View a PDF of the paper titled Not all Chess960 positions are equally complex, by Marc Barthelemy

Abstract:We analyze strategic complexity across all 960 Chess960 (Fischer Random Chess) starting positions. Stockfish evaluations show a near-universal first-move advantage for White ($\langle E \rangle = +0.30 \pm 0.14$ pawns), indicating that the advantage conferred by moving first is a robust structural feature of the game. To quantify decision difficulty, we introduce an information-based measure $S(n)$ describing the cumulative information required to identify optimal moves over the first $n$ plies. This measure decomposes into contributions from White and Black, $S_W$ and $S_B$, yielding a total opening complexity $S_{\mathrm{tot}} = S_W + S_B$ and a decision asymmetry $A=S_B-S_W$. Across the ensemble, $S_{\mathrm{tot}}$ varies by a factor of three, while $A$ spans from $-2.5$ to $+1.8$ bits, showing that some openings burden White and others Black. The mean $\langle A \rangle = -0.25$ bits indicates a slight tendency for White to face harder opening decisions. Standard chess (position \#518, \texttt{RNBQKBNR}) exhibits above-average asymmetry (91st percentile) but typical overall complexity (47th percentile). The most complex opening is \#226 (\texttt{BNRQKBNR}), whereas \#198 (\texttt{QNBRKBNR})is the most balanced, with both evaluation and asymmetry near zero. These results reveal a highly heterogeneous Chess960 landscape in which small rearrangements of the back-rank pieces can significantly alter strategic depth and competitive fairness. Remarkably, the classical starting position-despite centuries of cultural selection-lies far from the most balanced configuration.