High-Precision Estimation of the State-Space Complexity of Shogi via the Monte Carlo Method
This work provides a more accurate estimate for the complexity of Shogi, addressing a specific problem in game theory and AI for researchers and developers, though it is incremental in refining existing bounds.
The paper tackled the problem of estimating the state-space complexity of Shogi, a challenging task with previous estimates varying widely, and achieved a high-precision statistical estimate of 6.55 × 10^68 reachable positions with a 3σ confidence level, also applying the method to Mini Shogi.
Determining the state-space complexity of the game of Shogi (Japanese Chess) has been a challenging problem, with previous combinatorial estimates leaving a gap of five orders of magnitude ($10^{64}$ to $10^{69}$). This large gap arises from the difficulty of distinguishing Shogi positions legally reachable from the initial position among the vast number of valid board configurations. In this paper, we present a high-precision statistical estimation of the number of reachable positions in Shogi. Our method combines Monte Carlo sampling with a novel reachability test that utilizes a reverse search toward a set of "King-King only" (KK) positions, rather than a single-target backward search to the single initial position. This approach significantly reduces the search effort for determining unreachability. Based on a sample of 5 billion positions, we estimated the number of legal positions in Shogi to be $6.55 \times 10^{68}$ (to three significant digits) with a $3Ï$ confidence level, substantially improving upon previously known bounds. We also applied this method to Mini Shogi, determining its complexity to be approximately $2.38 \times 10^{18}$.