 + 1}^{(k + 1)} \right) + p_{i \mod 3 + 1}^{(k + 1)} \left( p_{i \mod 3 + 1}^{(k + 1)} + 2 p_{i}^{(k + 1)} \right) \right) \\ &\quad + \left( q_{i \mod 3 + 1}^{(k)} \left( q_{i \mod 3 + 1}^{(k)} + 2q_{i + 1 \mod 3 + 1}^{(k)} \right) \right) \left( p_i^{(k + 1)} \left( p_i^{(k + 1)} + 2 p_{(i + 1) \mod 3 + 1}^{(k + 1)} \right) \right) \end{align*}\]