Appendix 1. Quiescent Two Ply, Three Ply and Four Ply Search Likelihood Function

Opponent is the opponent at one ply, and Player is the player at one ply.

Some definitions:

$$\begin{eqnarray*}P_{i_1 i_2 \ldots i_d} & = & \hbox{position resulting from $P$... .... and also makes early checkmates better than late ones);} } \\ \end{eqnarray*}$$

For $h\ge1$, define

$$v_h(P_{i_1 i_2 \ldots i_h}) = \left\{ \begin{array}{l} 0, ~~~... ... i_h}$\space is neither won nor quiescent.} \end{array}\right.$$

At h-ply distant from P the function $v_h(\_)$ assigns the values to positions. The ordinary evaluation function, where the value is calculated from the $\lambda$ values and the attribute values is given by $v(\_)$. The summations are over all moves i_h+1 that can be made in position $P_{i_1 i_2 \ldots i_h}$. Note that $v_d(\_)$ never equals the last of the possibilities listed above, since all non-won non-quiescent positions at d-ply are evaluated using $v(\_)$. This is purely because a certain maximum search depth has been set.

The move from position P is chosen probabilistically according to

$$\mbox{Pr(move $i^*(P)$ )} = \frac{\exp\left(\sigma_1(P) v_1(P... ...ight)} {\sum_{i=1}^{m} \exp\left(\sigma_1(P) v_1(P_i)\right)}.$$

In the advanced method of minimum one ply maximum four, d=4 is used as the maximum, and P itself is always fully searched to at least the minimum of depth 1 (so is treated as non-quiescent).

This gives our likelihood for a single position P. For a set of positions, $\mathcal{P}$, we use the likelihood

$$\prod_{P \in \mathcal{P}} \mbox{Pr(move $i^*(P)$ )} .$$